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588424 | Convergence of a Difference Scheme for Conservation Laws with a Discontinuous Flux. | Convergence is established for a scalar finite difference scheme, based on the Godunov or Engquist--Osher (EO) flux, for scalar conservation laws having a flux that is spatially dependent through a possibly discontinuous coefficient. Other works in this direction have established convergence for methods employing the solution of 2 2 Riemann problems. The algorithm discussed here uses only scalar Riemann solvers. Satisfaction of a set of Kruzkov-type entropy inequalities is established for the limit solution, from which geometric entropy conditions follow. Assuming a piecewise constant coefficient, it is shown that these conditions imply L1-contractiveness for piecewise C1 solutions, thus extending a well-known theorem. | Introduction
. The subject of this paper is a nite dierence algorithm for
computing approximate solutions of the Cauchy problem for scalar conservation laws
of the form
1). The
ux k(x)f(u) has a possibly discontinuous spatial
dependence through the coe-cient k, which is allowed to have jump discontinuities.
A simple physical model corresponding to 1.1 is the Witham model of car tra-c
ow
on a highway [11]. The spatially varying coe-cient k corresponds to changing road
conditions. Equations of this type have been addressed by several authors in recent
years, often as the simplest examples of nonstrictly hyperbolic systems [6], [7], [20].
This approach is based on the observation that 1.1 can be written as a 2x2 system:
The eigenvalues associated with this system are The system
fails to be strictly hyperbolic if f 0 can vanish, in which case the system is described
as resonant.
Even when k and the initial data u 0 are smooth, solutions to 1.1 develop discon-
tinuities, and so weak solutions are sought. A weak solution is a bounded measurable
function
Z
RR
Z
R
for all 2 C 1
(R [0; 1)).
This paper considers the conservation law 1.1 within the framework of scalar
nite dierence schemes. The spatial domain R is divided into cells I
with centers at the points x Z. Similarly, the time
domain [0; 1) is discretized via t nt for resulting in time strips
I j;n be the characteristic function for the rectangle R n
October 30, 1999 jtowers@cts.com
J.D. Towers
The nite dierence scheme then generates, for each mesh size
constant approximation
The initial data is discretized via the piecewise constant approximation
x
Z
I j
where j (x) is the characteristic function for the interval I j . The approximations u n
are generated by the explicit, conservation-form, time marching algorithm
Here, 1is based on a two point monotone (nondecreasing with
respect to the second argument, nonincreasing with respect to the rst) numerical
ux, h(v; u), consistent with the actual
ux, i.e., h(u; In order to maintain
the monotonicity of the scheme, the arguments of the numerical
ux are transposed
when the coe-cient k is negative, so that
In particular, this paper focuses on the closely-related Godunov [3] and Engquist-
Osher (EO) [2]
uxes. The Godunov
ux is derived by applying the exact solution
operator for 1.1, with constant to piecewise constant initial data. The numerica
ux that results is then h(v; Here u G (v; u) is the similarity
solution of the resulting Riemann problem with right and left states v and u, evaluated
anywhere along the vertical half line t > 0 in (x; t)-space where the jump in the
initial data occurs. The following formula for the Godunov
ux is derived in [14]:
min [u;v] f(w); if u v
The Godunov
ux is Lipschitz continuous, and for f 2 C 1 it has piecewise continuous
partial derivatives. With the notational convention that a = min(a; 0),
it follows from 1.6 that
The EO
ux, which is dened by
is also Lipschitz continuous, but is smoother than the Godunov
ux. For f 2 C 1 , it
has continuous partial derivatives satisfying
(1.
Dierence Scheme for a Discontinuous Flux 3
It follows from 1.7 and 1.9 that kf 0 k1 serves as a Lipschitz constant for both
uxes.
In the case where the
ux f is convex or concave, with a single stagnation point,
the EO and Godunov
uxes are identical, except for the single case where the two
arguments are joined by a sonic shock, i.e., f 0 (v) < 0 < f 0 (u).
In the scheme 1.4, the coe-cient k is approximated at each cell boundary, resulting
in a discretized version of k,
where j+ 1(x) is the characteristic function for the interval I j+ 1= [x j ; x j+1 ). The
discretized version k has jumps at cell centers, as opposed to cell boundaries, i.e., the
discretization of k is staggered with respect to that of u. This results in a reduction
in complexity, compared with the approach where the two discretizations are aligned.
When the discretizations of the conserved quantity and
ux coe-cient are aligned,
more complicated 2x2 Riemann problems arise, and thus a 2x2 Riemann solver is
required. The concept of reducing complexity via mesh staggering is well known.
In [19], Temple used the Glimm scheme, solving 2x2 Riemann problems, to establish
existence and uniqueness for a 2x2 resonant system of conservation laws. The
singular function , used to prove compactness in x3 of this paper, originated in
[19], in a slightly dierent form. (Unlike the system 1.2, the system studied was not
reducible to a scalar conservation law, so the scalar scheme 1.4 discussed in this paper
is not applicable to that problem.) The authors of [12] and [13] studied the 2x2
Godunov method as it applies to
They adjoined the equation k treated 1.10 as a 2x2 system, for the purpose
of modeling a resonant system. Using to establish compactness, they proved that
the 2x2 Godunov method is convergent (modulo extraction of a subsequence) to a
solution. They established a bound on the total variation of a function that corresponds
to the function z of this paper, but their method of analysis was dierent,
entailing a study of the various waves arising from the 2x2 Riemann problems. In
[13] it was observed that time independent bounds on derivatives (measured via )
had been achieved for the 2x2 Glimm and Godunov schemes, but not for any scalar
scheme that applies to 1.10. Section 3 of this paper provides a bound of this type
for equations of the form 1.1, which constitute a nontrivial subset of 1.10, and are
sometimes used to model the more general 1.10 [6], [7], [20] .
Outside of the dierence scheme methods, the front tracking approach has been
eective, computationally, and as an analytical tool for studying solutions to 1.1.
Reference [7] establishes existence, uniqueness, and asymptotic behavior of the solution
of 1.1, under essentially the same concavity assumptions about the
ux f as this
paper. The singular function , in the form appearing in x3 of this paper, is used
in [7] also. A bound on the variation, measured via , is achieved by studying the
wave interactions that arise. Additionally, satisfaction of a wave entropy condition
is established for the limit of the front tracking scheme, and uniqueness is shown to
follow from this entropy condition. In [6], it is shown that solutions of 1.1 and 1.10
depend continuously on k.
Reference [20] discusses computational di-culties that can arise with schemes for
1.1 and 1.10 that are based on the solution of 2x2 Riemann problems. Specically, the
4 J.D. Towers
random choice method and front tracking method can fail, due to variation blow-up,
and non-existence of solutions to 2x2 Riemann problems. The 2x2 Lax-Friedrichs
scheme does not experience these failures, as is shown in [20] by examples, since it
avoids solving Riemann problems. The scheme 1.4 discussed in this paper, which
is based on scalar Riemann solvers, is also unaected by these di-culties. In the
example provided in [20] where a solution to the Riemann problem fails to exist, the
source of this failure is a sign change in k. This provides the motivation for addressing
the case of indenite k in x1 and x2 of this paper.
This paper is organized as follows. Section 2 discusses the monotonicity and L 1 -
contractiveness properties of the scheme 1.4, including the case of indenite k. Section
3 focuses on the case of positive k, and establishes the main result of the paper,
convergence of a subsequence to a weak solution of the conservation law. Section
4 addresses the related issues of uniqueness and entropy satisfaction for the limit
solution.
2. Monotonicity. For the case of constant k, the theory of monotone schemes
has been essentially complete for many years [1], [5], [9], [17]. In that setting, approximations
generated by monotone schemes are well-known to share many of the
properties of the actual entropy weak solution to the conservation law 1.1, including
monotonicity, L 1 -contractiveness, and satisfaction of a discrete entropy inequality.
Their major drawback is that they are at best only rst order accurate even in regions
where the solution is smooth [5]. Nevertheless, they provide the starting point
for many of the modern higher order accurate schemes, some of which are constructed
by modifying the two-point monotone
ux function with higher order correction terms,
and then applying
ux limiters [4], [15]. The
ux limiters damp out spurious oscillations
that the correction terms often generate in regions of rapid transition.
A nite dierence scheme such as 1.4 is monotone if
When k is constant, monotonicity of the scheme follows from monotonicity of the
numerical
ux h under suitable CFL conditions. It follows that, in addition, the
computed approximations u n
remain within the convex hull of the initial data for all
n. The following proposition provides an analog of these properties for the variable k
situation.
Proposition 2.1. With the CFL condition 2kkk1 kf 0 k1 1 both the Godunov
and EO versions of the scheme are monotone. If k is nonnegative or nonpositive the
CFL condition can be relaxed to kkk1kf 0 k1 1. If the initial data u 0 (x) lies within
the
interval
j0 , for all j, and all n 0.
Proof. Let
. Expressing the three-point scheme as u n+1
the proof proceeds by showing that the partial derivatives @G=@u j+i are each non-
negative, from which monotonicity of the scheme follows. That the partial derivative
of G with respect to u j 1 is nonnegative is clear from
similar formula for @G=@u j+1 shows that it is also nonnegative. For @G=@u j , there
are four cases, depending on the signs of k j 1and k j+ 1. If k
Dierence Scheme for a Discontinuous Flux 5
The case where k j 1 0 and k j+ 1 0 is similar, and the calculation is omitted. For
the case where k j 1< 0 and k j+ 1> 0,
The case where k j 1> 0 and k j+ 1< 0 is similar and is omitted. The stated invari-
ance
of
with respect to the computed solution u
j follows from the monotonicity
of the scheme, along with the fact that f vanishes at u and u. Specically,
For the remainder of this paper it will be assumed that u
loc
denotes the space of locally integrable functions w
having bounded total variation, denoted TV (w). Also, it will be assumed that k is
constant for large x, specically, that there are constants k(+1), k(1), and X such
that . With the cell-average
type discretizations discussed in x1 for u 0 and k, the discretized versions also satisfy
these assumptions, and the various norms of the discretized quantities, u
are bounded uniformly in by the corresponding norms for u 0 and k [1].
Proposition 2.2. With the CFL conditions in Proposition 2.1 and the assumption
that the initial data satises u
is constant for
large x, the scheme 1.4 is L 1 -contractive, i.e., the inequality
holds for a pair of approximate solutions u n
generated by the scheme. The
following inequality also holds:
Proof. Both solutions u n
remain in L 1
. The L 1 bound follows from
Proposition 2.1. The L 1 bound follows from the nite range of in
uence of the initial
data, along with the assumption that k is constant for large x. These observations
along with the fact that the scheme is monotone (due to the CFL condition) and
conservative, add up to the hypotheses of the Crandall-Tartar Lemma [1], giving
2.1. The inequality 2.2 follows from L 1 -contractiveness, applied inductively, to two
successive time steps, u n
and u n+1
j , of a single computed solution.
6 J.D. Towers
3. Convergence. In the case where the coe-cent k is constant, monotonicity
implies that the scheme is Total Variation Decreasing (TVD). However, even for
smooth, but nonconstant, k, the total variation of the solution to the conservation
law generally increases, at least initially, and so there is no hope that the numerical
scheme for variable k will be TVD.
As in [7], [12], and [19], the approach to convergence in this paper is to use the
singular mapping rst used by Temple in his study of a resonant hyperbolic system
[19]. Some assumptions about the
ux function f(u) will be required before dening
. The
ux function f 2 C 2 [0; 1] is assumed to be strictly concave on [0; 1], i.e.,
the interval (0; 1), f(u) > 0, with a
single maximum f at u 2 (0; 1). These are similar to the conditions imposed on f
in [6] and [7]. Then the singular function is dened by
f
Z u
Here In the remainder of this paper, k will be
assumed to be bounded and strictly positive: 0 < k k(x) k. Then 1.5 becomes
The convergence result of this paper remains valid (with a more
restrictive CFL condition, and a larger bound on the variation of z ) if k is allowed
to have nitely many points x where k(x )k(x In order to avoid obscuring
the main idea of the paper, the analysis of this more general case is not presented.
For each value of k in [k; k], (; is an increasing, 1 1
mapping. It is regular everywhere, except at the stagnation point u , where both f 0
and @ =@u vanish. The following Lipschitz continuity relationships in u and k follow
directly from the denition of and the conditions imposed on the
kf 0 k1
The function maps a function w(x; t) into a new function z(x; t) via z(x;
(w(x; t); k(x)). The following elementary facts concerning z are readily veried.
Proposition 3.1. Let w
1. For each t 0, z(;
2. For each t 0, z(;
loc (R), and
R +B
It is now possible to state the main result of this paper.
Theorem 3.2. Let k 2 L 1
loc
taking constant values for large
x, and assume that 0 < k k(x) k. Let f 2 C 2 [0; 1], f 00 < 0,
in (0; 1), with a single maximum f at u 2 (0; 1). Let
Let the mesh size ! 0 with xed and satisfying the CFL
condition resulting in the sequence of approximations u . Then,
there is a weak solution u of 1.1 and a subsequence u i such that u
in L 1
loc (R [0; 1)).
The rest of this section carries out the analysis required to prove Theorem 3.2.
The basic approach is standard as in [17] and [18], with the exception that the quantity
of immediate interest is z instead of the conserved variable u . Compactness
for z follows in the usual way from bounds on the total variation and the
along with a time-continuity estimate. Once the existence of a subsequential
limit z has been established, the invertibility of then allows the corresponding
Dierence Scheme for a Discontinuous Flux 7
solution u to be recovered from the limit z, with u ! u guaranteed by the
continuity of . Finally, a version of the Lax-Wendro Theorem [10] demonstrates
that the limit u is a weak solution to 1.1.
Lemma 3.3. Let
For both the Godunov and EO versions of the scheme,
elsewhere.
Proof. Like (u; k), is a strictly increasing function of u, so
assume that u j > u j+1 . For concave f , and u j u j+1 , the Godunov
and EO
uxes are identical, so 1.9 holds for both
uxes. Also, for either
ux,
Z
h u
Z u j+1
Using the following decomposition of j j+1 ,
Z
Z
Z
it su-ces to show that the following two inequalities are satised:
Z u j+1
Z u j+1
For 3.3, if are to the left of u , so the integral
vanishes, and the inequality holds. So assume that
R u j+1
R
h u
R
h u In either case 3.3 holds, since then
Z
h u
Z u j+1
Z u j+1
Z u j+1
For the integral vanishes. If
Z u j+1
h u
Z u j+2
Z u j+1
h u
Z u j+1
8 J.D. Towers
and denote forward and backward dierence operators. For example,
With the assumptions stated in
Theorem 3.2, uniformly, for all n 0, and all > 0. Specically,
Proof. The superscript n is suppressed in the proof, except where two time levels
are of interest. Taking into account the jumps in z at the cell boundaries (due to jumps
in u ) and the jumps at the cell centers (due to jumps in k ), the total variation of
z(x) is
where u
The second sum in 3.6, due to jumps in k , is bounded by TV (k), using 3.1. For the
rst sum, let Summing over all of the jumps, and using the
fact that z(x) ! k(1) as x ! 1, results in
u
It follows from 3.7 that
Then,
The following identity will be useful in estimating
f
By Lemma 3.3,
Dierence Scheme for a Discontinuous Flux 9
The last inequality uses the fact that jh j+ 1j f , which follows directly from the
conditions on f , along with 1.6 and 1.8. Substituting this into 3.8, shifting indices,
and applying L 1 -contractiveness gives
f
f
The nal step in the proof is to estimate the term
j. For the jth term in
this sum,
Summing over j results in
Substituting this into the last inequality in 3.9 yields
When all of the terms are collected, the bound 3.5 stated in the theorem results.
Lemma 3.5. With the assumptions stated in Theorem 3.2, there is a constant L,
independent of the mesh size , such that for n > m 0,
Z
R
Proof. Taking into account the constant values of z in each half-cell, the integral
on the left side of 3.11 is
An application of 3.2 and L 1 -contractiveness yields
Z
R
f
kf 0 k1
f (n m)
J.D. Towers
The proof is completed by bounding
independently of , using 3.10.
The following is essentially the Lax-Wendro theorem [10].
Theorem 3.6. Let be a sequence of approximations
computed via the scheme 1.4 which converges in L 1
loc (R [0; 1)) and with
uniformly bounded, to u 2 L 1
loc (R [0; 1))
(R [0; 1)). Then u is a weak
solution of 1.1.
Proof. Let
(R [0; 1)) and t n
.
Jx such
jxj B. As in [10], multiply the scheme 1.4 by
n 0, then sum by parts to get
where h j+ 1= h(u arguments [11], the sums
involving
j converge to their integral counterparts in 1.3, as
It remains to show that
xt
Z
RR
kf(u) x dxdt:
The proof of 3.12 reduces to verifying that
Z
I j
tx
The sum appearing in 3.13 does not exceed
Z
R
xZ
R
which proves 3.13. The expression on the left side of 3.14 is bounded by
Z
The rst integral in 3.15 tends to zero by assumption, and the following estimate
tx
Z
tx
Dierence Scheme for a Discontinuous Flux 11
shows that the second integral also approaches zero.
It is now possible to prove Theorem 3.2.
Proof. The CFL condition guarantees that the computed solutions u remain
within [0; 1]. An application of Proposition 3.1 gives uniform (with respect to both
t and ) bounds on kz (; t)k 1 and kz (; t)k compact interval
Theorem 3.4 provides a uniform bound on TV (z (; t)). Finally, from
Lemma 3.5, it follows that
where the constant C is independent of and t. By standard compactness arguments
applied to the sequence z , there is a subsequence, z i , which converges in
loc (R [0; 1)) to some function z 2 L 1
loc (R [0; 1))
(R [0; 1)). There is
a further subsequence, also denoted z i , which converges a.e. to z. Let u(x;
Due to the strict monotonicity of (; k), the function u is well-
dened a.e.,
loc (R [0; 1))
(R [0; 1)). Dropping
the subscript on , it remains to show that u ! u a.e. and in L 1
loc (R [0; 1)).
Using the fact that u
ju u jdxdt
The rst integral tends to zero by the bounded convergence theorem, due to the
continuity of 1 as a function of its rst argument. For the second integral, an
estimate of j 1 (z ; dierentiation gives
@k
@k
R u
When the numerator and denominator are each expanded in a Taylor series about u ,
the result is
@ 1
@k
2k
kf 00 k1
ju u j:
With this estimate, convergence of the second integral follows from the fact that
Having established convergence in L 1
loc (R [0; 1)), there is
yet a further subsequence u , which converges to u a.e., as well as in L 1
loc (R[0; 1)).
Theorem 3.6 proves that u is a weak solution of the conservation law.
4. Entropy satisfaction. Even for constant k, solutions of 1.1 are not necessarily
unique. Additional conditions, usually referred to as entropy conditions, are
required to single out the physically relevant solution. When k 2 C 1 the Kruzkov
entropy condition applies [8]. Specically, uniqueness is guaranteed if
Z
RR
dxdt 0
holds for all c 2 R, and for all nonnegative 2 C 1
0 (R R
condition 4.1 is not directly applicable if k is discontinuous, as was observed in [7].
12 J.D. Towers
For k discontinuous, the authors of [7] proved uniqueness within the class of solutions
that satisfy a wave entropy condition. The wave entropy approach has the advantage
of not requiring that the solution satisfy additional regularity conditions.
The approach in this section is to concentrate on the Godunov version of the
scheme, and assume, as in [6] and [7], that k has nitely many jumps. Proposition 4.1
ensures that the limit solution u satises the Kruzkov entropy inequalities 4.1 locally,
away from the jumps in k. Proposition 4.3 provides Kruzkov-type entropy inequalities
that apply when the test function has support which intersects one or more jumps in k.
Theorem 4.5 shows that these entropy inequalities imply geometric entropy conditions
for piecewise C 1 solutions, and Theorem 4.6 applies the geometric entropy conditions
to show uniqueness with the additional assumption that k is piecewise constant.
Proposition 4.1. In addition to the conditions stated in Theorem 3.2, let k
be piecewise C 1 , with a bounded derivative, jk 0 (x)j for all x, and with nitely
many jumps (in k and k 0 ) , located at 1 < . Let u be a convergent
subsequence generated by the scheme 1.4 using the Godunov
ux, converging to u,
as in Theorem 3.2. Then the Kruzkov entropy inequalities 4.1 hold for every real
number c, and every smooth test function 0 with compact support in t > 0,
g.
To avoid complications arising from the discontinuity in V 0 the
following entropy inequality is established for smooth V before proceeding with the
proof of Proposition 4.1.
Lemma 4.2. In addition to the assumptions of Proposition 4.1, let (V; F ) be
a convex entropy pair for 1.1, i.e., V is convex and F assume that
1]. For every smooth test function 0 with compact support in t > 0,
g, and every c 2 R, the following inequality holds:
Z
RR
Z
RR
Proof. Write the scheme 1.4 as
Let u G
the Godunov discrete entropy
ux dened by H j+ 1=
j+ 1), the discrete entropy inequality
holds for w n
. After rearranging terms, a discrete entropy inequality for the scheme
results:
Multiplying by a smooth, nonnegative test function with compact support in t > 0,
g, and proceeding as in the proof of Theorem 3.6 gives
xt
xt
Dierence Scheme for a Discontinuous Flux 13
As approaches zero, the rst sum in the top line of 4.2 converges to
in the proof of the Lax-Wendro theorem. For the second sum in the rst line of 4.2,
convergence to
reduces to establishing
tx
where J , N , B, and T , are dened as in the proof of Theorem 3.6. Using the fact
that kV 0 k1 kf 0 k1 is a Lipschitz constant for H j+ 1, the expression on the left side of
4.3 is bounded by
Z u G
The rst term in 4.4 tends to zero. Using jF and the fact that u G
1lies
between
and u n
j+1 , the second term does not exceed
Z
The estimate 3.16 proves that this term tends to zero also. Consider the second line
of 4.2. Since k 2 C 1 within the support of ,
xt
Z
RR
The last term in 4.2 to be dealt with is xt
))=t. Expanding
the divided dierence in a Taylor series yields
Due to L 1 -contractiveness, and the time continuity estimate 3.10, the sums involving
the last two terms on the right side of 4.5 approach zero as ! 0. The rst term on
the right side of 4.5 approaches V 0 (u)f(u)k 0 in L 1
loc (R R + ), which gives
xt
Z
RR
It is now possible to prove Proposition 4.1.
Proof. As in [8], approximate the convex function ju cj by a sequence of twice
continuously dierentiable convex functions V i . Apply Lemma 4.2 for each V i , and
14 J.D. Towers
Let -(x i ) denote the Dirac measure with support located at i .
Proposition 4.3. With the same assumptions as in Proposition 4.1, the following
inequality holds for the special case where c = u , for all nonnegative 2
0 (R R
Z
RR
ju
Z
RR
dxdt 0:
Proof. Proceeding as in the proof of Lemma 4.2 gives an inequality of the form
4.2, with V and the corresponding versions of F (u) and H j 1. The
sums in the rst line of 4.2 converge to their integral counterparts, as in the proof of
Lemma 4.2. For the second line of 4.2, there are n
such that
Using the fact that h j 1= f(u G
With this inequality, 4.2 becomes
xt
xt
The term in the bottom line of 4.7 converges to the integral in the second line of 4.6.
This can be veried by breaking the spatial portion of the sum in the bottom line of
4.7 into sums over intervals where k is dierentiable, and isolating the nite number
of cells where the jumps in k are concentrated.
Let u be a piecewise smooth solution to 1.1. It follows from a standard test
function calculation that the Rankine-Hugoniot condition across a jump in k at one
of the points
where the subscripts L and R refer to limits from the left and right, respectively, at
the jump in k.
Lemma 4.4. Let F )). For a pair of states uL , uR
satisfying the Rankine Hugoniot condition 4.8,
kRF
Proof. Take kL kR ; the other case is similar. By considering the various cases
for a pair of states (u L ; uR ) which satisfy 4.8, the following relationships result:
Dierence Scheme for a Discontinuous Flux 15
1. uR > u > uL ) kRF
2.
3.
4. uR < u < uL ) kRF
The rst three cases cover the situation where the equation on the right side of 4.9 is
satised. In each of those cases, the inequality on the left side of 4.9 also holds. Case
4 is the only case where the right side of 4.9 fails. In that situation, the left side of
4.9 also fails, since uL > u ) fL < f , and so
The entropy inequalities 4.1 and 4.6 yield the following fact concerning the satisfaction
of geometric entropy conditions.
Theorem 4.5. In addition to the assumptions of Proposition 4.3, assume that
the limit solution u is piecewise C 1 . Suppose that k jumps from kL to kR at
and u is C 1 in some neighborhood of the point each side of
the notation
For a discontinuity located at away from the jumps in k, assume that u is
some neighborhood of each side of a C 1 curve
(t). With
the following standard entropy condition holds:
where the shock speed
Proof. For 4.10, let N( a neighborhood of the type described in the
statement of the theorem. Let 0 be a smooth test function with support in the
rectangle
centered at extending backward and forward
in time from t 0 by an amount . Assume that and are small enough
that
standard test function calculation, applied to 4.6, gives
Z
The integral over
can be made arbitrarily small by shrinking the width of the rectangle. It follows
that kRF j. The entropy condition 4.10 then follows from
Lemma 4.4. The geometric entropy condition 4.11 follows directly from Proposition
4.1. For k 2 C 1 , it is well known that if 4.1 holds for all c 2 R, then the geometric
entropy condition 4.11 is satised [8].
The geometric entropy condition 4.11 is the usual geometric entropy condition
satised by shocks in the smooth k case. It requires that characteristics on both sides
of the shock extend toward the x-axis when followed backward in time from the shock.
The geometric entropy condition 4.10 requires that the characteristics on at least one
side extend toward the x-axis.
In the constant-k setting, the author of [16] established L 1 -contractiveness of
solutions which satisfy the geometric entropy condition 4.11. The following
theorem is presented as evidence that limits of the scheme 1.4 are the physically
J.D. Towers
relevant solutions. It extends the theorem in [16] to the case of piecewise constant k,
assuming that the additional geometric entropy condition 4.10 is satised.
Theorem 4.6. In addition to the previous assumptions, let k be piecewise con-
stant, with nitely many jumps, located at 1 < Let u and v be
solutions of 1.1 satisfying the geometric entropy conditions 4.10
and 4.11. Assume that the initial data u 0 and v 0 are piecewise C 1 , and u
Then, for t 0,
Proof. Following [16], the approach is to show that the time derivative of the
integral on the left side of 4.12 is nonpositive. That integral is broken up into integrals
over segments where u v does not change sign. This decomposition yields
which is constant within the interval Dierentiating
4.13 with respect to time results in terms of the form
d
dt
_
In [16], it is noted that 4.14 holds even if there are shocks in the interior of
For essentially the same reason, i.e., the Rankine-Hugoniot condition 4.8, the relationship
4.14 holds even if one or more of the points j lie within
(or x i+1 ) away from the jumps in k, the contribution to 4.14 from x i (or x i+1 ) is non-
positive, since the argument in [16] applies in this case, due to the geometric entropy
condition 4.11. The only case remaining is where u v changes sign at a jump in k,
say x
assume that kL < kR , the other case
being similar. There are contributions to 4.13 from two consecutive terms of the form
4.14. Taking into account that _
along the line the total contribution is
The situation where can be eliminated, since then the Rankine-Hugoniot
condition 4.8 implies that both i and i+1 vanish. Take the case where
1. The other case is
similar. Then 4.15 becomes
In order for there to be a sign change, at least one of uR < uL , v R > v L must hold.
Take the case where uR < uL . With kL < kR and uR < uL , the geometric entropy
condition 4.10 and Rankine-Hugoniot condition 4.8 require that uL < u . This is
easily veried by viewing being determined by
the intersection of a horizontal line with the graphs of kL f(u) and kR f(u). Since
vL < uL u , and f is increasing in (0; u ), it is clear that f(vL ) < f(uL ). Then
using 4.8, the expression 4.16 is equal to
Dierence Scheme for a Discontinuous Flux 17
Next, take the case where v R > v L . With kL < kR and v R > v L , the Rankine-Hugoniot
condition 4.8 requires that v R > u . If v L u , then uL > v L ) uL > u .
The entropy condition 4.10 then requires that uR u . Then, with 4.8, uL > v L )
uR > v R , which is a contradiction, so it must be that v L < u . If also uL < u , then
case 4.16 reduces to
If uL > u , the entropy condition 4.10 requires that uR u . Then, uR <
In this case 4.16 equals
2kR
--R
Monotone Di
One sided di
On a class of high resolution total-variation-stable nite dierence schemes
Stability of conservation laws with discontinuous coe-cients
Convex conservation laws with discontinuous coe-cients
First order quasilinear equations in several independent variables
Accuracy of some approximate methods for computing the weak solutions of a
Systems of conservation laws
Numerical methods for conservation laws
A comparison of convergence rates for Godunov's method and Glimm's method in resonant nonlinear systems of conservation laws
Suppression of oscillations in Godunov's method for a resonant non-strictly hyperbolic system
the entropy condition
High resolution schemes and the entropy condition
An example of an L 1
On convergence of monotone di
Shock waves and reaction-diusion equations
Global solution of the cauchy problem for a class of 2x2 nonstrictly hyperbolic conservation laws
The solution of nonstrictly hyperbolic conservation laws may be hard to compute
--TR
--CTR
Michael Herty , Mohammed Sead , Anita K. Singh, A domain decomposition method for conservation laws with discontinuous flux function, Applied Numerical Mathematics, v.57 n.4, p.361-373, April, 2007
Sebastian Noelle , Normann Pankratz , Gabriella Puppo , Jostein R. Natvig, Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows, Journal of Computational Physics, v.213
Adimurthi , Siddhartha Mishra , G. D. Veerappa Gowda, Conservation law with the flux function discontinuous in the space variable-II, Journal of Computational and Applied Mathematics, v.203 n.2, p.310-344, June, 2007
Raimund Brger , Anbal Coronel , Mauricio Seplveda, On an upwind difference scheme for strongly degenerate parabolic equations modelling the settling of suspensions in centrifuges and non-cylindrical vessels, Applied Numerical Mathematics, v.56 n.10, p.1397-1417, October 2006
S. Berres , R. Brger , K. H. Karlsen, Central schemes and systems of conservation laws with discontinuous coefficients modeling gravity separation of polydisperse suspensions, Journal of Computational and Applied Mathematics, v.164-165 n.1, p.53-80, 1 March 2004 | conservation laws;difference approximations;discontinuous coefficients |
588428 | Long-Time Energy Conservation of Numerical Methods for Oscillatory Differential Equations. | We consider second-order differential systems where high-frequency oscillations are generated by a linear part. We present a frequency expansion of the solution, and we discuss two invariants of the system that determine the coefficients of the frequency expansion. These invariants are related to the total energy and the oscillatory harmonic energy of the original system.For the numerical solution we study a class of symmetric methods that discretize the linear part without error. We are interested in the case where the product of the step size with the highest frequency can be large. In the sense of backward error analysis we represent the numerical solution by a frequency expansion where the coefficients are the solution of a modified system. This allows us to prove the near-conservation of the total and the oscillatory energy over very long time intervals. | Introduction
. Long-time near-conservation of the total energy and of adiabatic
invariants in numerical solutions to Hamiltonian differential equations is important
in a wide range of physical applications from molecular dynamics to nonlinear
wave propagation. Backward error analysis [BG94, HaL97, Rei98] has shown that
symplectic numerical integrators approximately conserve the total energy and adiabatic
invariants over times that are exponentially long in the step size; more precisely,
over times of length exp(c=h!) where ! is the highest frequency in the system. Such
a result is meaningful only for h! ! 0, which is often not a practical assumption.
For example, in spatially discretized wave equations h! is the CFL number, which is
not chosen small in actual computations. Recently, in [GSS99, HoL99] new symplectic
or symmetric time-stepping methods have been studied which admit second-order
error bounds on finite time intervals independently of the frequencies of the dominant
linear part of the system. In particular for such "long-time-step methods", the case
is of no computational interest. The situation is reminiscent of stiff versus
nonstiff differential equations, where stiff integrators are not appropriately analyzed
by considering only the limit behavior h ! 0. In the stiff case, much insight has been
gained by studying the behavior of numerical methods on well-chosen, rather simple
linear and nonlinear stiff model problems.
As a first step towards an understanding of the numerical energy behavior in
Hamiltonian systems when the product of the step size and the highest frequency is
not a small quantity, we consider in this article the nonlinear, highly oscillatory model
problem
x
where
Dept. de math'ematiques, Universit'e de Gen'eve, CH-1211 Gen'eve 24, Switzerland
y Mathematisches Institut, Universit?t T-ubingen, Auf der Morgenstelle 10, D-72076 T-ubingen,
Germany (E-mail: lubich@na.uni-tuebingen.de)
(with blocks of arbitrary dimensions), and where the nonlinearity
has a Lipschitz constant bounded independently of !. This situation arises in the celebrated
Fermi-Pasta-Ulam model, for which we will present numerical experiments.
Clearly, in the model problem (1.1) we take strong restrictions in that the high frequencies
are confined to the linear part of the problem, and that the linear part has
a single high frequency. The diagonal form
of\Omega is not essential, since the numerical
methods are invariant under a diagonalization of the matrix.
We study the long-time energy behavior of a class of symmetric numerical methods
which are used with step sizes h such that the product h! is bounded away from zero
and can be arbitrarily large. The methods integrate the linear part of (1.1) exactly
and reduce to the St-ormer/Verlet method for The class includes the methods
of [GSS99, HoL99]. Classical symmetric methods such as the St-ormer/Verlet method,
the trapezoidal rule, or Numerov's method are not considered in this article. However,
using the results of the present paper, their energy behavior on (1.1) for h! in the
range of linear stability is analyzed in [HaL99].
Our approach to the near-conservation of the energy is based on a frequency
expansion of the solution x(t) of (1.1),
which is an asymptotic series where the coefficient functions y(t) and z k (t) together
with their derivatives are bounded independently of !. It turns out that the system
determining the coefficient functions has two (formally exact) invariants. One of these
is close to the total energy
according to the partitioning of \Omega\Gamma The other invariant is close to
which represents the oscillatory energy of the system.
For the numerical solution we derive a similar frequency expansion which is valid
on grid points Under a non-resonance assumption on h!,
the equations determining the coefficient functions have a similar structure to those
of the continuous problem. This allows us to obtain two almost-invariants close to H
and I , and rigorous estimates for the near-conservation of the total and the oscillatory
energy over time intervals of size CN h \GammaN . The only restriction on N comes from the
above non-resonance condition. The analysis uses only the symmetry of the methods
and does not require symplecticness.
In Sect. 2 we describe the numerical methods and we present numerical experiments
with the Fermi-Pasta-Ulam problem. These experiments illustrate the long-time
conservation of the total and the oscillatory energy in non-resonance situations,
which will later be completely explained by the theory. We also show the energy behavior
of the methods near resonances. This behavior depends strongly on properties
of the filter functions that determine the numerical method. We identify conditions
that yield satisfactory energy conservation near resonances and the correct energy
exchange between highly oscillatory components.
Sect. 3 gives a complete analysis of the two-dimensional linear case of (1.1) over
the whole range of non-resonant, near-resonant and exactly resonant cases. This
gives already much insight into conditions determining the energy conservation in the
general situation.
The frequency expansion of the analytical solution of (1.1) is introduced in Sect. 4,
that of the numerical solution in Sect. 5. The numerical invariants are derived in
Sect. 6. The main result on the numerical long-time conservation of energy for (1.1)
is formulated and proved in Sect. 7.
2. Numerical methods and numerical experiments. In this section we
present the numerical methods and we illustrate the main results of this paper with
the Fermi-Pasta-Ulam problem.
2.1. The discretization. We consider the differential equation (1.1),
is a symmetric and positive semi-definite (not necessarily diagonal) real matrix, and
we assume that initial values x 0 and -
x 0 are given at t By the variation-of-
constants formula, the exact solution of (1.1) satisfies
t\Omega \Gamma\Omega sin
t\Omega cos
t\Omega
s)\Omega
ds
(observe
t\Omega is well-defined also for
singular\Omega\Gamman It is therefore natural to
consider, for a fixed step size h, the explicit discretization
h\Omega xn
\Gamma\Omega sin
real functions OE(-), depending smoothly on - 2 . For
method integrates the problem (1.1) without error.
If the method is consistent of order 2. For
fixed\Omega and for h ! 0, second-order convergence follows from classical results. In this
article we are mainly interested in the situation where
h\Omega can take large values.
For long-time integrations, symmetric and/or symplectic methods are expected
to have favorable properties. By exchanging n in (2.2)-(2.3), it
is seen that the method is symmetric for all g(x) if and only if
(where sinc sin -). It can be shown by direct verification that (2.2)-(2.3) is
a symplectic discretization if, in addition to (2.4), also holds. This
condition will not be required for the analysis of our paper.
Since the nonlinearity g(x) in (1.1) does not depend on -
x, we can eliminate -
xn in
(2.2) with the help of (2.3). In the case of a symmetric discretization we thus get the
two-step recurrence
The starting value x 1 is obtained from (2.2) with For the
recognize the well-known St-ormer method.
stiff
harmonic
soft
nonlinear
Fig. 1. Alternating soft and stiff springs
Methods of the type (2.5) or (2.2)-(2.3) have been proposed and studied by several
authors. Gautschi [Gau61] suggests to take (-=2). With this choice, (1.1)
is integrated exactly even for const . Deuflhard [Deu79] discretizes the integral
in (2.1) by the trapezoidal rule and thus arrives at (2.5) with
recently, Garc'ia-Archilla, Sanz-Serna and Skeel [GSS99] introduce
a function OE(-) in the argument of g, and they consider the case where the method
is symplectic, so that Hochbruck and Lubich [HoL99] consider
OE(-). The papers [GSS99] and [HoL99] derive error
bounds on finite time intervals which are independent of ! and of the smoothness of
the solution.
In this article, we consider general functions OE; / with
have no zeros except possibly at integral multiples of -. Since we are interested in
the energy conservation of the numerical solution, we need also an approximation to
the derivative if we use the two-term recurrence relation (2.5). This can be obtained
by the relation (2.3) or, in the case of a symmetric method, also by the formula
This is possible if h! is not a nonzero integral multiple of -. We obtain (2.6) by
subtracting (2.5) from twice the formula of (2.2). For a symmetric method we obtain
a formula for -
by exchanging n in (2.3). Subtracting the
resulting formula from (2.3), we obtain the two-step recurrence
\Gamma2\Omega sin
Formulas (2.5) and (2.7) give a symmetric two-step method even if (2.4) is not satis-
fied. If -, then this method is exact for
const . The choice / 1 been considered in [HoL99].
2.2. Experiments with the Fermi-Pasta-Ulam problem. We consider a
chain of springs, where soft nonlinear springs alternate with stiff harmonic springs
(see [GGMV92] and Fig. 1). The variables x
for the displacements of end-points of the springs. The movement is described by a
Hamiltonian system with
Using the symplectic change of variables u
2,
we get a new Hamiltonian system with
(a) H
I
I
I
(a) H
I
I
I
(a) H
I
I
(c) H
I
100 200 3001
(a) H
I
100 200 300
I
100 200 300
(c) H
I
Fig. 2. Energy exchange of stiff springs
where This is exactly of the form (1.1).
For our numerical experiments we consider the case shown in Fig. 1)
As initial values we take
and zero for the remaining initial values. We apply the method (2.2)-(2.3) with the
following data:
(a)
(c)
with / 0 (-) and / 1 (-) given by (2.4).
We study the total energy (2.8) and the oscillatory energy
along the numerical solution on the interval 0 - t - 400. With the chosen initial
values we have In Fig. 2 we have
plotted, for three different step sizes and for all three methods, the numerical values
for I 1 ; I 2 ; I 3 ; I and H \Gamma 0:8. We see that an exchange of energy takes place, going
from the first stiff spring with energy I 1 to the second stiff spring and later to the
third one. For the smallest step size we have also plotted in gray the numerical values
for perturbed initial values obtained by adding 10 \Gamma8 to u 1 (0), -
illustrates that the solution is very sensitive to perturbations.
6 E. HAIRER AND CH. LUBICH
In all cases we see that H and I are well preserved over the whole interval, even
for step sizes where the numerical solution is completely wrong. Further experiments
have shown that such a preservation holds for much longer intervals (we tested up to
An explanation of this phenomenon is the main objective of this paper.
2.3. Numerical experiments in near-resonant situations. When the product
of the step size and the frequency h! is close to a multiple of -, then the different
methods show widely different behavior. Energy is conserved only for some choices of
/ and OE. Satisfactory numerical behavior also in near-resonance situations is obtained
if the numerical method satisfies the following additional conditions:
These conditions yield long-time energy conservation for all values of h! with the
exception of h! in intervals of width O(h) near integral multiples of 2-. The total
energy appears to be conserved uniformly for arbitrary values of h! if
The necessity of these conditions is seen from an analysis of the linear case, which is
given in Sect. 3.
When h! is close to 2m- with a positive integer m, the condition (2.9) requires
a double zero of / at 2m-. Similarly, for h! close to an odd multiple of -, condition
there requires a simple zero of /. For the choice
HoL99] the condition (2.9) is obviously satisfied for all values of h!, but (2.10) is
violated near odd multiples of -. For condition (2.10) is trivially
satisfied for all h!, but condition (2.9) fails near even multiples of -. The choice
satisfies the three conditions (2.9)-
for all h!. Condition (2.12) is not satisfied by any of the methods previously
proposed in the literature.
Table
Methods used for the numerical experiments of Sect. 2.3
A sinc (-) 1 2k- p
Let us illustrate the effect of the conditions (2.9), (2.10) and (2.11) on the numerical
solution when h! is close to a multiple of -. We consider the Fermi-Pasta-Ulam
problem of Sect. 2.2 with the same initial values, and we apply six different methods.
Their characteristics are given in Table 1. The sign p
indicates that the corresponding
condition on / and OE is satisfied. If a condition is not satisfied for all values of
-22 (C)
out of scale
-22 (D)
Fig. 3. Energy conservation of different methods for
Fig. 4. Energy conservation of different methods for
Fig. 5. Energy conservation of different methods for
h!, we give the values close to which it is violated. For each of the methods (B), (C),
(D) only one of the conditions (2.9)-(2.11) is not fulfilled.
In Fig. 3 we show the errors of the Hamiltonian over the interval [0; 1000]. We
have used the step size such that
a maximal error of size 396497, because / 1 (-) given by (2.4) has a singularity at -.
.2
.2
.2
I
.2
I
Fig. 6. Error in the total and oscillatory energies as a function of h! for the FPU problem
Methods (A) and (D) show a clear drift from the constant value of the Hamiltonian.
Only the methods (B), (E), and (F), for which all three conditions are satisfied close to
-, conserve the Hamiltonian very well. For the pictures corresponding to these three
methods, we have changed the scale so that the small oscillations become visible.
Fig. 4 shows the same experiment, where this time ! is chosen such that
2:0000001 \Delta -. For both situations, we get the same qualitative behavior when we plot
the oscillatory energy instead of the Hamiltonian. The results of these experiments
confirm that the conditions (2.9), (2.10), (2.11) cannot be omitted if we are interested
in long-time energy estimates that hold uniformly in h!.
In Fig. 5 the numerical results of the nonresonant case are included.
All methods give satisfactory results. The most accurate results are obtained by the
method (C).
In the upper pictures of Fig. 6 we plot the maximal errors in the Hamiltonian
as a function of h!, and we take step sizes 0:025. The
picture to the right corresponds to the method (E) of Table 1. The picture to the left
is obtained with method (F) which satisfies (2.12). Uniform convergence of the error
can be observed only in this case. The lower pictures of Fig. 6 show the analogue
for the deviations in the oscillatory energy. Close to integral multiples of 2- this
deviation is large for both methods. The same phenomenon can be observed already
for linear problems (see Fig. 8), for which a complete analysis is given in Sect. 3.
2.4. Energy exchange. The energy exchange between stiff components takes
place on time intervals of length O(!). In Fig. 2, this is reproduced qualitatively
correctly for large h! only in the case where 1. The numerical
frequency expansion of Sect. 5 shows that the condition
.2
HH
.2
HH
.2
II
.2
II
Fig. 7. Same experiment as in Fig. 6 with method (2.5),
with method (2.16) (right pictures)
is needed for the approximation of the energy exchange between stiff components
when h! is bounded away from zero (compare the equations for the z 2 component in
(4.8) and (5.8)). This is a severe condition which excludes all methods considered so
far with the exception of the above-mentioned method
the other hand, we have seen that this method has rather poor energy conservation
properties. We therefore extend the class of methods (2.5) to
(h\Omega\Gamma2 For consistency, the functions / k , OE k must satisfy
Conditions (2.9) and (2.10) are now needed for
condition (2.13) is
replaced with
For example, the method
with (2.6), or equivalently in one-step form
h\Omega xn
\Gamma\Omega sin
shows the correct energy exchange
for large h!, as for method (b) in Fig. 2. In Fig. 7, we compare the energy
conservation for the method (2.5) with and the method
for the same step sizes as in Fig. 6. For (2.16), the total and oscillatory
energies are well conserved over long times except for h! in intervals of length O(h)
around integral multiples of -.
For ease of presentation, the following analysis will be done for methods of class
(2.5), but the arguments extend in an obvious way to the class (2.14).
3. Long-time energy conservation for linear problems. We start our analysis
with the case where
with a two-dimensional symmetric matrix A
satisfying a 11 ? 0, so that
is linear. This gives already a lot of insight and illustrates the importance of the
conditions (2.9)-(2.12). In this situation, the differential equation (1.1) becomes
x
The total energy H given by (1.5) is an invariant of the system. In the following we
assume that H
\Delta is bounded uniformly in !. This requires x 2
3.1. Analytical solution. The exact solution of (3.1) is given by
O(! \Gamma2 )'
i are the eigenvalues
so that
a 11 +O(! \Gamma2
For given initial values x(0), -
Consequently, we have
be
This implies
that the quantity
remains O(! \Gamma1 )-close to the constant value I
\Delta for all times t.
3.2. Numerical solution. We search for functions b
v, such that xn :=
x(nh) satisfies the numerical scheme (2.5) with This implies
so that has to be an eigenvalue of
and v a corresponding eigenvector. In this section we use the short notation
and OE(h!). Since the off-diagonal elements of (3.3) are small, the eigenvalues -
are close to ff
respectively. The corresponding eigenvectors are
'fl/
\Gammafl OE'
the general solution can thus be written as
where the complex coefficients are computed from the initial values b
obtained from (2.2).
Inserting
To study the long-time near-conservation of H
\Delta and I
\Delta we distinguish
two cases.
Case I: well-separated eigenvalues. We assume that one of the conditions
a 11 or
is satisfied. This covers nearly all choices of h!. Only values in intervals of length
O(h) are excluded.
Theorem 3.1. Consider the numerical method (2.2)-(2.3) applied to (3.1) with
a Under one of the restrictions (3.5) or (3.6) on the
step size, and under the conditions (2.9), (2.10), (2.11) on the numerical method, the
energies H and I of (1.5) and (1.6) along the numerical solution satisfy
for all n - 0. The constants symbolized by O(\Delta) are independent of !, h and n.
Proof. The characteristic polynomial of the matrix (3.3) is
Its zeros can be computed explicitly. Each of the conditions (3.5)
or (3.6) together with (2.11) implies that
for both eigenvalues - i of (3.3). Hence, for sufficiently small h, - 1 and - 2 are real and
in the interval [\Gamma1; 1]. The angles - i , defined by - are therefore also real
and satisfy
a
We next estimate the coefficients a; a; b; b in (3.4). Since the - i and v i are real, the
coefficients a; b are the complex conjugates of a; b. In both situations, (3.5) and (3.6),
we have under the assumption (2.9), we further have
Consequently, the condition b
of using b more convenient to work with b x(h) \Gamma cos
cos
From the estimates
sinc (h!=2)), which all follow from (2.9) and (2.10), we then get
be
The second relation is a consequence of the fact that
(which follows from -
and (2.11)). The statement (3.8) is now an immediate consequence of (3.9) and of the
because the modulus of c = be i-2 t is independent
of t.
The near-conservation of the Hamiltonian can be seen similarly. If one of the
conditions (3.5) and (3.6) is satisfied, we get flOE sin(- 2 so that
This implies jbx 0
Const
all t, this
together with (3.8) proves the statement (3.7) for the total energy.
Case II: nearly collapsing eigenvalues. We now consider the complementary case4 h 2 a
so that the two eigenvalues of (3.3) are very close.
Theorem 3.2. Consider the numerical method (2.2)-(2.3) applied to (3.1) with
a Under the condition (2.12) on the numerical method,
viz.
for all n - 0 and uniformly in h! - c ? 0. The constants symbolized by O(\Delta) are
independent of !, h and n.
Proof. Under the condition (2.12) the numerical method satisfies (2.9), (2.10),
so that Theorem 3.1 is applicable. It therefore remains to consider the situation
where h! is restricted by (3.10).
The condition (3.10) implies 1
2a 11 for sufficiently small
h. We now assume that OE/ - 0, which is satisfied by the choice (2.12). This guarantees
that the eigenvalues of (3.3) are real, that -
OE/), and that
a 11 +O(h 3
For the special choice (2.12) we further have jflj - 1= p
O(!), so that the coefficients of (3.4) satisfy a = O(1) and
(3.
.2
.4
.2
.4
.2
.4
I
.2
.4
I
Fig. 8. Error in the total and oscillatory energies as a function of h! for a linear problem
holds (recall that
This is a consequence of the estimate
OE/) and of the fact that jflj - 1=
OE/.
The relations (3.12) and (3.13) thus yield
a 11
Together with (3.4) this implies
To prove that the t-dependent term is small, we need the relation (2.12) between OE
and /. Using also (3.12) and (3.13) we obtain
a
a
sin
This completes the proof of Theorem 3.2, because
The proof above shows that in general
I
In the situation of Theorem 3.1 we have so that the first term in
the right-hand expression of (3.14) becomes negligible, and long-time conservation
of I can be concluded. If a 12 6= 0 and cos(h!) -
14 E. HAIRER AND CH. LUBICH
OE/ and fl - 1=
OE/, so that fl/ -
/=OE and flOE -
OE=/. The
initial condition b
so that none of the terms in (3.14) can be neglected. If - is different from an
integral multiple of 2-, the expression (3.14) cannot remain close to a constant value.
This result is intuitively clear, because in the situation (3.10) the two frequencies are
indistinguishably close for the numerical method.
The upper pictures of Fig. 8 show the maximal error in the Hamiltonian H(xn ; -
in dependence of h! for the problem (3.1) with a
initial values
. The three curves correspond
to the step sizes 0:05. The picture to the left it obtained
with a method satisfying (2.12). Uniform convergence of the error can nicely be
observed. The picture to the right corresponds to the method (E) of Table 1. The
lower pictures of Fig. 8 plot the maximal deviation of the oscillatory energy I(xn ; -
as a function of h!. It confirms the analysis above, which shows that for h! satisfying
a 11 the oscillatory energy cannot be well conserved.
4. Frequency expansion of the analytical solution. The main tool of our
analysis for nonlinear problems is a decomposition of the solution x(t) of (1.1) into
a smooth part and into highly oscillatory terms with smoothly varying amplitudes.
This decomposition is valid over finite time intervals. We show the existence of two
almost-invariants for the coefficients of this decomposition, which are related to the
total energy and the oscillatory energy of the system. A repeated use of these almost-
invariants then allows us to prove the long-time near-conservation of the oscillatory
energy.
4.1. The frequency expansion. We assume the nonlinearity g in (1.1) analytic
on an open set D, and we consider solutions of (1.1) which satisfy
where K is a compact subset of D. We assume further that the initial values have
limited harmonic energy:2 k -
where E is independent of !.
Theorem 4.1. Under the assumptions (4.2) and (4.1) for 0 - t - T , the solution
x(t) of Eq. (1.1) has for arbitrary N - 2 an expansion of the form
e ik!t z k (t) +RN (t);
where the remainder term and its derivative are bounded by
The real functions and the complex functions z
are bounded,
together with all their derivatives, by
and we have z . They are unique up to terms of size O(! \GammaN \Gamma2 ). The constants
symbolized by the O-notation are independent of ! and t with
on E, N , T , and on the order of the derivative).
Proof. To determine the smooth functions
we put
e ik!t z k (t);
insert this function into (1.1), expand the nonlinearity around y(t) and compare the
coefficients of e ik!t . With the notation g (m) (y)z
the following system of differential equations:
z k
z k
Here the sums range over all m - 1 and all multi-indices
integers ff j satisfying which have a given sum
For large !, the dominating terms in these differential equations are given by
the left-most expressions. However, since the central terms involve higher derivatives,
we are confronted with singular perturbation problems. We are interested in smooth
functions that satisfy the system up to a defect of size O(! \GammaN ). In the spirit
of Euler's derivation of the Euler-Maclaurin summation formula (see e.g. [HaW96])
we remove the disturbing higher derivatives by using iteratively the differentiated
equations (4.7)-(4.9). This leads to a system
z k
are formal series in powers of ! \Gamma1 . Since we get formal algebraic relations
for , we can further eliminate these variables in the functions F
.
We finally obtain for y the algebraic relations
z k
z k
and a system of real second-order differential equations for y 1 and complex first-order
differential equations for z
At this point we can forget the above derivation and we can take it as a motivation for
the ansatz (4.10)-(4.11), which we truncate after the O(! \GammaN ) terms. Inserting this
ansatz and its first and second derivatives into (4.7)-(4.9) and comparing like powers
recurrence relations for the functions F k
jl . This shows that these
functions together with their derivatives are all bounded on compact sets.
We determine initial values for (4.11) such that the function e x(t) of (4.6) satisfies
e
x(0). Because of the special structure of the ansatz (4.10)-
(4.11), this gives a system
which, by the implicit function theorem, yields (locally) unique initial values y 1 (0),
(0). The assumption (4.2) implies that z 2 It further follows
from the boundedness of F 2l that z 2 looking closer at
the structure of the function G k
jl it can be seen that it contains at least k times the
factor z 2 . This implies the stated bounds for all other functions.
We still have to estimate the remainder RN x(t). For this we consider
the solution of (4.10)-(4.11) with initial values (4.12). By construction, these functions
satisfy the system (4.7)-(4.9) up to a defect of O(! \GammaN ). This gives a defect of size
O(! \GammaN ), when the function e x(t) of (4.6) is inserted into (1.1). Hence on a finite time
To obtain the
slightly sharper bounds (4.4), we apply the above proof with N replaced by N 2.
4.2. The Hamiltonian of the frequency expansion. Consider now the situation
so that (1.1) is a Hamiltonian system
x
with Hamiltonian
where U(x) is assumed to be analytic. Let v k
that by (4.7)-(4.9) these functions satisfy
Here, the sum is again over all m - 1 and all multi-indices
integers ff which have a given sum
and we write
Further we denote
From the above it follows that the vector (y; V ) satisfies the system
y
which, neglecting the O(! \GammaN ) terms, is Hamiltonian with
Theorem 4.2. Under the assumptions (4.2) and (4.1) for
The constants symbolized by O(\Delta) are independent of ! and t with
depend on E, N and T .
Proof. Multiplying (4.17) and (4.18) with -
y T and ( -
respectively, gives
dt
Integrating from 0 to t and using v
By the bounds of Theorem 4.1, we have for
On the other hand, we have from (4.14) and (4.3) that
Using
it follows from
Inserted into (4.22) and (4.23) this yields the statement (4.21).
4.3. Another almost-invariant. Besides the Hamiltonian H(y; -
V ), the
coefficients of the frequency expansion have another almost-invariant. It only depends
on the oscillating part and it is given by
This almost-invariant turns out to be close to the energy of the harmonic oscillator,
Theorem 4.3. Under the assumptions (4.2) and (4.1) for
The constants symbolized by O(\Delta) are independent of ! and t with
depend on E, N and T .
Proof. With the vector holds that U(y;
Differentiating the identity
with respect to t yields
z \Gammak
because
The proof of Theorem 4.3 is now very similar to that of Theorem 4.2. We multiply
the relation (4.18) with \Gammai!k(v \Gammak ) T instead of ( -
Summing up yields, with the
use of (4.28),
\Gammai!
The derivative of I(V; -
by (4.24), is
d
dt
(v \Gammak
In the sums
the terms with k and \Gammak cancel.
Hence, the statement (4.26) follows from (4.29) and (4.30).
Using -
from the bounds of
Theorem 4.1 that
On the other hand, using the arguments of the proof of Theorem 4.2, we have
This proves the second statement of the theorem.
Corollary 4.4. If x(t) 2 K for
The constants symbolized by O(\Delta) are independent of ! and t with
depend on E and N .
Proof. With a fixed T ? 0, let V j denote the vector of frequency expansion terms
that correspond to starting values (x(jT ); -
x(jT )). For
we have by (4.27)
Vn ('T
Vn ('T
Vn
We note that I(V j+1 (0); -
by the uniqueness
statement of Theorem 4.1, we have V j+1
we have the bound (4.26) of Theorem 4.3. The same argument
applies to I(Vn ('T ); -
Vn ('T
Vn (0)). This yields the result.
Remark. It is already known from the article [BGG87] that the oscillatory energy
x(t)) is nearly preserved over long times. The proofs in [BGG87] are completely
different. They use coordinate transforms from Hamiltonian perturbation theory and
show that I is nearly preserved over time intervals which grow exponentially with !.
By carefully tracing the N-dependence of the constants in the O(! \GammaN )-terms, it is
possible to obtain near-conservation of I over exponentially long time intervals also
within the present framework of frequency expansions.
5. Frequency expansion of the numerical solution. In this section we show
that the numerical solution (2.2), (2.3) for nonlinear problems (1.1) has a frequency
expansion similar to that of the analytical solution. Following the idea of backward
analysis and motivated by the results of Sect. 4 we look for a function
e ik!t z k (t)
(with smooth y(t) and z k (t) depending on h) 1 such that, up to a small defect,
We assume throughout this section that
and that the numerical solution \Phix n remains in a compact subset of the region where
g(x) is analytic, i.e.,
5.1. Functional calculus. For the computation of the functions y(t) and z k (t)
the following functional calculus is convenient. Let f be an entire complex function
bounded by jf(i)j - C e fljij . Then,
converges for every function x which is analytic in a disk of radius r ? flh around t.
We note that (hD) k
are two such entire functions, then
whenever both sides exist. In particular, we have
To avoid an overloaded notation with hats, we use the same letters y and z k as for the analytical
solution. We hope that this does not cause confusion.
We therefore introduce the operator
h\Omega
sin
which, for h ! 0, is an approximation to h 2 (D 2
We next study the application of such an operator to functions of the form e i!t z(t).
By Leibniz' rule of calculus we have (hD) k e i!t z(t). After a
short calculation this also yields
f(hD)e i!t
5.2. Modified equations for the coefficient functions of the frequency
expansion. With the operator L(hD) of (5.5) the condition (5.2) becomes
Inserting the ansatz (5.1), expanding the right-hand side of (5.7) into a Taylor series
around \Phiy(t), and comparing the coefficients of e ik!t yields for the functions y(t) and
z k (t)
Here, multi-index as in the proof of Theorem 4.1,
ff is an abbreviation for the m-tupel (\Phiz ff To get
smooth functions y(t) and z k (t) which solve (5.8) up to a small defect, we look at the
dominating terms in the Taylor expansions of L(hD) and L(hD ik!h). With the
abbreviations
2 kh!) we have
(ihD)
The situation is now more complicated than in (4.7)-(4.9) for the frequency expansion
of the analytical solution, because several of the coefficients in (5.9) may vanish due
to numerical resonance. We here confine the discussion to the non-resonant case. We
assume that h and ! \Gamma1 lie in a subregion of the small parameters
for which there exists a positive constant c such that
2:
LONG-TIME ENERGY CONSERVATION 21
The condition excludes that h! is o(
close to integral multiples of -. For given h
and !, the condition imposes a restriction on N . In the following, N is a fixed integer
such that (5.10) holds.
Theorem 5.1. Under the limited-energy condition (4.2), under the non-resonance
condition (5.10), under the conditions (5.3), (5.4), and under the conditions (2.9) and
(2.10) on the numerical method (2.2)-(2.4), the numerical solution is of the form
uniformly for where the functions
satisfy (5.8) up to a defect of O(h N+2 ) in their first components, and O(/(h!)h N+2 )
in their second components. Together with all their derivatives these functions are
bounded by
, and the constants symbolized by the O-notation
are independent of ! and h, but depend on E, N , and T .
Proof. Under assumption (5.10), the first non-vanishing coefficients in (5.9) are
the dominant ones, and the derivation of the defining relations for y and z k is the
same as for the analytical solution in Theorem 4.1. We insert (5.9) into (5.8) and we
eliminate recursively the higher derivatives. This motivates the following ansatz for
the computation of the functions y and z k :
z k
z k
where the functions depend smoothly on the variables y 1 , -
and on the bounded
parameters
/(h!). Inserting this ansatz and its derivatives into
(5.8) and comparing like powers of
h yields recurrence relations for the functions
jl . The functions g k
jl (for k - 1) contain at least k times the factor OE(h!)z 2 , and
f 2l contains at least once this factor. Since the series in (5.12) need not converge, we
truncate them after the (
We next determine the initial values y 1 (0), -
x(h) of (5.1) coincide with the starting values x 0 and x 1 of the numerical scheme (x 1 is
computed from x 0 and -
x 0 via the formula (2.2) with Using the non-resonance
22 E. HAIRER AND CH. LUBICH
assumption (5.10), the condition b
The formula for the first component of (2.2), x
with b
implies that
For the second component we have x
x
(2.2), and b x 2
\Delta , which after division by h sinc h! yields
O
The four equations (5.13), (5.14), (5.15) constitute a nonlinear system for the four
quantities y 1 (0), -
z
. By the implicit function
theorem and using the limited-energy assumption (4.2), we get a locally unique
solution for sufficiently small h, if the conditions (2.9) and (2.10) are satisfied.
The initial value for z 2 satisfies z 2 it follows from (2.10) that
by (5.12). This implies z 2
for . The other estimates (5.11) are directly obtained from (5.12). Conse-
quently, the values b x(nh) inserted into the numerical scheme (2.5) yield a defect of
size O(h N+2
Standard convergence estimates then show that on bounded time intervals
is of size O(t 2 h N ) in the first component and of size O(/(h!)t 2 h N ) in the second
component. This completes the proof of Theorem 5.1.
5.3. Frequency expansion of the derivative approximation. Under the
condition (5.10) we have h! 6= k- for integer k, so that the derivative approximation
xn is given by (2.6). We now define b x 0 (t) by the continuous analogue
Using condition (2.10), Theorem 5.1 implies that
on bounded time intervals. We next write the function b x 0 (t) as
e ik!t z 0k (t):
Inserting the relation (5.1) into \Gammai
which is equivalent
to (5.17), and comparing the coefficients of e ik!t we obtain
sinc (ihD) -
In particular, we get for z 1
2 that
z 01
cos !h
sin !h
Theorem 5.2. Under the assumptions of Theorem 5.1, the numerical solution
xn , given by (2.6), satisfies
uniformly for where the functions y
together with all their derivatives are bounded by
z 01
The constants symbolized by the O-notation are independent of ! and h, but depend
on E, N and T .
Proof. The estimates follow from (5.19) and from Theorem 5.1. For y 0
1 and
z 01
2 we use the formulas (5.12) to get the sharper result.
5.4. Energy along the numerical solution. In the Hamiltonian case
\GammarU (y), the total energy H and the oscillatory energy I are related to the frequency
expansion coefficients as follows.
Lemma 5.3. If the coefficients of the frequency expansions for xn and -
xn satisfy
(5.11) and (5.21), respectively, then
Proof. By definition (4.25) we have I(bx; b
e i!t z 1
e i!t z 1
the statement (5.23) follows from the fact that jv . The formula
(5.22) can be proved in the same way.
6. Almost-invariants of the numerical frequency expansion. In this section
we show that, in the Hamiltonian case the coefficients of the
frequency expansion of the numerical solution have invariants that can be obtained as
in Sect. 4. We denote
z k are the coefficients of the frequency expansion (5.1). Similar to (4.16) we consider
the function
where the sum is taken over all m - 1 and all multi-indices
non-vanishing integral components for which
It
then follows from Theorem 5.1 that the coefficients y and v k satisfy
r y
The factor OE(h!) in the defect of (6.3) is due to the presence of the factor OE(h!)z 2 in
the relations (5.12) defining the z-components.
The similarity of these relations to (4.17), (4.18) allows us to obtain invariants
that are the analogues of H and I of Sect. 4.
6.1. First invariant. As in Sect. 4.2 we multiply (6.2) and (6.3) by -
y T and
respectively, and we thus obtain
dt
Since we know bounds on z k and on its derivatives (Theorem 5.1), we switch to the
quantities z k and we get the equivalent relation
dt
We shall show that the left-hand side is the total derivative of an expression that only
depends on y, z k and its derivatives. Indeed, the term -
y T y (2l) can be written as
dt
Similarly, we get for z \Gammak that
Re -
z T
z
dt
z T
z
Re z T z
dt
z T z
z T
z
dt
z T
z
Im z T z
dt
z T z
z T
Hence, there exists a function b
which depends on the values at t of the
functions y and of their first N derivatives, such
that (6.4) reads
d
dt
This yields immediately the first statement of the following result.
Theorem 6.1. Under the assumptions of Theorem 5.1, the coefficient functions
y and
for
Proof. The formula for b
H 0 is obtained from the formulas (5.9) for L(hD
together with the estimates of Theorem 5.1.
Remark. Symplectic discretizations have
6.2. Second invariant. As in the proof of Theorem 4.3 we have for the function
of (6.1) that
Consequently, it follows from (6.3) that
\Gammai!
Written in the z variables, this becomes
\Gammai!
As in Sect. 6.1, the left-hand expression can be written as the total derivative of a
function b I 0 [Z](t) which depends on the values at t of the function Z and its first N
derivatives:
d
dt
I
Theorem 6.2. Under the assumptions of Theorem 5.1, the coefficient functions
I
I
for
with -(h!) as in Theorem 6.1.
Proof. From (6.5) and the estimates of Theorem 5.1, we obtain
Because of condition (2.10), this yields the stated formula for b I 0 .
7. Long-time energy conservation of the numerical discretization. We
are now able to prove the main result of this paper. This shows that the total energy
H and the oscillatory energy I are nearly conserved over time intervals of length
CN h \GammaN , for any N for which the non-resonance condition (5.10) is satisfied.
For the convenience of the reader we restate our assumptions:
ffl the limited-energy condition
26 E. HAIRER AND CH. LUBICH
ffl the boundedness condition (5.4) for the numerical solution sequence: \Phix n stays in
a compact subset of the domain of analyticity of g;
ffl the condition (5.3): h! - d ? 0;
ffl the conditions (2.9) and (2.10) on the numerical method:
ffl the non-resonance condition (5.10): for some N - 2,
Theorem 7.1. Under the above conditions, the numerical solution of (1.1) obtained
by the method (2.2)-(2.4) satisfies
nh. The constants symbolized by O(\Delta) are independent of n and of h and !
satisfying the above conditions, but depend on N .
Proof. (a) If we consider the linear combinations b
I 0 and
I 0 =-, it follows from Theorem 6.1 and Theorem 6.2 that
Moreover, by Theorem 6.1 and Theorem 6.2 together with Lemma 5.3 we have
I
where again b x(t) is defined by the frequency expansion (5.1) with coefficients y(t) and
defined by (5.17). The relations (7.1) and (7.2) hold only on finite
time on which the frequency expansion is defined.
(b) We now apply the above relations repeatedly on intervals of length h, for
frequency expansions corresponding to different starting values. As long as
satisfies the limited-energy condition (4.2) (possibly with a larger constant E), Theorem
5.1 gives us frequency expansion coefficients yn (t); Zn (t) corresponding to starting
values
Because of the uniqueness (up to O(h N+1 )) of the coefficients of the
frequency expansion, the following diagram commutes up to terms of size O(h N+1
numerical
method
(up to O(h N+1
The construction of the coefficient functions via (5.12) shows that also higher derivatives
of (y n ; Zn ) at h and (y n+1 ; Zn+1 ) at 0 differ by only O(h N+1 ). We thus have
from (7.3) and (7.1)
Using this relation repeatedly, we obtain
Moreover, from (7.2) we have the following for the coefficient functions corresponding
to the starting values
xn ) and
by construction, and b x 0
Theorem 5.2, we
obtain
which gives the desired bound for the deviation of the total energy along the numerical
solution. The same argument applies to I(xn ; -
Acknowledgment
. We are grateful to Sebastian Reich for drawing our attention
to the Fermi-Pasta-Ulam problem.
--R
Realization of holonomic constraints and freezing of high frequency degrees of freedom in the light of classical perturbation theory.
On the Hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms
A study of extrapolation methods based on multistep schemes without parasitic solutions
On the problem of energy equipartition for large systems of the Fermi-Pasta-Ulam type: analytical and numerical estimates
Numerical integration of ordinary differential equations based on trigonometric polynomials
The life-span of backward error analysis for numerical integrators
Energy conservation by St-ormer-type numerical integrators
Analysis by Its History
A Gautschi-type method for oscillatory second-order differential equations
Dynamical systems
--TR
--CTR
M. Van Daele , G. Vanden Berghe, Geometric numerical integration by means of exponentially-fitted methods, Applied Numerical Mathematics, v.57 n.4, p.415-435, April, 2007
Ernst Hairer, Important Aspects of Geometric Numerical Integration, Journal of Scientific Computing, v.25 n.1, p.67-81, October 2005
J. M. Franco, New methods for oscillatory systems based on ARKN methods, Applied Numerical Mathematics, v.56 n.8, p.1040-1053, August 2006
Paul J. Atzberger , Peter R. Kramer , Charles S. Peskin, A stochastic immersed boundary method for fluid-structure dynamics at microscopic length scales, Journal of Computational Physics, v.224 n.2, p.1255-1292, June, 2007 | Fermi-Pasta-Ulam problem;frequency expansion;backward error analysis;second-order symmetric methods;oscillatory differential equations;long-time energy conservation |
588460 | Iterative Substructuring Preconditioners for Mortar Element Methods in Two Dimensions. | The mortar methods are based on domain decomposition and they allow for the coupling of different variational approximations in different subdomains. The resulting methods are nonconforming but still yield optimal approximations. In this paper, we will discuss iterative substructuring algorithms for the algebraic systems arising from the discretization of symmetric, second-order, elliptic equations in two dimensions. Both spectral and finite element methods, for geometrically conforming as well as nonconforming domain decompositions, are studied. In each case, we obtain a polylogarithmic bound on the condition number of the preconditioned matrix. | Introduction
. Since the late nineteen eighties, interest has developed in non-overlapping
domain decomposition methods coupling different variational approximations
in different subdomains. The mortar element methods, see [10], have been designed
for this purpose and they allow us to combine different discretizations in an optimal
way. Optimality means that the error is bounded by the sum of the subregion-by-
subregion approximation errors without any constraints on the choice of the different
discretizations. One can, for example, couple spectral methods of different polynomial
degrees, or spectral methods with finite elements, or different finite element methods
with different meshes. Also, the domain partitioning need not be geometrically con-
forming, i.e. the intersection of the closures of two neighboring subdomains may only
be parts of certain edges of these subdomains.
The basic ideas of the mortar method can be outlined as follows: the skeleton of
the decomposition (i.e. the union of the subdomains interfaces) is itself partitioned
into mortars. Each mortar is an entire edge of one of the subdomains; the mortars
are disjoint open sets. The chosen local discretizations may force the method to be
nonconforming and we only impose a type of weak continuity. For each
and for each nonmortar side \Gamma j
k of
@\Omega k , we introduce a carefully chosen discrete space
~
kh of functions supported on \Gamma j
k . Weak continuity, in this context, then means that
the
k )\Gammaprojection of the jump across \Gamma j
k into the space ~
kh vanishes. In the first
version of the mortar method, strong continuity constraints were also imposed at the
vertices of the subdomains but this turned out not to be necessary. A second version
of the mortar method, developed and analyzed by Ben Belgacem and Maday [6],[7],
does not require such constraints. In particular for problems in three dimensions, the
second version offers important advantages over the first and in what follows, we shall
exclusively work with this more recently developed method. We note that, in a finite
element context, similar nonconforming methods have been studied by Le Tallec et al
INSA Rennes, 20 Av des Buttes de Coesmes, 35043 Rennes, France and CMAP, Ecole Polytechnique
91128 Palaiseau, cedex France. Electronic mail address: achdou@cmapx.polytechnique.fr
y Laboratoire ASCI, B"atiment 506, Universit'e Paris Sud, 91405 Orsay and Universit'e Paris 6,
Paris, France. Electronic mail address: maday@ann.jussieu.fr
z Courant Institute of Mathematical Sciences, 251 Mercer St, New York, NY 10012. Electronic
mail address: widlund@cs.nyu.edu. URL: http://cs.nyu.edu/cs/faculty/widlund/index.html. This
work was supported in part by the CNRS, while this author was visiting Universit'e Paris 6, in part by
the National Science Foundationunder Grant NSF-CCR-9503408, and in part by the U. S. Department
of Energy under contract DE-FG02-92ER25127.
.
Mortar element methods offer many advantages:
ffl They increase the portability of spectral methods.
ffl In the context of finite elements, they provide flexibility in the construction of
the mesh. For example, they may be used in some cases to avoid updating the finite
element mesh (sliding meshes [5]) or, on the contrary, to simplify the adaption of the
meshes ([9]).
ffl They are well suited for parallel computing.
There has already been several implementations of the mortar methods, among
them [5] with sliding meshes, [23] for spectral element methods, [18] for a nonconforming
finite element method for elasticity problems, and [4] for the Navier Stokes
equation.
In the present paper, we propose algorithms for solving the algebraic linear systems
arising from the mortar methods. After the elimination, in parallel, of the degrees
of freedom internal to the subdomains, there remains to find the traces of the solution
on the subdomain boundaries, i.e. to solve the Schur complement system. In our
methods, we work only with the true unknowns of the Schur complement systems, i.e.
the unknowns associated with the mortars and the vertices of the subdomains. The
method presented here can be viewed as a generalization of an iterative substructuring
algorithm first introduced by Bramble, Pasciak, and Schatz [12], for two-dimensional
conforming discretizations and which was reinterpreted in terms of block-Jacobi methods
in [14]. The algorithm consists essentially of decomposing suitably the discrete
space into a direct sum of subspaces in such a way that the related block-Jacobi preconditioned
conjugate gradient method has a satisfactory rate of convergence. Each
mortar can be associated in a natural way with a subspace but, in addition, a global
coarse space must be included to deal with the low frequency error. We obtain a
polylogarithmic bound, in terms of the local number of unknowns, on the number of
iterations required for a given accuracy. Therefore, the proposed algorithm can be
considered as almost perfectly scalable.
Other algorithms have also been proposed. A Neumann-Neumann preconditioner
is studied and tested in [18]. In [2], a saddle point formulation of the system, as in
[6], is considered, and iterative methods based on a certain class of preconditioners is
suggested. A saddle point algorithm for which the internal degrees of freedom need
not be eliminated is proposed in [16]. In [13], a method based on a hierarchical basis
representation, cf. [24], is developed and tested for low order mortar finite elements
and geometrically conforming decompositions of the regions.
In three dimensions, the preconditioner of this paper is not satisfactory, and another
iterative substructuring method has been proposed; see [22]. In addition, an extension
of the theory for two-level Schwarz algorithms, using overlapping subregions,
has been completed for the mortar finite element case; see [26].
The paper is organized as follows. In Section 2, a brief review is given of the
mortar finite element method in the geometrically conforming case. An iterative substructuring
preconditioner for that case is proposed and studied in Section 3. The
geometrically nonconforming mortar finite element method is discussed in Section 4.
Finally, a generalization to the spectral element method in the geometrically conforming
and geometrically nonconforming cases is carried out in Section 5.
2. Mortar Element Methods in the Geometrically Conforming Case.
Let\Omega be a bounded polygonal domain of IR 2 , and let
k=1 be a partition
of\Omega
into K non-overlapping open quadrilaterals:
We make this restriction to polygonal domains and subdomains only to simplify the
presentation. The domain decomposition is called geometrically conforming if the
intersection of the closure of two subdomains is either empty, a vertex, or an entire
common edge of the two subdomains. For any 1 k 6= ' K, let \Gamma k' be the closed
straight segment, possibly degenerate, given by \Gamma k'
us also introduce
V as the set of crosspoints of the domain decomposition which are not on @
\Omega\Gamma and the
skeleton, defined by
We assume that the subdomains have uniformly bounded aspect ratios but there
is no need to assume that the subdomains form a quasiuniform coarse triangulation.
All what follows concerns the Dirichlet problem for Poisson's equation
(1)
but our results hold for any self-adjoint, elliptic, second order operator.
Families of finite element triangulations T k;h are associated with
K, which we assume satisfy the classical shape regularity assumption on the elements.
We denote by h k the maximum diameter of the elements of T k;h . To simplify our
analysis, we also assume that the meshes are quasiuniform for each
recall that quasiuniformity for a triangular mesh means that there exist two positive
constants and oe such that for all triangles T of T k;h , h k hT oeae T . Here hT is
the diameter of T , and ae T the diameter of the circle inscribed in T . Let X kh be the
related space of piecewise linear continuous finite element functions which vanish on
@
Denoting by T r k the trace operator
The product spaces X h and X h are defined by:
Y
Y
If j\Gamma k' j 6= 0, we also introduce X k;';h by
Clearly, X k;';h is a subspace of the space
X k;';h of the piecewise linear continuous
functions on the corresponding mesh of \Gamma k' (X
;). The
dimension of
denoted by N k' the number of nodes of T k;h on \Gamma k' .
We note that the meshes need not match at the interface between two subdomains.
Thus, in order to discretize the space H 1
0(\Omega\Gamma8 we have to introduce, for each 1 k !
space ~
W k;';h of Lagrange multipliers used to impose a weak
continuity constraint across \Gamma k' . A choice has to be made since this space of Lagrange
multipliers can be associated with either
X k;';h or
One strategy is always to
choose the one of largest dimension, but we emphasize that any other choice can also
be supported by existing theory, and that the same asymptotical error bound results
in all cases.
In the case where the Lagrange multiplier space ~
W k;';h is based on
be the shape functions of X k;';h associated with the nodes of
T k;h on \Gamma k' , with OE 0 and OE Nk'+1 associated with the endpoints of \Gamma k' . Then, ~
W k;';h is
Fig. 1. Shape functions of the spaces
chosen as the space spanned by (OE it is a
subspace of
X k;';h of codimension two. Figure 1 illustrates the construction of ~
from
It is now possible to define the subspace Y h of
R
oe
and the subspace Y h of
ae
Z
(v
oe
Consider an edge Assuming that the Lagrange multiplier space ~
is built from the mesh T kh , then the nodes of T are called
slave and master nodes, respectively, because the value of v h 2 Y h at any slave node is
completely determined by the values at the master nodes and crosspoints. Assuming
that j\Gamma k' j ? 0, then the edge
is said to be a slave, or nonmortar, and
master, or mortar, edge
respectively, if the space ~
W k;';h is based on the mesh
T kh and T 'h , respectively. We denote by NV the number of degrees of freedom at the
crosspoints of the domain decomposition, and by Nm and N s the number of master
and slave nodes, respectively. Then, the dimension of X h and Y h are NV
and NV +Nm , respectively.
IR; be the bilinear form:
Z\Omega
The discretized problem corresponding to (1) is: Find u h 2 Y h such that
Z\Omega
It is natural to introduce two subspaces which are orthogonal in the sense of this
energy inner product. The first, X ffi
consists of functions which vanish on the
interfaces, i.e. X ffi
0g. The other is the
subspace of the discrete harmonic extensions ~
i.e. the unique
solution ~ u h of
a(~
~
(2)
In order to reduce the size of the problem, it is possible to solve, in parallel, a discrete
Dirichlet problem for each subdomain, i.e. to find u
h such that
Z\Omega
Defining the bilinear form s corresponding to a discrete Poincar'e-
Steklov operator
there remains to find u h 2 Y h such that
The solution of (2) is then given by u
The goal of the next section is to find a basis of Y h for which a block diagonal
preconditioner for S h yields a condition number almost independent of the mesh
parameters.
3. Preconditioners for the Geometrically Conforming Mortar Element
Method. In the following c and C will denote positive constants uniformly bounded
away from 0 and 1, respectively. They are, in particular, independent of the H k and
the diameters of the
subdomain\Omega k and its elements, and in the spectral case, of
the degree of the polynomials.
3.1. Decomposition of the space Y h . The purpose of this section is to decompose
the vector space Y h into the direct sum of a coarse space YH of dimension
NV (the number of degrees of freedom associated with the crosspoints) and of a fine
space Y H
h of dimension Nm (the number of master nodes):
Here
A few notations will be needed in order to specify the coarse space YH . Let A be a
crosspoint and let KA denote the set
It is clear that
cardinal (KA ). For each crosspoint A, and for each k 2 KA ,
we define a basis vector e A;k 2 Y h , such that,
1. e A;k
2. for all vertices B 6= A
3. for all ' 6= k, and for all vertices B
4. e A;k is linear on master edges.
For a given crosspoint A 2
clearly vanishes on all edges except those
which have A as an endpoint. Let A and B be the endpoints of In
the case where \Gamma k' is a master side of
@\Omega k , the restriction of e A;k
k to \Gamma k' is the linear
function ~ e k;';A with the value 1 at A and 0 at B. The restriction of e A;k
l to \Gamma k' is the
unique function ~
';k;h such that
R
If conversely, \Gamma k' is a slave side of
@\Omega k , the restriction of e A;k
l to \Gamma k' is 0, while
the restriction of e A;k
k to \Gamma k' is the unique function ~ e k;';A in X k;';h such that,
~
Z
~
e k;';A
The coarse space YH is defined by
It is clear that the dimension of YH is NV .
In what follows, it will be necessary to have accurate estimates of certain Sobolev
norms of the basis vectors of the coarse space YH .
Lemma 1. Let A be a crosspoint and k; ' 2 KA , with j\Gamma k' j ? 0. Assume that
is a slave side
of\Omega k and let ~ e k;';A be the unique function in X k;';h defined
by (6). Then,
Proof. Let ~
E be the vector of coordinates of ~
e k;';A in the previously described basis
of shape functions of X k;';h associated to the nodes of T k;h which lie on
. Using the boundary values for ~
e k;';A , we find that the vector ~
satisfies:
~
~
l 13
l
l N kl \Gamma23
(l N kl \Gamma2
l N kl \Gamma13
l
Here l i is the length of the i-th mesh interval of \Gamma k' . Since the mesh T kh is quasiuniform,
~
B is spectrally equivalent to the diagonal matrix D j h k I and therefore the Euclidean
norm of the vector ~
E is of order 1, since the Euclidean norm of F is of order h k .
Therefore, (8) is proved. The next inequality, (9), now follows by using a well known
inverse inequality for quasiuniform meshes. Finally, (10) is obtained from (8) and
and the Gagliardo-Nirenberg interpolation inequality.
Remark 1. In the same way, we can also prove the same estimates for the
function ~ e ';k;A \Gamma ~
e k;';A when the side
is a master side
of\Omega k .
Remark 2. From (10) and Remark 1, it follows immediately that 8A 2 V; 8k 2
KA ,
Denoting by j:j 1=2; the product semi-norm on
useful to have bounds of je A;k j 1=2; for any A 2 V and k 2 KA . Three cases can be
1. Both sides
adjacent to A are slave sides.
2. Both sides
adjacent to A are master sides.
3. One side
adjacent to A is a slave side, the other a master side.
In the third case, we have the following result.
Lemma 2. Let A be a crosspoint and let k;
Assume that
is a slave side
of\Omega k and that
master side
of\Omega k . Then,
je A;k
Proof. From the quasiuniformity of the mesh T kh , there exists a constant C such
that for all ffl 2 (0; 1=2),
je A;k
Ch \Gamma2ffl
Let f k;m;A and f k;';A be the functions on
@\Omega k , which coincide with e A;k
respectively, and with 0 on
@\Omega k n\Gamma km and and
respectively. It is then
clear that at all mesh points which are not vertices,
e A;k
The semi-norm jf k;m;A j 2
can be computed explicitly, because f k;m;A is
piecewise linear, and the following bound is obtained:
It now follows from (8) and an inverse inequality that
Ch 2ffl
Choosing
combining (13) and (14), we obtain the desired
result by using (12).
The next lemma is proved in the same way as Lemma 2.
Lemma 3. Let A be a crosspoint and let k; ' 2 KA , with j\Gamma k' j ? 0. Assume that
is a master side
of\Omega k . Then,
je A;k
It is also possible to prove the following result for the first and second cases:
Lemma 4. Let A be a crosspoint and let k;
Assume that the sides
are either both master or
both slave sides
of\Omega k . Then,
je A;k
C:
Proof. The result is very easy when both sides are master sides, because e A;k
is then continuous and piecewise linear on
@\Omega k . When both sides are slave sides, it
follows from Lemma 1 that
and the proof is completed by using an inverse inequality.
Lemmas 2-4 can be summarized in the following corollary:
Corollary 1. Let A be a crosspoint and let k 2 KA . Then,
3.2. A block-Jacobi preconditioner.
Let ~
S be the matrix of S h in the new basis described above. The matrix ~
S can be
written as
~
ShH
~
~
In order to design a preconditioner for ~
S, we replace the block ~
ShH by 0, and the
block ~
S hh by its block diagonal part with one block for each mortar. The resulting
preconditioner is "
S with
In this section, we will develop bounds for the condition number of the preconditioned
S. For that purpose, the following well known result will prove useful:
There exist two constants c and C such that
see, e.g., [11], in particular the discussion of an extension theorem for finite element
spaces. Let " s h be the bilinear form corresponding to the matrix "
S. The following
lemma gives an upper bound for the eigenvalues of "
Lemma 5. There exists a positive constant C such that
Proof. Consider an element v h 2 Y h . There then exists a unique pair (v H
h \Theta YH such that
Obviously,
Observing that for any x 2 \Gamma, there is a uniform bound on the number of subspaces
with elements which do not all vanish at x, we deduce that there exists a constant C
such that
In addition,
To find a lower bound for the eigenvalues of "
S, the following lemma is needed:
Lemma 6. There exists a constant C such that
is the coarse space component of v h .
Proof. Consider a vector v h 2 Y h , and let (v H
h \Theta YH be given as in
(17). Then,
Consider specifically the
subdomain\Omega ' and denote by fV i g 1iNV;' the vertices of
and by
1iNV;' the subdomains adjacent
to\Omega ' . We choose a numbering such
that
joins the crosspoints V i and V i+1 .
It is clear that
vH
Denote by w'H the continuous piecewise linear function on
@\Omega ' which interpolates
v 'h at the V i . We can then write v'H as
\Gamma k' is a slave side of
(v kh
Here ~ e ';k;V i is defined by (5). Proceeding exactly as in [12], we can prove that
In addition, since v h
denotes the mean value of v kh over the edge \Gamma k' . Therefore,
k' is a slave side of
But, see, e.g., [15],
and
In addition, from Lemma 3,
which gives the desired result
since\Omega ' has a uniformly bounded number of neighbors.
We can now prove a lower bound for the eigenvalues of "
Theorem 1. There exists a constant C such that
Proof. Consider a vector v h 2 Y h , and let (v H
h \Theta YH be given by (17).
It is clear that
C
We now focus on the term jv
. Using exactly the same arguments as
in [12],[15], it is possible to bound this expression by
which completes the proof of the theorem. To make our paper more self contained, we
will outline a proof of this result.
Assume that \Gamma k' is the segment (0; H). By the definition of the H 1=2
x
dx:
Clearly,
and it is possible to use Lemma 6.
Since the last two terms of (21) are very similar, we concentrate on the first. As
in [15], this integral is split into two, over (0; h k ), respectively. It is easily
seen that
x
and that
Z hkjv kh
x
From (11), it follows that
Thus, from (22),(23), and (24),
We now use the following very important property of the projection into the coarse
space: the component v kH
depends only on v kh
and v lh
, and, for any c 2 IR,
c is associated through this mapping to v kh
Recalling
that
, and choosing c =! v kh
, we find,
ck 2
ck 2
as in (20).
We can now obtain a bound on the condition number of "
Theorem 2. There exists a constant C such that
Remark 3. In order to design a convenient and inexpensive preconditioner,
we should replace the blocks of "
S hh in a suitable way. The preconditioners defined
above can be simplified in two ways: first the fine space blocks can be replaced by
more convenient matrices by using for instance hierarchical bases as described in [24]
and [13]. Another possible simplification is crucial for parallelism: it makes sense to
replace the block "
S hh of the preconditioner, related to the fine space, by a matrix
corresponding to a bilinear form s
h \Theta Y H
constructed as follows: Each
h is mapped to v h 2 X h given by
on the mortar sides,
on the nonmortar sides,
For the resulting preconditioner, it is easy to prove, by using the stability result of Ben
Belgacem [6], Lemma 1, that the condition number estimate (25) remains valid in the
geometrically conforming case. A full discussion will be given, in Subsection 4.3, of
the geometrically nonconforming case.
4. Preconditioners for the Geometrically Nonconforming Mortar Element
Methods.
4.1. The geometrically nonconforming mortar element method. In this
section, we turn to the mortar element method in the case when the decomposition
is no longer geometrically conforming. We will still assume that the aspect ratios
of the subdomains are bounded by a positive constant, and we recall that H k is the
diameter of the
subdomain\Omega k . We also assume that there exists a constant c such
that if
Before formulating the discrete problem, we will adapt some of our previous notations
and introduce some new ones. For
denote the edges
of
@\Omega k . Among the set of all edges
we select a family
of mortars ffl m g 1mM , satisfying the following three conditions:
1. [
2. 8(m; n)
3. 8m 2 there exists k(m); j(m) such that
k(m) .
Denoting by X j
kh the vector space of the traces on \Gamma j
k of the functions of X kh , we
introduce the vector space W h
Y
As in the geometrically conforming case, let us introduce the space
kh of the piecewise
linear continuous functions on the corresponding mesh of \Gamma j
k . Then ~
kh denotes the
subspace of
kh of the functions which are constant in the two end segments of T kh "\Gamma j
k .
The nonconforming approximation of H 1
0(\Omega\Gamma is given by the space
if 9m such that (k;
else
R
We can also introduce the trace space Y h
As in Section 2, the edge
k is called a mortar or master side
of\Omega k if there exists
Mg such that (k; and a nonmortar or slave side
otherwise.
Again, the unknowns interior to each subdomain can be eliminated by solving, in
parallel, one Dirichlet problem for each subdomain, and we are led to the problem of
solving (4). As in the previous section, the goal is to find a basis of Y h for which a
block-Jacobi preconditioner leads to condition numbers which are almost independent
of the size of the subdomains and elements. As in Section 3, the preconditioner will
consist of a coarse space block and a block for each mortar.
4.2. Decomposition of the vector space Y h . As in Subsection 3.1, we decompose
the vector space Y h into the direct sum of a coarse space YH of dimension NV
(the number of degrees of freedom associated with the crosspoints) and a fine space
h of dimension Nm (the number of master nodes). Thus,
where
A basis of YH is defined as follows. For each vertex A, and for each k 2 KA , the basis
vector e A;k 2 YH is fully determined by the following four conditions:
1. e A;k
2. for all vertices B 6= A
3. for all ' 6= k, for all vertices B
4. e A;k is linear on the master edges.
As in Subsection 3.1, the coarse space YH is defined by
Consider first a vertex A
k is a slave side
of\Omega k and let B be the other
end point of
k . Then,
e A;k
e A;k
R
e A;k
Exactly as in Lemma 1, we can prove that
C
Cp
C:
Assume now that \Gamma j
k is a master side
Let ~
e A;k
be the trivial extension of e A;k
k . Just as in Lemma 1, and
Remark 1, we can prove that
To give a flavor of the proof, let us consider the case depicted in Figure 2:
Let C 0 and C 1 be the endpoints of \Gamma i
' and let D 0 and D 1 be the endpoints of
the mesh segment containing the crosspoint A. We introduce the continuous function
~
e A;k;' defined on \Gamma i
' , which is piecewise linear on the mesh of \Gamma i
' , and satisfies
~
k on (D
linear on (D
l
Fig. 2.
In turn, ~ e A;k;' is split into the sum of two piecewise linear functions ~ e A;k;'
1 and ~ e A;k;'such that
~
e A;k;'
e A;k;'
where
~
e A;k;'
1 linear on (D
Let h be the L 2 -projection onto X i
'h . It is clear that
e A;k
Therefore,
It is clear that k~e A;k
' and from the L 2 stability of h , the first
term of the right hand side of the inequality above is bounded by C
h ' . Then, an argument
as in Remark 1 yields the same bound for the second term k(I \Gamma h )~e A;k;'
From the observation (30), it is possible to prove the following lemma in the same
way as Lemma 3.
Lemma 7. Let A be a crosspoint and let \Gamma j
k be a master side
of\Omega k with an end
point A. Let ' 6= k, and let
je A;k
As in Subsection 3.2, let ~
S be the matrix of S h in the new basis described above.
Again, the matrix ~
S can be described by formula (15) and it is possible to define a
block diagonal preconditioner "
S by (16). The bilinear form related to "
S is called
An upper bound for the eigenvalues of "
S is given by a counterpart of Lemma 5,
which is proved exactly as for the geometrically conforming case. To find a lower
bound, we have to prove an analogue of Lemma
Lemma 8. There exists a positive constant C such that
where v H is the projection of v h on YH along Y H
h .
Proof. Consider a vector v h 2 Y h , and let (v H
h \Theta YH be given by (17).
One can check that
We focus on one subdomain denoted
wH be the
continuous piecewise linear function on
@\Omega 0 taking the same values as v 0h at the
vertices of
@\Omega 0 . v 0H can be rewritten as
0 is a slave side of
~
where the functions ~
H will be specified below.
Let us focus on a single slave side \Gamma j
0 of
which we call
convenience. The related space of Lagrange multipliers ~
0h is denoted ~
W flh . Let
be the subdomains such that jfl "
in such a way that
and\Omega k+1 are adjacent. In the rest of the proof, we assume that k(fl) ? 1, but the
results also hold for slight modification, which will not be discussed
here. Also denote by the side of
adjacent to fl, as shown in Figure
3:
Fig. 3. The side fl of
@\Omega 0 and adjacent subdomains
The length of fl and fl k are called d fl and d fl k , respectively.
A calculation shows that
~
where
and
where the function wH has been extended linearly outside fl. Let us further focus
on the term fi k e Ck ;k
0 can be estimated in the same way. For
can be rewritten as
(v kh (C k
For simplicity only, we assume that for 0 k k(fl), the intersection of fl and fl k
contains the support of at least one basis function of the Lagrange multiplier space
related to fl; this hypothesis can be eliminated by decomposing the quantities v kh (C k )\Gamma
it is possible to choose
a nonnegative h 2 ~
W flh supported in fl k , such that
R
R
R
R
R
R
Therefore, as in the proof of Lemma 6,
Additionally, because of (28), we can prove the following estimate:
je Ck ;k
which is slightly stronger than (31).
For k(fl), the ratio d(A;Ck )
is smaller than one and the estimate
follows from (32) and (33). For the situation is somewhat more difficult
because d(A;Ck )
may be large; however, in this case je C k(fl) ;k(fl)
is bounded
by C
h0 )). Therefore, we find that
Exactly as for the geometrically conforming case, it is now possible to prove the
following result:
Theorem 3. There exists a constant C such that
4.3. The fine space block "
S hh . To simplify the implementation, it makes sense
to replace the block "
S hh of the preconditioner, related to the fine space, by a matrix
corresponding to a bilinear form s
h \Theta Y H
constructed as follows: Each
h is mapped to v h 2 X h given by
on the mortar sides,
on the nonmortar sides,
and
The related preconditioner is called
S, and it is easy to prove that
To obtain a condition number estimate, we first state the following lemma.
Lemma 9.
Let\Omega 0 be a subdomain and let fl be a nonmortar side of
@\Omega 0 . Let
1kk(fl) be the mortars adjacent to fl; here fl k is a whole side of
@\Omega k . Then,
there exists a constant C such that 8v h 2 Y H
Proof. Let the function ~ v h be defined on [ 1kk(fl) fl k by the restrictions of v kh to
Let be the mortar projection which maps ~ v h to v 0h j fl . As in the proof of Lemma
8, we set w be the piecewise linear continuous
function which interpolates ~ v h at (C k ) 0kk(fl) , A and B. Of course, w vanishes at all
the C k and thus depends linearly only on the two parameters v 1h (A) and v k(fl)h (B).
When necessary, we shall use the notation w(\Delta; \Delta).
It is clear that
Let us first bound w. Using the same arguments as in Lemma 7, we find that the
square of the H 1=2(fl)-norm of w(0; 1) and w(1; are bounded by C(1
Additionally,
Therefore,
There remains to estimate (~v h \Gamma w). From the stability of the operator , given in
Lemma 1 of [6], we have
C
The right hand side of (36) can be bounded using the same argument as in the proof
of Theorem 1, and we obtain
The proof of the lemma follows by using (35), (36), and (37).
The following corollary is an analogue of Lemma 5:
Corollary 4.1. There exists a positive constant C such that
The next result follows from Corollary 4.1, the analogue of Theorem 1 for the
geometrically nonconforming case, and (34):
Theorem 4. There exists a constant C such that
5. Preconditioners for the Mortar Spectral Method.
5.1. The geometrically conforming case. For simplicity only, we shall now
assume that the
subdomains\Omega k are rectangles with sides parallel to the coordinate
axes. We first give a brief review of the mortar spectral method; see [7] for more
details. For a review on the analysis of the Legendre spectral methods, see, e.g., [8].
Consider a family of integers fN k g k2f1:::Kg , all greater than two. Let X kN be the
space of polynomials
on\Omega k of degree N k in each space variables, which vanish on
\Omega\Gamma and let X kN be the space of traces of functions of X kN on
@\Omega k . A product
space is defined by
Y
For each
subdomain\Omega k , a quadrature formula is given, by the Gauss-Lobatto-
Legendre formula on (\Gamma1; 1), an affine transformation, and tensorization. We denote
the corresponding nodes and weights by i;j;k and ! i;j;k , and by
GL;k the quadrature
GL;k
The nodes i;j;k define a grid T kN
on\Omega k . A discrete bilinear form is given by
GL;k
Consider two adjacent
and\Omega ' and let L k' be equal to N k or N ' . Let
and denote by ~
W k;';N the space of polynomials of degree N k' . The
subspace YN of XN is now introduced by
R
oe
and the subspace YN of XN by
R
(v
oe
Slave and mortar sides are introduced exactly as in Section 2, and the Poincar'e-Steklov
bilinear form s can be introduced in terms of the problem of finding
YN such that
GL;k
f ~
Here
N is obtained by solving the analogue of (3) and ~ vN is the discrete harmonic
extension of vN .
As before, we look for a decomposition of the space YN into the direct sum of
a coarse space YH , of dimension NV , and a fine space Y H
N . For each crosspoint A
and for each k 2 KA , the basis vector e A;k of YH is fully determined by the following
conditions:
1. e A;k
2. for all vertices B 6= A
3. for all ' 6= k, for all vertices B
4. e A;k is linear on the master edges.
Suppose first that \Gamma k' is a slave edge of
@\Omega k . We can then check, using coordinates
such that \Gamma with the value H corresponding to A, that
e A;k
e A;'
Here Ln is the Legendre polynomial of degree n. From (39) and (40), it is straight-forward
to show that C: The next lemma provides estimates of the
and H 1=2
\Gammanorms of e A;k .
Lemma 10.
log(N k' )
Proof. Without loss of generality, we can assume that \Gamma k' and to simplify
the notations, we also set . Recalling that
it is easy to show that
In order to evaluate the H 1=2 -norm of ~
e k;';A , we compute the H 1=2 -norm of Ln . By
the definition of the norm, we have to estimate
dxdy:
Integration by parts, in the case where n is even, leads to
x\Gammay
x\Gammay
x\Gammay
1\Gammay
\Gamma1\Gammay
x\Gammay
and similarly, in the case where n is odd,
Hence,
dxdy
1\Gammay 2 dy n even;
1\Gammay 2 dy n odd:
In order to estimate the last terms in this formula, we use Gaussian quadrature based
on the roots (i i ) 1in of Ln . It is well known that there exists positive quadrature
weights (! i ) 1in such that
We therefore obtain
We now recall, see [25],(thm 6.21.3 & (15.3.14)) that
and that there exist two positive constants c and C such that
c
We find that
dy
We recognize this last expression as a Riemann sum of ( 1
dy Clog(n):
The same argument leads to
dy Clog(n);
which in turn leads to
x\Gammay
x\Gammay dxdyj Clog(n):
Finally, we remark that
Here (Ln (x)\GammaL n (y)\Gamma(x\Gammay)L 0
(x\Gammay) 2 is a polynomial of degree ! n in both x and y. The
orthogonality of the Legendre polynomials then leads to
By using (41), we find that
dxdy Clog(n):
Thus, the square of the H 1=2 -norm of Ln is bounded by Clog(n) and so is the square
of the H 1=2 -norm of ~
The next result is the analogue of Lemma 2:
Lemma 11. Let A be a crosspoint and let k;
Assume that
is a slave side
of\Omega k and that
master side
of\Omega k . Then,
je A;k
Proof. Since we are interested in the H 1=2
is possible to
rescale the problem and assume that the edge \Gamma k' is (\Gamma1; 1). We again set
We first use the following inverse inequality,
je A;k
Cn 4ffl je A;k
see [8].
Let us denote by f k;m;A and f k;';A the functions on
@\Omega k which are equal to e A;k
respectively, and vanish on
@\Omega k n\Gamma k' , respectively. It
is clear that, almost everywhere,
e A;k
The semi-norm jf k;m;A j 2
can be computed explicitly, because f k;m;A is
piecewise linear, and the following estimate is obtained:
To evaluate the contribution of f k;';A , it is enough to estimate the following quantity
Here is the characteristic function of (\Gamma1; 1). It can be proved, as in Lemma 5.1,
that the contribution of
jx\Gammayj 2\Gamma2ffl is bounded by C(1+log(n)). There
remains to estimate
This is done as follows:
Hence,
je A;k
Choosing
log(n)
yields
je A;k
Let ~
S be the matrix of s N in the new basis described above. The matrix ~
S can
be written as
~
~
~
In order to design a preconditioner for ~
S, we replace the block ~
SNH by 0, and the block
~
SNN by its block diagonal with one block for each edge. The resulting preconditioner
is
s N be the bilinear form corresponding to the matrix "
S. The bound,
is obtained exactly as in Lemma 5.
The following lemma will also be needed:
Lemma 12. There exists a constant C such that 8v kN 2 X kN ,
Similarly, if x 0 is a point of
Proof. To prove (43), we first rescale so that the diameter
then prove that
This formula is true for any function and can be derived by using a Fourier transform
argument. From an inverse inequality for polynomials, we also have
Choosing
yields the desired result. Finally, (44) is obtained by a standard
quotient space argument.
Given Lemma 12, analogues of Lemma 6 and Theorem 1 can be obtained as in
the finite element case. The only notable difference is when estimating the integral
\GammaH
cf. (21)-(24) in the proof of Theorem 1. The integral is again split into two, over
respectively. Here the ( i ) i2f0;Nkg are the Gauss-Lobatto
points in [\Gamma1; 1]. We note that 1
) and that an inverse inequality for
polynomials of degree n is given by
Finally, since all necessary technical tools are at hand, the proof of the following
main result can be obtained
Theorem 5. There exists a constant C such that
5.2. The geometrically nonconforming case. The previous mortar spectral
element method can be generalized straightforwardly to the case where the assumption
of geometrical conformity is relaxed. The analysis of our preconditioner requires, in
Lemma 5.5 below, a bound on the relative size of the intersection of two edges; in
order to provide estimates which only depend polylogarithmically on the parameter
N , we make the assumption that
c;
a condition less standard and more stringent than (28). The mortar method is based
on the definition of the master and slave sides as described in Section 4 in the finite
element context. The projection that gives slave values in terms of those on the master
edges is defined as in Subsection 5.1. The iterative substructuring algorithm now
considered is the same as that of Section 4. We now have the new tools of Subsection
5.1 at our disposal, and in order to prove a result equivalent to Theorem 3 for the
spectral approximation, we additionally only have to derive an analogue of Lemma 7.
We do so after some preliminary work.
The definition of the mortar condition naturally leads to the introduction of a
projection operator
N (\Gamma1; 1); given by
This projection operator has already been analyzed in [7] and certain stability and
approximation properties are given in that paper. While we do not now know how to
prove a uniform bound for the H 1=2
00 -norm of this operator, the following results will
allow us to derive a strong result on our preconditioner.
Lemma 13. Let a be any real number in a denote the piecewise
linear, discontinuous function on that vanishes on linearly
from 1 and 0 over the interval [a; 1]. There then exists a constant C such that
log(N
Proof. Let 'N be a polynomial of degree N in one variable that vanishes at \Sigma1.
It is easy to see that it can be written as
an
Recalling that the Legendre polynomials Ln are L 2 (\Gamma1; 1)-orthogonal, with norm
and that they satisfy the differential equation
d
dx
we can first show that
a 2
Using now the integral relation
and the differential equation once more, we find that
an
Here we have used the L 2 \Gammaorthogonality of the Ln and the formula
an
(a
Let us denote by [IP N (\Gamma1; 1)"L 2 (\Gamma1; 1); IP N (\Gamma1; 1)"H 1
the interpolation
space, with index ', between the space of polynomials IP N (\Gamma1; 1) of degree N provided
with the L 2 -norm and the H 1
respectively. By a main result of interpolation
theory [20], it follows from our two estimates of the norms of 'N that there exists a
constant C such that, for all 'N 2 IP N (\Gamma1; 1),
C
We recall now that, according to [21], [IP N (\Gamma1; 1)"L 2 (\Gamma1; 1); IP N (\Gamma1; 1)"H 1
coincides topologically with IP N (\Gamma1; 1) provided with the H '
-norm. In particular, we
can conclude that
C
In order to complete the proof, we now evaluate the coefficients an of the decomposition
of given by (46). By using the definition of a , we find that
a (x)L 0
a
a Ln (x)
Combining the integral and the differential formulae for the Legendre polynomials,
we deduce
(a)
(a)
From [1], (22.14.9), we know that
jLn (a)j C
As another consequence of the integral formula, we find that
which yields
(a)
Finally from (48),(49), and (50), we deduce that
By using (46), we obtain
Thus,
an
and hence
By introducing this bound into formula (47), we can conclude that
log(N
It is now an easy matter to state an analogue of the Lemma 7.
Lemma 14. Assume that (45) holds, let A be a crosspoint, and let \Gamma j
k be a master
side
of\Omega k with A as an end point. Let ' 6= k,
je A;k
Proof. Let us denote by ' the function obtained from by a scaling which maps
' with the further property that '
B are the end points of \Gamma i
k . By construction, e A;k
continuous and it even
belongs to H 3
2 \Gamma" for any positive ". We decompose e A;k
' into the following sum
e A;k
supplies the desired bound for the first term j '
(@\Omega l ) . The bound
on the second, j '
(@\Omega l ) , is obtained as in the geometrically conforming
case.
Lemma 14, we are ready to prove the following result using the same arguments
as in Section 4.
Theorem 6. Under the assumption (45), there exists a constant C such that
--R
Handbook of mathematical functions
Substructuring preconditioners for the Q 1 mortar element method
M'ethodes it'eratives de sous-structuration pour les 'el'ements avec joints
A fast solver for Navier-Stokes equations in the laminar regime using mortar finite element and boundary element method
On the mortar element method: generalization and implementation
The mortar finite element method with Lagrange multipliers
A spectral methodology tuned to parallel implementations
Approximations Spectrales de Probl'emes aux Limites Ellip
Raffinement de maillage en 'el'ements finis par la m'ethode des joints
A new nonconforming approach to domain de- composition: the mortar element method
Iterative methods for the solution of elliptic problems on regions partitioned into substructures
The construction of preconditioners for elliptic problems by substructuring
A hierarchical preconditioner for the mortar finite element method
Some domain decomposition algorithms for elliptic problems
Domain decomposition algorithms with small overlap
Efficient iterative solvers for elliptic finite element problems on nonmatching grids
Domain decomposition methods in computational mechanics
Domain decomposition with non matching grids: Augmented Lagrangian approach
Probl'emes aux Limites non Homog'enes et Applications
Rel'evement de traces polynomiales et interpolations hilbertiennes entre espaces de polyn
In preparation.
Nonconforming discretizations and a posteriori error estimators for adaptive spectral element techniques
A domain decomposition algorithm using a hierarchical basis
--TR
--CTR
C. Cinquini , P. Venini, A mortar approach for the analysis and optimization of composite laminated plates, Computational structures technology, Civil-Comp press, Edinburgh, UK, 02
Ilaria Perugia , Dominik Schtzau, On the Coupling of Local Discontinuous Galerkin and Conforming Finite Element Methods, Journal of Scientific Computing, v.16 n.4, p.411-433, December 2001
Dan Stefanica, FETI and FETI-DP Methods for Spectral and Mortar Spectral Elements: A Performance Comparison, Journal of Scientific Computing, v.17 n.1-4, p.629-638, December 2002
Micol Pennacchio, The Mortar Finite Element Method for the Cardiac Bidomain Model of Extracellular Potential, Journal of Scientific Computing, v.20 n.2, p.191-210, April 2004
Feng , Catherine Mavriplis , Rob Feng , Rupak Biswas, Parallel 3D Mortar Element Method for Adaptive Nonconforming Meshes, Journal of Scientific Computing, v.27 n.1-3, p.231-243, June 2006
T. P. Mathew , G. Russo, Maximum norm stability of difference schemes for parabolic equations on overset nonmatching space-time grids, Mathematics of Computation, v.72 n.242, p.619-656, 1 April | domain decomposition;mortar finite element method;iterative substructuring |
588473 | Analysis of Numerical Errors in Large Eddy Simulation. | We consider the question of "numerical errors" in large eddy simulation. It is often claimed that straightforward discretization and solution using centered methods of models for large eddy motion can simulate the motion of turbulent flows with complexity independent of the Reynolds number and dependence only on the resolution ``$\delta$'' of the eddies sought. This report considers this question analytically: Is it possible to prove error estimates for discretizations of actually used large eddy models whose error constants depend only on $\delta$ but not Re? We consider the most common, simplest, and most mathematically tractable model and the most mathematically clear discretization. In two cases, we prove such an error estimate and try to explain why our technique of proof fails in the most general case. Our analysis aims to assume as little time regularity on the true solution as possible. | Introduction
The laminar or turbulent
ow of an incompressible
uid is modeled by solutions (u; p) of the
incompressible Navier-Stokes equations:
in
r
in
in
@
Z
(1)
Here
is a bounded, simply connected domain with polygonal boundary
d is the
uid velocity, p
R is the
uid pressure,
f(x; t) is the (known) body force, u 0 (x) the initial
ow eld and Re the Reynolds number.
Unfortunately, when Re is large the resulting turbulent
ow is typically so complex that, so
called, direct numerical simulation of (u; p) is not practically feasible.
One conjecture of Leray is that \turbulence" in nature is associated with a breakdown
of uniqueness of weak solutions to (1). It is known that, for example, weak solutions to (1)
are unique for
There numerous generalizations of this basic result, [GHR00], [Lad69]. With this in mind,
solutions u to (1) with kruk are frequently described as \laminar". Thus,
the L p {regularity in time which can be reasonably assumed is of critical importance.
There are numerous approaches to the simulation of turbulent
ows in practical settings.
One of the most promising current approaches is large eddy simulation (LES) in which
approximations to local spacial averages of u are calculated. A spacial length scale - is
selected and the velocity scales of O(-) and larger are approximated directly while the eects
of those smaller than O(-) on the O(-) and larger eddies are modeled. In computational
turbulence studies using LES it is often reported that the resulting computational complexity
is independent of the Reynolds number (but dependent on the resolution sought, -). There
has been little or no analytical support for this observation however. The goal of this report
is to begin numerical analysis in support of this claim.
To be more specic, a smooth, nonnegative function g(x) with
Z
R d
is selected and the mollifer g - (x) is dened in the usual way:
One common example is a Gaussian, where the summation
convention is used. The spacial averaging/ltering operation is now dened by convolution
In LES, approximations to (u; p) are sought rather than to (u; p). The usual procedure is to
rst lter the Navier-Stokes equations:
in
r
in
where the \Reynolds' stress tensor" T is
Closure is addressed by a modeling step in which T is written in terms of u. The resulting
(closed) space ltered Navier-Stokes equations are solved numerically. In this procedure,
there are three essential issues:
1. The \modeling error" committed in approximating T.
2. The \numerical error" in solving the resulting system.
3. Correct boundary conditions for the
ow averages.
In this report, we study the numerical error analytically. Since there are many models
in LES (see, e.g., [IL00], [GL00], [HMJ00], [FP99], [BFR80], [Lia99], [Sag98], [Par92] for
examples) and few analytical studies, we take herein the simplest model commonly in use
presented, for example, in Ferziger and Peric [FP99, Section 9.3].
To describe the model, let D (u) be the deformation tensor associated with the indicated
velocity eld by:
The Reynolds stresses are thought of as a turbulent diusion process based upon the Boussinesq
assumption or eddy viscosity hypothesis that \turbulent
uctuations are dissipative in
the mean", [IL00], [Fri95], [MP94], [Par92]. We will accordingly consider a model of the
where turb
turbulent viscosity or eddy viscosity. This turbulent
viscosity's determination can be very complex, involving even solutions of accompanying
systems of nonlinear, partial dierential equations. In the simplest case, the turbulent
viscosity depends on the mean
ow u through the magnitude of the deformation
of u; (D (u)), with a functional dependence. Under the Boussinesq assumption,
r T should act like a physical viscosity. Following the reasoning of Ladyzhenskaya [Lad70],
thermodynamic considerations imply that the Taylor series of turb (D ) should be dominated
by odd degree terms. The simplest case is of linear dependence upon jD j
turb (jD
where jD j denotes the Frobenius norm of D . For specicity and for accord with the most
commonly used Smagorinsky [Sma63] model, we take a 0 (-) 0 and a 1 Other
scalings are possible, [Lay96], though less tested, as are many other subgridscale models
typically either chosen to be around 0.1 or taken to be a function
extrapolated as in the \dynamic subgridscale model" of Germano,
[GPMC91].
With the model (2) the resulting system of equations for the approximations (w; q) to
in
in
in
Z
Boundary conditions must be supplied for the large eddies. It is physically clear that large
eddies do not adhere to solid walls. (For example, tornadoes and hurricanes move while
touching the earth and lose energy as they move.) Therefore, in [GL00], [Sah00], (see also
[Par92] for the use of similar boundary conditions in a conventional turbulence model), it
was proposed that the large eddies w should satisfy a no-penetration condition and a slip
with friction condition on @
@
where ~ t is the Cauchy stress vector on , for background information see Serrin [Ser59],
is the friction coe-cient (calculated explicitly in [Sah00]), ^ n the outward unit
normal
j an orthonormal system of tangent vector's on . The friction coe-cient
can be calculated once a specic lter is chosen, [Sah00]. It has the property ([Sah00]) that
no slip conditions are recovered as - ! 0:
A Dirichlet boundary condition
ow on 0 , is appropriate if 0 is an in
ow boundary
upon which u can be calculated by extending the known, in
ow velocity eld upstream.
The Cauchy stress vector ~ t includes the action of both the viscous stresses and Reynolds
stresses and is given by:
Standard properties of convolution operators imply that the
ow averages (u; p) are
in space, have bounded kinetic energy
Z
Z
have no solution scales smaller than O(-) and converge to u as On the
other hand, it is not obvious, nor has it been proven yet, that solutions (w; q) to the large
eddy model approximating (u; p) share any of these properties! Nevertheless, the spacial
regularity of solutions (w; q) we shall consider to be a modeling issue (beyond the scope of
this report studying numerical errors in LES). The time regularity of solutions (w; q) is still
an important consideration. For example, we shall show that solutions of this model satisfy
Z Tkrwk 3
uniformly in Re. One goal is to keep the assumed time regularity as close to L 3 (0; T ) as
possible and below L 4 (0; T ). The fundamental error analysis of Heywood and Rannacher
[HR82] for the Navier-Stokes equations is based, in part, on a laminar-type assumption
62 Weakening this to an assumption of the form ru 2 L 3 (0; T; L
3((as we seek to do herein) is nontrivial.
Preliminaries
This section sets the notation used in the report, describes the function spaces employed and
collects several useful inequalities. The notation used is standard for the most part. The
norms, for p 6= 2, are denoted explicitly as kfk L p . Sobolev spaces W k;p
are dened
in the usual way, [Ada75]. The associated norm is denoted k k k;p . If the domain in question
is
not
(e.g.,
will be explicitly indicated. If norms will be
for the W
norm and k k k; for the W k;p () norm and k k and k k ,
respectively, for the L
and L 2 () norms. Suppose the polygonal boundary =
@
is
composed of faces
The spaces associated with the boundary conditions (4) are:
d
Z
The boundary condition in X is dened to hold in the sense of the trace theorem on each
is the outward unit normal to = @
The L
and L 2 () inner products are
denoted (; ) and (; ) respectively.
denotes the usual deformation tensor, dened in the introduction. The
denotes an orthonormal system of tangent vectors on . Whenever
it will be understood that the term is to be summed over the two tangent vectors if
For example:
The following dual norms are dened in an equivalent but slightly nonstandard way:
kfk := sup
kfk W 1;q
R t(f; v)dt 0
R tkD (v)k q
Lemma 2.1 [Inf-Sup Condition]. Let ~
Jg: The velocity-pressure spaces ( ~
X;Q) satisfy the inf-sup condition:
sup
kk
C > 0: (5)
Proof: The trace theorem [Gri92] and Korn's inequality together show that (5) is
implied by the usual inf-sup condition
sup
d
kk krvk C > 0:
Lemma 2.1 implies that the space of weakly divergence free functions
is a well dened, nontrivial, closed subspace of ~
X. The inf{sup condition (5) is used to get
a bound on kqk once a bound on kD (w)k is proved. The bound on kD (w)k follows from
Lemma 3.1 below. An inf{sup condition with ~
replaced by X and kD (w)k by kD (w)k L 3
would also su-ce to get the bound on kqk.
Remark 2.2 Since is not C 1 discontinuities in j and ^
have forced modications in the
norms to piecewise denition. For example,
but
The conforming nite element method for this problem begins by selecting nite element
spaces X h X and Q h Q, where \h" denotes as usual a representative meshwidth for
satisfying the usual approximation theoretic conditions required of nite element
spaces. The condition that X h X imposes the restriction that v h ^
. For intricate boundaries, this could possibly be onerous so it is interesting to
consider imposing
with penalty or Lagrange multiplier methods, following, e.g.,
the work in [Lia99]. Nevertheless, there is already considerable computational experience
with imposing this condition in nite element methods, see, e.g., [GS00], [ESG82], so we
shall not focus on the interesting detail of the treatment of corners. Without these additional
regularizations in the numerical method, it is useful in the analysis to assume that (X h
satises the discrete analogue of (5):
sup
kD (v h
where C > 0 is independent of h. The next lemma shows, in essence, that if the computational
mesh follows the boundary and if the velocity space restricted to no-slip boundary
conditions and the pressure space satisfy the usual inf-sup condition, then (6) holds.
Lemma 2.3 [Discrete Inf-Sup Condition]. If (X
sup
d
then (6) holds.
Proof: By trace theorem [Gri92] and the Poincare-Friedrichs inequality, for any h 6= 0,
kD (v h
Thus, (6) will be assumed throughout this report. Under (6), the space of discretely
divergence free functions
is a nontrivial closed subspace of X h , [GR86], [Gun89].
We shall frequently use Young's inequality in the form:
ab
a
The generalization of Holder's inequality:
Z
r
is also useful. We shall frequently use the Sobolev embedding theorem, often, but not always,
in the form that in 3 dimensions W
The nonlinear form in the subgridscale term: for v; w 2 W
(jD (w)jD (w); D (v))
is of p-Laplacian type (with Thus, it is strongly monotone and locally Lipschitz
continuous in the sense made precise in the following well-known lemma, see, e.g., [Lay96],
[DG91].
Lemma 2.4 [Strong Monotonicity and Local Lipschitz-Continuity]. There are constants C
and C such that for all
6 d and
(jD
(jD
Korn's inequalities relate L p norms of the deformation tensor D (v) to those same norms
of the gradient for 1 < p < 1, see Galdi [GHR00], Gobert [Gob62, Gob71], Temam [Tem83]
or Fichera [Fic72], and fail if
Theorem 2.5 [Korn's Inequalities]. There is a C > 0 such that for 1 < p < 1
for all v 2 (W
1 d .
Further, if
(v) is a semi-norm on L
which is a norm on the constants, then
holds for 1 < p < 1 and for all v 2 (W
2 d .
As a consequence of Korn's inequality it follows that, taking
for all v 2 fv 2 W 1;p
We will often use Poincare's inequality, which holds since
p. 56.,
kvk
krvk; for all v 2 X:
We shall use the Gagliardo-Nirenberg inequality in W 1;p
\ X to reduce the time
regularity required for w. This inequality [Ada75], [Nir59], [GHR00], [DiB93] states that
provided satises a weak regularity condition (holding in particular for polygonal domains)
and
kvk L q Ckrvk a
d
where,
for
if
s
In particular, note that taking
kvk L
Ckrvk 2=3
The following combination of this and Korn's inequality will be useful in Section 4.
Lemma 2.6 Let
and
Proof: This follows immediately from (7) and Korn's inequality.
3 Finite Element Formulation
This section develops the nite element method for the LES model. The stability of the
model is also studied. In particular, we show w and w h 2 L
1(uniformly in Re. Lastly, the error in an equilibrium projection is considered.
The variational formulation is derived in the usual way by multiplication of (3) by (v;
(X; Q) and applying the divergence theorem. The boundary integral terms require careful
treatment (following, e.g., [Lia99]) on account of the slip with friction condition on . Let
0 be a constant. The formulation which results is to nd w
(w
and w(x; X. For compactness, dene the nonlinear and trilinear form:
(w
It is a simple index calculation to check that for v 2 X; w 2 V (since such functions have
zero normal components on ) (w rw; Thus, the variational formulation
can be rewritten as: nd (w;
(w
for all (v;
Using Lemma 2.4, it is easy to prove that the LES model (3), (4) satises the analogue
of Leray's inequality for the Navier-Stokes equations.
Lemma 3.1 [Leray's inequality for the LES model]. A solution of
Z t" J
Proof: in (9) and use Lemma 2.4.
Remark 3.2 1. Because of the slip with friction boundary conditions (4), it is important to
choose the formulation of the viscous terms, as in (8), (9), involving the deformation tensor.
2. Leray's inequality immediately implies stability in various norms (which we will de-
velop) and is the key, rst step in proving existence of weak solutions to (3), (4). This last
question is fully investigated (under dierent boundary conditions) in remarkable papers by
Ladyzhenskaya [Lad67], Pares [Par92] and Du and Gunzburger [DG91].
The continuous-in-time nite element method for (3) uses the variational formulation
as follows. First, velocity-pressure nite element spaces X h X \ W
satisfying (6) are selected. Next, the least squares parameter 0 is selected.
The nite element approximations to (w; q) are maps (w h ; q h
(w h
for all (v h ; h is an approximation to w(x;
It is straightforward to verify that Leray's inequality holds for w h as well as w.
Lemma 3.3 [Leray's inequality for w h ]. Any solution of
Using various inequalities in the right hand side, stability bounds for w h follow from
Lemma 3.3.
Proposition 3.4 [Stability of w h ]. The solution w h of (10)
J
Proof: Inequality (11) follows by applying Young's inequality to Lemma 3.3. The
bound (12) follows from the denition of the dual norm and ab a 3 applied
in the same manner.
For in (10), use Lemma 2.4 and apply the Young's
inequality on the right hand side. This gives:
d
dt
Inequality now follows by using an integrating factor.
In the analysis of the error in the approximation of the time dependent problem, it is
useful to have a clear description of the error in the Stokes projection under slip with friction
boundary conditions, [Lia99]. It is also necessary that any dependence on Re; - and be
made explicit.
Under the discrete inf-sup condition, the Stokes projection
dened as follows. Let (w;
q), where ( ~
(r (w ~
((w ~
(r (w ~
This is equivalent to the following formulation provided Given (w; q),
nd ~
(r (w ~
((w ~
for all v h 2 V h and h 2 Q h . Under the discrete inf-sup condition, it is well-known that
q) is a quasi-optimal approximation of (w; q). The dependence of the stability and error
constants upon Re and is important to the error analysis. That dependence is
described in the next lemma and proposition.
Lemma 3.5 [Stability of the projection ~
w]. Let w 2 V be given. Then, ~
w satises if > 0:
kr ~
in the second formulation of the Stokes projection. This gives
immediately:
kr ~
(w
from which the rst result follows. If the term (q; r ~
w) is bounded by noting that
r ~
trace (D ( ~
w)) so that
(q; r ~
Proposition 3.6 Suppose the discrete inf-sup condition (6) holds. Then, ( ~
exists uniquely
in
kr (w ~
k(w ~
C inf
Proof: The proof follows standard arguments, carefully tracking the dependence of the
constants upon Re and .
Note that the use of least squares penalization of the divergence allows an error estimate
for the Stokes projection whose constants are essentially independent of the Reynolds number
in a suitably weighted norm.
4 The Convergence Theorem
Let us rst note that for standard piecewise polynomial nite element spaces it is known
that, e.g., the L 2 -projection of a function in L p ; p 2, is in L p itself and the L 2 -projection
operator is stable in L
w denote an approximation of w in V h \ W
5 for example,
the L 2 -projection under the conditions of [CT87]. We assume that each norm of ~
w can be
bounded by the same norm of w times a constant which is independent of Re and h. This
assumption is proven for many piecewise polynomial nite element spaces in [CT87].
The error is decomposed as
w) (w h ~
w and
An error equation is obtained by subtracting (9) from (10) and using
the fact that w 2 V . This gives, for any v h 2 V h \ W
and h 2
This is rewritten, adding and subtracting terms and setting v
The monotonicity lemma (Lemma 2.4) implies that
and with r := maxfkD (w)k L 3 ; kD ( ~
Remark 4.1 If ~
w is taken to be the Stokes projection of (w; q) into V h then, e.g., the term
kD ()k" on this last right hand side does not occur.
Inserting these two bounds in (15) and using the Cauchy-Schwarz and Young's inequalities
d
dt
s r 3=2 kD ()k 3=2
Picking collecting terms gives2
d
dt
This is the basic dierential inequality for the error. Three cases will be considered, revolving
around the treatment of the rst term on the RHS of (16).
Remark 4.2 If completely analogous estimate holds with the pressure term modied
to be either nonuniform in Re (e.g., \C(2Re 1 or nonuniform in -.
Consider the convection terms
The terms containing \" shall be bounded rst. Consider Using
the inequalities in Section 2 appropriately gives
1
1h
The term b(; w h ; h ) is bounded similarly as follows:
1
Korn's inequality and the stability bounds of Section 3, (12) and (13), immediately imply
that D (w h uniformly in Re so that krw h k 2
uniformly in Re. The
imbedding theorem and Korn's inequality also imply kwk 2
uniformly in
Re. Thus, these bounds su-ce for a later application of Gronwall's inequality.
The rst term containing only h ; b(w; h ; h ), is zero due to skew symmetry. Thus, there
only remains the term Estimating the term is the essential, core
di-culty in obtaining an error bound which is uniform in Re. There are only a few natural
ways to bound this using Holder's inequality and the Sobolev embedding theorem. There
are two cases in which the analysis is successful
very regular, rw 2 L
1(There is one important case in which the analysis fails:
(iii) a 0
3(To highlight subsequent analysis and, hopefully, spur further study, we shall rst present the
case (iii) and explain the failure of the analysis.
4.1 The case rw 2 L 3 (0;
8 and a 0
If we assume only that rw 2 L 3 (0; T; L
84 there is no need to add and subtract terms
since a priori bounds on krw h k L 3 (0;T;L 3 ) have been proven which are uniform in Re. Thus,
we can use Holder's inequality to write:
picking s using the embedding
and Poincare's inequality give
Remark 4.3 Using Lemma 2.6 instead of the embedding of W 1;3 ! L 6 changes the critical
exponent on k h k \3=2" to 12=7 in the rst term of (20) but not the nal conclusion.
Combining (18), (19), (20), with gives an initial bound on the convection term's
dierence
kk 3=2
Inserting (21) into (16), applying Korn's inequality and collecting terms gives2
d
dt
Thus, pick such that
d
dt
Consider the three terms bracketed on this right hand side. The rst is approximation
theoretic; the second is an L 1 function multiplying k h (t)k 3=2 the third is an L 1 function
multiplying k h (t)k 2 . Let y(t) := k h (t)k 2 . This inequality may then be written as:
d
dt
(nonnegative terms) C(t)h
where
The nal step would normally be to apply Gronwall's inequality to deduce y(t) =2
to be bounded by its initial values and approximation theoretic terms. Unfor-
tunately, the term y 3=4 is not Lipschitz, so the argument fails at this last step.
Tracing the inequalities backward, the problem term arises from the steps used to bound
to obtain Re independence. The error analysis in the successful cases (i) and
(ii) centers therefore on alternate bounds for this term. We shall rst consider case (i).
Remark 4.4 If the estimate in (20) is improved as noted in Remark 4.3, the term y(t) 3=4
is changed to y(t) 6=7 but the nal conclusion still holds.
4.2 The case rw 2 L 3 (0;
8 and a 0 (-) > 0
Theorem 4.5 Assume > 0 and a 0 (-) > 0. Let
a
Then, there is a C independent of Re and h, such that
Further, there is a C independent of Re and h, such that
Then, the error w w h satises for T > 0
~
F(w ~
with
F(w ~
k(w ~
k(w ~
+kr (w ~
wk 3=2
Proof: This analysis follows closely the previous discussion except for the treatment of
the and the nal application of Gronwall's inequality.
Consider therefore Integration by parts and using the fact that h ^
on gives:
Using the embedding H 1=2 ,! L 3 in d = 2; 3 and Young's inequality give
1
Consider now the last term on the above right hand side. By Holder's inequality, we obtain
The Sobolev embedding theorem implies that for any s; 1 s < 1 in 2 or 3 dimensions
C(s;
kD (w h )k 2
This implies that for any r 0 > 2
Consider the last term on the above right hand side. The Sobolev embedding theorem also
implies
(The nal result is not improved by applying here instead the Gagliardo-Nirenberg inequal-
ity.) As r inequality. Thus, picking r close enough to 2
implies, using an embedding inequality and Korn's inequality,
for any t > 0. Thus, for these values of r 0 and s
for any t > 0. For conjugate exponents inequality, we then have
Picking these values of r 0 and s:
Using this bound, (22) and (23) gives nally
Remark 4.6 It appears on rst consideration that this last term (r h ; w h h ) can be
agreeably bounded more directly and easily by:
This bound, while certainly true, is not su-cient because of the condition that inevitably
arises from using it that w h or w 2 L
1 . The extra work in the bound we use
reduces the time regularity requirements arising from this term to w h 2 L
1;3((which is bounded uniformly in Re by problem data in Section 3).
Substituting this bound for b( in the derivation of the upper estimate (21) for
the dierence of the convection terms gives
3
To proceed further, (24) is inserted in the right hand side of (16). This yields the
dierential inequality2
d
dt
Pick These choices simplify (25) to:2
d
dt
a
Before applying Gronwall's inequality, let us rst verify that it will indeed give us an
error bound that is uniform in the Reynolds number by considering the coe-cients on the
right hand side of (26).
By the stability estimates kwk 2 uniformly in Re. Thus,
kwk 3=2 kk 3=2
Consider the (critical) bracketed coe-cient of the last term on the right hand side. We
must show this coe-cient is in L 1 (0; T ) uniformly in Re. Indeed, by the stability estimates
uniformly in Re. Since T < 1, L 3 (0; T )
thus the rst factor of the last term is in L 1 (0; T ) uniformly in Re. In addition,
note that r CkD(w)k L 3.
Hiding all constants in generic C's, Gronwall's lemma now implies for almost all t 2 [0; T
Z tkD (w)k 3=2
Z t1
Note that by the Cauchy{Schwarz inequality in L 2 (0; T ) and the stability estimates
Z tkD (w)k 3=2
C(-)kD ()k 3=2
Now, the essential supremum of t 2 [0; T ] is applied on both sides of the inequality. As
the triangle inequality completes the proof of Theorem 4.5.
4.3 The case rw 2 L 2 (0; T ; L
8 and a 0 (-) 0
We now consider the case of smoother w, i.e.,
9 uniformly in Re;
allowing for the case a 0 (-) 0. This case is primarily of interest because many tests involve
\academic"
ow elds given in closed form (as in Section 5). These are typically smooth and
bounded. In this case Theorem 4.7 gives an error estimate with constants independent of Re
(but depending on - and ). It is noteworthy in this estimate that multiplicative constants
depend on - but the rate constant in the (inevitable) exponential term takes the form
with no explicit dependence on -.
Theorem 4.7 Suppose a 0 (-) 0; > 0 and w 2 L
9 uniformly in Re. Let
then there is a C (w) such that
be such that
Then, the error w w h satises:
~
F(w ~
with
F(w ~
k(w ~
k(w ~
kr (w ~
kD (w ~
Proof: In this case, the dierence in the nonlinear terms is decomposed a bit dierently
as:
Consider the individual terms on the right hand side of (27):
1
(w h
Combining these three estimates gives
The term kw h k L 6 is bounded using the Gagliardo-Nirenberg inequality
Since kw h k is bounded uniformly in and h by (12) or (13), it follows
This bound, together with (28) is now inserted in the right hand side of (16) giving2
d
dt
To apply Gronwall's inequality we need4
in other words w 2 L
The term on the right hand side of this inequality
containing r 3=2 is treated as in the proof of Theorem 4.5. In the nal result of Gronwall's
lemma, we must also verify that the resulting terms containing kD (w h )k L 3 are bounded
uniformly in Re. To this end, apply Holder's inequality:
Z TkD (w h )k 4=3
1. From the stability estimates, we clearly must take q such that 4q=3 3.
Accordingly, take 9. This gives
Z TkD (w h )k 4=3
Similarly, for q and q 0 conjugate exponents:
Z TkD (w h )k 2
kD (w h )k 2
The stated error estimate now follows from Gronwall's inequality and the triangle inequality
as in the proof of Theorem 4.5.
5 A Numerical Example
To give a numerical illustration several decisions must be made, mainly to work on an 'aca-
ow problem with a known exact solution or a more realistic
ow problem containing
the accompanying uncertainties. Since our aim is to illustrate a convergence theorem we have
chosen the former. (To assess a model or study the limitations of an algorithm we would
naturally have chosen the latter.) Accordingly, we have selected the vortex decay problem
of Chorin [Cho68], used also by others, e.g., Tafti [Taf96]. The domain
is
choose
For the relaxation time Re this is a solution of the Navier-Stokes equation consisting
of an array of opposite signed vortices which decay as t ! 1. The right hand side f ,
initial condition and non-homogeneous Dirichlet boundary conditions are chosen so that this
is the closed form solution of (3).
Since we are studying convergence as h ! 0 for - xed and Re varying we have accordingly
chosen
Relaxation time
Vortex conguration
Final time
Eddy Scale
It is signicant that
n so that the vortices are larger than O(-) and hence
should be \visible" to the model. Tables 2 to 4 give the error in L
6 and show
uniformity in Re and in L
5 and show weak dependence on Re (since this term
in the error estimate is scaled by Re 1=2 ).
The least squares constant is chosen to be zero. Time discretization is by the fractional
step -method with the indicated time steps. The tables show a decrease of the error as h
decreases until it reaches the error introduced by the time stepping procedure. The spacial
discretization is done on a uniform square mesh with the Q 2 =P disc
denoted in
element with the associated meshwidths indicated. The viscous term is treated not
as (rw h ; rv h ) but using the deformation tensor formulation, (D (w h ); D (v h )) as analyzed
herein. Both the Smagorinsky subgridscale model and the convection term are treated
implicitly.
Using the above elements and meshes the calculations involved the numbers of degrees
of freedom listed below in Table 1.
mesh width velocity d.o.f. pressure d.o.f. total d.o.f.
1=8 578 192 770
Table
1: Mesh widths and degrees of freedom (d.o.f.) in space.
These are certainly not extremely large numbers of degrees of freedom. However, their
importance is only relative to the Reynolds numbers chosen
the resolution sought Again, LES is focused on situations in which the number
of degrees of freedom is small relative to Re. Thus, the chosen values of h and Re seem
appropriate.
The
Tables
{ 4 present the L norms of the errors. Note that the trends are
exactly as anticipated by the theory; there is none to minimal degradation in the error as
Re increases from 10 2 to 10 5 and the error plateau's as h decreases at a value which seems
to be the induced error in the time stepping procedure.
Re n h 1/8 1/16 1/32 1/64
Table
2: 0:02. Note the uniformity in Re.
Re n h 1/8 1/16 1/32 1/64
Table
3: observe the uniformity in Re.
Re n h 1/8 1/16 1/32 1/64
Table
4: 0:005: Note the uniformity in Re and the reduction of the
minimal error as t decreases.
The
Tables
{ 7 present the errors in L These are not predicted to be uniform
in Re so some error degradation is expected as Re increases. Very mild degradation is indeed
observed. The degradation is mild, possibly because
are predicted to have the same uniform in Re convergence rates. Thus, the theory forecasts
increase as Re increases until it reaches the (slower) uniform convergence
rate predicted by the kD (e)k L 3 (0;T;L 3 ) bound.
Re n h 1/8 1/16 1/32 1/64
Table
5: kD (e)k L 2 (0;T;L 2
Re n h 1/8 1/16 1/32 1/64
Table
Re n h 1/8 1/16 1/32 1/64
Table
7: kD (e)k L 2 (0;T;L 2
--R
spaces.
Improved subgrid models for large eddy simulation.
Numerical solution for the Navier-Stokes equations
The stability in l p and w 1
Analysis of a ladyzhenskaya model for incompressible viscous ow.
Degenerate parabolic equations.
The implementation of normal and/or tangential boundary conditions in
Existence theorems in elasticity.
Computational Methods for Fluid Dynamics.
Turbulence, the Legacy of A.
An Introduction to the Mathematical Theory of the Navier-Stokes Equations
Fundamental Directions in Mathematical Fluid Mechanics.
Approximation of the larger eddies in uid motion ii: A model for space
Une inequation fondametale de la th
Sur une in
A dynamic subgrid-scale eddy viscosity model
Finite Element Methods for Navier-Stokes equa- tions
Singularities in Boundary Value Problems
Incompressible Flow and the Finite Element Method.
Finite Element Methods for Viscous Incompressible Flows.
Large eddy simulation and the variational multiscale method.
Finite element approximation of the nonstationary navier-stokes problem i: Regularity of solutions and second order error estimates for spacial discretization
An assessment of models in large eddy simulation.
"Al.I.Cuza"
New equations for the description of motion of viscous incompressible uids and solvability in the large of boundary value problems for them.
The Mathemetical Theory of Viscous Incompressible Flow.
A nonlinear
Weak imposition of boundary conditions in the Stokes and Navier- Stokes equation
Analysis of the K-Epsilon Turbulence Model
On elliptic partial di
New perspectives on boundary conditions for large eddy simulation.
Mathematical principles of classical uid mechanics.
General circulation experiments with the primitive equations.
Comparison of some upwind-biased high-order formulations with a second-order central-dierence scheme for time integration of the incompressible Navier-Stokes equations
Problemes Mathematiques en Plasticite.
--TR
--CTR
Volker John, An assessment of two models for the subgrid scale tensor in the rational LES model, Journal of Computational and Applied Mathematics, v.173 n.1, p.57-80, 1 January 2005
Faisal A. Fairag, Analysis and finite element approximation of a Ladyzhenskaya model for viscous flow in streamfunction form, Journal of Computational and Applied Mathematics, v.206 n.1, p.374-391, September, 2007 | navier-stokes equations;turbulence;large eddy simulation;finite element methods |
588514 | Superconvergence of the Local Discontinuous Galerkin Method for Elliptic Problems on Cartesian Grids. | In this paper, we present a superconvergence result for the local discontinuous Galerkin (LDG) method for a model elliptic problem on Cartesian grids. We identify a special numerical flux for which the L2-norm of the gradient and the L2-norm of the potential are of orders k+1/2 and k+1, respectively, when tensor product polynomials of degree at most k are used; for arbitrary meshes, this special LDG method gives only the orders of convergence of k and k+1/2, respectively. We present a series of numerical examples which establish the sharpness of our theoretical results. | Introduction
. In this paper, we derive a priori error estimates of the Local
Discontinuous Galerkin (LDG) method on Cartesian grids for the following classical
model elliptic problem:
@n
where\Omega is a bounded domain of R d and n is the outward unit normal to its boundary
we assume that the (d \Gamma 1)-measure of \Gamma D is non-zero.
Recently, Castillo, Cockburn, Perugia and Sch-otzau [3] obtained the first a priori
error analysis of the LDG method for purely elliptic problems. Meshes consisting
of elements of various shapes and with hanging nodes were considered and general
School of Mathematics, University of Minnesota, Vincent Hall, Minneapolis, MN 55455
(cockburn@math.umn.edu). Supported in part by the National Science Foundation (Grant DMS-
9807491) and by the University of Minnesota Supercomputing Institute.
y Institut f?r Angewandte Mathematik, Universit?t Heidelberg, INF 293/294, 69120 Heidelberg,
Germany (guido.kanschat@na-net.ornl.gov). This work was supported in part by the ARO
DAAG55-98-1-0335 and by the University of Minnesota Supercomputing Institute. It was carried
out when the author was a Visiting Professor at the School of Mathematics, University of Minnesota.
z Dipartimento di Matematica, Universit'a di Pavia, Via Ferrata 1, 27100 Pavia, Italy
(perugia@dimat.unipv.it). Supported in part by the Consiglio Nazionale delle Ricerche. This work
was carried out when the author was a Visiting Professor at the School of Mathematics, University
of Minnesota.
x School of Mathematics, University of Minnesota, Vincent Hall, Minneapolis, MN 55455
(schoetza@math.umn.edu). Supported by the Swiss National Science Foundation (Schweizerischer
Nationalfonds).
2 B. Cockburn, G. Kanschat, I. Perugia and D. Sch-otzau
numerical fluxes were studied. It was shown that, for very smooth solutions, the
orders of convergence of the L 2 -norms of the errors in ru and in u are k and k
respectively when polynomials of degree at most k are used. On the other hand,
Castillo [2] and Castillo, Cockburn, Sch-otzau and Schwab [4] proved that, for one-
space dimension transient convection-diffusion problems, the order of convergence of
the error in the energy norm is optimal, that is, k that the so-called
numerical fluxes are suitably chosen. In this paper, we extend these results to the
LDG method on Cartesian grids for the multi-dimensional elliptic model problem
(1.1); we show that the orders of convergence in the L 2 -norm of the error in ru and
respectively, when tensor product polynomials of degree at
least k are used. Our proof of this super-convergence result is a modification of the
analysis carried out in [3]; it takes advantage of the Cartesian structure of the grid
and makes use of a key idea introduced by LeSaint and Raviart [10] in their study of
the original DG method for steady-state linear transport.
Since our analysis is a special modification of that of [3], in order to avoid unnecessary
repetitions, we refer the reader to [3] for a more detailed description of the framework
of our error analysis. The organization of this paper is as follows. In Section 2, we
briefly display the LDG method in compact form, introduce the special numerical flux
on Cartesian grids and present and discuss our main result. In Section 3, the detailed
proofs are given and in Section 4, we present several numerical experiments showing
the optimality of our theoretical results. We end in Section 5 with some concluding
remarks.
2. The main results. In this section we recall the formulation of the LDG
method and identify the special numerical flux we are going to investigate on Cartesian
grids. Then we state and discuss our main results. As pointed out in the introduction,
we refer to [3] for more details concerning the formulation of the LDG method.
2.1. The LDG method. We assume that the problem
domain\Omega can be covered
by a Cartesian grid. To define the LDG method, we rewrite our elliptic model problem
(1.1) as the following system of first-order equations:
\Gammar
Next, we discretize the above problem on a Cartesian grid T . To obtain the weak
formulation with which the LDG is defined, we multiply equations (2.1) and (2.2)
by arbitrary, smooth test functions r and v, respectively, and integrate by parts over
the d-dimensional rectangle K 2 T . Then we replace the exact solution (q; u) by its
approximation (q N ; uN ) in the finite element space MN \Theta VN , where
and
of degree at most k in each variable on Kg:
The LDG method on Cartesian grids for elliptic problems 3
The method consists in finding (q VN such that
Z
Z
Z
Z
Z
Z
for all test functions (r; v) 2 S(K) d \Theta S(K), for all elements . The functions
uN and b q N in (2.5) and (2.6) are the so-called numerical fluxes. These are nothing
but discrete approximations to the traces of u and q on the boundary of the elements
K and are defined as follows. Consider a face e of the d-dimensional rectangle K. If
e lies inside the
domain\Omega\Gamma we define
uN
- ffq N gg
and, if e lies on the boundary of \Omega\Gamma
g N on \Gamma N ;
and b u :=
g D on \Gamma D ;
Moreover, the stabilization parameter C 11 and the auxiliary parameter C 12 are defined
as follows:
where i is a positive real number and v is an arbitrary but fixed vector v with non
zero components; see Fig. 3.1.
2.2. Error analysis on Cartesian grids. To state our main result, we need
to recall some notation and to introduce new hypotheses. We restrict our analysis
to
domains\Omega such that, for smooth data, the solution u of problem (1.1) belongs to
(\Omega\Gamma5 and such that when f is in L
2(\Omega\Gamma and the boundary data are zero, we have the
elliptic regularity result k u Grisvard [8] or [9]. Since the
domain\Omega
will be triangulated by means of a Cartesian grid, the above requirements hold only
if\Omega is a d-dimensional rectangle.
We denote by hK the diameter of an element K, and set, as usual h := maxK2T hK .
We denote by E I the set of all interior faces of the triangulation T , by ED the set of
faces on \Gamma D , and by EN the set of faces on we assume that \Gamma
e. The Cartesian triangulations we consider are regular, that is, if ae K
denotes the radius of the biggest ball included in K,
ae K
Finally, we denote by EN
ae\Omega a closed set containing the intersection between the
and the set fx 0g. Moreover, we assume
that the triangulation T is such that
where K e denotes, from now on, an element containing the face e.
4 B. Cockburn, G. Kanschat, I. Perugia and D. Sch-otzau
We are now ready to state our main result.
Theorem 2.1. Assume that the solution (q; u) of (2.1)-(2.4) belongs to H k+1
H k+2 for k - 0; assume also that if the intersection between \Gamma N and fx
non-empty, u belongs to W k+1;1 (EN ). Assume that the Cartesian
grid T is shape-regular, (2.12), and that it satisfies the condition
VN be the approximation of (q; u) given
by the LDG method with k - 0 and numerical fluxes defined by (2.9), (2.11) and by
(2.10).
Then we have
and
where the constant C solely depends on i, k, d, oe and on the norms k u k k+2 and
Several important remarks are in order before we prove this result in the next Section.
Remark 2.2. This theorem is an extension to the bounded domain case of the
corresponding result by Cockburn and Shu [7] for the LDG method for transient
convection-diffusion problems. It is also an extension to the multi-dimensional case of
the results obtained by Castillo, Cockburn, Sch-otzau and Schwab [4] in the one-space
dimension case. The key ingredient of its proof is a super-convergence result of LeSaint
and Raviart [10] used in their study of the original DG method for steady-state linear
transport in Cartesian grids.
Remark 2.3. Note that Theorem 2.1 holds true in the case
approximate solution is piecewise constant. In [3], all the error estimates obtained for
the corresponding LDG method on general grids are valid only for k - 1; moreover,
no order of convergence is numerically observed for
Remark 2.4. From an approximation point of view, the order of convergence in q,
namely, k+1=2, is suboptimal by one half; however, it is confirmed to be sharp by our
numerical experiments in Section 4. For general numerical fluxes and unstructured
grids, an order of convergence in q of only k is obtained; see [3].
Remark 2.5. If we take the more general case
e
e
are constants, we might conceive the possibility that a suitable
tuning of the value of ff could improve the order of convergence in q. However, this
is not true, as will be made clear in the proof of Theorem 2.1 displayed in the next
section. See also [3] for other results about the influence of the value of ff on the
orders of convergence of the general LDG method.
Remark 2.6. In Theorem 2.1 an extra regularity condition on the exact solution u
on the closed set EN containing part of the Neumann boundary is required. If this
condition is dropped, and if 0g is not empty, only an order
of convergence of k in the error in q can be proved by using our technique which
represents a loss of 1=2. Note that whenever it is possible to choose v in such a way
that regularity assumption on the exact
solution is required.
The LDG method on Cartesian grids for elliptic problems 5
3. Proofs. This section is devoted to the proof of Theorem 2.1. For simplicity,
we consider only the case
and\Omega rectangle; see Fig. 3.1. All the arguments we
present in our analysis rely on tensor product structures and can be easily extended
to the case d ? 2.
\GammaE
+Fig. 3.1. The Cartesian grid T and the auxiliary vector v used to define the numerical fluxes.
To prove Theorem 2.1, we follow the approach used by [3]. Thus, we start, in Section
3.1, by briefly reviewing the setting of our error analysis. We proceed in Section
3.2, by introducing the projections \Pi and \Pi which generalize to several space
dimensions the projections used by Castillo, Cockburn, Sch-otzau and Schwab [4] in
their study of the LDG method for transient convection-diffusion problems in one-
space dimension. Then, in Section 3.3, we derive the expressions of the functionals
KA and KB needed in the setting of [3] to get error estimates. To do so, we make
use of a super-convergence result essentially due to LeSaint and Raviart [10], and
whose proof is presented in Section 3.4. The proof of Theorem 2.1 is completed in
Section 3.5.
3.1. The framework of the error analysis. All the following results are collected
from [3]. First, we start by reviewing that, by summation over all elements,
the LDG method can be written in the compact form: Find (q
such that
for all (r; v) 2 MN \Theta VN , by setting
6 B. Cockburn, G. Kanschat, I. Perugia and D. Sch-otzau
with
a(q; r) :=
Z
K2T
Z
ur
Z
e
ds \Gamma
Z
e
c(u; v) :=
Z
e
ds
e2ED
Z
e
C 11 uv ds:
The linear forms F , G are defined by
F (r) :=
e2ED
Z
e
Z
e2ED
Z
e
ds
Z
e
We also introduce the semi-norm j (q; u) j 2
A that appears in a natural way in the
analysis of the LDG method and is defined as
Z
e
ds
e2ED
Z
e
To prove error estimates for the LDG method, we follow [3] and introduce two func-
tionals, KA and KB , which capture the approximation properties of the LDG method;
the functionals are related to two suitably chosen projections \Pi and \Pi onto the FE
spaces MN and VN , respectively. Namely, we require KA and KB to satisfy
for any (q; u); (\Phi;
2(\Omega\Gamma3 and
for any (r; v) 2 MN \Theta VN and (q; u) 2 H
By Galerkin orthogonality, all the error estimates can then be solely expressed in
terms of KA and KB as can be seen in the following result.
Lemma 3.1 ([3]). We have
A (q; u; q; u) +KB (q; u):
Furthermore,
KA (q; u; \Phi; ')
with ' denoting the solution of the adjoint problem
@n
and
The LDG method on Cartesian grids for elliptic problems 7
3.2. Projections. In this section we define the projections \Pi and \Pi we are
going to use to prove Theorem 2.1 and list their properties. To this end, we start by
introducing one-dimensional projections. Let I be an arbitrary interval,
and let P k (I) be the space of the polynomials of degree at most k on I . We denote
by - the L 2 (I)-projection onto P k (I), i.e., for a function w 2 L 2 (I) the projection -w
is the unique polynomial in P k (I) satisfying
Z
I
Furthermore, for w
we define the projections - \Sigma w 2 P k (I) by the following
I
On a rectangle we define the following tensor product operators:
with the subscripts indicating the application of the one-dimensional operators - or
- \Sigma with respect to the corresponding variable.
Finally, we define the projections \Pi and \Pi as
In our error analysis, we use key properties of these projections displayed in the
following result.
Lemma 3.2. With the notation indicated in Figure 3.1, we have
Z
Z
2:
We also need several approximation results which we gather in the lemma below.
Lemma 3.3 (Cf. [5]). Let
Furthermore, for any edge e i parallel to the x i -axis, we have
Finally, if u
8 B. Cockburn, G. Kanschat, I. Perugia and D. Sch-otzau
3.3. The functionals KA and KB . In this subsection, we obtain the functionals
KA and KB introduced in Section 3.1.
We consider the stabilization parameter C 11 , defined by (2.14), in order to highlight
the fact that any choice of ff 6= 0 in (2.14) deteriorates the rates of convergence of the
estimates of Theorem 2.1.
In [3, Corollary 3.4], KA has been investigated for general projection operators \Pi and
\Pi satisfying the approximation results in Lemma 3.3 with 1. Thus, we we just
report here the final result.
Lemma 3.4 ([3]). Let u
to be given by (2.14). Then, if we set the approximation
property (3.2) holds true with
KA (q; u; \Phi;
\Theta
Furthermore, in the particular case where (\Phi; there holds
KA (q; u; q;
\Theta
In [3], the functional KB was only studied in the case where \Pi and \Pi are L 2 -
projections. Next, we show that a better result for KB can be obtained on Cartesian
grids for the projections defined by (3.4) and the numerical fluxes defined by (2.11).
To obtain such a result, we use the following standard inverse inequality.
Lemma 3.5 (Cf. [5]). There exists a positive constant C solely depending on k, d
and oe such that for all s 2 MN we have
for all K 2 T , e being any side of K.
We set
the sides of \Gamma. We are now ready to state our main lemma.
Lemma 3.6. Let u
to be given by
and let \Pi and \Pi be the operators defined by (3.4). Then, for any (r; w) 2
MN \Theta VN , the approximation property (3.3) holds true, with KB given by
where the constant C solely depends on k, d and oe.
Proof. In order to be able to distinguish the many parts of \Gamma and facilitate the proof
of the above result, we introduce the following notation:
and define E
these boundaries are indicated in Fig. 3.1.
We set - q :=
The LDG method on Cartesian grids for elliptic problems 9
and estimate each of the forms separately.
a. Estimate of T 1 . We have
K2T
Z
K2T
K2T
K2T
b. Estimate of T 2 . We can write
K2T
Z
Z
e
ds
e2ED
Z
e
ds
Taking into account the definition of the fluxes in (2.11) and the properties of the
projection \Pi in Lemma 3.2, we conclude that
Z
Z
e
Z
e
Consequently,
Z
e
ds
Multiplying and dividing each term of the sum by C2
11 , and using the approximation
properties of \Pi, we have
Z
e
ds
0;e
0;e
s+1;Ke
Note that we have used the shape-regularity assumption (2.12) to bound C \Gamma1
11 by
Ke .
c. The estimate of T 4 . We have
Z
e
ds
e2ED
Z
e
ds
e2ED
0;e
e2ED
0;e
K2T
0;e
K2T
G. Kanschat, I. Perugia and D. Sch-otzau
d. Estimate of T 3 . This estimate cannot be obtained as easily as the previous ones
since it is here that the key idea introduced by LeSaint and Raviart [10] has to be
suitably applied.
We start by writing
K2T
Z
Z
e
(ff- u gg +C 12 \Delta [[- u ds \Gamma
Z
e
ds
K2T
Z
K2T
I
Z
e
(ff- ds
Z
e
ds
Again with (2.11), we see that the contribution of an interior element K to this
expression is
Z
Z
ds \Gamma
Z
where the superscript 'out' denotes the traces taken from outside K. Since u out
and [\Piu] out
for the corresponding one-dimensional projection -
i , this
contribution can be written as
Z
Z
ds \Gamma
Z
ds
Z
ds \Gamma
Z
For boundary elements, we add and subtract corresponding terms to obtain
K2T
Z
e
ds
Z
e
ds
ds
K2T
Z
e
ds
Z
e
ds
Z
e
with ZK (r; u) defined in (3.5).
We start by bounding the contributions to T 3 stemming from a boundary edge e
parallel to the x i -axis, 2. Since u 2 H s+2
implies
see [9], by the property (3.3) and the inverse inequality in Lemma 3.5, we get
Z
ds -
The LDG method on Cartesian grids for elliptic problems 11
Here, K e i denotes again the element containing the edge e i . Consequently, the global
contribution to T 3 of the boundary edges belonging to E i n can be bounded
by
0;Ke
For the edges e in EN "E \Gamma , we have to use a different argument. Thus, by Lemma 3.5,
we have
Z
e
ds -k - u k L 1
Hence, by the Cauchy-Schwarz inequality,
sup
and so
Finally, we estimate the contribution ZK (r; u), by using the following super-convergence
result, essentially due to LeSaint and Raviart [10], whose proof is postponed to Section
3.4.
Lemma 3.7. Let ZK (r; u) be defined by (3.5). Then we have for s - 0
By combining the result of Lemma 3.7 with the above estimates of the contribution
of boundary edges, we obtain
Conclusion. The result now follows by simply gathering the estimates for T i ,
obtained above. This completes the proof.
12 B. Cockburn, G. Kanschat, I. Perugia and D. Sch-otzau
3.4. Proof of Lemma 3.7. We can write
where
Z
Z
Z
and
Z
Z
Z
The proof of the approximation results for Z K;1 and Z K;2 are analogous; therefore,
we just present the one for Z K;1 , essentially following the same arguments as in [10].
First, we consider Z K;1 on the reference square (\Gamma1; 1) 2 . We claim that
To prove (3.6), fix r are polynomial preserving operators,
holds true for every u 2 Q k (K). Therefore, we just have to consider the cases
.
Let us start with
1 . On
2 we have
1, and on
2 we have
is a polynomial of degree at most k \Gamma 1 in x 1 , we
obtain Z
Z
Thus, Z K;1 (r
1 .
In the case u(x 1
2 , we integrate by parts and obtain
Z
Z
Z
Z
due to the special form of u, we conclude
that Z K;1 (r
2 . This completes the proof of (3.6).
For fixed r 1 2 Q k (K), the linear functional u 7! Z K;1 (r 1 ; u) is continuous on H s+2 (K)
with norm bounded by Ckr 1 k 0;K . Due to (3.6), it vanishes over P s+1 (K) for
k. Hence, by applying Bramble-Hilbert's Lemma (see [6, Lemma 6], for instance), we
obtain for
This proves the assertion on the reference element (\Gamma1; 1) 2 . The general case follows
from a standard scaling argument.
The LDG method on Cartesian grids for elliptic problems 13
3.5. Proof of Theorem 2.1. If the exact solution of our model problem, (q; u),
belongs to H k+1 2 \Theta H k+2
KA (q; u; q; u) - h 2k+1+ff kuk 2
and
with
. The estimate of the error j follows now from
Lemma 3.1. Notice that gives the best order of convergence in h equal to k
.
Our assumptions on the domain imply that the solution ' of the adjoint problem in
Lemma 3.1 belongs to H 2
(\Omega\Gamma and that we have k'k 2 - Ck-k 0 ; see [8, 9]. Hence, we
conclude that
The estimate of ku \Gamma uN k 0 thus follows from Lemma 3.1. Notice that
again the best order of convergence in h which is k + 1.
4. Numerical Experiments. In this section, we display a series of numerical
experiments showing the computed orders of convergence of the LDG method; we
show (i) that the orders given by our theoretical results are sharp, (ii) that they
can deteriorate when the stabilization parameter C 11 is not of order one, (iii) that
the exact capture of the boundary conditions induces an unexpected increase of 1in
the order of convergence of the gradient, and (iv) that the orders of convergence are
independent of the dimension.
In all experiments, we estimate the orders of convergence of the LDG method as
follows. We consider successively refined Cartesian grids T ' , ' - 0, consisting of 2 d '
uniform d-dimensional cubes with corresponding mesh size 2 \Gamma'+1 ; we present results
in two and three space dimensions. If e(T ' ) denotes the error on the '-th mesh, then
the numerical order of convergence is computed as follows:
log
The results have been obtained with the object-oriented C++ library deal.II developed
by Bangerth and Kanschat [1].
4.1. The sharpness of the orders of convergence of Theorem 2.1. We
consider the two-dimensional model problem (1.1) on the
f and boundary conditions chosen in such a way that the exact solution is given by
We consider two cases: In the first, we impose inhomogeneous
Dirichlet boundary conditions on the whole boundary, and in the second, we also impose
inhomogeneous Neumann boundary conditions on the edge f\Gamma1g \Theta (\Gamma1; 1). The
results are contained in Tables 4.1 and 4.2 where the numerical orders of convergence
in the L 2 - and L 1 -norm in u, q 1 and q 2 of the LDG method with Q k elements for
are shown. We take C and the coefficients C 12 as in (2.11) with
14 B. Cockburn, G. Kanschat, I. Perugia and D. Sch-otzau
In
Table
4.1, we report the results for Dirichlet boundary conditions imposed on the
whole boundary. Note that, because of the symmetry of the problem, the orders of
convergence are exactly the same for q 1 and q 2 . For we see the optimal order
of convergence of 1 in the L 2 -norm of the error of both u and q; note that Theorem
2.1 predicts an order of convergence of 1
2 only for q. However, for k - 1 the L 2 -rates
are of order k +1 in u and k
2 in q, in full agreement with Theorem 2.1. The orders
on convergence in the L 1 -norm of the error in u and q appear to be k
respectively.
The results displayed in Table 4.2 are those for the case of inhomogeneous Neumann
boundary conditions on part of the boundary. We see that the orders of convergence
in this case are the same as the ones in the previous case.
Thus, the above experiments show that the orders of convergence given by Theorem
2.1 are sharp.
Table
Orders of convergence for the LDG method with C 11
element
6 0.9683 0.9624 0.9724 0.3856
6 2.9661 2.8316 2.4678 1.9658
6 3.9661 3.8249 3.4676 2.9724
4.2. The effect of the choice of C 11 . Next, we test the effect of the choice of
the coefficients C 11 on the orders of convergence of the LDG method. We consider
the same problem as in the previous experiments, case use Q 1 and Q 2
elements. We only show the numerical orders of convergence for the finest grids.
The LDG method on Cartesian grids for elliptic problems 15
Table
Orders of convergence of the LDG method with C 11
element
6 0.9795 0.9555 1.0303 0.9283 0.9954 0.6270
5 2.9563 2.9428 2.5042 1.9392 2.4631 1.9392
6 2.9770 2.8316 2.5044 1.9658 2.4806 1.9658
6 3.9805 3.8264 3.5024 2.9722 3.4815 2.9721
The results are displayed in Tables 4.3 and 4.4. We must compare all these results
with those with C obtained in the first set of experiments. We see that when
C 11 is of order h \Gamma1 , the order of convergence in u remains but the order of
convergence in q degrades from k
2 to only k, as predicted by our analysis; see
section 3.5.
We also see that taking C at the outflow boundary and C 11 of order one
elsewhere only results in a slight reduction of the L 1 -orders of convergence.
In the remaining cases, we take C 11 to be of order h in all the domain and then in all
but the outflow boundary where it is taken to be of order h \Gamma1 . We observe a slight
degradation of all the orders of convergence.
These results indicate that the best choice of the stabilization parameter C 11 for the
LDG method is to take it of order one, as predicted by our analysis.
4.3. Piecewise polynomial boundary conditions. The purpose of these numerical
experiments is to show that if the boundary data are piecewise polynomials
of degree k, the order of convergence of the L 2 -norm of the error in q is optimal, that
1, and not only k
2 as predicted by Theorem 2.1 and shown to be sharp in
sub-section 4.1.
B. Cockburn, G. Kanschat, I. Perugia and D. Sch-otzau
Table
Orders of convergence of the LDG method with Q 1 elements.
1=h 5 1.9607 1.9550 1.1409 0.8816
6 1.9792 1.9057 1.1019 0.9366
1=h on
elsewhere 6 1.9646 1.7914 1.4605 0.9268
6 1.8603 1.7887 1.4564 0.9701
1=h on
elsewhere 6 1.8563 1.7887 1.4556 0.9319
Table
Orders of convergence of the LDG method with Q 2 elements.
1=h 5 2.9555 2.9541 2.2223 1.8475
6 2.9754 2.9584 2.1685 1.9228
1=h on
elsewhere 6 2.9634 2.7424 2.4663 1.9427
6 2.8240 2.5482 2.4656 1.9742
1=h on
elsewhere 6 2.8211 2.4554 2.4643 1.9365
We consider two test problems. In the first, we take homogeneous Dirichlet boundary
conditions and f such that the exact solution is
In the
second, we take piecewise quadratic Dirichlet boundary conditions and f such that
the exact solution is
The results of the first problem are reported in Table 4.5 where we can see that the
optimal order of convergence of k for the L 2 - and L 1 -norms of the errors in both
u and q are obtained; the results for are displayed.
The results of the second problem are reported in Table 4.6, where we see that the
optimal order of convergence of k for the L 2 - and L 1 -norms of the errors in both
u and q are obtained for k - 2, as claimed. For 2, the order of convergence in the
-norm of the error in q is
2 only which nothing but the order of convergence
predicted by Theorem 2.1.
To better understand this phenomenon, we plot the errors in q 1 for Q 1 and Q 2 elements
in Figs. 4.1 and 4.2, respectively; the triangulation has 16 \Theta 16 elements and
corresponds to the index We immediately see the oscillatory behavior of the
error typical of finite element methods. In Fig. 4.1, we see that the error obtained
with elements is bigger at the boundary than at the interior. This, together with
the fact that the order of convergence in L 2 is 3
2 whereas the order of convergence in
L 1 is only 1, suggests that the error at the boundary is a factor of order
bigger
The LDG method on Cartesian grids for elliptic problems 17
than the error at the interior of the domain. On the other hand, the behavior of the
error with Q 2 elements is rather different, as can be seen in Fig. 4.2. Indeed, the
error behaves in the same way at the boundary and at the interior; this is further
confirmed by the fact that both the the order of convergence in L 2 and the one in L 1
are equal to k + 1.
These experiments justify our contention that the optimal order of convergence in q
can be reached if the boundary conditions are piecewise polynomials of degree k. Our
theoretical analysis does not explain this phenomenon.
Table
Orders of convergence for the LDG method with C
element
6 0.9456 0.9658 0.9662 0.9483
6 2.0213 1.9878 2.0003 1.9858
6 2.9815 3.0150 2.9855 3.0161
6 4.0247 3.9918 4.0041 3.9748
Table
Orders of convergence for the LDG method with C 11
element
6 2.0015 1.9775 1.4976 1.0091
6 2.9815 3.0150 2.9855 3.0162
6
B. Cockburn, G. Kanschat, I. Perugia and D. Sch-otzau
-2e-022e-026e-02Fig. 4.1. The error in the first component of the gradient for
Fig. 4.2. The error in the first component of the gradient for
The LDG method on Cartesian grids for elliptic problems 19
4.4. A three-dimensional example. In this experiment, we consider the model
problem (1.1) on the three-dimensional
We take Dirichlet
boundary conditions and f such that the exact solution is
The results are displayed in Table 4.7; the computation on level 5 with Q 2 did not
fit into the computers available to us. We can see that the orders of convergence
are similar to those obtained in the corresponding two-dimensional test problem in
the previous sub-section, cf. Table 4.6. This gives an indication that the orders of
convergence of the LDG method in three space dimension behave in the same way
they do in the two-dimensional case.
Table
Orders of convergence for the LDG method with C
element
3 2.9204 2.8642
5. Concluding remarks. In this paper we have shown that the LDG method
on Cartesian grids and with a special numerical flux super-converges; the proof of
this result is based on suitable defined projections \Pi and \Pi exhibiting a tensor product
structure. This work extends the corresponding result by LeSaint and Raviart
[10] for the DG method for linear hyperbolic problems and that by Castillo [2] and
Castillo, Cockburn, Sch-otzau and Schwab [4] for the LDG method applied to the
one-dimensional transient convection-diffusion. Extensions of this work to more general
elliptic and both steady and transient convection-diffusion problems can easily
be made.
--R
Concepts for object-oriented finite element software - the deal
An optimal error estimate for the local discontinuous Galerkin method
An a priori error analysis of the local discontinuous Galerkin method for elliptic problems
An optimal a priori error estimate for the hp-version of the local discontinuous Galerkin method for convection-diffusion prob- lems
The finite element method for elliptic problems
General Lagrange and Hermite interpolation in R n with applications to finite element methods
The local discontinuous Galerkin finite element method for convection-diffusion systems
Elliptic problems in nonsmooth domains
On a finite element method for solving the neutron transport equa- tion
--TR
--CTR
Paul Castillo, A review of the local discontinuous Galerkin (LDG) method applied to elliptic problems, Applied Numerical Mathematics, v.56 n.10, p.1307-1313, October 2006
Ilaria Perugia , Dominik Schtzau, On the Coupling of Local Discontinuous Galerkin and Conforming Finite Element Methods, Journal of Scientific Computing, v.16 n.4, p.411-433, December 2001
Ilaria Perugia , Dominik Schtzau, Thehp-local discontinuous Galerkin method for low-frequency time-harmonic Maxwell equations, Mathematics of Computation, v.72 n.243, p.1179-1214, July
Bernardo Cockburn , Chi-Wang Shu, RungeKutta Discontinuous Galerkin Methods for Convection-Dominated Problems, Journal of Scientific Computing, v.16 n.3, p.173-261, September 2001 | finite elements;elliptic problems;discontinuous Galerkin methods;cartesian grids;superconvergence |
588515 | Superlinear Convergence of Conjugate Gradients. | We give a theoretical explanation for superlinear convergence behavior observed while solving large symmetric systems of equations using the conjugate gradient method or other Krylov subspace methods. We present a new bound on the relative error after $n$ iterations. This bound is valid in an asymptotic sense when the size $N$ of the system grows together with the number of iterations. The bound depends on the asymptotic eigenvalue distribution and on the ratio $n/N$. Under appropriate conditions we show that the bound is asymptotically sharp.Our findings are related to some recent results concerning asymptotics of discrete orthogonal polynomials. An important tool in our investigations is a constrained energy problem in logarithmic potential theory.The new asymptotic bounds for the rate of convergence are illustrated by discussing Toeplitz systems as well as a model problem stemming from the discretization of the Poisson equation. | Introduction
. The Conjugate Gradient (CG) method is widely used for solving
systems of linear equations with a positive denite symmetric matrix A.
The CG method is popular as an iterative method for large systems, stemming e.g.
from the discretisation of boundary value problems for elliptic PDEs. The rate of
convergence of CG depends on the distribution of the eigenvalues of A. A well-known
upper bound for the error e n in the A-norm after n steps is
where e 0 is the initial error and the condition number is the ratio
of the two extreme eigenvalues of A. In practical situations, this bound is often too
pessimistic, and one observes an increase in the convergence rate as n increases. This
phenomenon is known as superlinear convergence of the CG method. It is the purpose
of this paper to give an explanation for this behavior in an asymptotic sense.
The error bounds are derived from the following polynomial minimization prob-
lem. For any compact set S R, we dene
p2Pn
where Pn is the set of polynomials p of degree at most n with 1. The standard
convergence analysis of the CG method leads to
Laboratoire d'Analyse Numerique et d'Optimisation, UFR IEEA { M3, UST Lille, F-59655
Villeneuve d'Ascq CEDEX, France, e-mail: bbecker@ano.univ-lille1.fr
y Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, B-3001
Leuven, Belgium, e-mail: arno@wis.kuleuven.ac.be
2 B. BECKERMANN AND A. B. J. KUIJLAARS
where (A) is the spectrum of A. The usual way to analyze (1.3) is to include the
spectrum into a 'continuous' compact set S, so that
The quantity En (S) can be estimated using notions from potential theory, since
lim
log En
where gS (z) is the Green function for the complement of S with pole at 1. Thus one
arrives at
as an upper estimate for the error. For example, if one chooses
the Green function evaluated at 0 is known to be
min
min
which leads to the asymptotic estimate
in terms of the condition number which is in agreement with (1.1).
We refer the reader to the survey paper [DTT98] of Driscoll, Toh, and Trefethen for an
excellent account on the interaction between iterative methods in Numerical Linear
Algebra and logarithmic potential theory.
The estimate (1.7) is typically accurate at early stages of the iteration. The
reason for this is that for small n, a polynomial p 2 Pn that is small on (A) has
to be uniformly small on the full interval [ min ; max ] as well. When n gets larger,
however, a better strategy for p is to have some of its zeros very close to some of the
eigenvalues of A, thereby annihilating the value of p at those eigenvalues, while being
uniformly small on a subcontinuum of S only. Then the right-hand side of (1.7) may
become a great overestimation of the error. This eect is the reason for the superlinear
convergence behavior of the CG iteration, observed in practical situations.
As an illustration we look at the case of a matrix A with 100 equally spaced
100g. The error curve computed for this example is the
solid line in Figure 1. See also [DTT98, page 560]. The classical error bound given
by (1.1) with is the straight line in Figure 1. For smaller values of n, the
classical error bound gives an excellent approximation to the actual error. The other
curve (the one with the dots) is the new asymptotic bound for the error that we nd
in Corollary 3.2 below. This curve follows the actual error especially well in the region
of superlinear convergence (for n 40).
The phenomenon of superlinear convergence has been understood for compact
operators, see [Win80, Mor97, Nev93]. Also, the above heuristic for the convergence
behavior of CG for large matrices has been discussed and further analyzed by several
authors [VSVV86, Gre89, SlVS96, DTT98]. To our knowledge, a formula for
SUPERLINEAR CONVERGENCE OF CONJUGATE GRADIENTS 3
-5log(rel.
iterations
CG error energy norm
Classical bound
Our asymptotic bound
Fig. 1. The CG error curve versus the two upper bounds for the system
solution x, and initial residual r Our new asymptotic
bound is given in formula (3.11).
the relative error improving (1.7) and explaining the superlinear convergence is still
lacking.
Our goal in this paper is to provide a better understanding of the superlinear
convergence of CG iteration, and in particular to explain the form of the error curve
as seen in Figure 1. We will argue that for a large NN matrix A, the error En
in the polynomial minimization problem (1.2) is approximatelyn
log En ((A)) < t
decreasing family of sets, depending
on the distribution of the eigenvalues of A. The sets S() have the following inter-
pretation: S() is the subcontinuum of [ where the optimal polynomial of
degree uniformly small.
From (1.3) and (1.8) we nd the improved estimate
with
Note that t depends on n, since . As the sets S() are decreasing as
increases, their Green functions g S() (0), evaluated at 0, increase with . Hence the
numbers t decrease with increasing n (see also Remark 2.3 below), and this explains
the eect of superlinear convergence.
4 B. BECKERMANN AND A. B. J. KUIJLAARS
The phenomenon of superlinear convergence may also occur for other Krylov
subspace methods applied to a system A is no longer symmetric
positive denite. For instance, for symmetric but not positive denite A one usually
employs iterative methods like MINRES. Also, the method GMRES may be applied
in case of a general matrix A. Supposing that A is diagonalizable, i.e.,
with D a diagonal matrix containing the (possibly complex) eigenvalues of A, the nth
relative residual may be bounded for these methods by
(see, e.g., [Saa96, Proposition 6.15]). In particular, for symmetric or more generally
normal matrices, V is unitary, and thus again we may give bounds for the relative
residual by describing the (asymptotic) behavior of En ((A)). Indeed, for the ease of
presentation our results are stated for real spectra, but they remain equally valid for
complex spectra (see also Remark 2.4 below).
The paper is organized as follows: In x2, we describe the (sequence of) matrices
AN under considerations. We explain the potential-theoretic origin of our sets S(t),
and establish in Theorem 2.1 the estimate (1.8). Under some stronger assumption
concerning the clustering of eigenvalues, we prove in Theorem 2.2 that estimate (1.8)
is sharp. Section 3 contains a description of eigenvalue distributions where our sets
are explicit intervals. Subsequently, we give an analysis of the plot of Figure 1.
In x4 it is shown that our assumptions are valid for a large class of symmetric positive
denite Toeplitz matrices. Our ndings are illustrated by considering some Toeplitz
matrix occurring in time series analysis. The discretized two-dimensional Poisson
equation on a uniform grid is analyzed in x5. Finally, a lemma used in the proof of
Theorem 2.1 is proved in the appendix.
We should mention that our results concerning the two applications above are
more of theoretical nature since in the present paper neither preconditioning nor
nite precision arithmetic are considered. The main aim of this paper is to illustrate
that some recent results in logarithmic potential theory may help to understand better
a classical phenomenon in Numerical Linear Algebra (see also [BeSa98, Kuij99]).
2. The main result. Properly speaking, the concept of superlinear convergence
for the CG method applied to a single linear system does not make sense. Indeed, in
the absence of roundo errors, the iteration will terminate after N steps if N is the
size of the system. Also the notion that the eigenvalues are distributed according to
some continuous distribution is problematic when considering a single matrix.
Therefore we are not going to consider a single matrix A, but instead a sequence
(AN ) N of symmetric positive denite (or more generally invertible symmetric) matri-
ces. The matrix AN has size NN , and we are interested in asymptotics for large N .
These matrices need to have an asymptotic eigenvalue distribution. By this we mean
that there exists a positive Borel measure with compact support supp() such that
the following condition is satised.
Condition (i) The spectra (AN ) are all contained in a xed compact set S R,
and for every function f continuous on S we have
lim
Z
SUPERLINEAR CONVERGENCE OF CONJUGATE GRADIENTS 5
This condition is equivalent to the weak convergence of the normalized eigenvalue
counting measures N dened by
where is the unit point mass at , to . As all the AN have spectra contained in
S, the measure is supported on S. The total mass kk is at most one, and it can be
strictly less than one if the matrices AN have many coinciding eigenvalues. Note that
in the sum in (2.1) each in (AN ) is taken only once, regardless of its multiplicity
(see also Remark 2.5 below).
For the use of the potential theory in what follows, we need to impose a condition
on . The logarithmic potential of a Borel measure with compact support is the
function
U
Z
logj
This is a superharmonic function on C taking values in (1;1]. In particular it is
lower semi-continuous. We refer the reader to [Ran95, SaTo97] for detailed accounts
of logarithmic potential theory. Our assumption is the following.
Condition (ii) The logarithmic potential U of the measure from (i) is a continuous
real-valued function on C .
The condition (ii) is not very restrictive. For example, if is absolutely continuous
with respect to Lebesgue measure with a bounded density then (ii) is satised. It
is also satised if the density has only logarithmic-type or power-type singularities
at a nite number of points. On the other hand, condition (ii) is not satised if
has point masses. A consequence of (ii) is that for any measure satisfying ,
the potential U is also continuous. Indeed, U is lower semi-continuous, and since
continuous and U lower semi-continuous, U is also
upper semi-continuous; hence U is continuous.
There is a third condition we impose on the sequence (AN ) N .
Condition (iii) The limit (2.1) also holds for
Notice that (iii) follows from (i) if 0 62 S, or even if the (in modulus) small eigenvalues
of AN do not approach zero too fast. If (iii) would not hold, then the matrices AN
are too ill-conditioned and the estimate (2.9) given below may very well fail.
In many practical applications, the family (AN ) N of matrices appears as discretizations
of a continuous operator, and then (i){(iii) are natural conditions, see for
instance the discussion in x4 and x5 below.
The sets S(t) that were announced in (1.8) depend only on the asymptotic eigenvalue
distribution . They are determined by the solution of an energy minimization
problem which we describe now.
The logarithmic energy of a Borel measure with compact real support is the
double integral
Z
U ()
ZZ
logj
For every t 2 (0; kk), we dene the class
is a Borel probability measure on R :
6 B. BECKERMANN AND A. B. J. KUIJLAARS
and we let t be the unique measure minimizing the logarithmic energy (2.2) in the
class M(t; ) (compare [Rak96], [DrSa97, Theorem 2.1]). Thus
The minimizer t depends on t and . The minimization problem (2.3) is a constrained
problem, since measures in M(t; ) are dominated by the constraint =t. It
is known that the minimizer t is characterized by the following variational conditions
associated with (2.3). There exists a constant F t such that
see [Rak96, Theorem 3] and [DrSa97, Theorem 2.1]. From these variational conditions
one obtains
Finally, the sets S(t) which are crucial in our ndings are dened by
The extremal problem (2.3) has been studied before in connection with the
asymptotic behavior of discrete orthogonal polynomials, see, e.g., [Rak96], [DrSa97],
[KuVA98], [Beck98], [BeSa98], and [Joh99]. In particular, the monic analogue of (1.2)
is covered by these results, that is, the study of
where P
n denotes the class of monic polynomials of degree n. Notice that if S [0; 1)
(as for instance for symmetric positive denite matrices), then E
n (S) and En (S) are
realized (up to scaling) by the same polynomial, namely the generalized Chebyshev
polynomial.
Our main result is the following.
Theorem 2.1. Let (AN ) N be a sequence of symmetric invertible matrices, AN
of size N N , satisfying the conditions (i), (ii) and (iii) for some measure . Let the
measures t , the constants F t , and the sets S(t) be dened by (2.3), (2.4){(2.5), and
(2.7), respectively. Then for t 2 (0; kk), we have
lim sup
n=N!tn
log En ((AN
Proof. Let t 2 (0; kk), and let depend on N in such a way that
Our goal is to construct for every large N a polynomial pN in
Pn which is suciently small on (AN ), so as to obtain the estimate (2.9).
We x > 0, and dene
SUPERLINEAR CONVERGENCE OF CONJUGATE GRADIENTS 7
Since U t is a continuous function (cf. the discussion following condition (ii)), the set
K is closed. It is disjoint from S(t) because of (2.4) and (2.7). Thus (K
By choosing a smaller if necessary, we may assume that (@K
possible to nd for every large N , a set ZN such that
(3) for all continuous functions f ,
lim
Z
For the proof that this is indeed possible, we refer to Lemma A.1 in the Appendix.
We write for N large,
Y
Then pN 2 Pn by property (1). We want to estimate max 2(AN ) jp N ()j. Let
be such that
Since pN vanishes on (AN ) \ K by property (2) and the denition (2.13), we have
and the latter is a bounded set. Passing to a subsequence, if necessary, we may assume
that the sequence (N ) converges as N !1 with limit
We have by (2.13),n
log jp N (N )j =n
log
log jj: (2.16)
From property (3), we have that the normalized counting measures of ZN , i.e.,
converge in weak sense to t t . The principle of descent, see [SaTo97, Theorem I.6.8],
and (2.15) then imply that
U N (N
t, this gives
lim sup
log
8 B. BECKERMANN AND A. B. J. KUIJLAARS
and thus by (2.15)
lim sup
log
The principle of descent also implies
By property (2) we have N N , where N is the normalized counting measure of
(AN ). Since N ! by condition (i), we nd that ( N N ) N is a sequence of
positive measures that converges to t t in weak sense. Applying the principle of
descent once more, we obtain
U N N (0): (2.19)
Also the condition (iii) gives
U
The relations (2.18){(2.20) easily imply that
U N (0)
and this is equivalent to
lim
log
Combining (2.16) with (2.17) and (2.21), we obtain
lim sup
log jp N (N )j F t
By (2.14) and the denition (1.2) of En , we then see
lim sup
En ((AN
The number > 0 can be chosen arbitrarily close to 0. Hence (2.9) follows.
To obtain (2.10) we need to show that
To establish this and the inclusion property
claimed in the introduction, we recall the connection of the constrained minimization
problem (2.3) with the energy problem in the presence of an external eld. For a
SUPERLINEAR CONVERGENCE OF CONJUGATE GRADIENTS 9
continuous function Q sucient growth at 1, the logarithmic energy
of a measure in the presence of the external eld Q is
I
ZZ
log 1
Z
The minimizer Q
s for the extremal problem
Z
exists and is unique if
log jj (2.24)
and it is characterized by the conditions
U Q
s
U Q
for some constant G s , see, e.g., [SaTo97, Theorem I.1.3]. Buyarov and Rakhmanov
[BuRa99, Theorem 2] proved the following formula for Q
s
where !S is the equilibrium measure for the set S. In fact, the authors consider
external elds where the limit on the right-hand side of (2.24) equals +1, and s 2
(0; +1). However, from their proof (see the last paragraph of [BuRa99, Section 2])
it becomes clear that (2.27) remains valid as long as (2.24) holds.
Now, in our situation with the constraint , we take
By comparing the conditions (2.4){(2.5) with (2.25){(2.26) we can easily check that
for s; t > 0 with s
In particular, ( Q
converges in weak sense to for s ! kk. Then the Buyarov{
Z t! S() d: (2.29)
From (2.29) we obtain the inequality t t for < t, and thus (2.23) holds. In
order to show (2.22), notice that the Green function is connected with the potential
of the equilibrium measure by the formula
A. B. J. KUIJLAARS
where cap denotes the logarithmic capacity. Combining this with (2.29), we obtain
for
Z tlog cap (S()) d: (2.30)
For 2 S(t), the left-hand side of (2.30) vanishes according to (2.4). Also, by (2.23),
each 2 S(t) belongs to S() for all < t, so that the integral in (2.30) involving the
Green functions vanishes for 2 S(t). Consequently,
Z tlog cap (S()) d;
and the equation (2.22) follows from (2.30). This completes the proof of Theorem 2.1.
Under additional conditions the inequality (2.9) can be improved to give equality
lim
n=N!tn
log En ((AN
These additional conditions are related to the separation of the eigenvalues. If many
eigenvalues are very close to each other then the inequality (2.9) may be strict. For the
extremal problem (2.8), various separation conditions were considered by Rakhmanov
[Rak96], Dragnev{Sa [DrSa97], Kuijlaars{Van Assche [KuVA98], and Beckermann
[Beck98], see also [KuRa98] for a survey.
If one of these conditions holds in the present situation, the limit (2.31) can be
proved. Indeed, according to Theorem 2.1, we only require a sharp lower bound for
En ((AN )). For sets S with positive capacity, lower bounds for En (S) are usually
obtained by applying the Bernstein-Walsh inequality. In our discrete setting, some
analogue of the Bernstein-Walsh inequality in terms of the extremal measure t exists,
see [KuVA98, Lemma 8.1 and Corollary 8.2] and [Beck98, Theorem 1.4(c)], implying
(2.31).
We will give here a proof using the separation condition of Beckermann [Beck98],
which was rst conjectured by Rakhmanov [KuRa98]. For a nite subset Z C , we
introduce
I (Z) :=(#Z) 2
logj
which may be thought of as the discrete energy of a system of #Z particles each
having a charge 1=#Z. Beckermann's condition is:
Condition (iv) With I() as in (2.2) we have
lim
I ((AN
It can be shown that lim inf I ((AN
already follows from condition (i).
Notice also that the separation conditions of [Rak96, DrSa97] imply (iv).
SUPERLINEAR CONVERGENCE OF CONJUGATE GRADIENTS 11
It is not dicult to prove using for instance [Beck98, Lemma 2.2(b)] that condition
(iv) is equivalent to the fact that
I (XN
whenever (XN ) is a sequence of sets satisfying XN (AN ) for each N , lim(#XN )=N
In this form the condition (iv) will be used.
Theorem 2.2. Suppose that the assumptions of Theorem 2.1 are satised and
that in addition the condition (iv) holds. Then for every t 2 (0; kk), the limit (2.31)
holds.
Proof. Let t 2 (0; kk). As in the proof of Theorem 2.1 we assume that
depends on N in such a way that n=N ! t.
For every N 2 N, let N be a set of n points in (AN ). That is, N
has denoted by maximizes the product
Y
among all n 1-point subsets of (AN ). Equivalently, N minimizes the discrete
energy I (N ). Since N (AN ), it is clear that
En ((AN
Our rst goal is to show that the normalized counting measures of the Fekete
points tend to t , that isn
as N ! 1. Since the sets N are all contained in the compact S, Helly's theorem
asserts that from any subsequence of the sequence of normalized counting measures
of the Fekete points, we may extract a further subsequence having a weak limit
(which clearly is an element of M(t; )). Our claim (2.35) follows by showing that
. According to (2.33), we nd that along an appropriate subsequence we then
have
I (N
Let (ZN ) N be a sequence of sets satisfying
It follows from Lemma A.1 in the Appendix that such a sequence exists. Again by
(2.33), we nd that
I (ZN
12 B. BECKERMANN AND A. B. J. KUIJLAARS
by (2.3), and I (ZN ) I (N ) by the denition of Fekete
points, we may conclude that I( by the uniqueness of
the minimizer in (2.3). This proves the claim (2.35).
Next, we dene for N 2 N and
Y
Then P k;N has degree n, and any polynomial p 2 Pn can be written in the form
a k
with coecients a k satisfying
k=0 a
and
a k
P k;N (0)
P k;N (0)
ng be such that it maximizes
P k;N (0)
among all k 2 f0; ng. Then it follows from (2.38) that
Since this holds for every p 2 Pn , we nd
We write shorter
~
with normalized counting measure
SUPERLINEAR CONVERGENCE OF CONJUGATE GRADIENTS 13
Because of (2.35) we see that (N ) has the weak limit t for N ! 1. From the
principle of descent [SaTo97, Theorem I.6.8] it follows that
lim sup
log jP kN ;N
Also, we will show below that
lim inf
log jP kN ;N ( kN ;N )j F t (2.41)
(compare with [Beck98, Lemma 2.6]). Combining (2.40) and (2.41) with (2.34) and
(2.39), we may conclude that
lim inf
log En ((AN
log
which in view of Theorem 2.1 is the inequality required for the proof of Theorem 2.2.
Finally, in order to establish (2.41), we note that by the denition of Fekete points
we have for every 2 (AN ),
Taking logarithms, and adding the inequalities for 2 (AN
N , we obtain
log jP kN ;N ()j
and therefore
log jP kN ;N ( kN ;N )j 1
log 1
One easily veries that the right-hand of (2.42) side equals2
which according to (2.33) converges to2t 2
Z
where for the last equality we have used the variational condition (2.4). Since
(#(AN assertion (2.41) follows from (2.42), and Theorem 2.2
is proved.
Remark 2.3. We have shown in Theorem 2.1 that, for n; N !1, the quantity
log En ((AN )) is asymptotically bounded above by
log n
14 B. BECKERMANN AND A. B. J. KUIJLAARS
and this bound is sharp (under some additional assumptions) according to Theorem
2.2. This conrms our claims (1.9) and (1.10) of the introduction. The graph
of Nt log t for xed N and varying is drawn in the plots of Figures 1, 2,
and 4. From (2.43) one sees that N t log t is dierentiable, up to at most a countable
number of points, with derivative
d
Thus it follows that (2.43) is decreasing. Also because of (2.23) one sees that g S(n=N) (0)
is increasing with n, and therefore the graph of (2.43) is concave.
If S is a compact set containing all the spectra (AN ), then S(t) S, for every
one easily checks that
In other words, the bound (1.9) is sharper than (1.6). The equality
holds if and only if which again is true if and only if the
equilibrium distribution !S of S is less then or equal to =t. This may be translated
by saying that, roughly, about tN out of the eigenvalues of AN are asymptotically
distributed like the equilibrium distribution of S.
Remark 2.4. Theorems 2.1 and 2.2 are equally valid for complex discrete
sets (AN ), here supp() may be a subset of the complex plane. Indeed, the energy
problems with constraint have been studied in a complex setting (see, e.g., [DrSa97]),
and it is possible to show that the representation of F t U t in terms of Green
functions remains equally true. Furthermore, all other arguments used in the proofs
of Theorems 2.1 and 2.2 still apply for complex sets (AN ). As a consequence, our
Theorems can also be used for bounding the relative residual while solving systems of
linear equations with normal matrices AN via methods like MINRES or GMRES.
Remark 2.5. In many applications (as for instance for symmetric Toeplitz ma-
trices) it is dicult to know in advance the multiplicities of the eigenvalues, and one
only obtains a measure ~
dened by a modication of condition (i) where multiple
eigenvalues are counted according their multiplicities. We will refer to this modi-
cation as condition (i'). Condition (ii) with this (possibly) new measure ~
will be
called (ii'), and accordingly (iii) becomes (iii'), where again we count multiplicities.
Notice that Theorem 2.1 remains valid if assumptions (i),(ii),(iii) are replaced by
(i'),(ii'),(iii') (and is replaced by the new measure ~ ).
In case of, e.g., real S, conditions (i') and (iii') have an interesting interpretation
in terms of asymptotics of determinants: Indeed,N
log jdet(I N AN
log 1
(in this formula we count multiplicities), and from logarithmic potential theory we
know that relation (2.1) holds for every function f continuous on S if and only if
lim
log jdet(I N AN
SUPERLINEAR CONVERGENCE OF CONJUGATE GRADIENTS 15
for all 2 C n S. Furthermore, it is sucient that (2.44) holds for 2 where
C has a nite accumulation point outside of S. Notice that condition (iii') may
be rewritten as (2.44) with Finally, condition (i') is known to hold i
lim
trace
for all 2 , C as above. Using (2.45), one can for instance easily show that
condition (i') remains valid (with the same measure) if AN is perturbed by some
matrix BN , with sup N kBN k < 1, and rank(BN )=N ! 0.
3. Equidistant eigenvalues. In order to apply Theorems 2.1 and 2.2 we have
to calculate the sets S(t) from the eigenvalue distribution . This is a problem in
itself. In general, the sets S(t) can have a complicated form. They may consist of
several intervals, or even have a Cantor-like structure. The easiest case would be
if all S(t) are single intervals. This would also be the most convenient case for the
computation of the Green function at 0, since for an interval [a; b], we have
a
a
Lemma 3.1. Suppose that is supported on the interval [a; b] and has a density
w() with respect to Lebesgue measure. We write
~
w() :=
(a) Suppose ~
w is strictly increasing on (a; b). Then S(t) is an interval containing
b for every t 2 (0; kk). More precisely, we have
and
is the unique solution in (a; b) of the equation
Z r
a
r
r
w() d: (3.2)
(b) Suppose ~
w is strictly decreasing on (a; b). Then S(t) is an interval containing
a for every t 2 (0; kk). More precisely, we have
w(b
and
w(b
is the unique solution in (a; b) of the equation
r
r
a
r
w() d:
B. BECKERMANN AND A. B. J. KUIJLAARS
(c) Suppose ~
w is symmetric with respect to the midpoint m := (a + b)=2 and
strictly decreasing on (m; b). Let t 2 (0; kk). Then
w(b
and
w(b
is the unique solution in (0; (b a)=2) of the equation
a
w()d:
Proof. (a) We consider as in the proof of Theorem 2.1 the external eld
a
log
Let Q
s be the extremal measure with external eld Q and normalization s 2 (0; kk)
(cf. the paragraph preceding formula (2.26)). In [KuDr99, Theorem 2] it was proved
that the support of Q
s is an interval of the form [r; b] if Q and w are related as in (3.3)
and if ~
w() increases on [a; b]. In [KuDr99] this is stated under the assumption that
Q is dierentiable with a Holder continuous derivative. An inspection of the proof,
however, shows that this assumption is not necessary. It was also assumed that s = 1.
This is also not essential. Since
s by (2.28), it thus
follows that S(t) is an interval containing b for every t.
We show that
w(a+). For t ~
w(a+), we have from
the fact that ~
w is strictly increasing,
(b )( a)
Thus the equilibrium measure ! [a;b] of [a; b], i.e.,
(b )( a)
d
belongs to the class M(; t). Since ! [a;b] minimizes the energy (2.2) among all probability
measures on [a; b], it is then also the minimizer over M(; t). Thus
and it follows from (3.4) that
Conversely, if is a probability measure on [a; b] whose potential
is constant on [a; b] by (2.4). This implies that Hence t! [a;b] and (3.4)
holds. Thus t ~
w(a+).
For the rest of the proof we assume that t 2 ( ~
w(a+); kk). Then
with a < r(t) < b. From [DrSa97, Corollary 2.15] we know that t
is the balayage (see [SaTo97, Section II.4]) of onto the interval [r; b].
Consequently, according to [SaTo97, Eqn. (II.4.47)], t t has the density
w() if 2 (a; r),
Z r
a
s
SUPERLINEAR CONVERGENCE OF CONJUGATE GRADIENTS 17
For 2 (r; b), we rewrite the density as
Z r
a
Since a < r < b and 0 v() w() < 1 for 2 (a; b), we must have
lim
In view of (3.5), the relation (3.2) follows.
To show that there is only one r satisfying (3.2), we rewrite the right-hand side
of (3.2) as
Z r
a
r
r
Z r
a
~
w()
d
Z =2~
where for the second equality, we used the change of variables
Since ~
w is strictly increasing, it is then clear that (3.6) strictly increases for r 2 (a; b).
This completes the proof of part (a).
(b) The proof of part (b) is similar.
(c) Part (c) follows using a quadratic transformation, compare with [Kuij99, Proof
of Theorem 5.1].
Lemma 3.1 allows us to determine the sets S(t) in a number of situations. We
consider here the case of equidistant eigenvalues.
Suppose AN has N equidistant eigenvalues . Multiplying the matrix
by a positive constant does not change the numbers En ((AN )). We multiply AN by
1=N and so we consider instead matrices with spectrum
These matrices have an asymptotic eigenvalue distribution
and the conditions (i){(iv) are satised.
The explicit solution of the energy minimization problem (2.3) with given by
(3.7) is due to Rakhmanov [Rak96]. We show how the sets S(t) can be determined
from Lemma 3.1. The assumptions of Lemma 3.1(c) are clearly satised with a = 0,
Therefore we have for 0 < t < 1,
d
=1=2 r
B. BECKERMANN AND A. B. J. KUIJLAARS
Thus
and
11p
Using (3.1) and (3.8) we nd after a little calculation
log
Hencet
Z tlog
d
Theorem 2.2 and (3.10) then give the following result.
Corollary 3.2. For every t 2 (0; 1) we have
lim
n=N!tn
log En (f1;
Corollary 3.2 gives the theoretical justication for our CG bound in the case of
equidistant eigenvalues as given in Figure 1. Notice that, already for 100, the
approximation for log En (f1; obtained by multiplying the right-hand side
of (3.11) by n is quite accurate.
To conclude this section, we note that Lemma 3.1(c) also applies to the case of
ultraspherical eigenvalue distributions. The corresponding sets S(t) were determined
in [Kuij99].
4. Applications to Toeplitz matrices. Toeplitz matrices provide interesting
examples for our results. Toeplitz systems arise in a variety of applications, such
as signal processing and time series analysis, see [ChNg96] and the references cited
therein.
be an integrable function with Fourier coecients
Z
()e ik d;
We assume is bounded and not equal to a constant. The Nth Toeplitz matrix with
symbol is given by
TN
SUPERLINEAR CONVERGENCE OF CONJUGATE GRADIENTS 19
Then TN () is a Hermitian matrix and it is well known that ( inf ; sup ) is the smallest
interval containing the spectrum of TN () for every N , where inf and sup denote
the essential inmum and essential supremum of , respectively. Thus, since is
non-negative, all eigenvalues 1;N 2;N N;N of TN () are strictly positive,
and the matrix TN () is positive denite.
A classical result of Szeg}o, see e.g. [GrSz84, pp. 63-65], [BoSi99, Theorem 5.10
and Corollary 5.11], says that
lim
Z
for every continuous function f on [ min ; max ]. It follows that the sequence (TN ())
satises the condition (i') (see Remark 2.5) with the measure given by
Z
Z
Then is a probability measure and its support is equal to the essential range of .
Another result of Szeg}o (see [Sz67, Eqn. (12.3.3)] or [GrSz84, p. 44 and p. 66]) is
that
lim
Z
log () d
provided that satises the Szeg}o condition
Z
log
Notice that this condition can be rewritten as U (0) < +1. It follows from (4.2),
(4.3) that
lim
log det TN ()
Z
log
Z
log d() 2 R;
and the condition (iii') is satised.
Consequently, for Toeplitz matrices TN () with non-negative, integrable, and
bounded symbol and continuous real-valued potential U , the conditions (i')-(iii')
are satised, and we may apply Theorem 2.1.
We will discuss an example of Kac, Murdock and Szeg}o [KaMuSz53, p. 783]
with
1). Toeplitz matrices with this symbol (or with a multiple of this
arise as covariance matrices of rst-order autoregressive processes [ChNg96,
Section 4.6.1]. The corresponding Fourier coecients are given by
Suppose without loss of generality that
> 0. Then the measure from (4.2) has
support [a; b] where
20 B. BECKERMANN AND A. B. J. KUIJLAARS
Since is even we have2
Z
Making the substitution = (), we obtain after some calculations2
Z
a
d
Thus the measure has density
a < < b: (4.4)
with respect to Lebesgue measure. From (4.4) it is easy to show that U is continuous,
so that Theorem 2.1 applies in this case.
Now we apply Lemma 3.1(b) in order to compute S(t). Notice that for r 2 [a; b),
r
r
a
r
r
d
r a
r
Consequently, by Lemma 3.1(b), we have
r a
a
if a < t < 1:
In particular, we get from (1.10) and (3.1) the convergence rate
log
whereas for a < t < 1, we have
t log
a
log
a
a
d
a log(
a
log
d
a log(
a
log
It is quite interesting that, in the superlinear range, we obtain (up to some linear
transformation) the same function as for equidistant nodes.
Numerical experiments for the symmetric positive denite Toeplitz matrix T 200 of
order 200 of Kac, Murdock and Szeg}o are given in Figure 2. The four dierent plots
correspond to the choices
19=20g of the parameter. Notice that the
CG error curve (solid line) of the last two plots is clearly aected by rounding errors
leading to loss of orthogonality, whereas the GMRES relative residual curves (dotted
line) behave essentially like predicted by our theory. In particular, the classical bound
SUPERLINEAR CONVERGENCE OF CONJUGATE GRADIENTS 21
log(
rel.
residual
log(
rel.
residual
iterations
iterations
Fig. 2. The error curve of CG (solid line) and GMRES (dotted line) versus the classical upper
bound (crosses) and our asymptotic upper bound (circles) for the system T 200
solution x, and initial residual r Here TN is the Kac, Murdock and Szeg}o matrix of
x4, with parameter
19=20g.
(1.1), (1.11) (crosses) does no longer describe correctly the size of the relative residual
of GMRES for n 20 and
19=20g. Experimentally we observe that the
range of superlinear convergence starts in the dierent examples approximately at
the iteration indices 50, 30, 20, and 10, respectively. This has to compared with
the predicted quantity N a which for the dierent choices of
approximately takes
the values 66, 40, 29, and 5, respectively. Though theses numbers dier slightly,
we observe that the new bound (1.9), (1.10) re
ects quite precisely the shape of the
relative residual curve, and in particular allows to detect the ranges of linear and of
superlinear convergence.
Let us nally mention the Toeplitz matrices occuring in the context of the rst-
order moving average process [ChNg96, Section 4.6.1], where the symbol is given by
Here the eigenvalues are asymptotically distributed like the equilibrium distribution
on
and therefore there will be no superlinear convergence in
22 B. BECKERMANN AND A. B. J. KUIJLAARS
this case.
5. The Model Problem. Consider the two dimensional Poisson equation
for (x; y) in the unit square 0 < x; y < 1, with Dirichlet boundary conditions on the
boundary of the square. The usual ve-point nite dierence approximation on the
uniform grid
leads to a linear system of size N N where After rescaling, the coecient
matrix of the system may be written as a sum of Kronecker products
Bm
where
. 1
mm
and I m is the identity matrix of order m. It is well known and easy to verify that the
eigenvalues of Bm are
and that the eigenvalues j;k of AN are connected with the eigenvalues of Bm via
most of the eigenvalues have multiplicity at least 2. Also, j;m+1 j
and the eigenvalue 4 has multiplicity m. We suspect that
which is conrmed by our numerical
experiments presented below.
To calculate the asymptotic distribution of the eigenvalues j;k as
rst note that the eigenvalues k of Bm are in [0; 4] and have the asymptotic density
(4
1. The asymptotic density of the is then given by the
convolution of v with itself, i.e.,
SUPERLINEAR CONVERGENCE OF CONJUGATE GRADIENTS 23
Fig. 3. Bar chart of the eigenvalue distribution of the matrix A 1600 (without counting multi-
plicities) resulting from discretizing the 2D Poisson equation on a uniform grid with
points. The solid line corresponds to the asymptotic density function.
where the factor 1=2 is added in accordance with the multiplicities of the eigenvalues
of AN . The density w is symmetric around 4. For 2 (0; 4), we have from (5.4) and
In (5.6) we put make the change of variables
to obtain
w(4 4x) =8 2
Z 11
By the Euler integral representation for hypergeometric functions, (5.7) is
w(4 4x) =8 F (1=2; 1=2;
is a Pochhammer symbol. It is interesting to observe
that 4 2 w(4 4x) equals the complete elliptic integral of the rst kind, evaluated at
Eqn. 7.3.2.(75)]. Since w is symmetric around 4 and the
right-hand side of (5.8) is even in x, the formula (5.8) holds for 1 < x < 0 as well.
B. BECKERMANN AND A. B. J. KUIJLAARS
In the series in the right-hand side of (5.8) each term is clearly decreasing as
increases. Thus w() is increasing for 2 (0; 4), which is also clear from
Figure
3. Then also
increases for 2 (0; 4), and therefore the assumptions
of Lemma 3.1(c) are satised. From Lemma 3.1(c) we then conclude that,
for every t 2 (0; 1=2), the set S(t) associated with w()d is
with r 2 (0;
w() d: (5.9)
Putting in (5.9) we have
x
w(4 4x) dx: (5.10)
Inserting the series (5.8) for w(4 4x) and interchanging integration and summation,
we nd
x
For each k, the integral is easily transformed to a beta-integral, and it follows that
x
see also [Kuij99]. Inserting this in (5.11), we obtain
This is a known series expansion for the arccos function
see [PrBrMa90, Eqn. 7.3.2.(76)]. Inverting this we obtain the remarkably simple
and so
-5log(rel.
iterations
2D Poisson discretized, 150 2 inner gridpoints
CG error energy norm
Classical bound
Our asymptotic bound
Fig. 4. The CG error curve versus the two upper bounds for the system AN resulting
from discretizing the 2D Poisson equation on a uniform grid with points. We have
chosen a random solution x, and initial residual As predicted by (5.15), we obtain
superlinear convergence from the beginning. Notice that the classical upper bound for CG is far too
pessimistic for larger iteration indices. Similar plots are obtained for other mesh sizes.
Finally, after a small calculation using (1.10) and (3.1) we obtain the convergence
factor
log t =t
Z tlog(tan(
Notice that, for small t, the set S(t) of (5.14) approximately equals the set obtained for
equidistant eigenvalues on [0; 8], compare with x3. This observation is in accordance
with the behavior of the eigenvalues of AN at the endpoints of
One might be curious about what CG error curve is obtained if other boundary
conditions are imposed. In this case, we need to modify O(m) rows of AN , and
such \small rank" perturbations have been covered in Remark 2.5. However, since
multiplicities are in general not preserved by such modications, we need to have a
closer look in order to obtain sharp error bounds.
In our case we can be more precise since again the eigenvalues can be computed
explicitly for a number of congurations (see, e.g., [ChEl89]). For instance, in case of
periodic boundary conditions, most of the eigenvalues are of multiplicity 8. Thus, in
accordance with [ChEl89], the convergence behavior for Dirichlet boundary conditions
with mesh size h is similar to the one obtained for periodic boundary conditions with
mesh size h=2. In case of \no-
ow" Neumann boundary conditions on the vertical
boundaries discretized by a rst order scheme, the corresponding eigen-values
are given by (3.4) plus the m eigenvalues of Bm . Here we may expect the same
convergence behavior as for Dirichlet boundary conditions.
Appendix
A. In the appendix we state and prove a lemma that is used in the
proof of Theorem 2.1.
26 B. BECKERMANN AND A. B. J. KUIJLAARS
Lemma A.1. Let be a nite Borel measure on R with compact support. Suppose
(N ) N is a sequence of sets, all contained in a xed compact set, such that
lim
Z
for all continuous functions f on R.
let be a Borel probability measure such that t . Let
#N such that n=N ! t. Then there exists a sequence of sets (ZN ) N such
that
(a)
(b) ZN N , and
(c) for all continuous functions f ,
lim
Z
Furthermore, if K is a closed set such that then the
sets ZN can be chosen such that in addition to (a), (b) and (c), we also have for N
large enough,
(d) N \ K ZN .
Proof. We have to prove that for some sets ZN satisfying (a) and (b) the normalized
counting measures
converge in weak sense to t. To show this, we proceed in three steps.
Step 1 Suppose we have a nite partition of R consisting of measurable sets
j. Since the normalized counting
measures of the sets N tend to , we then have for every j,
lim
then possible to choose, for every j and N , a subset
ZN;j N \ U j such that
lim
The sets ZN;j are disjoint and for their union
Z
we have Z
Hence
lim
#(Z
Then also
lim
lim
#(Z
so that #Z
1. The set Z
N may not have exactly n elements.
By adding or deleting o(N) elements, we obtain from Z
N a set ZN with exactly n
elements. If we add elements, we choose them from N . Then ZN N and the
limits
lim
hold.
Now assume we have a nite collection U j , of measurable
sets such that (@U j j. The sets U j are not necessarily disjoint. For each
I
I =@ \
U jA \@ \
(R
The sets V I with I ranging over all subsets of a partition of R. By
Step 1, see (A.1), there exist sets ZN such that
lim
Since every U j is a nite disjoint union of some of the V I , it also follows that
lim
be a basis for the topology of R, chosen such
that (@U j j. From Step 2 we get for each k, a sequence of sets (Z (k)
such that #Z (k)
N N and
lim
see (A.2). Then by a diagonal argument, it is possible to nd a sequence (k N ) tending
to innity, such that the sets ZN dened by
lim
We also have
28 B. BECKERMANN AND A. B. J. KUIJLAARS
so that (a) and (b) hold.
if N is the normalized counting measure of ZN , then by (A.3) we have for
every j,
lim
Since the U j form a basis for the open sets, it follows that the measures N tend in
sense to t. Thus (c) holds.
Next, assume that K is a closed set such that
Then
lim
#(N \ K)
and since also we have because of (c)
lim
#(ZN \ K)
we then have
Then we modify ZN by adding the elements of (N n ZN ) \ K to ZN and removing
o(N) arbitrary elements from ZN n K. This is always possible for N large enough.
Then clearly (d) is satised, while (a), (b) and (c) continue to hold.
This completes the proof of the lemma.
--R
On a conjecture of
The sensitivity of least squares polynomial approxima- tion
Families of equilibrium measures with external
Fourier analysis of iterative methods for elliptic problems
Conjugate Gradient Methods for Toeplitz systems
Constrained energy problems with applications to orthogonal polynomials of a discrete variable
From potential theory to matrix iteration in six steps
Comparisions of splittings used with the conjugate gradient algorithm
Which eigenvalues are found by the Lanczos method?
Equilibrium problems associated with fast decreasing polynomials
Zero distributions for discrete orthogonal poly- nomials
Extremal polynomials on discrete sets
A note on the superlinear convergence of GMRES
Convergence of Iterations for Linear Equations
Integrals and Series
Potential theory in the complex plane
Equilibrium measure and the distribution of zeros of the extremal polynomials of a discrete variable
Iterative methods for sparse linear systems
Logarithmic potentials with external
Further results on the convergence behavior of conjugate-gradients and Ritz values
The rate of convergence of conjugate gradients
Some superlinear convergence results for the conjugate gradient method
--TR
--CTR
A. L. Levin , D. S. Lubinsky, Green equilibrium measures and representations of an external field, Journal of Approximation Theory, v.113 n.2, p.298-323, December 2001
S. Helsen , M. Van Barel, A numerical solution of the constrained energy problem, Journal of Computational and Applied Mathematics, v.189 n.1, p.442-452, 1 May 2006 | superlinear convergence;conjugate gradients;toeplitz systems;krylov subspace methods;logarithmic potential theory |
588536 | Integral Operators on Sparse Grids. | In this paper we are concerned with the construction and use of wavelet approximation spaces for the fast evaluation of integral expressions. The spaces are based on biorthogonal anisotropic tensor product wavelets. We introduce sparse grid (hyperbolic cross) approximation spaces which are adapted not only to the smoothness of the kernel but also to the norm in which the error is measured. Furthermore, we introduce compression schemes for the corresponding discretizations. Numerical examples for the Laplace equation with Dirichlet boundary conditions and an additional integral term with a smooth kernel demonstrate the validity of our theoretical results. | Introduction
. A naive Galerkin discretization of an integral operator
Z
with global kernel K leads to a dense stiness matrix. Hence, on a uniform full grid
with O(2 nJ ) unknowns (n dimension, J maximal level in a multiscale discretization),
the discrete operator has O(2 2nJ ) entries. This makes matrix vector multiplications,
as they are needed in iterative methods, prohibitively expensive for large n or large J .
A standard strategy to reduce this cost is to exploit the decay properties of the kernel
that, for singular kernels, are typically of the form
@
x @
y K(x; y)
C
x y
f(;)
Such decay properties do hold for pseudo-dierential operators [24]. Together with
(isotropic) biorthogonal wavelets with a su-cient number of vanishing moments, most
entries in the corresponding stiness matrices are then close to zero and can be replaced
by zero without destroying the order of approximation [4, 9, 12].
Another approach to reduce the cost of the integral evaluation is to replace the full
grid approximation space by the so-called sparse grid space [30]. The idea is to use
an anisotropic tensor-product basis together with a specic subset of the full grid
approximation space. It has been shown that under some assumptions on the approximation
and smoothness properties of the underlying basis functions and provided
that certain additional regularity assumptions are fullled, the resulting approximation
spaces exhibit the same order of approximation as the full grid space, while having
less dimension, see for example [32].
The sparse grid approach (that also appeared under the names hyperbolic cross approximation
or boolean blending schemes) is well established in approximation and
interpolation theory, see e.g. [1, 11, 13, 30, 33, 34]. First approaches to use sparse
grids for integral operators can be found in [16, 18, 20, 27]. In [20] it has been observed
that a discretization with adaptive sparse grid spaces leads to good approximation
rates for the single layer potential equation on a square. In [18] theoretical results
on the discretization of elliptic operators of arbitrary order with sparse grids and
biorthogonal wavelets have been presented. In both papers sparse grids are used as
test- and ansatz-spaces in a Galerkin discretization. Hence, without further com-
pression, the resulting discrete systems are dense. This rises the question, whether
additional compression schemes along the lines of [10, 29] are applicable for this type
of tensor-product discretizations.
For integral operators with smooth kernel, sparse grids may be used directly for the
approximation of the kernel. The theoretical background to this approach is that the
smoothness of the kernel translates into decay properties of its multiscale coe-cients.
Then one can construct (modications of) sparse grid spaces such that the evaluation
of integral expressions of the type (1.1) can be performed with much lower cost but
the same accuracy as with the full grid approximation space. This approach has
already been used for complexity estimates of Fredholm integral equations of the
second kind with smooth kernels for the special case with isotropic Sobolev smoothness
or dominating mixed smoothness for K und u [15, 16, 27].
In this paper we develop sparse grid type approximation spaces for the approximate
evaluation of (1.1). We start with smoothness classes including isotropic Sobolev
spaces as well as spaces of dominating mixed derivative, and construct sparse grid
type approximation spaces for the kernel that are adapted to the smoothness of K
and u from (1.1), as well as the norm in which the error of the integral evaluation is
measured. Moreover we develop compression schemes for anisotropic tensor-product
discretizations for kernels that obey (1.2). Note that although these two types of
compression are based on dierent properties of the kernel, it is instructive to present
both approaches in a somewhat unied framework.
The remainder of the paper is as follows. Section 2 introduces the necessary notation
and summarizes the basic facts about biorthogonal wavelet bases, tensor-product
spaces, norm equivalences and the smoothness classes we consider. They are certain
intersections of classes of functions with dominating mixed derivatives, see (2.1) below.
In Section 3 we investigate strategies for the fast evaluation of (1.1). We introduce our
new approximation spaces and give estimates on the order of approximation obtained
with these spaces and discuss compression schemes for anisotropic tensor-product
discretizations for kernels that obey (1.2). In Section 4 we present numerical examples
for the integral evaluation and for the numerical solution of the Laplace equation with
Dirichlet boundary conditions and an additional integral term with smooth kernel.
2. Preliminaries. Multi-indices (vectors) are written boldface, for example
(j Inequalities like l t or l 0 are to be understood componentwise. We
exist independent of any parameters x or y may depend on
such that C 1 y x C 2 y: In the rest of the paper C denotes a generic constant
which may depend on the smoothness assumptions and on the dimension n of the
problem. Moreover, for
By dist(x; y) we denote the
euclidian distance between x and y.
2.1. Sobolev spaces. Let us denote by H t (I n ); t 2 IR, a scale of standard
Sobolev spaces on I the space of L 2 -integrable functions
on I n .
Now, we dene the Sobolev spaces H t;l
mix . They x the smoothness assumptions we
consider.
Definition 1. Let t 2 IR; l 2
the
i-th unit-vector in IR n . Then we dene
mix
mix
mix
mix
H kn (I) for k 2 IR n . For l < 0 we dene H t;l
mix
as dual of H t; l
mix
This denition includes the class of functions of dominating mixed derivative H t;0
mix
as well as the standard isotropic Sobolev spaces H t
mix additional
information on these spaces, specically on the denition via Fourier transform, the
treatment of boundary conditions and the connections between spaces of bounded
mixed derivative and these anisotropic tensor-product spaces see [18, 22] and [21].
When the meaning is clear from the context, we will sometimes drop I n in H t;l
mix
and write H t;l
mix .
2.2. Biorthogonal wavelet bases. The approximation spaces considered here
stem from anisotropic tensor-products of univariate function spaces. We start from
a one-dimensional multi-resolution analysis
and we assume that the
complement spaces W are spanned by some multiscale
basis functions such that we have W
set dened from the subdivision rate of successive renement levels. We will consider
dyadic renement throughout the paper. Moreover, we assume that f jk
forms a Riesz-basis of W j and that there exists a dual system f ~
such that
holds. We assume k jk k L . In the following let
jk and ~
jk have N and ~
vanishing moments, respectively. Moreover we write
~
for the complement spaces spanned by the dual wavelets.
Under these conditions every u 2 L 2 has unique expansions
To simplify things we assume in the following the validity of the norm equivalences
where
0 and an analogous relation for the dual
wavelet (with t 2 (
r)). Such two-sided estimates can be inferred from the validity
of direct estimates (estimates of Jackson type) and inverse estimates (Bernstein
inequalities) for the primal and the dual wavelets as a consequence of approximation
theory in Sobolev spaces together with interpolation and duality arguments, see e.g.
[8, 25].
For the higher-dimensional case n > 1, let j 2 IN n be given, and consider the tensor-product
partition with uniform step size 2 j i into the i-th coordinate direction. By
we denote the corresponding tensor-product function spaces W j := W
W jn and ~
~
basis of W j is then given by [k2 j f jk
Under the assumption of the validity of (2.3) and an analogous relation for the dual
wavelet, it is then straightforward to prove the following norm equivalences
mix
r) and
mix
~
r), see [18]. For we regain the (standard) norm equivalences
for the isotropic Sobolev space H t and the Sobolev space with dominating mixed
derivative H t;0
mix . The dierent factors 2 2tjjj 1 and 2 2ljjj 1 in these equivalences re
ect
the dierent smoothness requirements.
One of the merits of the validity of a norm equivalence for H s is the fact that it leads
directly to optimal preconditioning for H s -elliptic problems and hence to fast iterative
methods with convergence rates independent of the number of unknowns, see [8].
Moreover, (2.4) and (2.5) may be used to analyse the approximation of the embedding
and to construct optimized sparse grid type Finite Element approximation
spaces for elliptic variational problems including integral operators [18]. However,
for integral operators the resulting stiness matrices are dense. In the next Section
we develop schemes to avoid the additional cost due to the denseness of the matrices.
3. Tensor-product approximation and compression of integral opera-
tors. The aim of this Section is to e-ciently approximate the coe-cients h jk of
To this end, we represent K and u from (1.1) in the dual and primal basis, respectively,
(j;k)2IN 2n
(l;m)2 jk
a jklm ~
jl (x) ~
where a
jk := j k . Here we assume that eventual boundary conditions are implemented
into the denition of the primal and dual wavelets, compare Section 4.
Given an index set I IN 2n we consider the approximation space
~
I :=
(j;k)2I
~
~
Typical examples are the full grid spaces
~
~
~
and the sparse grid (or hyperbolic cross) spaces
~
~
~
(3.
Figure
3.1. Index sets corresponding to the full (left) and the sparse grid space (right) with
maximal level in one dimension
from [30]. These are associated with rectangular index sets and with the index sets
respectively, compare Figure 3.1. The dimensions of the full grid and
the sparse grid approximation space are O(2 2nJ ) and O(2 J J 2n 1 ), respectively, see
[14, 30]. Note that the dimension of the sparse grid space compares favourably with
the dimension of the full grid space.
Given an additional index set comp
jk jk we dene the following two approximations
of the kernel K,
K I (x; y) :=
(j;k)2I
(l;m)2 jk
a jklm ~
jl (x) ~
km (y)
and
I (x; y) :=
(j;k)2I
(l;m)2 comp
a jklm ~
jl (x) ~
In (3.5) the kernel K is approximated by tensor-product approximation and by setting
all coe-cients a jklm with (j; I to zero. The second approximation step (3.6)
consists of (additionally) setting some of the coe-cients a jklm to zero.
The corresponding integral operators are the uncompressed operator A from (1.1) and
the two operators
A I u(x) :=
Z
K I (x; y) u(y)dy and A comp
I u(x) :=
Z
I (x; y) u(y)dy:
Now, to obtain schemes for an e-cient evaluation of the integral expression Au, we
have to choose the index sets I and comp
jk in such a way, that on the one hand a
good approximation of the integral is obtained, and on the other hand, the cost of the
integral evaluation remains as small as possible. The rest of this Section is devoted
to this task.
3.1. Tensor product approximation. The following Theorem estimates the
approximation error k(A A I )ukH s for a given index set I.
Theorem 1. Let
mix . We assume that the norm
equivalences (2.4) and (2.5) hold. Moreover let the parameters ~
r from (2.4) and
(2.5) be such that ~
holds. Then,
(j;k)62I
mix
mix
Proof: We use the biorthogonality between the dual and the primal wavelets and the
validity of (2.4) and (2.5). It holds
(j;k)62I
(l;m)2 jk
a jklm ~
jl (x) ~
km (y)A
p2o
uop op (y)
(j;k)62I
(l;m)2 jk
a jklm ukm ~
k:(j;k)62I
a jklm ukmA ~
k:(j;k)62I
a jklm ukmA max
(j;k)62I
k:(j;k)62I
(j;k)62I
jklm
km
(j;k)62I
mix
mix
:Now, for a given bound of the error k(A A I )ukH s we determine minimal index sets
I. To simplify things, we assume in the rest of this Section l; t;
we use in (3.7) for I the index set f(j;
which corresponds to the standard full grid approximation space (3.3) for the kernel
K, we obtain
mix
mix
Note that the order of approximation in (3.8) does not explicitely depend on the
smoothness of u (p and q only restrict the range of l and t). For example for a
kernel with product structure K(x; the smoothness of Au is only
dependent on the smoothness of k 1 .
A closer look on (3.7) and (3.8) reveals that one could discard indices from the full grid
index set without destroying the order of approximation from (3.8). This motivates
the following denition of index sets.
Figure
3.2. Index sets I100 (0; 0;
from left to right.
Definition 2. We introduce the parametrized index sets
I J (s; t; l; p; q) := f(j;
According to (3.2) the related approximation spaces are
~
(j;k)2IJ (s;t;l;p;q)
~
~
When the meaning is clear from the context, we sometimes drop the parameters
(s; t; l; p; q) and write I J instead of I J (s; t; l; p; q). For
we regain the index sets corresponding to the sparse grid
spaces (3.4) and the standard full grid spaces (3.3), respectively, compare Figure 3.2
(rst and second) and Figure 3.1. Additional smoothness in u, that is p > 0 or q > 0,
further reduces the number of entries in the index sets I J . Figure 3.2 (third and
fourth) gives two examples for the one-dimensional case.
Due to the biorthogonality of the primal and dual wavelets, not all coe-cients of the
wavelet representation of u are required for the evaluation of A I u or A comp
I u. For
example, the coe-cients of h IJ := A IJ u read
a
Therefore, we need the coe-cients ukm only if 9k 2 IN I J . In this way, I J
also induces approximation spaces for u,
VUI J :=
k2UI J
where
Similarly, for Au the denition of I J implies the index set
Note that this index set is optimal for the approximation of functions from H t;l
mix if
the error is measured in the H s -norm, compare [18]. Moreover, it depends only on
s l
t .
The following Lemma summarizes the approximation properties of our new approximation
spaces. It is a direct consequence of (3.7) and the denition of I J in (3.9).
Lemma 1. Let under the assumptions of Theorem 1 it holds
mix
Inequality (3.12) shows that the use of the index set I J (s; t; l; p; q) leads to the same
rate of convergence as the use of the full grid index set, compare (3.8), although in
most cases the number of elements in I J (s; t; l; p; q) is much smaller.
Note that the validity of (3.7) and (3.12) may be extended to the maximal range
because of the wider range of the -estimate in
(2.3), see [29] (with eventual additional logarithmic terms in the extremal case and
special cases of the index sets).
If the optimal approximation order is bounded by the approximation order of the
wavelets and not by the smoothness of K or u (i.e. for example for the case t > ~
then the optimal index set I J is obtained by setting
Then, for example for the L 2 -norm, i.e. the optimal index set is I J (0; ~
Together with an estimate of the number of elements in I J (s; t; l; p; q) (which may
be obtained from general results in [14]), it is straightforward to obtain estimates
for the cost of approximating Au. For example in the case t > 0, i.e. for a kernel
with dominating mixed smoothness, the number of elements N needed to obtain an
approximation with bounded by
O( 1
O( 1
O
s l
for s l < 0:
In the case of isotropic smoothness, i.e. the number of elements needed
to obtain an approximation A IJ u with k(A A IJ )ukH s is bounded by
O( n
O( n
O
2n
s l q
for
The above estimates have to be compared with the corresponding number of elements
when using the full grid approximation spaces.
Hence, for a kernel with dominating mixed smoothness and s > l, an -approximation
may be obtained with a cost that is asymptotically independent of the dimension n,
compare (3.13). For s l, there appears a slight n-dependence. This is similar to
the approximation of functions from H t;l
mix in H s . Indeed, in this case, the integral
evaluation is asymptotically not more expensive than the approximation of a function
from H t;l
mix , compare [18].
For K and u with isotropic smoothness, the cost is dependent on n, compare (3.14).
But in any case, the integral evaluation with the above scheme is less costly than the
evaluation with the use of full grid approximation spaces. See also [26, 27], where
special cases of this phenomenon have been described previously.
Note however, that without further compression the above sparse grid approach still
leads to dense discrete operators.
3.2. Additional compression. Additional compression may be obtained via a
special choice of the index sets comp
jk from (3.6). The idea is to drop small entries
in the stiness matrix without destroying the order of approximation. Here, simple
thresholding, i.e. dropping those entries in the stiness matrix that are under a certain
threshold, is in general not feasible, as this approach requires the computation of all
entries. Moreover, this approach does not take into consideration the smoothness of
u and the norm in which the error is measured.
Special compression schemes for isotropic discretizations that take these considerations
into account can be found in [10, 29]. Then for isotropic tensor-product wavelets
on a full grid the number of entries in the stiness matrix after compression is only
of order O(J k 2 Jn ) with some k 2 IR + or even O(2 Jn ). Corresponding investigations
concerning anisotropic tensor-product discretizations seem still to be missing, especially
together with sparse grid type discretizations. Along the lines of [10] we obtain
the following results.
Decay properties of the kernel such as (1.2) introduce decay properties of the stiness
matrix entries. Specically, if the decay property (1.2) holds for all ; N
Taylor expansion of the kernel together with the cancellation properties of the primal
wavelets shows
for That is, the size of the entries depends
on the distance of the supports of the basis functions lk , l 0 k 0 and on their scales.
The rate of the decay of the entries is governed by the number of vanishing moments.
The following Lemma estimates the error of setting entries in the stiness matrix to
zero.
Lemma 2. Let
I
mix . Moreover let the norm equivalences
(2.4) and (2.5) hold. The parameters ~
r from (2.4) and (2.5) shall full
~
I A comp
I )uk 2
(j;k)2I
(l;m)2 jk n comp
mix
The proof of Lemma 2 is similar to the proof of Theorem 1. It uses again the biorthogonality
of the primal and the dual wavelet and the stability of the dual wavelets in
H s , see (2.5), and of the primal wavelets in H p;q
mix , see (2.4).
Now, we need to dene comp
jk in such a way that the order of approximation does not
deteriorate because of this compression. Assume for a moment that we have
A I )ukH s ' O(2 RJ ) with some R 2
. Then we require that k(A I A comp
I )ukH s
is of the same order, i.e. k(A I A comp
I )ukH s ' O(2 RJ ). According to Lemma 2 it
is su-cient to require
(j;k)2I
(l;m)2 jk n comp
jklm O 2 2RJ
The following Theorem tells us how to dene the index sets comp
jk . Note that the choice
of the comp
jk has to balance the overall complexity and the error after compression in
an optimal way.
Theorem 2. For (j;
where J := maxfjj; Ig. We dene
comp
Under the above assumptions ((3.15) and Lemma 2) we then have
I A comp
I )ukH s C 2 RJ kuk H p;q
mix
Proof: Combination of (3.15), Lemma 2 and the Denition of comp
jk shows (for
shortness we use the abbreviation
I A comp
I )uk 2
(j;k)2I
mix
C
(j;k)2I
(l;m)2 jk
mix
Now we apply
(l;m)2 jk
I A comp
I )uk 2
(j;k)2I
mix
(j;k)2I1
A kuk 2
mix
mix
:The resulting stiness matrix will in general be non-symmetric. A symmetric operator
may be derived by taking the maximum of B jk and B kj in the denition of comp
jk .
At this point we need an estimate of the number of remaining non-zero elements in the
compressed stiness matrix. A full analysis of this problem is presently missing. Note
however, it must be possible to obtain a compression with anisotropic wavelets which
is optimal up to a logarithmic factor: Each compactly supported univariate scaling
function on level j has a representation by O(j) compactly supported univariate
wavelets. Hence, each isotropic tensor product wavelet on level j has a representation
by at most O(j product wavelets. Therefore, the compression
schemes of [10, 29] translate into compression schemes for anisotropic wavelets which
result in stiness matrices with O(J 2n 1 J k 2 Jn ) entries.
4. Numerical examples. In this Section we present two numerical examples
in 2D which show the e-ciency of our scheme for the approximate integral
evaluation and for the solution of integro-dierential equations. Here we use orthogonal
Daubechies wavelets on the interval with
e.g. [7, 19]. Now, there
is a certain coarsest level such that the approximation space V j0 cannot be
further decomposed into V j0 1 and W j0 . In the multivariate setting of our numerical
experiments, this leads to some modications in the denition of the index sets I J ,
U IJ for the approximation of K and u. Here, we use
I J := f(j;
for the optimized scheme and
I f
for the classical full grid scheme. Except for the correction terms 6j 0 and 2j 0 , the
index sets I J and U J correspond to the index sets I J (0; N; 0; N; 0); U I2J (0;N;0;N;0)
which are optimized for smooth K and u (where the order of approximation is limited
only by N) and for k:k L 2-norm estimates of the error. In our experiments K and u
are arbitrarily smooth, see subsections 4.1 and 4.2. Table 4.1 shows the dimension
of the approximation spaces ~
I for the dierent index sets. It holds dim( ~
and dim( ~
Table
Dimension of approximation spaces for and the index sets from (4.1) and (4.2).
In the previous Sections the exact projections K I and
were used for the analysis. However, in numerical applications K I and u I are usually
not explicitly known and have to be approximated by, say, K 0
I and u 0
I . This is
an essential part in applications and needs to be considered also for accuracy and
complexity estimates. For full grid approximation spaces one usually employs tensor
products of one-dimensional quadrature schemes, see [31], to approximate the scalar
products of K and u with the scaling functions on the nest level of renement. In
all our experiments we used a quadrature scheme of order Then, by the
wavelet transform one obtains the coe-cients of K 0
I and u 0
I . For the sparse grid
spaces ~
IJ we use Smolyak's blending scheme [2, 11, 17, 30]. Then, for example, for
mix and U I2J Jg, one can show
I2J
mix
For
mix and the index set I J (0; N; 0; N;
obtain
mix
Note, that this error is of order O(J 2n 1 2 (M+1=2)J=2 ). However, since the
overall error is dominated by k(A A IJ )ukH s and not by the approximation of K IJ
and u I2J . Similar estimates (without the logarithmic terms) hold for the full grid
case.
For the calculation of errors the nodal values of the numerical solution h I and the exact
solution h are computed on a
grid
with mesh size 2 12 for the x- and y-direction.
Then, the exact (semi-) norms kh h I k L 2 and jh h I j H 1 are approximated using the
discrete (semi-) norms
These discrete norms yield accurate estimates of the true norms of the error.
4.1. Integral evaluation. We consider the integral evaluation (1.1) with
K(x; y) := sin
Y
and
The exact result is
0:577::: is Euler's constant. Furthermore, Si and
Ci denote the Sinus and Cosinus Integralis. See Figure 4.5 for a plot of u and h.
Figure
4.1 shows the (discrete) L 2 - and H 1 -norms of the error. The new scheme
with index sets (4.1) yields approximately the same accuracy as the classical full
grid scheme { with a much smaller work count, of course. Furthermore, the rate
of error reduction agrees very well with the rate predicted by Theorem 1. Plots of
error vs. number of degrees of freedom are given in Figure 4.2. The L 2 -optimized
scheme leads to superior convergence rates in both norms. In all experiments the
calculation of a 0
jklm was by far the most expensive part of the algorithm. Table 4.2
lists cpu-times for the numerical quadrature, the wavelet transforms involved in the
blending scheme and the matrix-vector product (3.10) for the L 2 -optimized scheme.
The measurements were carried out on a SGI O2000 with a 195 MHz R10k processor.
Due to the blending scheme the numerical work count and, therefore, the cpu-times
are only asymptotically proportional to dim( ~
This explains the growth of the
normalized cpu-time shown in the fth column of Table 4.2.
error
full grid I J
-optimized J
error
full grid I J
-optimized J
Figure
4.1. Discrete L 2 - and H 1 -errors against the maximal level J for the integral evaluation.
degrees of freedom
error
full grid I J
-optimized I J
degrees of freedom
error
full grid I J
-optimized I J
Figure
4.2. Discrete L 2 - and H 1 -errors against dim( ~
I ) for the integral evaluation.
Table
-optimized scheme: cpu-times in [s] for the calculation of the coe-cients a 0
jklm (quadratures
(Q), wavelet transforms (WT)), the multiplications (M) according to (3.10) and cpu-times for Q+
WT and M normalized with the dimensions of ~
I J .
4.2. Solution of integro-dierential equations. We consider the numerical
solution of the integro-dierential equation
Z
Here, K is given by (4.3) and the right hand side f is chosen such that the exact
solution is (4.4), see Figure 4.5. The trial and test functions for the Galerkin discretization
are Daubechies wavelets with homogeneous Dirichlet boundary conditions
and vanishing moments. A BiCGStab(2) iterative solver was used for the
solution of the resulting linear system. The number of iterations required to reduce
the residual to machine precision was between 40 and 50 (essentially independent of
the maximal level J).
As shown in Figure 4.3 the optimized scheme yields approximately the same accuracy
as the classical scheme and the theoretical convergence rates are matched well. Again
the new scheme leads to dramatically reduced errors when compared to the full grid
scheme, see Figure 4.4.
error
full grid I J
-optimized J
error
full grid I J
-optimized J
Figure
4.3. Discrete L 2 - and H 1 -errors against the maximal level J for the solution of (4.6).
degrees of freedom
error
full grid f
-optimized I J
degrees of freedom
error
full grid f
-optimized I J
Figure
4.4. Discrete L 2 - and H 1 -errors against dim( ~
I ) for the solution of (4.6).0.5010.010.030.50126
Figure
4.5. Left: u, (4.4). Middle: h, (4.5). Right: f , (4.6)
Acknowledgements
: The authors would like to thank the referees for their useful
comments. The second author has been supported by the Deutsche Forschungsge-
meinschaft, GR 1144/7-2.
--R
Approximation by trigonometric polynomials in a certain class of periodic functions of several variables
On the adaptive computation of integrals of wavelets
Fast wavelet transforms and numerical algorithms I
Wavelets on the interval and fast wavelet transforms
Stability of Multiscale Transformations
Wavelet and Multiscale Methods for Operator equations
Theory 34
Compression of Wavelet decompositions
Constructive Approximation 14
Number of integral points in a certain set and the approximation of functions of several variables
Complexity of local solution of multivariate integral equations
Information complexity of multivariate Fredholm equations in Sobolev classes
Numerical Integration using Sparse Grids
Optimized tensor-product approximation spaces
Orthogonal Wavelets on the Interval
Hyperbolic cross approximation of integral operators with smooth kernel
Approximation und Kompression mit Tensorprodukt-Multiskalen-Raumen
Wavelets: Calderon-Zymund and multilinear operators
On discrete norm estimates related to multilevel preconditioners in the
On the complexity of
On the complexity of
analysis of the combination technique
Quadrature and interpolation formulas for tensor products of certain classes of functions
Quadrature formulae and asymptotic error expansions for wavelet approximations of smooth functions
Approximation of periodic functions
Parallel Algorithms for Partial Di
--TR | sparse grids;optimized approximation spaces;biorthogonal wavelets;hyperbolic cross approximation;compression;integral equations;boolean blending |
588564 | An A Priori Error Analysis of the Local Discontinuous Galerkin Method for Elliptic Problems. | In this paper, we present the first a priori error analysis for the local discontinuous Galerkin (LDG) method for a model elliptic problem. For arbitrary meshes with hanging nodes and elements of various shapes, we show that, for stabilization parameters of order one, the L2-norm of the gradient and the L2-norm of the potential are of order k and k+1/2, respectively, when polynomials of total degree at least k are used; if stabilization parameters of order h-1 are taken, the order of convergence of the potential increases to k+1. The optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains. | Introduction
. In this paper, we present the first a priori error analysis of the
Local Discontinuous Galerkin (LDG) method for the following classical model elliptic
problem:
@n
where\Omega is a bounded domain of R d and n is the outward unit normal to its boundary
for the sake of simplicity, we assume that the (d \Gamma 1)-dimensional measure
of \Gamma D is non-zero.
The LDG method was introduced by Cockburn and Shu in [25] as an extension to
general convection-diffusion problems of the numerical scheme for the compressible
Navier-Stokes equations proposed by Bassi and Rebay in [6]. This scheme was in turn
an extension of the Runge-Kutta Discontinuous Galerkin (RKDG) method developed
Scientific Computing Program, School of Mathematics, University of Minnesota, Vincent Hall,
Minneapolis, MN 55455, email: castillo@math.umn.edu.
y School of Mathematics, University of Minnesota, Vincent Hall, Minneapolis, MN 55455, email:
cockburn@math.umn.edu. Supported in part by the National Science Foundation (Grant DMS-
9806956) and by the University of Minnesota Supercomputing Institute.
z Dipartimento di Matematica, Universit'a di Pavia, Via Ferrata 1, 27100 Pavia, Italy, email:
perugia@dimat.unipv.it. Supported in part by the Consiglio Nazionale delle Ricerche. This work
was carried out when the author was a Visiting Professor at the School of Mathematics, University
of Minnesota.
x School of Mathematics, University of Minnesota, Vincent Hall, Minneapolis, MN 55455, email:
schoetza@math.umn.edu. Supported by the Swiss National Science Foundation (Schweizerischer Na-
tionalfonds).
P. Castillo, B. Cockburn, I. Perugia and D. Sch-otzau
by Cockburn and Shu [19, 21, 23, 24, 26] for nonlinear hyperbolic systems. The LDG
method is one of several discontinuous Galerkin methods which are being vigorously
studied, especially as applied to convection-diffusion problems, because of their applicability
to a wide range of problems, their properties of local conservativity and high
degree of locality, and their flexibility in handling adaptive hp-refinement. The state
of the art of the development of discontinuous Galerkin methods can be found in the
volume [20] edited by Cockburn, Karniadakis and Shu.
Let us give the reader familiar with (classical and stabilized) mixed and mortar finite
element methods for elliptic problems an idea of what kind of method is the LDG
method.
ffl The LDG is obtained by using a space discretization that was originally applied to
first-order hyperbolic systems. Hence, to define the method, we rewrite our elliptic
model problem as a system of first-order equations and then we discretize it. Thus,
we introduce the auxiliary variable obtain the equations
\Gammar
Since these are nothing but the equations used to define classical mixed finite element
methods, the LDG method can be considered as a mixed finite element method.
However, the auxiliary variable q can be eliminated from the equations which is usually
not the case for classical mixed methods.
ffl In the LDG method, local conservativity holds because the conservation laws (1.2)
and (1.3) are weakly enforced element by element. In order to do that, suitable
discrete approximations of the traces of the fluxes on the boundary of the elements
are needed which are provided by the so-called numerical fluxes; these are widely
used in the numerical approximation of non-linear hyperbolic conservation laws and
are nothing but the so-called approximate Riemann solvers; see Cockburn [17]. As in
the case of non-linear hyperbolic conservation laws, these numerical fluxes enhance
the stability of the method and hence the quality of its approximation. This is why the
LDG method is strongly related to stabilized mixed finite element methods; indeed,
the stabilization is associated with the jumps of the approximate solution across the
element boundaries.
ffl The LDG method allows general meshes with hanging nodes and elements of several
shapes since no inter-element continuity is required. This is also a key property
of the mortar finite element method. However, in the LDG method there are no
Lagrange multipliers associated to the continuity constraint; instead, the Lagrange
multiplier is replaced by fixed functions of the unknowns which are nothing but the
above mentioned numerical fluxes.
ffl In the LDG method, on each element, both the approximation to u as well as
the approximations to each of the components of q belong to the same space, which
is very convenient from an implementational point of view. Moreover, the lack of
continuity constraints across element boundaries in the finite element spaces renders
the coding of the hp-version of the LDG method much simpler than that of standard
mixed methods.
us briefly describe the recent work on error analysis of DG methods in order
to put our results into perspective. Analyses of the LDG method in the context of
An analysis of the LDG method for elliptic problems 3
transient convection-diffusion problems have been carried out by Cockburn and Shu
[25], by Cockburn and Dawson [18], by Castillo [14] and more recently by Castillo,
Cockburn, Sch-otzau and Schwab [15].
The DG method of Baumann and Oden [7, 8, 9, 32] has also been analyzed by several
authors. Oden, Babu-ska and Baumann [31] studied this method for one dimensional
elliptic problems and later Wihler and Schwab [41] proved robust exponential rates
of convergence of the Oden and Baumann DG method for stationary convection-diffusion
problems also in one space dimension. Rivi'ere, Wheeler and Girault [35] and
Rivi'ere and Wheeler [34] analyzed several variations of the DG method of Baumann
and Oden (involving interior penalty techniques) as applied to non-linear convection-diffusion
problems and, finally, S-uli, Schwab and Houston [38] synthesized the self-adjoint
elliptic, parabolic, and hyperbolic theory by extending the analysis of these
DG methods to general second-order linear partial differential equations with non-negative
characteristic form.
As applied to purely elliptic problems, the LDG method and the method of Baumann
and Oden are strongly related to the so-called interior interior penalty (IP) methods
explored mainly by Babu-ska and Zl'amal [3], Douglas and Dupont [27], Baker [4],
Wheeler [39], Arnold [2] and later by Baker, Jureidini and Karakashian [5], by Rusten,
Vassilevski, and Winther [36] and by Becker and Hansbo [10]. All of these DG methods
for elliptic problems can be recast within a single framework as shown by Arnold,
Brezzi, Cockburn and Marini [1]; this framework should provide a basis for a better
understanding of the connections among them and lead to a unified error analysis of
these methods. As a contribution to this effort, we present in this paper an a priori
error analysis of the LDG method for purely elliptic problems.
We show that if polynomials of degree at least k are used in all the elements, the rate
of convergence of the LDG method in the L 2 -norm of u and q are of order k
and k, respectively, when the stabilization or penalization parameter C 11 is taken to
be of order one. When the stabilization parameter C 11 is taken to be of order h \Gamma1 ,
the order of convergence of u is proven to be k + 1, as expected. Indeed, this is what
happens for the interior penalty methods and for the modifications of the method of
Bassi and Rebay [6] studied by Brezzi, Manzini, Marini, Pietra and Russo [13]; the
penalization parameters of these methods are also of order h \Gamma1 . These results are
summarized in the Table 1.1.
Table
Orders of convergence for k - 1.
method penalization
interior penalty O(h
Brezzi et al. O(h
Finally, let us point out that the order of convergence of u for the DG method for
purely convective problems is k +1=2. This order of convergence was proven by Johnson
and Pitk-aranta [30] and later confirmed by Peterson [33] to be sharp. Whether
4 P. Castillo, B. Cockburn, I. Perugia and D. Sch-otzau
or not a similar phenomenon is actually taking place for the LDG method, with the
stabilization parameter C 11 of order one, as applied to elliptic problems remains to
be investigated. Our numerical experiments for the LDG method have all been performed
on structured and unstructured triangulations without hanging nodes and give
the optimal orders of convergence of k for u and q, respectively, with C 11
of order h \Gamma1 and, remarkably, with C 11 of order one.
The organization of the paper is as follows. In section 2, we present the LDG methods
and state and discuss our main a priori error estimates. We also give a brief sketch
of the proofs in order to display the ideas of our analysis. The analysis is carried out
in full detail in section 3 and several possible extensions are indicated in section 4.
In section 5, we present several numerical experiments testing the sharpness of our
theoretical results. We end in section 6 with some concluding remarks.
2. The main results. In this section, we formulate the LDG method and show
that it possesses a well-defined solution. We then state and discuss our main result
and, finally, we display the main ideas of our error analysis.
We assume, to avoid unnecessary technicalities, that the exact solution u of our model
problem (1.1) belongs to H 2
(\Omega\Gamma and that the solution of the so-called adjoint problem
satisfies the standard ellipticity regularity property. Extensions to more general
situations are discussed in section 4.
2.1. The LDG method. To introduce our LDG method, we consider a general
discontinuous Galerkin (DG) method of which the LDG method is a particular but
important case. We consider a general triangulation T with hanging nodes whose
elements K are of various shapes.
To obtain the weak formulation with which our DG method is defined, we multiply
equations (1.2) and (1.3) by arbitrary, smooth test functions r and v, respectively,
and integrate by parts over the element K 2 T to obtain
Z
Z
ur
Z
Z
Z
Z
Note that the above equations are well defined for any functions (q; u) and (r; v) in
Next, we seek to approximate the exact solution (q; u) with functions (q in the
finite element space MN \Theta VN ae M \Theta V , where
and the local finite element space S(K) typically consists of polynomials. Note that
for a given element K, the restrictions to K of uN and of each of the components of q N
belong to the same local space; this renders the coding of these methods considerably
simpler than that of the standard mixed methods, especially for high-order polynomial
An analysis of the LDG method for elliptic problems 5
local spaces. In order to ensure the existence of the approximate solution of the DG
method, we require the following local and quite mild condition:
Z
Other than these properties, there is complete freedom in the choice of the local spaces
since no inter-element continuity is required at all.
The approximate solution (q defined by using the above weak formula-
tion, that is, by imposing that for all K 2 T , for all
Z
Z
Z
Z
Z
Z
where the numerical fluxes b uN and b q N have to be suitably defined in order to ensure
the stability of the method and in order to enhance its accuracy.
As pointed out in the introduction, we can see that the numerical fluxes b uN and b q N
are nothing but discrete approximations to the traces of u and q on the boundary of
the elements. To define these numerical fluxes, let us first introduce some notation.
be two adjacent elements of T ; let x be an arbitrary point of the set
which is assumed to have a non-zero (d \Gamma 1)-dimensional measure,
and let n + and n \Gamma be the corresponding outward unit normals at that point. Let
(q; u) be a function smooth inside each element K \Sigma and let us denote by (q
the traces of (q; u) on e from the interior of K \Sigma . Then, we define the mean values
ff\Deltagg and jumps [[\Delta]] at x 2 e as follows:
Note that the jump in u is a vector and the jump in q is a scalar which only involves
the normal component of q.
We are now ready to introduce the expressions that define the numerical fluxes in
(2.2) and (2.3). If the set e is inside the
domain\Omega\Gamma we take
b u
ffugg
where the auxiliary parameters C 11 , C 12 and C 22 depend on x 2 e and are still at
our disposal. The boundary conditions are imposed through a suitable definition of
(b q; b u), namely,
g N on \Gamma N ;
and
b u :=
where the superscript + denotes quantities related to the element the edge we are
considering belongs to, and We remark that the definition of (b q; b u) on
6 P. Castillo, B. Cockburn, I. Perugia and D. Sch-otzau
the boundary
@\Omega is still of the form (2.4) if the exterior traces are taken to
be
and C 12 is chosen such that C 12
Let us stress once more that the numerical fluxes we just defined are nothing but a
particular case of the so-called approximate Riemann solvers widely used in numerical
schemes for non-linear hyperbolic conservation laws.
This completes the definition of our DG method. The LDG method is obtained when
C 22 j 0; in this case, the function q N can be locally solved in terms of uN and hence
eliminated from the equations, as can be easily seen from (2.3). This local solvability
gives its name to the LDG method.
That this DG method actually defines a unique approximate solution depends in a
crucial way on the coefficients C 11 and C 22 . Indeed, we have the following result.
Proposition 2.1 (Well posedness of the DG method). Consider the DG method
defined by the weak formulation (2.2) and (2.3), and by the numerical fluxes in (2.4)
and (2.5). If the coefficients C 11 are positive and the coefficients C 22 are non-negative,
the DG method defines a unique approximate solution (q
Notice that the above result, which we prove in the next subsection, is independent
of the auxiliary vector parameter C 12 . The choice C
symmetry and stability of the DG method. Finally, let us point out that the role
of the auxiliary parameters C 11 and C 22 is to enhance the stability and hence the
accuracy of the method.
2.2. The classical mixed setting. The study of our DG method is greatly
facilitated if we recast its formulation in a classical mixed finite element setting. To
do that, we need to introduce some notation. We denote by E i the union of all interior
faces of the triangulation T , by ED the union of faces on \Gamma D , and by EN the union of
we assume that \Gamma
Now, we sum equations (2.2) and (2.3) over all elements and obtain, after some
simple manipulations, that the DG approximation (q N ; uN ) is the unique solution of
the following variational problem: find (q VN such that
\Gammab(v;
for all (r; v) 2 MN \Theta VN . Here, the bilinear forms a, b and c are given by
a(q; r) :=
Z\Omega
Z
C 22 ds
Z
K2T
Z
ur
Z
ds \Gamma
Z
c(u; v) :=
Z
ds
Z
ED
C 11 uv ds:
The linear forms F , G are defined by
F (r) :=
Z
ED
Z
C 22 (g N \Delta n)(r \Delta n) ds;
Z\Omega
Z
ED
ds
Z
An analysis of the LDG method for elliptic problems 7
Note that these two linear forms contain all the data of the problem. In particular,
they contain both the Dirichlet and Neumann data, which is not the case for the
classical mixed finite element methods.
Equations (2.6) and (2.7) can be rewritten in a more compact form as follows:
by setting
We end this section by proving Proposition 2.1.
Proof of Proposition 2.1. Due to the linearity and finite dimensionality of the problem,
it is enough to show that the only solution to the equations (2.6) and (2.7) with
taking
adding the two equations, we get
which implies q
As a consequence, equation (2.6) becomes
after integration by parts, the form b(\Delta; \Delta) can be rewritten as
K2T
Z
Z
ds
Z
ED
we get that
K2T
Z
Hence, owing to (2.1), on every K 2 T and since uN is a continuous function
equal to zero on the Dirichlet boundary, we get that uN j 0. This completes the proof
of Proposition 2.1. 2
2.3. A priori error estimates. In this section we state and discuss our a priori
error bounds for the DG method. As pointed out at the beginning of this section,
we restrict our analysis to
domains\Omega such that, for smooth data, the solution u of
problem (1.1) belongs to H
We also assume that when f is in L
2(\Omega\Gamma and the
boundary data are zero, we have the elliptic regularity result k u
Grisvard [28] or [29].
We assume that every element K of the triangulation T is affine equivalent, see [16,
Section 2.3], to one of several reference elements in an arbitrary but fixed set; this
allows us to use elements of various shapes with possibly curved boundaries. For each
by hK the diameter of K and by ae K the diameter of the biggest ball
included in K; we set, as usual, h := maxK2T hK . The triangulations we consider can
8 P. Castillo, B. Cockburn, I. Perugia and D. Sch-otzau
have hanging nodes but have to be regular , that is, there exists a positive constant oe
such that
ae K
see [16, Section 3.1]. Moreover, we let the maximum number of neighbors of a given
element K be arbitrary but fixed. To formally state this property, we need to introduce
the set hK; K 0 i defined as follows:
interior of @K " @K 0 otherwise:
Thus, we assume that there exists a positive constant 1 such that, for each element
These three hypotheses allow for quite general triangulations and are not restrictive
in practice.
The only assumptions we use for the local space S(K) are that it contains the space
of polynomials of degree at most k on K and satisfies (2.1).
Next, we introduce a semi-norm that appears in a natural way in the analysis of
these methods. We denote by H k (D), D being a domain in R d , the Sobolev spaces of
integer orders, and by k \Delta k k;D and j \Delta j k;D the usual norms and semi-norms in H k (D)
and H k (D) d ; we omit the dependence on the domain in the norms whenever
We define j (q; u) j 2
A := A(q; u; q; u), that is,
where
\Theta 2 (q; u) :=
Z
C 22 ds
Z
ds
Z
ds
Z
ED
We assume that the stabilization coefficients C 11 and C 22 defining the numerical fluxes
in (2.4) and (2.5) are defined as follows:
ih ff
-h fi
independent of the mesh-size and jC 12 j
of order one. Our main result will be written in terms of the parameters - ? and - ?
defined as follows:
An analysis of the LDG method for elliptic problems 9
We are now ready to state our main result.
Theorem 2.2. Let (q; u) be the solution of (1.2)-(1.5) and let (q N ; uN ) be the approximate
solution given by the DG method (2.2) and (2.3). We assume the hypotheses
on the local spaces and on the form of the stabilization parameters described
above. The triangulations are assumed to satisfy the hypothesis (2.11); if ff 6= 0 or
fi 6= 0, we also assume that hypothesis (2.12) is satisfied. Then we have that, for
where C solely depends on oe, ffi (not when
Let us briefly discuss the above result:
ffl We begin by noting that the orders of convergence depend on the size of the stabilization
parameters C 11 and C 22 only through the quantities - ? and - ? . This fact
has several important consequences:
ffi The same orders of convergence are obtained with either C 22 = 0 or C 22 of
order h. This means that there is no loss in the orders of convergence if
instead of penalizing the jumps of the normal component of q N with a C 22
or order h, no penalization at all (the LDG method) is used.
ffi The same orders of convergence are obtained with either the LDG method
of order one or C 11 of order h \Gamma1 and C 22 of order one.
In general, the same orders of convergence are obtained by taking (ff;
or by taking (ff;
ffi The most remarkable cases occur when \Gammaff; fi 2 f0; 1g since it is for those values
that - ? and - ? achieve their maximum and minimum. The corresponding
orders of convergence are displayed in Table 2.1 for k - 1.
Table
Orders of convergence for
ffl In the case 1 - k - s, that is, when the degree of the polynomial approximation
is less than needed to fit the smoothness of the exact solution, we see in Table 2.2
that the best orders of convergence for
P. Castillo, B. Cockburn, I. Perugia and D. Sch-otzau
respectively, are obtained for both C 11 and C 22 of order one. When C 22
is taken to be of order h or equal to zero, the stability of the method is weakened
and, as a consequence, a loss in the orders of convergence of 1=2 takes place. If
now C 11 is taken to be of order h \Gamma1 , the full order of convergence in the error of
the potential is recovered. The numerical experiments of section 5 show that these
orders of convergence are actually achieved. However, the expected loss in the orders
of convergence when C 11 is taken of order one is not observed, which shows that
in practice the LDG method is essentially insensitive to the size of the stabilization
parameter C 11 .
ffl The influence of the choice of the coefficients C 12 on the accuracy has not been
explored in this paper; we only assume those to be of order one. In [22] it is shown
that the LDG method, with a suitable choice of the coefficients C 12 , still gives the
orders of convergence of k
respectively, if Cartesian grids and tensor product polynomials of degree k in each
variable are used.
Table
Orders of convergence for
ffl For the case k - s + 1, that is, when the degree of the polynomial approximation
is more than needed to fit the smoothness of the exact solution, we see in Table 2.3
that the LDG method performs at least as well as all the other methods; it performs
better if C 11 is of order h \Gamma1 .
Table
Orders of convergence for
An analysis of the LDG method for elliptic problems 11
ffl In the case the DG method converges provided C 22 6= 0; in particular, for
constant coefficients C 11 and C 22 , we obtain estimates of order one for
. This is one of the few finite element methods for second-order
elliptic problems that actually converges for piecewise-constant approximations.
When C 22 = 0, that is, for the LDG method, our numerical results, which we do not
report in this paper, show that there is no positive order of convergence in this case,
as predicted by Theorem 2.2.
Finally, let us point out that the hypothesis (2.12) is not necessary when
2.4. The idea of the proof. The proof of Theorem 2.2 will be carried out in
section 3. The purpose of this section is to display as clearly as possible the basic
ingredients and the main steps of our error analysis. As usual, we express the error
as the following sum:
where \Pi and \Pi are projections from M and V onto the finite element spaces MN
and VN , respectively.
a. The basic ingredients. The basic ingredients of our error analysis are two. The
first one is, as it is classical in finite element error analysis, the so-called Galerkin
orthogonality property, namely,
This property is a straightforward consequence of the consistency of the numerical
fluxes.
The second ingredient is a couple of inequalities that reflect the approximation properties
of the projections \Pi and \Pi, namely,
for any (q; u); (\Phi;
for any (r; v) 2 MN \Theta VN and (q; u) 2 H
As we show next, all the error estimates we are interested in can be obtained solely
in terms of functionals KA and KB .
b. The estimate of the error in the A-semi-norm. We have the following result.
Lemma 2.3. We have
A (q; u; q; u) +KB (q; u):
Proof. j (\Delta; \Delta) j A is a semi-norm and, hence,
Since
by the definition of A, (2.9);
by assumption (2.19);
P. Castillo, B. Cockburn, I. Perugia and D. Sch-otzau
we have that
and so,
The estimate now follows from a simple application of the assumption (2.18). This
completes the proof.
c. Estimate of the error in u in non-positive order norms. To obtain an
estimate of k e u k \Gammat;D , where t is a natural number and D is a sub-domain of \Omega\Gamma
we only have to obtain an estimate of the error in the approximation of the linear
functional denotes the L 2 -inner product, by (u N ) since
In this paper, we are only interested in the case but we give here the general
argument to stress the fact that it is essentially the same for all natural numbers
t. Error estimates in negative order norms are very important, as we point out in
section 4 of this paper.
To obtain our estimate, we need to introduce the solution ' of the so-called adjoint
problem, namely,
@n
Lemma 2.4. Let t be a natural number. Then, we have
KA (q; u; \Phi; ')
with ' denoting the solution of (2.21)-(2.23) and
Proof. Since ' is the solution of the adjoint equation, it is easy to verify that if we
set
for all (s; w) 2 M \Theta V ; indeed, note that problem (1.1) can be rewritten as in (2.8).
Taking (s;
by the definition of A, (2.9);
, by the assumption (2.19) and the estimate (2.20), we
obtain
An analysis of the LDG method for elliptic problems 13
and hence,
The estimate now follows from a simple application of assumption (2.18), and from
the definition of a non-positive order norm. This completes the proof.
d. Conclusion. Thus, in order to prove our a priori estimates, all we need to do
is to obtain the functionals KA and KB ; this will be carried out in the next section.
Then, Theorem 2.2 will immediately follow after a simple application of Lemmas 2.3
and 2.4.
3. Proofs. In this section, we prove our main results. We proceed as follows.
First, we obtain the functional KA for general projection operators \Pi and \Pi. To
obtain the functional KB , the projections \Pi and \Pi are taken to be the standard L 2 -
projections, just as done by Cockburn and Shu [25] in their study of the LDG method
for transient convection-diffusion problems.
3.1. Preliminaries. The following two lemmas contain all the information we
actually use about our finite elements. The first one is a standard approximation result
for any linear continuous operator \Pi from H r+1 (K) onto S(K) satisfying
any w 2 P k (K); it can be easily obtained by using the techniques of [16]. The second
one is a standard inverse inequality.
Lemma 3.1. Let w \Pi be a linear continuous operator from
H r+1 (K) onto S(K) such that
for some constant C that solely depends on oe in inequality (2.11), k, d and r.
Lemma 3.2. There exists a positive constant C inv that solely depends on oe in inequality
(2.11), k and d, such that for all s 2 S(K) d we have
for all K 2 T .
We are now ready to prove our main result.
3.2. The functional KA . In this subsection we determine the functional KA
in (2.18), up to a multiplicative constant independent of the mesh-size. We start by
giving an expression for KA which is valid for coefficients C 11 and C 22 that vary from
face to face, for any regularity of the solution. Then we write KA for
the particular choice (2.15), (2.16) of C 11 and C 22 in Theorem 2.2.
Let \Pi and \Pi be arbitrary projections onto VN and MN , respectively, satisfying
(component-wisely) the assumptions in Lemma 3.1.
Lemma 3.3. Assume (q; u) 2 H s+1
the approximation property (2.18) holds true with
KA (q; u; \Phi; ') =X
14 P. Castillo, B. Cockburn, I. Perugia and D. Sch-otzau
where
K2T
K2T
K2T
22 h 2 minfs;kg+1
K2T
22 h 2 minft;kg+1
K2T
K2T
K2T
K2T
K2T
K2T
ii := supfC ii 2. The
positive constant C is independent of the mesh-size but depends on the approximation
constants in Lemma 3.1 and on the coefficients C 12 .
Furthermore, in the case where (\Phi;
KA (q; u; q;
Proof. We set, for convenience, - q :=
We start by writing
and then proceed by estimating each of the forms on the right-hand side separately.
The form a(\Delta; \Delta) can be written as
K2T
Z
C 22 (- q \Delta n)(- \Phi \Delta n) ds
Z
@Kn@\Omega C 22 (- q \Delta n)(- \Phi \Delta n) ds
Z
@Kn@\Omega C 22 (- out
ds
where the superscript 'out' denotes quantities taken on @K n
@\Omega from outside K. By
repeated applications of the Cauchy-Schwarz's inequality, we obtain that ja(- q ; - \Phi )j
is bounded by
K2T
22 - q \Delta nk 0;@K"\Gamma N kC2
22 - \Phi \Delta nk 0;@K"\Gamma N
22 - q \Delta nk
22 - out
22 - \Phi \Delta nk
K2T
K2T
An analysis of the LDG method for elliptic problems 15
K2T
22 k- q \Delta nk 2
K2T
22 k- \Phi \Delta nk 2
Now, a straightforward application of Lemma 3.1 yields
To deal with the second term, we first note that
K2T
r-
Z
ds
Z
ds
and obtain, after repeated applications of the Cauchy-Schwarz's inequality with suitably
chosen weights, that jb(- u ; - \Phi )j is bounded by
K2T
K2T
k-
K2T
K2T
k-
K2T
Once again, a straightforward application of Lemma 3.1 gives that
For the third term, we use the same arguments to get
Finally, proceeding as above, we get
K2T
K2T
This proves the first assertion. The second one immediately follows by taking into
account that
and the proof of the lemma is complete.
The following result is a straightforward consequence of the estimates in Lemma 3.3.
Corollary 3.4. Let (q; u) 2 H s+1 2 \Theta H s+2
be the exact solution of
0, be the solution of the dual problem (2.21)-(2.23),
P. Castillo, B. Cockburn, I. Perugia and D. Sch-otzau
and \Gammar'. Assume that coefficients C 11 and C 22 satisfy (2.15), (2.16) . Then
there exist a constant C that solely depends on oe, i, - , k and d such that
KA (q; u; \Phi;
for k - 1. Moreover,
KA (q; u; q;
Proof. From Lemma 3.3, we get
KA (q; u; \Phi; ') =C
\Theta h minfs;kg+1
and
KA (q; u; q;
\Theta
Note that the above results hold for arbitrary ff and fi. If now we restrict ourselves
to the case of Theorem 2.2, the result follows after simple algebraic manipulations.
3.3. The functional KB . In this subsection we determine the functional KB
satisfying (2.19), up to a multiplicative constant independent of the mesh-size. Here,
we take \Pi to be L 2 -projection and Again, we start by determining
expressions which are valid for varying coefficients C 11 and C 22 , and we conclude by
considering the particular case of Theorem 2.2. We proceed as follows. We show that
there exists a form j (\Delta; \Delta) j B , which is a semi-norm in both variables, such that for any
with C independent of the mesh-size. Then it is enough to determine KB such that
for any (q; u) 2 M \Theta V . In the following lemma we prove that (3.1) is satisfied by
defining the semi-norm j (\Delta; \Delta) j B as
Z
ED
ds
Z
ds
Z
C 22
where for each internal or Neumann boundary face e we set
C 22 (x) otherwise:
Note that only the function values along faces enter the j (\Delta; \Delta) j B semi-norm. As can
be inferred from the proof of Lemma 3.5 below, this is due to the particular choice of
\Pi and \Pi as L 2 -projections.
An analysis of the LDG method for elliptic problems 17
Lemma 3.5. Let \Pi and \Pi be the L
-projection and L
-projection onto VN
and MN , respectively, and j (\Delta; \Delta) j B be defined by (3.3). Then (3.1) holds true, with a
constant C that solely depends on oe, k and d.
Proof. Setting by the definition of the form
A in (2.9),
Using Cauchy-Schwarz's inequality and the fact that \Pi is the L
d -projection, we
obtain
'Z
C 22 ds
Z
ds
'Z
ds
Z
ds
Furthermore,
Z
ds
Z
ED
ds
Multiplying and dividing by C2
11 and then applying Cauchy-Schwarz's inequality, we
obtain
'Z
ds
Z
ED
ds
'Z
ds
Z
ds
Analogously,
Z
(ff- ds
Z
ds
'Z
ds
Z
ds
'Z
(ff- ds
Z
ds
The first factor can be estimated as follows:
Z
ds
Z
ds -
Z
C 22 ds
Z
ds
Z
ds
Z
ds
Z
ds
Z
P. Castillo, B. Cockburn, I. Perugia and D. Sch-otzau
. By the inverse
inequality in Lemma 3.2,
Z
ds
Z
ds -
K2T
Z
e
ds
K2T
K2T
@Kg. Thus, combining the above estimates, we get
Finally,
Z
ds
Z
ED
ds
'Z
ds
Z
ED
ds
'Z
ds
Z
ED
ds
To complete the proof, we simply have to gather the estimates of the terms T i ,
4, and apply once again the Cauchy-Schwarz's inequality.
The function KB can be easily defined by applying the estimates in Lemma 3.1 to
defined in (3.3).
Lemma 3.6. For any (q; u) 2 H s+1 2 \ThetaH s+2
the approximation property
(3.2) holds true with
K2T
/e
K2T
e
where e
and C is a
constant independent of the mesh-size and solely depending on the approximation and
inverse inequality constants (cf. Lemmas 3.1 and 3.2).
From this lemma, we immediately obtain the following result.
Corollary 3.7. Let (q; u) 2 H s+1 2 \Theta H s+2 , s - 0. Assume that the coefficients
C 11 and C 22 satisfy (2.15), (2.16). The triangulations are assumed to satisfy
the hypothesis (2.11); if ff 6= 0 or fi 6= 0, we also assume that hypothesis (2.12) is
Then there exists a constant C that solely depends on oe, ffi, i, - , k and d such that
the constant C is independent of ffi.
An analysis of the LDG method for elliptic problems 19
Proof. If we take the coefficients C 11 and C 22 as in Theorem 2.2, we get, after a simple
computation,
/e
and
where the parameter ffi is defined in (2.12), and b
Note that the left-hand sides of the above inequalities are trivially uniformly bounded
when otherwise, we must invoke the hypothesis (2.12) to ensure the
boundedness of these quantities. We emphasize that this is the only instance in which
this hypothesis is used.
Hence we obtain
\Theta
where C is independent of the mesh-size but depends on ffi and on the approximation
and inverse inequality constants, and
-h fi otherwise:
The result follows after simple algebraic manipulations.
3.4. The proof of Theorem 2.2. From Lemma 2.3 and Corollaries 3.4 and 3.7,
we get
and since minfPA ; the estimate
follows.
Next, consider the L 2 -norm of the error
=\Omega in
Lemma 2.4. From the elliptic regularity of the adjoint problem (2.21)-(2.23), we
have . The estimates of ku \Gamma uN k 0 directly follow
from substituting the expression of KA (q; u; \Phi; ') given by Corollary 3.4, and the expressions
of KB (q; u), KB (\Phi; ') given by Corollary 3.7 in (2.24), and bounding k\Phik 1
and k'k 2 by k-k 0 . Indeed, we get
and since minfQA j
follows with
This completes the proof of Theorem 2.2.
4. Extensions. In this section, we indicate how to the extend our main result
in several possible directions.
P. Castillo, B. Cockburn, I. Perugia and D. Sch-otzau
4.1. The case of polygonal domains. In the case of a non-convex polygonal
domain in two dimensions, our assumptions on the smoothness of the solution u of
our model problem (1.1) and on the elliptic regularity inequality are no longer true.
Indeed, if for instance the Neumann boundary is empty, the Dirichlet data is smooth
and f is in L 2
(\Omega\Gamma3 we have, see Grisvard [28], that u 2 H s+2
and ! is the maximum interior angle of @ Moreover, if the Dirichlet data is zero,
we have
see (1.7) in Schatz and Wahlbin [37] and the references therein. This is the elliptic
regularity result that we must use.
To prove our error estimates in this case, we proceed as follows. First, we note that
our main result Theorem 2.2 can be easily extended to this case; indeed, a simple
density argument shows that Lemmas 3.3 and 3.6 remain valid for s; t 2 (\Gamma1=2; 0).
Now we proceed as in subsection 3.4 and obtain the desired estimates by using the
above mentioned lemmas and the above described elliptic regularity inequality. The
estimate of the error in the j(\Delta; \Delta)j A -seminorm remains the same but the estimate of
the L 2 -norm of the potential has to be suitably modified.
For turns out that only for non-zero orders of
convergence for
respectively. The results for k - 1 are displayed in Table 4.1 for
smooth solutions
and in Table 4.2 for non-smooth solutions
simply write fl instead of
Table
Orders of convergence for
4.2. Estimates of the error in negative-order norms. It is very well known
that the error in linear functionals can be estimated in terms of the error in negative-
order norms. Moreover, Bramble and Schatz [11] showed how to exploit the oscillatory
nature of finite element approximations, captured in estimates of the error in negative-
order norms, to enhance the quality of the approximation by using a simple post-processing
on regions in which the exact solution is very smooth and the mesh is
locally translation invariant.
An analysis of the LDG method for elliptic problems 21
Table
Orders of convergence for
estimates of negative-order norms can be easily obtained for our general DG
method by following the argument described in subsection 2.4 and the technicalities
displayed in section 3.
4.3. Curvilinear elements. The analysis in section 3 covers the case of triangulations
of curvilinear elements affine-equivalent to fixed curvilinear reference elements.
The aim of this subsection is to show how our main result can be extended to the more
general case where such an affine equivalence can not be established anymore. This
is, for instance, the case when the problem domain has a boundary with a generic
curvature.
There are two distinctive possibilities to do that. The first one is to keep the finite
element spaces described in the introduction; in this case, the local space S(K) could
be taken to be simply P k (K), for example. For our main result to hold in this case,
only Lemmas 3.1 and 3.2 would have to be proven for these elements and for the case
in which \Pi is the L 2 -projection.
The other possibility is to consider elements obtained through the so-called Piola
transformation [12, Section III.1.3]. This transformation associates the function (q; u)
defined on K to the function (b q; b u) defined on b
K by
where FK denotes the mapping from b
K to K. With the above notation, our finite
element spaces are given by
It is easy to verify that the following properties are satisfied on each element K of
22 P. Castillo, B. Cockburn, I. Perugia and D. Sch-otzau
our triangulation:
Z
Z
Z
r
Z
Z
Z
This implies that with this choice of finite element spaces, our main result holds if Lemmas
3.1 and 3.2 hold for the reference element b
K and for the standard L 2 -projection,
provided the mappings FK are sufficiently smooth; see [12] and the references therein.
Indeed, the proof of section 3 holds in this case if we use the projections \Pi and \Pi
defined by
d \Piq := b
\Pi is the L 2 -projection into the space S( b
K) and b
\Pi). The only
slight modification of the proof occurs in section 3.3 in the definition of j (\Delta; \Delta)j 2
to which we have to add the term k q k 2
This implies that an extra term in the upper
bound of the term T 1 in the proof of Lemma 3.5 appears which is easily dealt with.
modification of the proof is required at all.
4.4. General elliptic problems. The extension of our main result to more
general elliptic problems which include lower order terms can be done in a straight-forward
way by applying our techniques to the formulation used by Cockburn and
Dawson [18].
4.5. Exponential convergence of hp-approximations. In the analysis of
the DG methods considered in this paper, we have only derived error estimates with
respect to the mesh-size h and we have not exploited the dependence of our estimates
on the approximation order k. However, this can be done by modifying Lemmas
3.3 and 3.6 correspondingly; see also the work of Houston, Schwab and S-uli [38]
and the references there. In addition, by using the proper mesh design principles
and by obtaining suitable approximation error estimates in the elements abutting
at solution singularities, exponential convergence of the DG method can be proved.
See, for example, the recent work of Wihler and Schwab [40] who showed exponential
convergence for a model elliptic problem on a polygonal
domain\Omega for the DG method
of Baumann and Oden with interior penalties.
5. Numerical results for the LDG method. The purpose of this section is
to validate our a priori error estimates for the LDG method (i.e., C 22 = 0) and to
assess how the quality of its approximations depends on the size of the stabilization
parameters C 11 . Since C 22 = 0, the function q N can be expressed locally in terms of
uN and hence can be eliminated from the equations. In our examples we solve the
resulting linear system for uN by using the standard Conjugate Gradient algorithm;
in order to obtain as much precision as possible, the stopping criterion is such that the
absolute residual norm is less than 10 \Gamma12 . The approximation q N is then recovered
in a post-processing step by using the local expression of q N in terms of uN .
We present numerical results using sequences of structured as well as unstructured
triangular meshes fT i g, where the mesh-size parameter of T i+1 is half
the one of T i . The numerical orders of convergence of the errors are computed for
An analysis of the LDG method for elliptic problems 23
polynomials of degree 1 to 6 in the L 2 -norm and A-semi-norm. These orders are
defined as follows. If e(T i ) denotes the error on mesh T i (in the corresponding norm),
then the numerical order of convergence r i is
In all our computations, we take C 12 normal to the edges and of modulus 1=2. The
stabilization coefficient C 11 is chosen to be of order h \Gamma1 . We emphasize, however,
that for all our experiments no significant difference has been observed in the errors
of the approximations when C 11 is of order one. We also remark that results for
are not included either, since no positive orders of convergence have been obtained,
as predicted in Theorem 2.2.
5.1. Smooth solutions. In our first example, we investigate the order of convergence
for smooth solutions. We solve the model problem (1.1)
with homogeneous Dirichlet boundary conditions and empty Neumann boundary. The
right hand side f is chosen such that the exact solution is given by
cos
The sequence of structured meshes used in this example is created from consecutive
global refinement of an initial coarse structured mesh; at each refinement, every triangle
is divided into 4 similar triangles. The number of triangles of the meshes are
16, 64, 256, 1024 and 4096. Since our analysis is valid for arbitrary meshes, we also
perform some tests with a sequence of unstructured meshes. It consists of a set of
meshes such that the maximum edge length is less than a certain value. This value
is reduced by a factor of two, from one mesh to the next. In this way, if we take
two consecutive meshes, one is not the global refinement of the other. The number of
elements of the meshes are 22, 88, 312, 1368 and 5404.
We show the orders of convergence in the L 2 -norm of the error in the gradient
in the A-semi-norm of the error of (q; u) and in the L 2 -norm of the error in u in
Tables
5.1, 5.2 and 5.3, respectively. For both types of meshes, we observe that the
optimal order of convergence predicted by our theory, see Table 2.2, is achieved. Note
that since machine precision is achieved for very fine grids and high polynomials, the
corresponding orders of convergence are meaningless and are replaced by a horizontal
line.
To give the reader a better idea of this phenomenon, in Figure 5.1, we display the
actual errors in the potential u whose orders of convergence appear in the left side
of
Table
5.3. Note how the very last part of the curve corresponding to polynomials
of degree bends as a consequence of having reached machine accuracy.
P. Castillo, B. Cockburn, I. Perugia and D. Sch-otzau
Table
Smooth solution; order of convergence of the L 2 error in the gradient q.
k order of convergence
Structured meshes Unstructured meshes
3 2.7216 2.9488 2.9924 3.0008 2.9303 2.4986 3.2825 2.9120
6 5.8878 5.9683 5.9820 - 6.4090 5.0744 6.4362 -
Table
Smooth solution; order of convergence of the A-semi-norm of the error in (q; u).
k order of convergence
Structured meshes Unstructured meshes
3 2.8380 2.9745 3.0018 3.0052 3.0120 2.5618 3.2984 2.9233
Table
Smooth solution; order of convergence of the L 2 error in the potential u.
k order of convergence
Structured meshes Unstructured meshes
6 7.0129 6.9889 6.8763 - 7.3003 6.
An analysis of the LDG method for elliptic problems 25
log||
Fig. 5.1. Smooth solution; the L 2 error in the potential u for the structured meshes.
26 P. Castillo, B. Cockburn, I. Perugia and D. Sch-otzau
5.2. An exact solution in H
5(\Omega\Gamma but not in H
We solve the
model problem (1.1) with exact Dirichlet boundary conditions in the convex domain
1). The right hand side is chosen such that the exact solution of
the problem is the function u ff defined by
ae cos
in (\Gamma1; 0) \Theta (\Gamma1; 1);
cos
This function belongs to H ff+ 1
but does not belong to H ff+ 1+"(\Omega\Gamma6 for all " ? 0.
In this test, ff = 4:5 and so u ff 2 H
5(\Omega\Gamma3 The predicted orders of convergence of the
-norm of the error in the gradient and that of the A-semi-norm of the error are both
5, and the predicted order of convergence of the L 2 -norm of the error in the potential
is 4; see
Tables
2.2 and 2.3. These are precisely the orders observed in Tables 5.4, 5.5
and 5.6, respectively. We use the sequence of structured meshes from the previous
test. Similar results not reported here are obtained using unstructured meshes.
Table
of convergence of the L 2 error in the gradient q.
k order of convergence
3 2.1363 2.8375 2.9531 2.9844
6 3.8556 3.9387 3.9710 3.9860
Table
of convergence of the A-semi-norm of the error in (q; u).
k order of convergence
3 2.5591 2.8822 2.9641 2.9882
6 3.9801 3.9664 3.9776 3.9876
Table
of convergence of the L 2 error in the potential u.
An analysis of the LDG method for elliptic problems 27
5.3. Smooth solution on an L-shaped domain. We solve the model problem
(1.1) in an L-shaped domain with Dirichlet boundary conditions. The exact solution
is the function u ff , described above, with 4:5. For this test we use a sequence
of unstructured meshes, created from a global refinement of an unstructured coarse
mesh. The number of elements of the meshes are 22, 88, 352, 1408 and 5632.
In
Tables
5.7, 5.8 and 5.9 below, we can see that we obtain the same order of convergence
as in the convex case even though the standard elliptic regularity result guarantees
an order of convergence for the L 2 -error of the potential smaller by
as indicated in Table 4.1.
A similar phenomenon takes place with the very smooth solution from the first test.
Table
5 -solution on L-shaped domain; order of convergence of the L 2 error in the gradient q.
k order of convergence
3 2.6595 2.8369 2.9260 2.9644
6 3.0742 3.9120 4.0307 4.1347
Table
5 -solution on L-shaped domain; order of convergence of the A-semi-norm of the error in (q; u).
k order of convergence
3 2.7984 2.8763 2.9379 2.9688
6 4.0916 3.9158 4.0313 4.1347
Table
5 -solution on L-shaped domain; order of convergence of the L 2 error in the potential u.
28 P. Castillo, B. Cockburn, I. Perugia and D. Sch-otzau
5.4. Non-smooth solution on an L-shaped domain. Finally, we present
numerical results for the classical L-shaped domain test with a singularity at the
reentrant corner. We consider the model problem (1.1) in an L-shaped domain with
zero right hand side and Dirichlet boundary conditions such that the exact solution
is given by
For conforming finite element methods, it has been shown that the orders of convergence
in the H 1 and L 2 norms are 2
respectively. The
numerical results for the LDG method on the sequence of unstructured meshes described
in the previous experiment are reported in Tables 5.10, 5.11 and 5.12. They
show that the rates of convergence predicted by Table 4.2 are achieved by the LDG
method. Observe that the same rates of convergence as in the conforming case are
achieved.
Table
Non-smooth solution on L-shaped domain; L 2 error in the gradient q.
k order of convergence
Table
Non-smooth solution on L-shaped domain; A-semi-norm of the error in (q; u).
k order of convergence
Table
Non-smooth solution on L-shaped domain; L 2 error in the potential u.
An analysis of the LDG method for elliptic problems 29
6. Concluding remarks. In this paper, we present the first a priori error analysis
for a general DG method that includes the LDG method and allows for triangulations
with hanging nodes and elements of several shapes.
We have proven that the orders of convergence of the approximations given by the
LDG method with the stabilization parameter C 11 of order h \Gamma1 are optimal; these
results have been confirmed by our numerical experiments which also indicate that
the quality of the approximation does not deteriorate when C 11 is taken to be of order
one. Theoretically, a loss of 1=2 in the orders of convergence can take place but this
phenomenon was not observed in the particular test problems we considered; as a
consequence, the sharpness of our error estimates in this case remains to be studied.
We have also theoretically shown that the effect of taking non-zero stabilization parameters
C 22 does not significantly improve the orders of convergence of the LDG
method. An exception is, of course, the piecewise constant case in which the LDG
method has an order of convergence of 0 whereas the DG method with C 11 and C 22
of order one do converge with orders of convergence of at least 1=2 and 1 in the error
of the gradient and potential, respectively.
In this paper, nothing has been said about how to chose the parameters C 12 . In
a forthcoming paper [22], it will be shown that, in the case of Cartesian grids and
tensor product polynomials, the orders of convergence of the LDG method can actually
increase if C 12 is suitably chosen.
Let us end by pointing out tha the implementation of codes for hp-adaptive versions
of the LDG method for general elliptic and transient convection-diffusion-reaction
problems is the subject of ongoing work.
--R
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A discontinuous Galerkin method applied to nonlinear parabolic equations
Improved energy estimates for interior penalty
Interior penalty preconditioners for mixed finite element approximations of elliptic problems
Maximum norm estimates in the finite element method in plane polygonal domains.
An elliptic collocation-finite element method with interior penalties
Exponential convergence of the hp-DGFEM for diffusion problems in two space dimensions
--TR
--CTR
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588569 | Numerical Approximation of the Maximal Solutions for a Class of Degenerate Hamilton-Jacobi Equations. | In this paper we study an approximation scheme for a class of Hamilton--Jacobi problems for which uniqueness of the viscosity solution does not hold. This class includes the eikonal equation arising in the shape-from-shading problem. We show that, if an appropriate stability condition is satisfied, the scheme converges to the maximal viscosity solution of the problem. Furthermore we give an estimate for the discretization error. | Introduction
Given a Hamilton-Jacobi equation, a general result due to Barles-Souganidis [3] says that
any "reasonable" approximation scheme (based f.e. on finite differences, finite elements,
finite volumes, discretization of characteristics, etc.) converges to the viscosity solution of
the equation. Besides some simple properties that the approximation scheme has to satisfy,
it is only requested that the equation satisfies a comparison theorem for discontinuous
solutions, which in particular implies uniqueness of the viscosity solution.
This result covers a wide class of first and second order Hamilton-Jacobi equations, yet there
are interesting examples of equations coming from the applications for which uniqueness of
the viscosity solution does not hold. A significant example is given by the Eikonal equation
on some open and bounded
domain\Omega ae R n coupled for example with a Dirichlet boundary
condition on @ This equation arises in the Shape-from-Shading problem in image analysis
and a large literature has been devoted to its study (see [4] for a description of the problem
This paper was written while the second author was visiting the Dipartimento di Matematica, Universit'a
di Roma "La Sapienza" supported by DFG-Grant GR1569/2-1. The research was partially supported by
the TMR Network "Viscosity solutions and their applications".
and [16] for a viscosity solution approach). It is well known that if f vanishes at some
points, there are infinite many viscosity solutions to (1.1) (see [15]). Nevertheless, among
these solutions, in general only one is the relevant solution (for example, from the physical
point of view, from the control theoretic one, etc.
In [6] (see also [14]), requiring a stronger condition for supersolution than that for the
standard viscosity solution, a Comparison Principle, which characterizes the maximal viscosity
solution of the problem, has been obtained for the following class of Hamilton-Jacobi
problems
Here\Omega is a bounded domain of R N , H and f are nonnegative continuous functions and f
can have a very general zero set (the Eikonal equation (1.1) fits into this class of equation).
It is worth noting that this maximal solution is the value function of a control problem
associated in a suitable way to (1.2)-(1.3).
There are, in general, two approaches to the discretization of problem (1.2)-(1.3).
A first possibility is to discretize problem (1.2)-(1.3) directly, but imposing some additional
condition which among the infinite many solutions singles out the one we want to approx-
imate: for example, in [17], it is assumed that the solution is known on the zero set of f ,
which is now a part of the boundary of the domain where the problem is discretized.
A second possible approach (see [4], [5] and references therein) is to discretize a regularized
version of problem (1.2)-(1.3), obtained by cutting from below f at some positive level
(note that for f ? 0 problem (1.2)-(1.3) has a unique viscosity solution). To prove
the convergence of the scheme, both ffl and the discretization step h have to be send to 0.
Since the limit problem does not have a unique viscosity solution, it is not possible to apply
the Barles-Souganidis theorem and, to our knowledge, there is no convergence theorem for
this class of schemes, at least for a general zero set of f . Furthermore, if ffl and h are not
related by some condition, the approximation scheme shows numerical instability and it is
not really known which solution is approximated (see [12] for some numerical tests in this
sense).
Aim of this paper is to describe an approximation scheme for which it is possible to prove
the convergence to the maximal solution of problem (1.2)-(1.3), without requiring any
additional assumptions.
The scheme is based on a two step discretization of the control problem associated to the
regularized problem: first in the time variable, discretization step h, and then in the space
variable, discretization step k (see [2], [13] for related ideas).
In the first part (Sections 3, 4), we study the approximation scheme obtained by discretization
in time. We show that, if ffl and h are related in an appropriate way, the scheme
converges to the maximal solution of (1.2)-(1.3) for ffl and h going to zero. This result is in
the spirit of [3], in the sense that it is based on stability properties of the maximal viscosity
solution and on its characterization given by the comparison theorem in [6]. Therefore,
the proof of the convergence theorem can be easily modified to manage other boundary
conditions instead of (1.3) or, also, different approximation schemes not necessarely based
on the control theoretic interpretation of the problem.
APPROXIMATION OF DEGENERATE HAMILTON-JACOBI EQUATIONS 3
In the second part (Section 5) we study the discretization error for the fully discrete scheme.
We show that, if the zero set of f is not too "wild", it is possible to estimate in terms of
ffl and of the discretization steps the L 1 -distance between the approximate solution and
the maximal solution of the continuous problem. This part deeply employs the control
theoretic interpretation both of the discrete problem and of the continuous one.
Continuous problem: assumptions and results
In this section we briefly recall the characterization of the maximal solution of problem
obtained in [6]. Here and in the remainder of the paper by (sub, super)solutions
we mean Crandall-Lions viscosity (sub, super)solutions (see [1] for a general treatment).
We first set the assumptions on the data of the problem. The hamiltonian H
:\Omega \Theta R N
is assumed to be continuous in both variables and to verify
lim
uniformly for x
strictly increasing for t 2 [0; 1]
for any (x; p)
2\Omega \Theta R N ,
and
is convex for any x
Note that the hypothesis (2.2) replaces the stronger one of convexity of H in p.
The function f
R is nonnegative, continuous
in\Omega . Moreover, defined K := fx 2
it is assumed that
Finally we assume g : R N ! R to be a continuous and bounded function.
We introduce the gauge function ae and the support function ffi of the convex set Z(x),
namely
for any (x; p)
2\Omega \Theta R N . Both these functions are convex and homogeneous in the variable
p, and are l.s.c. and respectively continuous
in\Omega (note that, if x 2 K, ae(x;
are related by the following equality
Example 2.1 Let / be a continuous function such that
strictly increasing. Consider the equation
In this case we have
We now define a nonsymmetric semidistance
on\Omega \Theta\Omega by
R T
and, for x
2\Omega and r ? 0, the open sets
fy
It can be shown that the family BL (x; r) induces a topology - L on \Omega\Gamma If K consists of
isolated points this topology is equivalent to the Euclidean topology and the problem can
be studied in the framework of viscosity solution theory (see [14]). In general, - L is weaker
than the Euclidean topology and, for x 2 K, the set of points having zero L-distance from
x is a subset of K.
To obtain the characterization of the maximal solution, the definition of viscosity solution
will be adapted to the topology - L .
Definition 2.2 Given a l.s.c. function v
continuous function OE is
called L-subtangent to v at x 0
2\Omega if, for some ffl ? 0,
The L-subtangent is called strict if OE(x) ! v(x) outside BL
0g.
We remark that the convexity assumption (2.2) allows us to use Lipschitz continuous test
functions instead of C 1 test functions as in the standard definition of viscosity solution.
For a Lipschitz continuous function OE, we denote by @OE(x) the generalized gradient of OE at
OE is differentiable at x n g:
Definition 2.3 A l.s.c. function v
is said to be a singular supersolution of
(1.2) if for any x 0
2\Omega and for any OE, L-subtangent to v at x 0 such that
there exists a sequence x n
2\Omega n K and a sequence p n 2 @OE(x n ) for
which
lim
and
lim
APPROXIMATION OF DEGENERATE HAMILTON-JACOBI EQUATIONS 5
It is worth noting that the definition of singular supersolution reduces to the standard
definition of viscosity supersolution if x 0
In fact, in this case, since the topology
- L and the Euclidean topology are equivalent in neighborhood of x 0 , L-subtangents at
x 0 coincide with standard subtangents. Moreover,
ae(x; p) - 1) if and only if H(x; p) - f(x) (resp. H(x; p) - f(x)).
In the following theorem, we compare viscosity subsolutions and singular supersolutions of
equation (1.2).
Theorem 2.4 Let u 2
USC(\Omega
LSC(\Omega ) be a viscosity subsolution and a singular
supersolution of equation (1.2), respectively, such that u - v on @ Then
Hypothesis (2.2) allows us to give a control theoretic interpretation of problem (1.2)-(1.3).
Let U be the value function of the control problem with dynamics
where x
2\Omega and q is any bounded measurable function from [0; +1) to R n such that
cost functional
Z
The dynamic programming equation associated to the control problem (2.10)-(2.11) is
sup
This equation turns out to be equivalent to equation (1.2), in the sense that any viscosity
sub or supersolution of equation (2.12) is also a viscosity sub- or supersolution of equation
(1.2) and vice versa.
In the following we will assume that the boundary datum g verifies the compatibility
condition
It is standard to show that, under hypothesis (2.13), U is a viscosity solution of (1.2) and
satisfies the boundary condition (1.3). Furthermore we have
Proposition 2.5 The value function U is a singular supersolution of equation (1.2) in \Omega\Gamma
Theorem 2.4 and Proposition 2.5 now allow us to characterize the maximal solution of
denote the set of functions v 2
USC(\Omega ) which are viscosity subsolutions
of (1.2) and which satisfy v - g on @
\Omega\Gamma From Theorem 2.4 and Proposition 2.5 it follows
that the value function U of the control problem (2.10)-(2.11) is the maximal element of S,
i.e. the maximal subsolution of problem (1.2)-(1.3). Moreover U is a singular supersolution
of (1.2) satisfying on @
hence it is the maximal solution.
6 FABIO CAMILLI AND LARS GR -
Remark 2.6 If H is convex in p, then U coincides with the value function of control
problem with dynamics (2.10) and cost functional
where H (x; \Delta) denotes the Legendre transform of H(x; \Delta), cp. [15]. Note, however, that
ffi(x; q) and f(x) +H (x; q) in general do not coincide pointwise.
We conclude this section stating a particular case of a general stability theorem proved in
[6] needed for the construction of the approximation scheme.
Proposition 2.7 Set f ffl be the sequence of viscosity solutions
of
Then
lim
ffl!0
uniformly in \Omega\Gamma where U is the maximal solutions of (1.2)-(1.3).
Note that for any ffl ? 0 fixed, since f ffl ? 0 in \Omega\Gamma problem (2.14) admits a unique viscosity
solution. Moreover this solution is given by the value function of the control problem with
dynamics (2.10) and cost functional
@\Omega and ffi ffl (x; q) is defined as ffi(x; q) with f ffl instead of f .
We introduce some notations we will use in the following. We define
Moreover, for ffl ? 0, we set
Note that, for any ffl ? 0,
(r) is bounded by !
rg.
3 The semidiscrete scheme
Let us introduce the semidiscrete approximation scheme, obtained by discretizing in time
the exit time control problem (2.10)-(2.15). For a fixed ffl ? 0, we choose a step in time
define discrete dynamics by the recursive sequence
APPROXIMATION OF DEGENERATE HAMILTON-JACOBI EQUATIONS 7
The cost is given by
where
(we assume the convention that
0). The value function for this control problem is
such that N ! +1g:
By a standard application of the discrete dynamic programming principle, the function u hffl
is a solution of the problem
The following result holds true
Proposition 3.1 There is a constant C (independent of h and ffl) such that
Moreover u hffl is the unique bounded solution of (3:1).
Proof: We first observe that it is always possible to assume, by adding a constant, that
- 0. It follows that u hffl - 0. Moreover
where M is as in (2.16).
be two bounded solution of (3.1) and set w i 2. Then
2\Omega
where
It follows that
with in R N n \Omega\Gamma
We conclude that for any ffl ? 0 and h ? 0 there exists at most one bounded solution of
(3.3) and therefore of problem (3.1). This solution is given by u hffl .
Remark 3.2 If we discretized the control problem (2.10)-(2.11) directly (which corresponds
to setting in the previous approximation scheme), the resulting approximating
equation does not have a unique bounded solution, similarly to what happens in
problem (1.2)-(1.3). This causes the drawback that any algorithm designed to solve that
approximating equation could not converge to the maximal viscosity solution and, in any
case, displays high numerical instability (see [12]).
4 Convergence of the semidiscrete scheme
In this section, we prove the convergence of the approximation scheme introduced in the
previous section to the maximal solution of (1.2)-(1.3).
Given a locally uniformly bounded sequence of functions v ffl
lim inf
ffl!0
ffl!0
lim sup
ffl!0
ffl!0
for any x 2 \Omega\Gamma The functions lim inf
ffl!0
ffl!0
are, respectively, l.s.c. and
u.s.c.
in\Omega .
Lemma 4.1 Let u hffl be a sequence of solutions of (3.1) and assume that is such
that
Then
ffl!0
2\Omega
is a singular supersolution of (1.2).
Proof: Because of (3.2), the function u is well defined in \Omega\Gamma Let OE
R be L-subtangent
to u at x 0 2 \Omega\Gamma It is possible to assume without loss of generality (see [6], Proposition 5.1)
that OE is a strict L-subtangent to u at x 0 .
Employing a standard argument in viscosity solution theory, we find a sequence x ffl of
minimum points for u tends to 0 + . Then
ae
oe
ae
oe
for some q ffl with jq ffl
From the Mean Value Theorem for Lipschitz continuous functions (see Clarke [7]), there
exist
Substituting (4.3) into (4.2), we get
Observe that x ffl 62 K, otherwise, since on K, we should have and from
APPROXIMATION OF DEGENERATE HAMILTON-JACOBI EQUATIONS 9
which is impossible since ffi ffl is strictly positive in \Omega\Gamma
Let By the homogeneity of ffi ffl (x; q) with respect to q, we have q ffl 2 fq 2
1g. Dividing (4.4) by recalling (2.6), we get
Since the sequence x ffl belongs
to hypothesis (4.1), that u is a singular supersolution of (1.2).
Theorem 4.2 Assume that either
or
\Omega is convex: (4.6)
If u hffl is a sequence of solutions of problem (3.1) and satisfies the assumption
(4.1), then
lim
ffl!0
uniformly
in\Omega , (4.7)
where U is the maximal solution of problem (1.2)-(1.3).
Proof: We set
ffl!0
ffl!0
These function are well defined because of (3.2).
From Proposition 4.1, it follows that u is a singular supersolution of equation (1.2). Moreover
it is standard to show that u is a subsolution of (2.12) and therefore of (1.2)
in\Omega (see,
f.e., [1] or [2]). If we show that u - u on @
\Omega\Gamma then Theorem 2.4 and Proposition 2.5 imply
that
in\Omega and therefore (4.7).
We will show that
To show that u(x) - g(x) on @
\Omega\Gamma we need an estimate on the behavior of u hffl in a neighborhood
of @
sufficiently small and
jg.
For x
@\Omega be such that d(x; @
for the
discrete control problem by
and, denoted by x n the corresponding discrete trajectory, let
62\Omega g.
Observing that Nh - jy \Gamma xj, we get
where M is as in (2.16) and ! g is a modulus of continuity of g. If x
@\Omega and x ffl
2\Omega is a
sequence converging to x 0 , we have either u hffl
@\Omega is such that d(x ffl ; @
converges to x 0 ,
we get u(x 0
To get the other inequality in (4.8), if g j 0, then u
in\Omega and therefore u - 0 on
@
If (4.6) holds, by adding a constant, we can always assume that g - 0.
For x
2\Omega , let q n be an j-optimal control for u hffl (x), x n the corresponding discrete trajectory
and N the exit time
we have
with C as in (3.2).
Let q(t) be a control law for the continuous problem obtained by setting
are respectively the trajectory and the
time corresponding to q(t), we have
R
where the estimate j-(T holds because of the convexity of \Omega\Gamma Since u ffl
for any x 2
@\Omega and the assumption (4.1) is satisfied, from (4.9) we easily get other inequality
in (4.8).
Remark 4.3 For the Eikonal equation (1.1) we have
condition (4.1) reduces to
f is the modulus of continuity of the function f on \Omega\Gamma
5 Discretization error for the fully discrete scheme
In this section we will discuss a fully discrete scheme derived from the semidiscrete one as
developed in the previous sections. In order to simplify the calculations we assume that
APPROXIMATION OF DEGENERATE HAMILTON-JACOBI EQUATIONS 11
the function g defining the boundary condition is uniformly Lipschitz with constant L g ,
and that the
domain\Omega is convex.
We will introduce a space discretization which transforms (3.1) into a finite dimensional
problem. For this purpose we choose a grid \Gamma
covering\Omega consisting of simplices S j with
nodes x i and look for the solution of (3.1) in the space
const on S j g
of piecewise linear functions on \Gamma. By the parameter k we denote the maximal diameter
of the simplices S j . For simplicity we assume that the boundary of the gridded domain
coincides with the boundary of \Omega\Gamma (In the general case we can always achieve an error
scaling linearly with the distance between these two boundaries due to the fact that g is
Lipschitz).
Thus we end up with the fully discrete scheme
ffl;h
ffl;h
for all nodes x i
2\Omega with the boundary condition u k
ffl;h for the nodes x i
62\Omega and
linear interpolation between the nodes.
Note that there exists a unique bounded solution of (5.1). The boundedness of any solution
of (5.1) follows from the fact that
ffl;h
ffl;h
holds for any q 2 R n with 1. Thus we can always choose q such that u k
ffl;h
depends on nodes which are closer to the boundary
@\Omega than x i and (if h !
but with a weight strictly less than one. Since the value in the boundary nodes is bounded
we obtain boundedness for each node by induction.
Due to the boundedness the existence of a unique solution u
ffl;h is now easily proved by
applying the Kruzkov transformation
ffl;h
as in the proof of Proposition 3.1.
Note that the function ffi ffl appearing in the scheme is defined implicitely via H and f ffl . In
order to solve the scheme we assume that we can compute this function analytically as
e.g. in Example 2.1. (In the case of a convex Hamiltonian one may alternatively use a
numerical approximation of the integrand from Remark 2.6 via the Legendre transform as
given e.g. in [10]. Note, however, that this procedure yields a different cost function than
in the following analysis.)
We will now start by estimating the discretization error ju ffl
ffl;h
we allow nonconstant boundary conditions we introduce the following auxiliary functions
which will be useful for the estimation of the error.
Definition 5.1 For each point x
2\Omega we define
where -(\Delta) is an optimal path for the initial value x and -(T
For each node x i of the grid pick a control q i minimizing (5.1) and let w 2 2 W be the
unique solution of
with the boundary condition w 2 interpolation between the nodes.
Finally we define
Remark 5.2 The existence of optimal paths follows from the continuous dependence of
the functional J(x; q) from the control function q using the weak -metric (as defined for
control functions e.g. in [9]), using the Gronwall Lemma as in [8, Proof of Lemma 3.4(ii)]
and the structure of ffi ffl . Note that the a-priori boundedness of the length of approximately
optimal trajectories - following from the positivity of ffi ffl - is crucial for this continuous
dependence. Thus in general the existence of optimal trajectories does not hold for the
non-regularized problem since there for any sequence of approximately optimal trajectories
the length of these trajectories may grow unbounded when we restrict
Note that we do not require uniqueness of the optimal paths in Definition 5.1. In the case
that there is no unique optimal path we may use one that minimizes w 1 .
Definition 5.1 defines functions which are 0 at
@\Omega and away from
@\Omega essentially grow like
ffl;h , respectively. More precisely we have that
and
ffl;h
ffl;h
for -(\Delta) and q i as used in the definition.
Note that in particular if g(x) j c is constant we obtain
Using this w we can give the following estimate for the discretization error.
Proposition 5.3 Let u k
ffl;h 2 W be the unique solution of (5.1). Then the estimate
fi(ffl)h
holds for each x
2\Omega and for all sufficiently small k ? 0 and h ? 0 with
and ff as
defined in (2.16)-(2.18),
and some constant C independent from ffl; h and k.
APPROXIMATION OF DEGENERATE HAMILTON-JACOBI EQUATIONS 13
The proof can be found in the appendix.
Remark 5.4 (i) Note that estimate (5.3) is stronger than the usual L1 estimate since
essentially the error scales with the function w(x) being 0 at @
\Omega\Gamma The reason for this
behaviour origins in the fact that the error is estimated along the optimal trajectories
whose length depends on the optimal value.
(ii) The constant ff(ffl) essentially depends on the growth of H in jpj, e.g. in Example 2.1 we
have (ffl). The constant fi(ffl) is determined by the difference between H(x; p) and
H(x; q) for In particular if H(x; p) 2 [C 1 jpj we have that fi(ffl) - C 1 =C 2
independently from ffl. Finally, ! ffi (which gives a bound for !
for combines the
continuity properties of H and f , i.e. in Example 2.1 we have that ! ffi
(iii) Note that the requirement on h ensuring the convergence of the fully discrete scheme is
thus it is consistent with condition (4.1) for the convergence
of the semidiscrete scheme.
(iv) The appearance of the value ff(ffl) in the denominator in (5.3) is due to the fact that
here we implicitely included the worst case, i.e. that the length of the optimal trajectories
may grow like 1=ff(ffl) for ff(ffl) ! 0. Since this is not necessarily the case in many practical
examples one can expect better convergence behaviour for ff(ffl) ! 0.
(v) A particular nice formulation of estimate (5.3) can be obtained if we consider the
Eikonal equation (1.1) (implying assume that f is uniformly
impose a homogeneous boundary condition, i.e.
(implying L 0). In this case the estimate becomes
for some constant C ? 0 independent from ffl; h and k. In particular this implies convergence
of the scheme
We will now turn to the discussion of the error obtained when equation (1.2) is replaced
by equation (2.14), i.e. the error introduced by the regularization of the problem.
Proposition 2.7 already implies that u " converges to U , where U is the maximal subsolution
of (1.2). Unfortunately, in general this convergence can be arbitrary slow. In the optimal
control interpretation this is due to the fact that the length of approximately optimal
trajectories may grow unbounded as the approximation gets better and better. Since
these long pieces of the trajectories can only appear in regions where f is sufficiently small
(otherwise the cost would be large contradicting the approximate optimality), we can derive
an estimate for the regularization error by defining a criterion for the sets where f is small
which in turn gives a bound on the length of approximately optimal trajectories.
The following definition is our main tool for this purpose.
Definition 5.5 Let B ae R d be a compact set. For each connected component B i of B we
define the inner diameter d(B i ) by
14 FABIO CAMILLI AND LARS GR -
where
and for B we define the inner diameter by
where the sum is taken over all connected components of B.
Using this definition we can state the following estimate for the regularization error.
Proposition 5.6 Let U be the maximal subsolution of (1.2) and let u ffl be the unique
viscosity solution of (2.14).
Then the estimate
holds where K ffl := fx
The proof can be found in the appendix.
Here the constant c(ffl) depends only on the sets Z ffl (x)g, i.e. on ffl
and on the Hamiltonian H. In fact an easy calculation shows that
Thus e.g. the estimate c(ffl) - C ffl fl for some constants C; fl ? 0 and all ffl ? 0 sufficiently
small holds if H(x; p) - (jpj=C) 1=fl for all x 2 K ffl , all ffl ? 0 sufficently small and all p 2 R n
with jpj sufficiently small. In particular for the Eikonal equation (1.1) this implies
Observe that if f is piecewise polynomial then bounded for all ffl ? 0 and hence
convergence with order c(ffl) follows for ffl ! 0. Piecewise polynomial maps are in particular
interesting since they include the case where f is obtained from experimental data by
some polynomial interpolation (e.g. using piecewise linear interpolations, multidimensional
splines.
The following theorem now gives the full a-priori estimates for the approximation error of
the whole numerical approximation.
Theorem 5.7 Let U be the maximal subsolution of (1.2) and let u k
ffl;h be the unique solution
of the numerical scheme (5.1).
Then the estimate
fi(ffl)h
holds for each x
2\Omega and the constants from the Propositions 5.3 and 5.6.
APPROXIMATION OF DEGENERATE HAMILTON-JACOBI EQUATIONS 15
Proof: Follows immediately from the Propositions 5.3 and 5.6.
Remark 5.8 (i) A possible modification of the scheme can be made if we allow smaller
time steps at the boundary @ i.e. for x i
2\Omega and x
62\Omega we use the restricted time
step
2\Omega g:
Although slightly more difficult to implement this modification usually gives better numerical
results. The proof of Proposition 5.3 also applies to this modified scheme.
(ii) Due to the structural similarity of the scheme described in this section with the scheme
considered in [13], the adaptive grid scheme developed there can also be applied here.
Similar convergence results as in [13] can be obtained for our scheme using the technique
from the proof of Proposition 5.3.
6
Appendix
Proof of the Propositions 5.3 and 5.6
In order to prove Proposition 5.3 we will first state a useful estimate for the local error
along the functional.
Lemma 6.1 For each measurable q(\Delta) with almost all t 2 [0; h] and the path
-(\Delta) with -
2\Omega for all t 2 [0; h] there exists p 2 R n with
that
Z
and
Conversely, for each p 2 R n with each x
2\Omega with x
2\Omega there exists a
measurable function q(\Delta) with
Z
and
2\Omega for all t 2 [0; h].
Proof: The convexity of ffi ffl in the second argument implies
Z
Hence by defining
Z hq(t)dt
the first assertion immediately follows from the continuity of ffi ffl which is measured by !
The second assertion follows directly from the continuity of ffi ffl setting q(t) j p and using
the convexity
of\Omega .
Proof of Proposition 5.3
We start giving some preliminary estimates.
First note that the error at the boundary can be estimated by
which simply follows from the Lipschitz property of g.
Furthermore it is easy to see that on each element S j of the grid we can estimate
for each two points
We show the estimate (5.3) by estimating seperately the quantities u k
ffl;h (x). First, we consider u
Observe that for any fl ? 0 there exists an j - 0 such that
which easily follows from the fact that w 1 - 0 and u k
ffl;h is bounded.
Now we fix some arbitrary fl ? 0 and choose j - 0 to be minimal with (6.3). If the
assertion immediately follows. Otherwise by the continuity of the functions and
the compactness
of\Omega we can conclude that there exists x
2\Omega such that
ffl;h
Now consider the element S j containing x . We can write x
where the x i are
the nodes of S j and the - i are nonnegative coefficients with
Using estimate (6.2) we obtain
ffl;h
i2I
ffl;h
Now for each of the nodes we distinguish three cases.
By (6.1) this implies
ffl;h
2\Omega and for the optimal path - i (\Delta) with - i Definition 5.1 there exists a
APPROXIMATION OF DEGENERATE HAMILTON-JACOBI EQUATIONS 17
@\Omega and -
(t). In this case by the convexity
of\Omega we can conclude
that there exists p 2 R n with
62\Omega such that jx i
we obtain
ffl;h
2\Omega and for the optimal path - i (\Delta) with - i 5.1 the equality
Z
holds
@\Omega and -
In this case Lemma 6.1 and the definition of u k
ffl;h imply
ffl;h
where by (6.8) we can estimate
and by Definition 5.1 thus also
Taking into account that the coefficients in (6.5) sum up to 1 we derive
i2I
ffl;h
and combining (6.3), (6.4), (6.5), (6.10) and (6.11) we obtain
(w 1
from which we conclude that
Estimating
(note that - j - ff(ffl)) this becomes
h-
Now we specify the assumption "h; k ? 0 sufficiently small" by choosing them such that
h-
for some constant C ? 0 and thus
fi(")h
h- j
which implies the desired estimate for w values in this
resulting inequality are independent from fl ? 0 this also implies the estimate for
The inequality for u ffl
ffl;h (x) follows with the same technique and the obvious modifications
using note that here the convexity
of\Omega is also needed in Lemma 6.1 used in
case (iii). Proceeding in this way we end up with the analogous estimate to (6.12)
which leads to the desired result here without using the assumptions on k and h.
Proof of Proposition 5.6
For any measurable and bounded q and any x
2\Omega denote the solution of (2.10) by
arbitrary and pick some x 2 \Omega\Gamma Then by the optimal
control representation of U (2.10)-(2.11) there exists a solution -
with
@\Omega and
We now divide the connected components K i
I of K ffl into two classes by defining
I 1 :=
and I Then by the continuity of H there exists a constant fl(ffl 2 ) with
as
Furthermore by the uniform continuity of f every set K i
has a volume bounded
from below by some uniform constant depending on ffl 2 and hence there are only finitely
many of these sets; we may number them by
Now we define for each of these K i
which is hit by the trajectory - ffl 1
times
by
" g and t i
where we omit those sets K i
" for which [t i
holds. This gives
us a finite number r of pairwise disjoint intervals [t i
which we assume to be numbered
according to their order, i.e. t i
For each trajectory piece - ffl 1
we have by (6.13) and by the fact that outside K ffl
the functions ffi and ffi ffl coincide the estimate
APPROXIMATION OF DEGENERATE HAMILTON-JACOBI EQUATIONS 19
For the points - ffl 1
yielding
ffl for all t 2 [0;
ffl
is possible by the definition of d(\Delta) and the structure of the dynamics (2.10). We now define
a sequence of times t i ,
and a measurable function ~ q(\Delta) by
~
This construction yields that
(t) for all
and
thus in particular it follows that -(t r ; x; ~
obtain
dt
dt
Now letting first ffl we obtain the assertion since u ffl - U is obvious
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588626 | Multilevel Boundary Functionals for Least-Squares Mixed Finite Element Methods. | For least-squares mixed finite element methods for the first-order system formulation of second-order elliptic problems, a technique for the weak enforcement of boundary conditions is presented. This approach is based on least-squares boundary functionals, which are equivalent to the H-1/2 and H1/2 norms on the trace spaces of lowest-order Raviart--Thomas elements for the flux and standard continuous piecewise linear elements for the pressure, respectively. Continuity and coercivity of the resulting bilinear form is proved implying optimal order convergence of the resulting Galerkin approximation. The boundary least-squares functional is implemented using multilevel principles and the technique is tested numerically for a model problem. | Introduction
. In the context of least-squares finite element methods for first-order
systems, boundary conditions can be enforced as essential boundary conditions
in the finite element spaces. This yields optimal order convergence of the Galerkin
approximations under suitable assumptions on the regularity of the problem (see, for
example, [18, 9, 10]). However, this approach constructs an approximation which
is much more accurate on the boundary than in the interior of the domain. For
least-squares finite element methods, a natural way of treating boundary conditions
is to enforce them weakly by adding boundary functionals. The boundary functional
approach is also a natural and simple way to handle the common situation that the
boundary conditions cannot be satisfied exactly by trial functions in the finite element
spaces (see, e.g., [16, 5, 13, 1], see also [4] for a different approach to handle boundary
conditions within a least-squares method). Nonlinear boundary conditions in the
modelling of flow in porous media (see, e.g., Bear [2, Chap. 7]) may also be handled
effectively by boundary functionals.
In this paper, we derive appropriate boundary functionals for the mixed formulation
of second-order elliptic problems. These are equivalent to the H 1=2 (\Gamma D )-norm
for the Dirichlet conditions and to the H \Gamma1=2 (\Gamma N )-norm on the Neumann boundary.
From the view of trace and extension theorems of Sobolev norms this gives the proper
balance between the interior and boundary least-squares functionals. These boundary
functionals are in some sense the weakest possible without losing the optimal order of
the Galerkin approximation. Related but different boundary functional methods in
connection to least-squares finite elements were suggested before in the literature. For
pure Dirichlet problems, a weighted L 2 (\Gamma) norm was used in [13]. This approach using
weighted L 2 norms was generalized to elliptic boundary value problems of A-D-N
type in [1].
Our aim in this paper is to justify our multilevel boundary functional approach
theoretically and by computational experiments for the mixed formulation of the Poisson
equation using lowest-order Raviart-Thomas elements. Clearly, the true potential
of this methodology lies in its applicability to more complicated problems, e.g., in
flow computations in porous media. The use of the least-squares mixed finite ele-
Fachbereich 6 (Mathematik und Informatik), Universit?t-GH Essen, 45117 Essen, Germany
ment approach for nonlinear boundary value problems arising in variably saturated
subsurface flow is studied in [20]. The techniques presented in this paper can also be
extended to least-squares formulations like, e.g., Maxwell, Stokes and Navier-Stokes
equations.
In the following section, we present the least-squares formulation including the
boundary functionals and prove coercivity and continuity of the corresponding bilinear
form. Section 3 reviews some results on the approximation properties of finite
element spaces including the lowest-order Raviart-Thomas elements. In Section 4, we
are concerned with an equivalent and computable boundary functional based on multilevel
principles. Section 5 gives a study of the effect of the boundary least-squares
functionals on the accuracy of the finite element approximation by computations for
a two-dimensional model problem.
2. Least-Squares Formulation with Boundary Functionals. We consider
the first-order system formulation of Poisson's equation,
div
oe
(\Omega\Gamma in a bounded polygonal
domain\Omega ae IR 2 . The boundary
of\Omega is
divided into
are prescribed. Actually, for \Gamma ' @
may be defined as the space of traces from H
1(\Omega\Gamma and H \Gamma1=2 (\Gamma) as the corresponding
dual norm with respect to L 2 (\Gamma) (see, for example, [11, Section I.1]).
Despite the fact that we restrict ourselves to two-dimensional problems for the
purpose of exposition, all the techniques presented below can be extended to higher
dimensions. Note that these techniques can also be extended to more general diffusion
problems like those arising in porous media flow.
Clearly, each solution (u;
1(\Omega\Gamma of this boundary value problem
also minimizes the least-squares functional
0;\Omega
kn
for any ff ? 0. With the corresponding bilinear form
0;\Omega
this is equivalent to finding (u; p) 2 H(div ; \Omega\Gamma \Theta H
1(\Omega\Gamma such that
for all (v;
We have the trace inequalities
kpk
1;\Omega for all
(see [11, Theorem 1.5]) with a constant c T and
kn
\Gamma1=2;@\Omega - kuk
div;\Omega for all u
MULTILEVEL BOUNDARY FUNCTIONALS FOR LEAST-SQUARES METHODS 3
(see [11, Theorem 2.5]).
Moreover, if \Gamma D is a curve of positive measure, the generalized Poincar'e-Friedrichs
inequality in [15, Theorem 1.9] gives
kpk
for all
(\Omega\Gamma with a constant c F . If \Gamma restrict ourselves to H
normalizing p to, for example, (p; 1)
This construction is necessary in order
to ensure uniqueness of p and we have a Poincar'e-Friedrichs inequality of the form
kpk
0;\Omega for all
which satisfy (p; 1)
(cf. [3, Section II.3]).
These tools allow us to prove coercivity and continuity of the bilinear form for
any ff ? 0.
Theorem 2.1. Under the assumptions above, for any ff ? 0, the bilinear form
B(\Delta; \Delta; \Delta; \Delta) is coercive and continuous with respect to H(div; \Omega\Gamma \Theta H
and
with positive constants c S and c E (which depend on ff, c T and c F ).
Proof. The above trace inequalities and repeated use of the Cauchy-Schwarz
inequality lead to
0;\Omega kdiv vk
div;\Omega kvk
1;\Omega kqk
which proves (2.9).
For the coercivity proof we consider two separate cases: (i) \Gamma
@\Omega a curve of positive measure.
Case
Cauchy-Schwarz inequality and
ff
kn
ff
1=2;@\Omega
lead to
0;\Omega
0;\Omega
ff
1=2;@\Omega
ff
1;\Omega
for any ffi 2 (0; 1). Combined with (2.8), this implies
\Gamma1=2;@\Omega
F
1;\Omega
F
ff
0;\Omega
F
ff
Choosing
F )g.
Case positive measure. The first step is as in case (i) and using (2.6)
and (2.5) we obtain
kpk
kpk
kn
ff
ff
1;\Omega
which holds for any ffi 2 (0; ff). This leads to
0;\Omega
Combining this with (2.7) gives
0;\Omega
kuk
ff
F
Choosing
proves (2.10).
We remark that in [18], coercivity and continuity of the bilinear form
0;\Omega
is shown for
MULTILEVEL BOUNDARY FUNCTIONALS FOR LEAST-SQUARES METHODS 5
Since more emphasis is put on enforcing the boundary conditions as ff increases, this
result can be regarded as the limiting case for ff !1. Standard finite element theory
implies that the Galerkin approximation in these spaces is of optimal order. There are
many ways to achieve this optimal order convergence by boundary functionals which
are stronger than the k \Delta k \Gamma1=2;\Gamma N and k \Delta k 1=2;\Gamma D norms, respectively. The importance of
using the norms k \Delta k \Gamma1=2;\Gamma N and k \Delta k 1=2;\Gamma D in (2.2) is that the least-squares functionals
on the boundary and in the interior are properly balanced.
3. Galerkin Approximation. For the numerical approximation we consider
finite element spaces
based on a quasiuniform sequence of triangulations fT l g l=0;1;::: of \Omega\Gamma Let h l denote
the mesh-size of fT l g given, for example, by the maximal diameter of the triangles.
We compute approximate solutions u l 2 V l and p l 2 W l for u and p, respectively, such
that F(u l ; p l ; f) is minimized among all u l 2 V l and p l 2 W l . This is equivalent to
the variational problem of finding l \Theta W l such that
for all (v l ; q l l \Theta W l . The simplest choice is to use the lowest-order Raviart-
Thomas space for V l and standard continuous piecewise linear functions for W l on the
triangulation T l .
Continuity and coercivity proved in Theorem 2.1 give us the usual quasi-optimality
of the Galerkin approximation
l 2V l
q l 2W l
Let us assume that f 2 H ff
for some ff 2 (0; 1] and that g; h and the boundary
are such that p 2 H 1+ff
and, consequently, u 2 (H
details on the conditions for such regularity results). For example,
holds
if
domain\Omega with the additional property
that the interior angles at boundary points separating \Gamma N from \Gamma D are at most -=2.
In order to simplify our notation, we write - l . j l to indicate that - l - cj l holds
with a constant c which is independent of l. We also write - l h j l to indicate that
both - l . j l and j l . - l are satisfied. Thus, [8, Proposition III.3.9] implies
l 2V l
div;\Omega . h ff
l [kfk
for the approximation by lowest-order Raviart-Thomas elements and standard finite
element interpolation results (cf., e.g., [7, Chapter 4]) give
q l 2W l
1;\Omega . h ff
l
In order to compute the solution of (3.1), we use the bases f\Phi (-)
l g M l
l and
l g N l
-=1 for W l . The variational problem (3.1) may then be formulated as a linear
system of equations
- A uu A up
A pu A pp
- u
6 GERHARD STARKE
l )] -=1;:::;N l ;-=1;::: ;M l
l )] -=1;::: ;N l
l
l ) \Gamma1=2;\Gamma N
l
and l are the basis representations of u l 2
Setting up the matrices A uu and A pp involves the computation of
(n \Delta \Phi (-)
l )
l
Since we cannot compute these inner products directly, we need to replace k \Delta k \Gamma1=2;\Gamma N
by equivalent norms such that the corresponding inner products are
computable. Alternatively, one could replace these norms by stronger norms, e.g.,
norms do not give the optimal balance between accuracy on the boundary and in the
interior.
4. Multilevel Implementation of the Boundary Functionals. In this sec-
tion, we derive computable norms which are equivalent to the norms k \Delta k \Gamma1=2;\Gamma N and
using additive multilevel decompositions in the spirit of [6]. To this end,
we define the L 2 (\Gamma)-orthogonal projection Q
@\Omega onto (the trace space of)
l . We will abuse notation and denote
this trace space W l wherever it is clear from the context. With the operator
l
we have (cf. [17, Theorem 15] or [14, Corollary 3.2.4])
l
for all q l 2 W l .
For a computable expression that replaces k \Delta k \Gamma1=2;\Gamma N
we use the following result.
Theorem 4.1. Let V l and W l be the lowest-order Raviart-Thomas spaces and
standard piecewise linear continuous finite element spaces, respectively, based on a
quasiuniform sequence of triangulations. Then, for u l 2 V l ,
kn \Delta u l k \Gamma1=2;\Gamma N h sup
z l 2W l ;z l 6=0
Proof. By definition (see [11, Section I.1]),
kzk
MULTILEVEL BOUNDARY FUNCTIONALS FOR LEAST-SQUARES METHODS 7
Clearly, since W l ae H 1=2 (\Gamma N ), this implies
kn \Delta u l k \Gamma1=2;\Gamma N
z l 2W l ; z l 6=0
For the upper bound, we use the L 2 (\Gamma N )-orthogonal projection Q l;\Gamma N to (the trace
space of) W l . It is well-known (cf. [21, Section 4]) that
for all z together with the trivial inequalities
implies
l kzk 1=2;\Gamma N and kQ l;\Gamma N zk 1=2;\Gamma N . kzk
We may write
sup
kzk
kzk
kzk
Using the fact that n \Delta u l is piecewise constant along the boundary, we may estimate
the second term on the right hand side using
zk
. h l
kzk
kzk 1=2;\Gamma N . h 1=2
l kn \Delta u l k \Gamma1=2;\Gamma N
kzk
where the last step follows from the inverse inequality k- l k 0;\Gamma N . h \Gamma1=2
l k- l k \Gamma1=2;\Gamma N
for piecewise constant functions - l on the subdivision of \Gamma N corresponding to T l . This
implies
kn \Delta u l k \Gamma1=2;\Gamma N
kzk
l kn \Delta u l k
Choosing l 0 such that 2Ch 1=2
l 0
- 1, we are led to
kzk
. sup
z l 2W l ; z l 6=0
From (4.2) and (4.3) we are led to a computable expression by
kn \Delta u l k \Gamma1=2;\Gamma N h sup
z l 2W l ; z l 6=0
z l ) 1=2
z l 2W l ; z l 6=0
(n \Delta u l ; C \Gamma1=4
where C
is the adjoint of C l;\Gamma N such that
(n \Delta u l ; C \Gamma1=4
z l )
holds for all u l 2 V l ; z l 2 W l .
This gives rise to the modified bilinear form
0;\Omega
The variational problem of finding l \Theta W l such that
0;\Omega
for all (v l ; q l l \Theta W l replaces (3.1).
The computation of the contributions of the boundary functionals to the Galerkin
matrix is based on the mass matrices M j;\Gamma in W j ,
on the mass matrices ~
M j;\Gamma for coupling V j and W j ,
~
and on the restriction matrices I j
l which, for j - l, compute moments
-=1 from s l = [(s; \Psi (-)
l
(note that I l
l is just the identity matrix in IR N l ). It is easy to see that
holds. Of course, the inverse of M j;\Gamma can only be formed on the subspace of nodal
basis functions which do not vanish on \Gamma (we keep using this notation for simplicity).
For the Dirichlet part, we have
l
l ;h 0
l
l
=h l
l
l
MULTILEVEL BOUNDARY FUNCTIONALS FOR LEAST-SQUARES METHODS 9
which implies
l
h l
l
I j
l M
In order to set up the matrix entries associated with the Neumann boundary
conditions, we start from
l )
l
I j
l ]M l;\Gamma N
l I l
l
I j
l M l;\Gamma N
This implies
l )
~
with B l;\Gamma N
l I l
l
I j
l
which leads to
l ); (C \Gamma1=4
l
~
Clearly, from the quasiuniformity of the sequence of triangulations we obtain
l ); (C \Gamma1=4
l
~
~
which is the form we actually use in our implementation.
The expressions for the computation of the boundary contributions in (4.5) and
(4.6) involve the inverses of mass matrices M j;\Gamma D
and M j;\Gamma N
These inverses are dense matrices on the subspace of unknowns associated with \Gamma D
respectively. The actual computation of M \Gamma1
should be avoided,
especially for three-dimensional problems, and replaced by an inner conjugate gradient
iteration for solving linear systems with M j;\Gamma D and M j;\Gamma N . It is well-known that
the number of iterations required to achieve a certain accuracy is bounded independently
of l for these systems. Alternatively, one can use biorthogonal wavelets for
the construction of an H \Gamma1=2 (\Gamma N )-equivalent boundary functional for the Neumann
conditions. In the two-dimensional case, i.e., \Gamma N consists of a finite number of line
segments, this is described in [19, Section 5.6].
Fig. 5.1. Pressure (left) and flux (right) approximation for Example 10.5 1 ComputationalExperiments. For our computational experiments, we consider
on the rectangular domain shown in Figure 5.1. For the first
example, the boundary conditions are on the right two thirds of the upper and
on the right boundary segment, n \Delta on the left third of the upper boundary
segment and n \Delta segments. This can be viewed as a
simple model problem for flow in porous media where water infiltrates at a prescribed
rate on the left part of the upper boundary segement. The left and lower boundary
are impermeable and on the remaining part of the boundary pressure is set to zero
(hydraulic potential equals gravitational potential). The solution for the flux u is indicated
by the arrows in Figure 5.1. For this example, we have
3=2\Gamma"(\Omega\Gamma for all
due to the jump from Neumann to Dirichlet boundary condition on the upper
boundary segment.
The coarsest triangulation T 0 consists of 74 triangles which are then uniformly
refined. Figure 5.1 shows the first refinement T 1 . Table 5.1 shows the error with
respect to the "exact solution", measured in the H(div ; \Omega\Gamma \Theta H
As an "exact solution" we use the Galerkin approximation (with exact enforcement
of the boundary conditions on an adaptively refined triangulation T ?
4 which
consists of 24042 degrees of freedom for the edges and 8133 for the nodes. The triangulation
4 was constructed from T 3 using four steps of the refinement algorithm
presented in [20] based on the least-squares functional as an a posteriori error esti-
mator. The least-squares functional is reduced by a factor of more than 10 on T compared to T 3 . This leads us to the conclusion that it is legitimate to use it as
reference solution in our experiments.
Table
Approximation error for the multilevel boundary functional for Example 1
MULTILEVEL BOUNDARY FUNCTIONALS FOR LEAST-SQUARES METHODS 11
Obviously, Table 5.1 shows that the approximation is slightly improved if the
boundary functional is weighted by instead of enforcing the boundary
conditions exactly 1). We need to compare the multilevel boundary functional
with the weighted L 2 (\Gamma) approach of [13] and [1]. In our situation, this means
replacing the least-squares functional (2.2) on V l \Theta W l by
kn
h l
This leads to the numbers listed in Table 5.2 which are slightly larger in most cases.
However, the difference between the multilevel and the weighted L 2 (\Gamma) functional is
rather marginal in this case.
Table
Approximation error for the weighted L 2 (\Gamma) boundary functional for Example 1
In our second example, the boundary conditions are chosen such that it is not
possible to satisfy them exactly with our finite element spaces. On the upper bound-
ary, we set linearly
interpolated for x 1 chosen to be smaller than the edge length of
our finest triangulation. The boundary conditions on the other boundary segments
are the same as in the first example. The numbers obtained with the multilevel and
the weighted L 2 boundary functionals are shown in Tables 5.3 and 5.4, respectively.
Table
Approximation error for the multilevel boundary functional for Example 2
Table
Approximation error for the weighted L 2 (\Gamma) boundary functional for Example 2
In conclusion, our computational results show the feasibility of using the multi-level
boundary functional approach. In particular, this method seems promising in
cases where it is not possible to enforce the boundary conditions exactly by the finite
element spaces. The construction and use of similar multilevel boundary functionals
for more complicated situations including nonlinear boundary conditions in porous
media applications is the focus of current research.
Acknowledgement
. I would like to thank Tom Manteuffel for many helpful discussions
related to the subject of this paper. The detailed comments of the anonymous
referees are also very much appreciated. In particular, I am thankful to one of them
for pointing out references [13] and [1] to me.
--R
Least squares methods for elliptic systems
Dynamics of Fluids in Porous Media
Cambridge University Press
A generalized Ritz-least-squares method for Dirichlet problems
Parallel multilevel preconditioners
The Mathematical Theory of Finite Element Methods
Mixed and Hybrid Finite Element Methods
Finite Element Methods for Navier-Stokes Equations
Elliptic Problems in Nonsmooth Domains
A least squares decomposition method for solving elliptic equations
Multilevel preconditioning - appending boundary conditions by Lagrange multi- pliers
Multilevel Finite Element Approximation
Two preconditioners based on the multi-level splitting of finite element spaces
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--CTR
Huo-Yuan Duan , Guo-Ping Liang, Nonconforming elements in least-squares mixed finite element methods, Mathematics of Computation, v.73 n.245, p.1-18, January 2004 | multilevel boundary functionals;least-squares mixed finite element method;raviart-thomas spaces |
588632 | Detection of Edges in Spectral Data II. Nonlinear Enhancement. | We discuss a general framework for recovering edges in piecewise smooth functions with finitely many jump discontinuities, where $[f](x):=f(x+)-f(x-) \neq 0$. Our approach is based on two main aspects--- localization using appropriate concentration kernels and separation of scales by nonlinear enhancement. To detect such edges, one employs concentration kernels, $K_\epsilon(\cdot)$, depending on the small scale $\epsilon$. It is shown that odd kernels, properly scaled, and admissible (in the sense of having small $W^{-1,\infty}$-moments of order ${\cal O}(\epsilon)$) satisfy recovering both the location and amplitudes of all edges. As an example we consider general concentration kernels of the form $K^\sigma_N(t)=\sum\sigma(k/N)\sin kt$ to detect edges from the first $1/\epsilon=N$ spectral modes of piecewise smooth f's. Here we improve in generality and simplicity over our previous study in [A. Gelb and E. Tadmor, Appl. Comput. Harmon. Anal., 7 (1999), pp. 101--135]. Both periodic and nonperiodic spectral projections are considered. We identify, in particular, a new family of exponential factors, $\sigma^{exp}(\cdot)$, with superior localization properties. The other aspect of our edge detection involves a nonlinear enhancement procedure which is based on separation of scales between the edges, where $K_\epsilon*f(x)\sim [f](x) \neq 0$, and the smooth regions where examples demonstrate that by coupling concentration kernels with nonlinear enhancement one arrives at effective edge detectors. | Introduction
We discuss a general framework for recovering edges from the spectral projections of piecewise smooth
functions. Our approach for edge detection is based on two fundamental aspects - localization to
the neighborhood of the edges using appropriate concentration kernels and separation of scales by
nonlinear enhancement. Both the location and amplitudes of all edges are recovered.
Let SN f(x) denote the spectral projection of a piecewise smooth f . Given SN f , one can accurately
reconstruct f away from its discontinuous jumps, e.g., [10],[14, x2.1], as well as up to the
discontinuities, [11]. In either case, an a priori knowledge on the location of the edges and their
amplitudes is required. This issue was treated in recent literature, consult [1], [5], [13], [15]. In [7],
we unified the previous treatments as special cases of appropriate concentration kernels. Here we
improve on these results in both generality and simplicity. To this end, let [f ](x) :=
denote the local jump function and let us consider a concentration kernel K ffl (\Delta), depending on a small
scale ffl. It is shown that odd kernels, properly scaled, and admissible (- in the sense of having small
W \Gamma1;1 -moments of order O(ffl), (2.6)), recover both the locations and the amplitudes of the jumps
so that
Thus, K ffl tends to "concentrate" near the singular support of f .
Differentiation of ffl-supported mollifiers is one example for local concentration kernels outlined in
x2.2.1. In x2 we also address the issue of detecting edges in global Fourier projections. Given the
first modes, we seek concentration kernels of the form
K oe
sin kt:
It is shown that if the concentration factors oe(-) j -) are normalized so that
K oe
N (t) is an admissible concentration kernel, K oe
and the following error estimate
holds
-(
log N
The non-periodic case is studied in x3. The analogous results for the Chebyshev case reads, consult
Corollary 3.2
log N
The special cases of Fourier concentration factors oe ff (- sin ff- and oe p were considered
earlier in [1],[7],[9],[13], and [15]. Our general framework motivates a new set of C 1
-exponential
concentration factors which yield superior localization properties away from the detected edges.
While (1.1) refers to the asymptotic behavior of the concentration kernel as a function of the
small parameter ffl # 0, it is essential to recover the exact locations of the edges of f for the accurate
reconstruction of f . In x4 we discuss another essential aspect of edge detection, namely nonlinear
enhancement. To this end, one introduces a critical threshold, J crit , for the amplitude of admissible
edges, and an enhancement exponent, p, to amplify the separation of scales in (1.1) between the
edges, where K ffl f(x) - [f ](x) 6= 0, and the smooth regions where K ffl
the enhanced kernel
Enhanced detection of edges in spectral data 3
ae
Clearly, with p large enough, one ends up with a sharp edge detector where K ffl;J [f at
all but O(ffl)-neighborhoods of the jump discontinuities. In this sense, the enhancement procedure
actually "pinpoints" the location jump discontinuities, allowing an accurate reconstruction of f . The
particular case corresponds to the quadratic filter studied in [12],[22], in the special context of
concentration kernels based on localized mollifiers.
Acknowledgment
. Research was supported in part by the Sloan Foundation (AG) and by NSF
Grant No. DMS97-06827 and ONR Grant No. N00014-1-J-1076 (ET).
Edge Detection by Concentration Kernels
2.1 Concentration Kernels
We want to detect the edges in piecewise smooth functions. Assume that f(\Delta) has jump discontinuities
of the first kind with well defined one-sided limits,
f(x\Gamma) denote the local jump function. By piecewise smoothness we mean 1
In practice one encounters functions f(x) with finitely many jump discontinuities, and (2.2)
requires the differential of f(x) on each side of the discontinuity to have bounded variation. For
example, if f 0 (x\Sigma) are well defined (for finitely many jumps), then (2.2) holds.
We will detect the edges in such piecewise smooth f 's using smooth concentration kernels,
depending on a small parameter ffl. Such kernels are characterized by
Thus the support of K ffl f(x) tends to "concentrate" near the edges of f(x). One recovers both the
location of the jump discontinuities as well as their amplitudes.
To guarantee the concentration property of K ffl , we seek odd kernels,
which are normalized so that Z
and which satisfy the main admissibility requirement
Z
Const
Remarks.
1 Here and below we use BV [a; b] to denote the space of functions with bounded variation, endowed with the usual
semi-norm kOEk BV [a;b] := R b
a jOE 0 jdx
A. Gelb and E. Tadmor
1. For example, if K ffl (t) concentrates near the origin so that its first moment does not exceed
Z
Const \Delta ffl; (2.7)
then it is clearly admissible in the sense that (2.6) holds. We note that our admissibility
condition also allows for more general oscillatory kernels, K ffl (t), where (2.7) might fail, yet
(2.6) is satisfied due to the cancelation effect of the oscillations, consult (2.18 below.
2. Observe that the admissibility requirement (2.6) generalizes both properties P 3 and P 4 in the
definition of admissible kernel [7, definition 2.1].
Our main result states that
Theorem 2.1 Consider an odd kernel K ffl (t), (2.4), normalized so that (2.5) holds, and satisfying
the admissibility requirement (2.6). Then the kernel K ffl (t) satisfies the concentration property (2.3)
for all piecewise smooth f 's, and the following error estimate holds
Const \Delta ffl: (2.8)
Proof. Using the fact that K ffl (t) is odd, we have
Z
Z
Z
Applying (2.5) yields
Z
By our assumption in (2.2), F x (t) is BV and it is therefore bounded. Consequently, in the particular
case that the moment bound (2.7) holds, the first term on the right of (2.9) is of order O(ffl), yielding
Const \Delta
Z
In the general case, F x (t) has bounded variation, and the admissibility requirement (2.6) implies
that the first term on the right of (2.9) is of order O(ffl), and we conclude
2.2 Examples of Concentration Kernels
2.2.1 Compactly supported kernels
Our first example consists of concentration kernels which 'concentrate' near the origin, so that (2.7)
holds. We consider a standard mollifier, OE ffl (t) := 1
), based on an even, compactly supported
bump function, OE 2 C 1
We then set
Enhanced detection of edges in spectral data 5
Clearly, K ffl is an odd kernel satisfying the required normalization (2.5)
Z
Z
In addition, its first moment is of order
Z
Z
and hence (2.7) holds. Theorem 2.1 then implies
Corollary 2.1 Consider the odd kernel K ffl
ffl (t), based on even OE 2 C 1
Then K ffl (t) satisfies the concentration property (2.3), and the following error estimate holds
2.2.2 The conjugate Dirichlet kernel
The conjugate Dirichlet kernel,
log N
~
sin kt;
is an example of an oscillatory concentration kernel. Clearly, KN (t) is an odd kernel. Moreover, the
normalization (2.5) holds with ffl - 1
log N ,
log N
log N
Finally, summing
~
sin
cos
sin t;
we find that the first moment of
DN (t)= log N does not exceed
Const \Delta ffl;
log
so that the requirement (2.7) is fulfilled.
Theorem (2.1) then yields the classical result regarding the concentration of conjugate partial
sums, [2, x42],[23, xII Theorem 8.13],
log N
~
log N
We note in passing that in the case of Dirichlet conjugate kernel, KN (t) does not concentrate near
the origin, but instead (2.7) is fulfilled thanks to its uniformly small amplitude of order O(1= log N ).
The error, however, is only of logarithmic order, consult [7, x2].
6 A. Gelb and E. Tadmor
2.2.3 Oscillatory kernels. general concentration factors
To accelerate the unacceptable logarithmically slow rate of Dirichlet conjugate kernel in (2.12), we
consider general form of odd concentration kernels
K oe
based on concentration factors, oe( k
N ) which are yet to be determined. Clearly K oe
N (t) is odd. Next,
for the normalization (2.5) we note that
K oe
Z 1oe(x)
x
dx:
In fact, the above Riemann's sum amounts to the midpoint quadrature, so that for oe(-)
one has Z -
K oe
Z 1oe(-)
and thus (2.5) holds for normalized concentration factors oe(-),
Z 1oe(-)
Consult [7] for further refinement concerning the assumed regularity of oe(\Delta) (We note that oe(\Delta) is
rescaled here with an additional factor of \Gamma- compared to [7]).
Finally, we address the admissibility requirement (2.6) (and in particular (2.7)). To this end, we
proceed along the lines of [7, Assertion 3.3], utilizing the identity (abbreviating -
This leads to the corresponding decomposition of K oe
K oe
sin t:
Here, R oe
N (t) consists of the first four terms on the right hand side of (2.16),
and it is easily verified that each one of these terms has a small first moment satisfying (2.7) (and
consequently, (2.6) holds), i.e.
jtR oe
Const
log N
Enhanced detection of edges in spectral data 7
For example, using the standard bound j sin(kt)=2 sin(t=2)j - minfk; 1=tg, the contribution corresponding
to the first term, I 1 (t), does not exceed
I 1 (t)
Similar estimates hold for the remaining contributions of I 2 ; I 3 and I 4 . In particular, since oe(- is
Finally, the admissibility of the fifth term on the right of (2.16) is due to standard cancelation
which guarantees that (2.6) holds,
sin tOE(t)dt
Const \Delta oe(1)
It is in this context of spectral concentration kernels that admissibility requires the more intricate
property of cancellation of oscillations. Summarizing (2.14), (2.17) and (2.18), we obtain as
a corollary an improved version of the main result in [7, Theorem 3.1] regarding spectral edge detection
using concentration kernels, K oe
N (t). In particular, since K oe
N (t) are N \Gammadegree trigonometric
polynomials, one detects the edges of the piecewise smooth function f(x) directly from its spectral
projection SN (f) :=
K oe
Corollary 2.2 Consider the odd concentration kernel (2.13)
K oe
sin kt; oe(-)
Assume that oe(\Delta) is normalized so that (2.15) holds
Z 1oe(-)
Then K oe
admits the concentration property (2.3), and the following estimate holds
Const \Delta
log N
Remark. One can relax the regularity on the concentration factor oe(\Delta), [7]. Corollary 2.2 is a
generalization of [7, Theorem 3.1] 2 ; in particular, the error estimate (2.19) is valid throughout the
interval, including at the location of the jump discontinuities.
Let us introduce few prototypical examples of concentration factors oe(\Delta) for the detection of edges
from spectral data. In this context we note that other detection methods of discontinuities in periodic
spectral data can be found in the works of Eckhoff [5], [6] and of Mhaskar & Prestin, e.g., [15] and
the references therein. We note that our results apply to the non-periodic expansions as discussed
in x3.2.2 below.
We note the different rescaling here of oe(\Delta) by an additional factor of \Gamma-, compared with the formulation in [7,
Theorem 3.1]
8 A. Gelb and E. Tadmor
1. Trigonometric factors. We consider concentration factors of the form
with the proper normalization Si(ff) :=
R ff(sin j=j)dj. The edge detector introduced originally
by Banerjee & Geer, [1] corresponds to oe -); the general case is found in [7, x3.2].
2. Polynomial factors. As a first example consider oe(-. In this case, K x
corollary 2.2 recovers Fej'er's result, [23, xIII Theorem 9.3], with the following error estimate
Const \Delta log N
This is the first member of a whole family of polynomial concentration factors, e.g., [7, x3.4],
which correlate to concentration kernels satisfying (2.4), (2.5), and (2.6). For odd p's, K oe p
N (f ); for even p's, K oe p
These edge detectors were introduced in [9] and were recently analyzed by
Kvernadze in [13]. Corollary 2.2 yields
i-
Const \Delta
log N
The last error estimate is (essentially) first order. It is sharp. It was noted in [7, x3.4], however,
that oe p 's with higher p's lead to faster convergence rate at selected interior points, bounded
away from the singularities of f . This leads us to the next example of
3. Exponential factors. Polynomial concentration factors (of odd degree) correspond to differentiation
in physical space; trigonometric factors correspond to divided differences in the physical
space - consult the original derivation in [1]. Our main result stated in Corollary 2.2 provides
us with the framework of general concentration kernels which are not necessarily limited to
a realization in the physical space. In particular, we seek concentration factors, oe(\Delta), which
vanish at to any prescribed order,
The higher p is, the more localized the corresponding concentration kernel, K oe
becomes.
Here is why.
Evaluating K oe
N (t) at the equidistant points t
K oe
sin 2-k'
we observe that K oe
coincide with the '-discrete Fourier coefficient of oe(\Delta); since oe(-) and
its first p-derivatives vanish with at both ends, there is a rapid decay of its (discrete)
Fourier coefficients, j-oe ' j - Const:' \Gammap ,
Enhanced detection of edges in spectral data 9
Thus, for t away from the origin, K oe
N (t) is rapidly decaying for large enough N 's. Moreover,
we claim an increasing number of moments of K oe
vanish. To this end we consider the odd
moments of K oe
N (\Delta) (- its even moments vanish, of course). With -
sin kt
\Gamma2
Z -N
Z -N
Integrate by parts - respectively, sum by parts the summation on the right of (2.23). Thanks
to (2.22) the boundary terms vanish and we have
sin(-N-)
d p\Gammaj
d- p\Gammaj
As an example, we consider the exponential concentration factors
oe exp Const
Z
exp
normalized so that
1. Here, the C 1
concentration factor oe exp (-)
vanishes exponentially at both ends, so that (2.22) holds for all p's. Figure 2.3, 2.4
confirms the improved localization of these exponential concentration factors.
4. Band pass filter. Bauer [3] have considered a family of what he termed as 'band pass filter',
supported in the range of middle frequencies, say suppj ae [1=4; 3=4]. We note in passing
that these are special cases of p-order admissible concentration factors, (2.22), although the
normalization used in ([3, eq. (1.35)]),
R
prevented the recovery of the
amplitude of the jumps.
To demonstrate the detection of edges by the concentration factors outlined above, consider the
following two examples of discontinuous f 's (defined on [\Gamma-]):
f a (x) := \Gammasgnx \Delta cos( x
In both cases, f a (x) and f b (x) are recovered from their Fourier coefficients using the Fourier partial
sums SN [f ](x), and we wish to recover their jump discontinuities
[f a
0; else.
0; else:
Figures
2.1 and 2.2 demonstrate the use of trigonometric and polynomial concentration factors
for the detection of edges from Fourier spectral data.
A. Gelb and E. Tadmor
-3.1 -2.1 -1.1 -0.1 0.9 1.9 2.9
x
-2.2
-1.2
-3.1 -2.1 -1.1 -0.1 0.9 1.9 2.9
x
Figure
2.1: Trigonometric concentration factor
for (left) f a (x) where the exact jump
value is where the exact jump values are [f ](\Sigma -
2.
-3.1 -2.1 -1.1 -0.1 0.9 1.9 2.9
x
-2.2
-1.2
-3.1 -2.1 -1.1 -0.1 0.9 1.9 2.9
x
Figure
2.2: Jump value obtained by the polynomial concentration factor oe p=1 (- for (left) f a (x)
and (right) f b (x).
Enhanced detection of edges in spectral data 11
Figure
2.3: Edge detection using the exponential concentration factor
oe exp vs. oe p=1 for S 40 f a (x) and (right) oe exp for SN f a (x) with modes.
-0.50.5s p1
Figure
2.4: Edge detection using the exponential concentration factor
oe exp vs. oe p=1 for S 40 f b (x) and (right) oe exp for SN f b (x) with modes.
As noted in [10, x3.4], polynomial factors of higher degree yields improved results away from
the jump discontinuities. Indeed, the corresponding concentration kernel, K oe p
N (\Delta) have additional
vanishing moments. In the limit, one arrives at exponential factors, K oe exp
Figures 2.3,2.4 demonstrate
the edge detection of these factors in Fourier expansions SN f a (x) and SN f b (x). The improved
localization is evident, due to the faster convergence rate in the smooth parts of f 's. In particular,
the superiority of the exponential factors is illustrated in the figures on the left, when compared
with the first-order accurate polynomial concentration factor, oe p=1 (-. At the same time, Gibbs
A. Gelb and E. Tadmor
oscillations can be noticed in the vicinity of the jump discontinuities.
3 Edge Detection in Non-periodic Projections
Consider a piecewise smooth f(\Delta). To simplify our presentation, we assume f experiences a single
jump discontinuity at The localization property of the appropriate concentration kernel in
the presence of a single jump applies to the case with finitely many jump discontinuities. We begin
with an alternative derivation of our results for the periodic case.
3.1 Revisiting the periodic case
If a 2-periodic f(\Delta) experiences a single jump, [f ](c), then it dictates the Fourier coefficients decay,
To extract information about the location of the jump from the phase of the leading term, we examine
the special concentration kernel, K oe
N with oe(-, where K -
ik
[f ](c) e ik(x\Gammac)
Here we used the concentration property of the Dirichlet kernel localized at
O
The same property applies to the class of concentration factors, oe(-), such that (2.15) holds,
-(
O
It then follows that the corresponding K oe
N in (2.13) is an admissible concentration kernel, so that
K oe
3.2 Non-periodic expansions
3.2.1 General Jacobi expansions
We begin with the Jacobi expansion of a piecewise smooth f(\Delta),
Enhanced detection of edges in spectral data 13
Here are the Jacobi polynomials - the eigenfunctions of the singular Sturm Liouville problem
with corresponding eigenvalues -
Different families of Jacobi polynomials are
associated with different weight functions To simplify the computations,
we assume that the P k 's are normalized so that kP k
As in the periodic case, integration by parts (against (3.28)) shows that a single jump disconti-
nuity, [f ](c), dictates the decay of the Jacobi coefficients,
To extract information about the location of jump, we consider the conjugate sum of the following
-(
\Theta P 0
corresponding to concentration factors oe(-). We shall focus our attention on the particular
case
\Theta P 0
This is the non-periodic analogue of the Fourier concentration kernel K -
with the additional
pre-factor weight of
We want to quantify the localization property of the last summation. To this end we note that
if fP (ff)
k (x)g are the Jacobi polynomials with respect to the weight function ! ff (x), then fP 0
k (x)g are
the Jacobi polynomials w.r.t. the modified weight function ! fi
-orthogonality follows from integration by parts of (3.28) against P (ff)
k . Thus,
The coefficients C k;fi are determined by normalization where by using (3.28) once more we find
and hence we set C
so that fP (fi)
is the orthonormal family w.r.t. ! fi weight. Inserted
into the leading term of (3.31), we end up with a Jacobi kernel associated with weight function
14 A. Gelb and E. Tadmor
\Theta
(c)P (fi)
\Theta
We rewrite this as
\Theta KN (c; x): (3.32)
By virtue of Christoffel-Darboux formula, e.g., [19, Theorem 3.2.2], the kernel KN (c; x) is given by
KN
and it remains to quantify the concentration property of KN (c; x). To this end we use the asymptotic
behavior of P (fi)
N which is stated as 3
denotes the separation between the interior and boundary
regions. Using this to upper bound KN (c; x) in (3.33), we find
\Theta 1
The upper bound on the right is in fact the leading term in the asymptotics of KN (c; x) for large
N 's as long as Similarly, the behavior at
The desired concentration property now follows, similar to the localization of the periodic Dirichlet
kernel DN (3.27). We restrict our attention to interior jumps, so that for
3 The first term on the right pf (3.34) follows from the classical asymptotic formula, e.g., [19, Theorem 12.1.4], which
tells us the behavior of the L 2
-normalized P (fi)
N (x) at the interior
The second term on the right of (3.34) reflects the fact that as x approaches the \Sigma1-boundaries, the L 2
-normalized
approaches to its maximal value e.g., [19, 4.7.3, 4.7.15]
r
Enhanced detection of edges in spectral data 15
large enough, c (3.35), (3.36) and (3.32) yield
\Theta KN (c; x) -
O
\Theta 1
We summarize by stating
Corollary 3.1 Let SN (f) denote the truncated Jacobi expansion (3.27) of a piecewise smooth f ,
associated with a weight function
the concentration property
Const \Delta log N
It is instructive to examine the above discussion for the special case of Chebyshev expansion
corresponding to
dx:
(Observe that except for Chebyshev expansion, the concentration bound (3.37) deteriorates as we
approach the boundaries, depending whether jxj - 1 for ff 1.) The
conjugate sum corresponding to (3.32) reads
k (c)
In this case, we can sum the corresponding Chebyshev kernel: setting
O( 1
[f
3.2.2 Chebyshev expansion
Our discussion above on edge detection in the non-periodic expansions is based on expansion of the
Jacobi coefficients to their leading order in (3.29). More precise information is obtained using the
general framework introduced in the main Theorem 2.1.
Corollary 3.2 Let f(\Delta) be a piecewise smooth function with Chebyshev expansion SN f(x) -
Consider the concentration factors, oe(-), with -(\Delta) normalized so that
A. Gelb and E. Tadmor
Then K oe
admits the concentration property (2.3), and the following estimate holds
-(
Const \Delta log N
Proof. With a piecewise smooth f(x) defined over the interval [\Gamma1; 1] we utilize the usual Chebyshev
transformation We consider the even extension f(cos '); \Gamma-.
Using Theorem 2.1 along the lines of Corollary 2.2, we find that the odd concentration kernel, K oe
recovers the jumps of f(cos '), i.e.,
log N
computation shows the sum on the left equals
-(
and the result follows.
We turn to numerical examples. The following tables summarize our results for the edge detection
in Legendre expansion , corresponding to ff = 0, and in Chebyshev expansion, corresponding to
\Gamma1=2. Scaled to the unit interval [\Gamma1; 1], we consider f a ( x
). The results confirm the
linear convergence rate stated in Corollary 3.1, both away from the jumps - consult Tables 3.1 and
3.2, as well as at the jump itself, Table 3.3.
N Legendre expansion Chebyshev expansion
Table
3.1: Pointwise error estimate j-
away from the jump
discontinuity at
We note that the critical threshold must be very high for to eliminate the artificial
jumps. This indicates that 40 nodes are not enough to resolve the jumps of f b (x) in either the
Chebyshev or Legendre case.
N Legendre expansion Chebyshev expansion
Table
3.2: Pointwise error estimate j-
away from the jump
discontinuities at
Enhanced detection of edges in spectral data 17
Legendre Chebyshev
Table
3.3: Pointwise error estimate j-
[f ](c)j at the point(s) of discontinuity,
It is clear from tables 3.1 and 3.2 that convergence is nonuniform at the boundaries. We have
observed in our numerical experiments, that the edge detector, -
experiences larger
oscillations near the boundaries which do affect the linear convergence rate there. In this context we
note the dependence of the error bounds on the smoothness of f(
The first-order convergence is re-confirmed, in table 3.4 below, when measuring the L 1 -error away
from the jumps discontinuities (and up to the boundaries) .
N Legendre Chebyshev
Table
away from discontinuities.
4 Nonlinear Enhancement
The detection of edges in Theorem 2.1 is based on separation of scales. Thus, consider for example a
piecewise smooth f with finitely many jump discontinuities at . If K ffl is an admissible
concentration kernel, then jK ffl f(x)j !! 1 for x away from these jumps, where as at
ae O(ffl); x
The last statement refers to the asymptotic behavior of the concentration kernel as a function
of the small parameter ffl # 0. In this section we outline a new, nonlinear enhancement procedure,
which is easily implemented to 'pinpoint' finitely many edges in piecewise smooth f 0 s.
To this end we enhance the separated scales in (4.41) by considering
Const
By increasing the exponent p ? 1, we enhance the separation between the vanishing scale at the
points of smoothness (- of order O(ffl p
)), and the growing scale at the jumps (- of order
Next one must introduce a critical threshold which will eliminate all the unacceptable jumps. Only
those edges with amplitudes larger than the critical threshold, [f ](x) ? J 1=p
crit
ffl, will be detected.
A. Gelb and E. Tadmor
Thus crit is a measure which defines the small scale in our computation of edge detection. We
note that data dependent and is typically related to the variation of the smooth part of
f .
Given this critical threshold, we form our enhanced concentration kernel K ffl;J [f
ae
Clearly, with p large enough, one ends up with a sharp edge detector where K ffl;J [f
at all but O(ffl) neighborhoods of the jumps In practical applications, a moderate
enhancement exponent, p - 5 will suffice. We consider two examples.
1. The quadratic filter. Consider the peaked concentration kernel (2.10) K ffl
ffl (t). Then, with
one finds the so called quadratic filter [12],[22], where
2. Enhanced spectral concentration kernels. We apply the procedure of nonlinear enhancement
in conjunction with spectral concentration kernels K
sin kt by
considering the corresponding enhanced spectral concentration kernel
K oe
ae
K oe
The enhanced spectral concentration kernel depends on four ingredients which are at our disposal
ffl The number of modes, N
ffl The enhancement exponent, p
ffl The critical threshold, J
ffl The concentration factor, oe(-).
Figures
4.5 and 4.6 demonstrate the enhancement procedure to the spectral detection of edges
depicted earlier in the corresponding Figures 2.2 and 2.1.
Enhanced detection of edges in spectral data 19
-3.1 -2.1 -1.1 -0.1 0.9 1.9 2.9
x
-2.2
-1.2
-3.1 -2.1 -1.1 -0.1 0.9 1.9 2.9
x
Figure
4.5: Jump value obtained by applying the polynomial concentration factor oe(- with
where the exact jump value is
and (b) f b (x) where the exact jump values are [f ](\Sigma -
2.
-3.1 -2.1 -1.1 -0.1 0.9 1.9 2.9
x
-2.2
-1.2
-3.1 -2.1 -1.1 -0.1 0.9 1.9 2.9
x
Figure
obtained by applying the trigonometric concentration factor oe 1
with modes and enhancement exponent where the exact jump value is
where the exact jump values are [f ](\Sigma -
2.
We conclude with non-periodic examples. In Figures 4.7 we show the detection of a single edge
in f a (x=-) from its Legendre expansion,
f a (x). The detection in Chebyshev expansion is
A. Gelb and E. Tadmor
shown in Figure 4.8 for f b (x=-). In both cases we used an enhancement factor critical
threshold
-0.5x
-0.5x
Figure
4.7: Detection of edges in Legendre expansion of f a (x=-) with exact jump value is [f a
(left) before and (right) after enhancement with
x
x
Figure
4.8: Detection of edges in Chebyshev expansion of f b (x=-) with exact jump value is [f b ](\Sigma
before and (right) after enhancement with
Concluding Remarks
Accurate reconstruction of piecewise smooth functions from their spectral projections is only plausible
when the location (and amplitude) of the underlying jump discontinuities are known, consult
Enhanced detection of edges in spectral data 21
[1],[7],[5],[6],[16],[11] and references therein.
Theorem 2.1, and its corollaries 2.2, and 3.1 provide the general framework for the detection
of edges from spectral data, in both periodic and non-periodic cases. The detection is based on
admissible concentration kernels which include as particular cases classical examples of Fej'er as
well as additional examples in recent literature, [1],[9],[13]. In particular, we introduce here a new
family of exponential concentration kernel, (2.24), with a superior convergence rate away from the
edges. A linear convergence rate is observed near the detected edges. We also introduce a nonlinear
enhancement (4.43) procedure which enables one to "pinpoint" edges with amplitude larger than a
critical threshold.
Recently the edge detection and enhancement method was applied to non-linear conservation
laws, [8], as a post-processing tool to improve the overall convergence rate of the spectral viscosity
solution. Since the edge detection occurs only at the post-processing stage, very little cost is added to
the procedure yet the results are dramatically improved. Future applications, in both one- and several
space dimensions, will also include image processing, where edge detection is needed to de-noise the
contamination by the O(1)-Gibbs' oscillations in the neighborhoods of the undetected edges.
--R
Exponential approximations using Fourier series partial sums
Treatise of
Band filters for determining shock locations
Introduction to the Theory of Fourier's Series and Integrals
Accurate reconstructions of functions of finite regularity from truncated series expansions
On a high order numerical method for functions with singularities
Detection of edges in spectral data
Enhanced spectral viscosity method for nonlinear conservation laws
Determination of the jump of a function of bounded p-variation by its Fourier series
"Progress and Supercomputing in Computational Fluid Dynamics"
On the Gibbs phenomenon and its resolution
Determination of the jump of a bounded function by its Fourier series
On the detection of singularities of a periodic function
The Fourier method for nonsmooth initial data
Multiresolution approximations and wavelets orthonormal bases of L 2 (R)
Convergence of spectral methods for nonlinear conservation laws
Family of spectral filters for discontinuous problems
Asymptotic behavior of quadratic edge filters
Cambridge University Press
--TR
--CTR
R. Pasquetti, On inverse methods for the resolution of the Gibbs phenomenon, Journal of Computational and Applied Mathematics, v.170 n.2, p.303-315, 15 September 2004
Anne Gelb, Parameter Optimization and Reduction of Round Off Error for the Gegenbauer Reconstruction Method, Journal of Scientific Computing, v.20 n.3, p.433-459, June 2004
Rick Archibald , A. Gelb, Reducing the Effects of Noise in Image Reconstruction, Journal of Scientific Computing, v.17 n.1-4, p.167-180, December 2002
Anne Gelb, A Hybrid Approach to Spectral Reconstruction of Piecewise Smooth Functions, Journal of Scientific Computing, v.15 n.3, p.293-322, Sept. 2000
Bernie D. Shizgal , Jae-Hun Jung, Towards the resolution of the Gibbs phenomena, Journal of Computational and Applied Mathematics, v.161 n.1, p.41-65, 1 December
Scott A. Sarra, The spectral signal processing suite, ACM Transactions on Mathematical Software (TOMS), v.29 n.2, p.195-217, June
Scott A. Sarra, Chebyshev super spectral viscosity method for a fluidized bed model, Journal of Computational Physics, v.186 | piecewise smoothness;concentration kernels;spectral expansions |
588637 | Multiplicative Schwarz Algorithms for the Galerkin Boundary Element Method. | We study the multiplicative Schwarz method for the h- and the p-version Galerkin boundary element method for a hypersingular and a weakly singular integral equation of the first kind. For both integral equations we prove that the contraction rate of the multiplicative Schwarz operator is strictly less than 1 for the h-version for the two level and the multilevel methods, whereas for the p-version we show that the contraction rate approaches one only logarithmically in p for the 2-level method. Computational results are presented for both the h-version and the p-version which support our theory. | Introduction
.
We study multiplicative Schwarz methods for the h and p versions of the Galerkin boundary
element method applied to hypersingular and weakly singular integral equations on open or
closed curves. These equations are integral reformulations for boundary value problems with
the Laplace equation and the Dirichlet or Neumann boundary conditions (see [4]). Application
of the Galerkin method to solve these integral equations yields linear systems with dense,
symmetric and positive-definite stiffness matrices. Since the condition numbers of these matrices
grow like h \Gamma1 in the h version and like p 2 in the p version (see [10]), the convergence rate of the
conjugate gradient algorithm approaches 1 (the non-convergent status of the iterative method)
like ch 1=2 as positive constant c. We shall
propose for both versions multiplicative Schwarz algorithms which significantly improve the
rate of convergence of the conjugate gradient method.
We shall analyze for the h-version a 2-level method which corresponds to a multigrid algorithm
using the Gau-Seidel algorithm for smoothing, and a multilevel method which corresponds
to a multigrid algorithm using the Jacobi smoother. We shall prove that both the
Institut f?r Angewandte Mathematik, University of Hannover, Germany
y Institut f?r Angewandte Mathematik, University of Hannover, Germany
z School of Mathematics, University of New South Wales, Sydney, 2052, Australia
2-level and multilevel methods yield an error reduction which is independent of the mesh sizes
and number of levels. This is an improved result compared with [8].
For the p-version we propose a 2-level method which has an error reduction factor approaching
one like
The analysis is based on the abstract framework of [3] (see also [13]) which requires two
main ingredients. The first ingredient is estimates for the extremum eigenvalues of the additive
Schwarz operator which is correspondingly defined from the multiplicative operator. For the
boundary integral operators considered in this paper, these estimates were recently obtained for
both versions [10, 11]. The second ingredient is a strengthened Cauchy-Schwarz inequality. We
will in this paper prove inequalities of this type for both versions for the hypersingular integral
equation, and for the p-version for the weakly singular equation. A strengthened Cauchy-Schwarz
inequality for the h-version of the Galerkin method applied to the weakly singular
integral equation is still an open question to us. It is noted that since the equations considered
in this paper yield dense stiffness matrices, the proofs of these inequalities are much more
complicated than those appear in the finite element method where differential operators are
considered which yield sparse matrices.
In Section 2 we introduce the gereral setting of multiplicative Schwarz methods and recall
the abstract analysis from [3, 13]. We prove in Section 3 strengthened Cauchy-Schwarz inequalities
for the hypersingular equation (Lemmas 3.2 and 3.12) and appropriately apply the result
of Section 2 to obtain estimates for the rates of convergence (Theorems 3.3, 3.9, and 3.13).
Similar treatment for the weakly singular equation is proceeeded in Section 4 (Lemma 4.2 and
Theorem 4.3). In Section 5 we present our numerical results which clearly underline the the-
ory. Some useful lemmas which are used in the proofs of the strengthened Cauchy-Schwarz
inequalities are proved in the Appendix.
For simplicity of notation, the integral equations considered in this paper are defined on the
interval (\Gamma1; 1). A generalization to a polygonal curve is straight forward.
General setting of multiplicative Schwarz methods.
Multiplicative (and additive) Schwarz methods are in general defined via a subspace decomposition
of the space of test and trial functions together with projections onto these subspaces.
More precisely, let
and let projections defined by
Here a(\Delta; \Delta) is a symmetric and positive-definite bilinear form on V . The multiplicative Schwarz
operator is then defined as
where I is the identity map and
is the error propagation operator. We note that
is the corresponding additive Schwarz operator.
By defining
we obtain
which in turn yields
We now present in this section some results which were mainly proved in [3] and [13]. We
include the proofs here for completeness. These results will be used in the analysis of the
following sections.
Lemma 2.1 For any v 2 V there holds
Proof. We have, for
or equivalently
Summing up we obtain the desired result. 2
The analyses in [3] and [13] suggest that we would have to prove a strengthend Cauchy-Schwarz
inequality corresponding to the decomposition (1). The following lemma shows that it
is possible to avoid the coarse grid subspace V 0 in this inequality and use instead a bound for
the maximum eigenvalue of the additive Schwarz operator.
Lemma 2.2 Let \Theta be an whose elements are
defined by
If there exist positive constants C 1 and C 2 such that
a(P AS v; v) - C 2 a(v; v) 8v 2 V; (3)
and that
then there holds for any v 2 V
a(P AS v; v) - C 1
Proof. Using (2) we obtain
implying
Cauchy-Schwarz's inequality implies
!1=2
On the other hand it follows from the definitions of ' ij and the k \Delta k 2 -norm of matrices that
!1=2
Inequalities (4), (5), and (6) yield
This inequality and (3) imply
The lemma is proved. 2
Theorem 2.3 If there exist positive constants C
and a constant C 1 satisfying
then there holds
a -
a is the norm given by the bilinear form a(\Delta; \Delta).
Proof. For any v 2 V we have using Lemma 2.2
Using Lemma 2.1 we deduce
which implies
a -
The theorem is proved. 2
Remark 2.4 If we use the multiplicative Schwarz method as a preconditioner for the CG-
scheme, we have to use the symmetrized version
The condition number is bounded by
a
a
Remark 2.5 C 0 is a lower bound for the minimum eigenvalue and C 2 is an upper bound for
the maximum eigenvalue of the additive Schwarz operator P AS .
Multiplicative Schwarz methods for the hypersingular integral
equation.
We consider the hypersingular integral equation
f:p:
Z
ds
where f.p. denotes a finite part integral in the sense of Hadamard. Let e
\Gamma be an arbitrary closed
curve containing \Gamma. We define, as in [7], the Sobolev spaces
e
loc
and H \Gamma1=2 (\Gamma) being the dual space of e
H 1=2 (\Gamma) with respect to the L 2 inner product on \Gamma. As
was shown in [4], D is continuous and invertible from e
H 1=2 (\Gamma) to H \Gamma1=2 (\Gamma). Moreover, D is
strongly elliptic, i.e., there exists a constant fl ? 0 such that
where h\Delta; \Deltai denotes the L 2 duality on \Gamma. Hence D defines a continuous, symmetric and positive-definite
bilinear form a(v;
3.1 The h-version.
We consider a uniform mesh of size h on \Gamma
and define on this mesh the space V h of continuous piecewise-linear functions on \Gamma which vanish
at the endpoints of \Gamma. We note that V h is a subset of e
(\Gamma). The h-version boundary element
method for Equation (7) reads as:
Find h such that
The stability and convergence of the scheme (9) was proved in [12]. It is known that the
condition number of the matrix system derived from (9) is N 2 . We show in this paper that
the multiplicative Schwarz method yields a preconditioned system which has convergence rate
strictly less than 1.
3.1.1 2-level method.
Let OE h;j , the hat functions forming a basis for V h . We then decompose
where VH is defined as V h with mesh size g. For notational
convenience we identify respectively. The projections
and the operators PMS and P AS are then well defined. Our task
is to verify the assumptions of Theorem 2.3.
The following result was proved in [11].
Lemma 3.1 There exist constants C independent of h such that for any v 2 V h
Let \Theta be an whose elements are defined by
It follows from the Cauchy-Schwarz inequality that 0 1. Therefore, in general there
holds
In view of Theorem 2.3 we will prove in the next lemma that the bound is indeed independent
of N . During the course of the proof, a strengthened Cauchy-Schwarz inequality is proved.
Lemma 3.2 There exists a positive constant C 3 such that
Proof. First we note that
u denotes the derivative of u with respect to the arc-length, and
Z
log
Let
z 2 log jzj for z 6= 0;
Then by using the formula
Z d
c
a
log
we can prove
where
This implies
log 2X
and hence it suffices to prove that
To do so, we define
to obtain
Since
f is a concave function on (1; 1), which results in G(m) ! 0 for m - 2. By the mean value
theorem there exist ' and ' 0 2 (0; 1) such that if
(For the mean value theorem argument used above, the reader is referred to [1, p. 101].) We
note that
Since
we deduce from (12) and (13)
for any m - 6;
which in turn yieldsX
x
x
dx +X
This completes the proof of the lemma. 2
As a consequence we obtain, due to the abstract Theorem 2.3,
Theorem 3.3 Let C 0 , C 2 , and C 3 be given by Lemmas 3.1 and 3.2, and let C g.
Then for any v 2 V h there holds
Remark 3.4 Due to the subspace decomposition (10) we see that this 2-level multiplicative
Schwarz algorithm corresponds to the unsymmetric 2-level multigrid algorithm using the Gau-
Seidel smoother.
3.1.2 Multilevel method.
We shall in this subsection design a multilevel method for the hypersingular equation. The
analysis for this method is slightly different from that of the general framework in Section 2.
The main reason for this difference is the non-availability of a strengthened Cauchy-Schwarz
inequality which can yield an estimate like Lemma 3.2.
Starting with a coarse mesh
we divide each subinterval into two equal intervals. Hence, if h l is the meshstep of N l ,
. For l be the spline space associated with N l
which contains continuous and piecewise-linear functions vanishing at the endpoints \Sigma1. Let
OE l
l
be the nodal basis for V l , where N is the dimension of V l . We then
decompose V l as
l
l
with V l
l . Eventually, V L is decomposed as
l
l
Let
l
for
i is defined for any v 2 V L by
and let
Y
The multilevel multiplicative Schwarz operator is now defined as
Analogously, we define
Letting
we deduce
which in turn yields
l
The following lemma follows easily in the same manner as Lemma 2.1.
Lemma 3.5 For any v 2 V L
The following two lemmas were proved in [11]:
Lemma 3.6 There exists a positive constant C 0 independent of h and L such that for any
there holds
Lemma 3.7 There exist constants such that for any v 2 V k where
there holds
The above lemma plays the role of Lemma 3.2 in the analysis of the multilevel method. We
also need the following technical lemma:
Lemma 3.8 For any there holds
where C 1 is the constant given by Lemma 3.7.
Proof. By using the Cauchy-Schwarz inequality for the symmetric and positive-definite
bilinear form a(T k \Delta; \Delta), and Lemma 3.7 we obtain
which implies
The lemma is proved. 2
Theorem 3.9 Let C 0 , C 1 , and fl be given by Lemmas 3.6 and 3.7. Then for any v 2 V L
where
Proof. The theorem is proved if we can prove that
By Lemma 3.5 it suffices to prove
We have from Lemma 3.6, the Cauchy-Schwarz inequality, the inequality 2ab - a noting
that
a(T l v; v)
It follows successively from (16), the Cauchy-Schwarz inequality,
Lemmas 3.7 and 3.8 that
Inequalities (18) and (19) yield (17) and theorem is proved. 2
Remark 3.10 Due to the subspace decomposition (14) this multilevel multiplicative Schwarz
algorithm corresponds to the unsymmetric multigrid algorithm using the Jacobi-smoother.
3.2 The p-version.
We shall in this subsection design a multiplicative method for the p-version of the Galerkin
boundary element method applied to the hypersingular integral equation.
We define on the mesh (8) the space V p of continuous functions on \Gamma whose restrictions
are polynomials of degree at most p, p - 1. In order to
guarantee that these functions belong to e
H 1=2 (\Gamma), we also require that the functions vanish at
the endpoints \Sigma1 of \Gamma. For the p-version of the Galerkin scheme, we approximate the solution
of (7) by functions in V p and increase the accuracy of the approximation not by reducing h
(which is fixed) but by increasing p. More explicitly, the p-version boundary element method
for Equation (7) reads as:
Find u
such that
The stability and convergence of the scheme (20) was proved in [9]. Note that the dimension of
Choosing a basis for V p , we derive from (20) a system of equations to
be solved for u
p . In practice, we use the following basis. Let OE k , defined
as hat functions satisfying
For we also define L q;j as the affine image onto \Gamma j of
where L q\Gamma1 is the Legendre polynomial of degree q \Gamma 1. We extend L q;j by 0 outside \Gamma j . It is
clear that
is a basis for V p . Solving the equation (20) amounts to solving
where the matrix AN has entries a(v; w) with v; w 2 B. The condition number of (21) grows
at least like p 2 and at most like p 3 . We will define a multiplicative Schwarz method to solve
instead of (21) a preconditioned system which has condition number growing significantly slower
than
We decompose V p as a direct sum
where
the space of continuous piecewise-linear functions on \Gamma vanishing at \Sigma1, and
The space V 0 serves the same purpose as the coarse grid space in the h-version. We note that
functions in V p
are supported in -
With the projections P j appropriately defined as in Section 2, we can define the multiplicative
and additive Schwarz operators as
where
Analogously to Lemma 3.1, we have
Lemma 3.11 [10] There exist constants C independent of p and N such that for any
there holds
Our next task is to show a strengthened Cauchy-Schwarz inequality for this version.
Lemma 3.12 Let V p
i be defined as in (23) and let
Then there exists a constant C independent of i; j; p such that
If 1-i;j-N , then there holds
where
Proof. Let
l=2 v l L l;j . Then we have
Due to Lemma A.6 (compare with [5, Lemma 1]) we have with
From (24) and (25) we obtain
\Theta
\Theta
oe
For
oe
oe
Due to Lemma A.1 we have (note the scaling)@ p
Using (26), (27) and (28) we obtain for
Due to the Cauchy-Schwarz inequality we have independent of p and N .
Therefore, for any there holds
Finally we have
This completes the proof of the lemma. 2
Using the abstract Theorem 2.3 we can prove
Theorem 3.13 Let C 0 , C 2 , and C 3 be given by Lemmas 3.11 and 3.12. If C 1
then for any v 2 V p there holds
Amultiplicative method for the weakly singular integral equation
We shall now in this section design a multiplicative algorithm for the p-version Galerkin method
applied to the weakly integral equation of the form
Z
log ds
As was shown in [4], V is continuous and invertible from e
H \Gamma1=2 (\Gamma) to H 1=2 (\Gamma). Here the Sobolev
space H 1=2 (\Gamma) is defined as the space of functions which are traces of functions in H 1
loc
e
H \Gamma1=2 (\Gamma) is its dual. It is known that there exists a constant fl ? 0 such that
e
where h\Delta; \Deltai denotes the L 2 duality on \Gamma. Hence V defines a continuous, symmetric and positive-definite
bilinear form a(v;
H \Gamma1=2 (\Gamma). We define on the mesh (8)
the space -
of piecewise continuous functions on \Gamma whose restrictions on \Gamma j :=
are polynomials of degree at most p, p - 1. For the p-version of the Galerkin
scheme, we approximate the solution of (29) by functions in -
increase the accuracy of
the approximation not by reducing h (which is fixed) but by increasing p. More explicitly, the
p-version boundary element method for Equation (29) reads as:
Find u
p such that
The stability and convergence of the scheme (30) was proved in [9]. Note that the dimension of
Choosing a basis for -
we derive from (30) a system of equations
to be solved for u
p . In practice, we use the following basis. Let OE k , be defined as
piecewise constant functions satisfying
For we also define L q;j as the affine image onto \Gamma j of the Legendre
polynomial L q of degree q. We extend L q;j by 0 outside \Gamma j . It is clear that
is a basis for -
. Solving the equation (30) amounts to solving
where the matrix AN has entries a(v; w) with v; w 2 B. The condition number of (31) grows at
least like p 2 and at most like p 3 . We will define a multiplicative Schwarz method to solve instead
of (31) a preconditioned system which has condition number growing significantly slower than
We decompose -
as a direct sum
where
the space of piecewise constant functions on \Gamma, and
The space -
serves the same purpose as the coarse grid space in the h-version. We note that
functions in -
are supported in -
With the projections P j defined appropriately as in Section 2 we can define the multiplicative
and additive Schwarz operators as
where
Similarly to Lemma 3.11, we have
Lemma 4.1 [10] There exist constants C 0 and C 2 independent of p and N such that for any
there holds
Again our next task is to prove a strengthened Cauchy-Schwarz inequality.
Lemma 4.2 Let -
i be defined as in (33), and let
there exists a constant C independent of i; j; p such that
Moreover, if
where C 3 := C- 2 =3.
Proof. Let
Due to Lemma A.6 (compare with [5, Lemma 1]) we have with
From (34) and (35) we obtain
l
l
\Theta
\Theta
oe
For
oe
oe
Due to Lemma A.3 we have for (note the scaling)
Using (36), (37) and (38) we obtain for
Due to the Cauchy-Schwarz inequality we have independent of p and N .
Therefore, for any there holds
Finally we have
This completes the proof of the lemma. 2
Using the abstract Theorem 2.3 we can prove
Theorem 4.3 Let C 0 , C 2 , and C 3 be given by Lemmas 4.1 and 4.2. If C 1
then for any v 2 -
there holds
5 Numerical results.
We consider the hypersingular integral equation (7) with the right hand side f(x) j 1. We
solve the Galerkin equations (20) for the p-version by the multiplicative Schwarz algorithm,
for different subspace decompositions and observe the expected behavior of the convergence
rate with respect to N and p (see Tab. 1). For the h-version we note the considerably lower
contraction rates of the 2-level method compared with the multilevel method in Tab. 3. But if
we take into account the high costs of solving a system of mesh width in each iteration
step we see that the multilevel method is the far superior method.
Analogously we consider the weakly singular integral equation (29) with the right hand side
We solve the Galerkin equations (30) for the p-version by the multiplicative Schwarz
algorithm. The numerical results in Tab. 2 show the expected behavior of the convergence rate.
In the case of the h-version we observe that the elements of the Galerkin matrix
x
f:p:
x
ds y ds x =: a i\Gammaj
depend only on the difference for an uniform mesh. Therefore we can reduce the memory
used to store the Galerkin matrix from O(N 2 ) to O(N ). In a more general case we have to use
a clustering or multipole technique to reduce the amount of memory needed. This also reduces
the amount of time for computing the Galerkin matrix. On a vectorcomputer SNI VPP 300/4
we have achieved a performance of 1000 MFlops/s.
In the case of the p-version we calculate the elements of the Galerkin matrix analytically.
Due to the smaller size of the Galerkin matrix we can store the full matrix in the main memory.
We have to note that our subspace decomposition in this case is actually a reordering of the
basis functions. Therefore the projections involved are simplyfied considerably.
Contraction rate
9 0.6245 0.6383 0.6524
Table
1: Hypersingular integral equation, p-version
Contraction rate
9 0.6401 0.6576 0.6745
Table
2: Weakly singular integral equation, p-version
A
Appendix
p. Then there exists a constant C
Contraction rate
2-level multilevel
Table
3: Contraction rates for the hypersingular integral equation, h-version
independent of u and p such that
~
Proof. From the definition of the interpolation norm k \Delta k ~
by the real K-method [2] we
have
~
Z 1'
u=v+w
We have
Therefore it is sufficient to take the infimum in H 1
~
Z 1t \Gamma2
dt: (40)
Due to v 2 H 1
0 (I) we can expand v also in antiderivatives of Legendre polynomials
For the norms in (40) there hold
and, due to Lemma A.2,
Let
l
and l 0
Using the the monotonicity of the square function and the inequality a when
we obtain from (40), (41), and (42)
~
Z 1t \Gamma2
dt
Z 1t \Gamma2 inf
dt
Z 1t \Gamma2 inf
dt
Z 1t \Gamma2 inf
(v k )2l /
dt
Z 1t \Gamma2 inf
(v k )2l 0/
dt
Z 1t \Gamma2
dt: (43)
Note that if
a+b . Therefore we obtain
from equation (43)
~
Z 1t \Gamma2
dt
Z 1t \Gamma2
dt
Z 1Ck \Gamma5 2
dt
-s
This completes the proof. 2
. Then there holds
Proof. With the normalized antiderivatives of Legendre polynomials defined by L
there holds due to the proof of [6, Lemma 3.1]
Due to
there holds
This completes the proof. 2
Lemma A.3 Let
(x=h). Then there exists a constant C independent of u,
h and p such that
~
Proof. Due to ~
kuk ~
For
there holds
kuk ~
Due to Lemma A.4 we have
This completes the the proof. 2
(x=h). Then there exists a constant C independent of v, h
and p such that
Proof. Let 1). From the definition of the interpolation norm k \Delta k H 1=2 (\Gammah;h) by the
real K-method [2] we have
Z 1'
v=w+a
Z 1i
Z 1t \Gamma2 inf w
Z 1t \Gamma2 inf w
Z 1t \Gamma2 inf w
Z 1t \Gamma2 inf w
By the orthogonality property of the Legendre polynomials we have
On the other hand
It follows from the definition of matrix norms that
Lemma A.5 and a straightforward calculation give
2: (50)
Inequalities (48), (49), and (50) yield
This inequality and (47) imply
Z 1t \Gamma2 inf w k
dt
Z 1t \Gamma2 inf w k
dt
Z 1t \Gamma2
Lemma A.5 Let k; m - 1. Then we have
Proof. From the recurrence formula of the Legendre polynomials we have
which results in
and
be the Legendre polynomial of degree p i linearly transformed onto the
open straight line I ae IR 2 . L p i ;I is supposed to be continued by 0 outside I on the entire line
containing I where necessary. Let J ae IR 2 be another open straight line with -
there exists a constant C such that
Proof. Let x 0 be the midpoint of I and y 0 be the midpoint of J . Then there holds using the
Taylor expansion of log jx \Gamma yj
log
y log jx
and
y log
with 1). Due to the orthogonal properties of the Legendre polynomials the first sum
vanishes and we have
Z
I
Z
ds y ds x
Z
I
Z
ds y ds x : (52)
Applying the Cauchy-Schwarz inequality two times we obtain
'Z
I
ds x
ds y
I
Z
ds y ds x
'Z
I
Z
ds y ds x
sup
Since
y log
there holds
sup
(jI
(jI
Inequalities (53) and (54) imply
This completes the proof. 2
--R
Mathematical Analysis: A Modern Approach to Advanced Calculus
Convergence Estimates for Product Iterative Methods with Applications to Domain Decomposition
Boundary Integral Operators on Lipschitz Domains: Elementary Results
Efficient Algorithms for the p-Version of the Boundary Element Method
A Multilevel Additive Schwarz Method for the h-p Version of the Galerkin Boundary Element Method
Multigrid Solvers and Preconditioners for First Kind Integral Equations
On the Convergence of the p-Version of the Boundary Element Galerkin Method
Additive Schwarz Algorithms for the p-Version Boundary Element Method
Additive Schwarz Methods for the h-Version Boundary Element Method
A Hypersingular Boundary Integral Method for Two-Dimensional Screen and Crack Problems
Iterative Methods by Space Decomposition and Subspace Correction
--TR
--CTR
Matthias Maischak, Multiplicative Schwarz algorithms for the p-version Galerkin boundary element method in 3D, Applied Numerical Mathematics, v.56 n.10, p.1370-1382, October 2006
T. Tran , E. P. Stephan, Two-level additive Schwarz preconditioners for the h-p version of the Galerkin boundary element method for 2-d problems, Computing, v.67 n.1, p.57-82, July 2001 | p-version Galerkin boundary element method;h-version Galerkin boundary element method;multiplicative Schwarz;multigrid algorithm |
588642 | Smoothing Methods and Semismooth Methods for Nondifferentiable Operator Equations. | We consider superlinearly convergent analogues of Newton methods for nondifferentiable operator equations in function spaces. The superlinear convergence analysis of semismooth methods for nondifferentiable equations described by a locally Lipschitzian operator in Rn is based on Rademacher's theorem which does not hold in function spaces. We introduce a concept of slant differentiability and use it to study superlinear convergence of smoothing methods and semismooth methods in a unified framework. We show that a function is slantly differentiable at a point if and only if it is Lipschitz continuous at that point. An application to the Dirichlet problems for a simple class of nonsmooth elliptic partial differential equations is discussed. | Introduction
. This paper considers the nonlinear operator equation
Y is a continuous mapping, X and Y are Banach spaces, and D
is an open domain in X . In a number of problems, the operator F is nondifferentiable.
For example, a class of such problems arising in optimal control problems for parabolic
partial differential equations with bound constraints on the control [16, 17]:
where K is a completely continuous map from L 1 to C
for some bounded
is the map on C given by
for given l and u in C : A paradigm for such problems is the Urysohn integral
equation of the second kind. Another class of nonsmooth equations related to MHD
(magnetohydrodynamics) equilibria [18, 37] will be discussed in Section 4. A paradigm
for this class is the Dirichlet problem for nonsmooth elliptic partial differential equations
as discussed in Section 4.
The nonsmoothness poses serious difficulties and challenges for devising for non-smooth
problems analogues of existing iterative methods, which use smoothness. For
Department of Mathematics and Computer Science, Shimane University, Matsue 690-8504, Japan
(chen@math.shimane.ac.jp). The work of this author was supported in part by the Japan Society
of the Promotion of Science Grant C11640119.
y Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, U.S.A.
(nashed@math.udel.edu). The research of this author was partially supported by a grant from the
National Science Foundation.
z Department of Applied Mathematics, Hong Kong Polytechnic University, Kowloon, Hong Kong.
(maqilq@polyu.edu.hk). School of Mathematics, University of New South Wales, Sydney 2052.
(L.Qi@unsw.edu.au). The work of this author was partially supported by the Australian Research
Council.
Nashed and L.Qi
example, the Newton method assumes that F is Fr'echet differentiable and is defined
by
where F 0 is the Fr'echet derivative of F . What are suitable analogues of Newton's
method when F is not smooth ?
Iterative methods for nondifferentiable equations have been studied for decades
[7, 10, 15, 16, 31, 34, 38, 40]. Among these methods, smoothing methods and semi-
smooth methods for nondifferentiable equations arising from variational inequalities
and complementarity problems in R n have been studied extensively in the last few
years [2, 4, 5, 9, 11, 14, 34].
Superlinear convergence analysis of semismooth Newton methods for equations
defining a locally Lipschitian operator in R n uses the notions of generalized Jacobian
[12] and semismoothness [32, 35], which are based on the Rademacher theorem. The
Rademacher theorem states that if F : R n locally Lipschitzian function,
then F is differentiable almost everywhere. For a locally Lipschitzian function F :
defined the generalized Jacobian [12] by
where DF is the set of points where F is differentiable. For nonsmooth equations
described by a locally Lipschitzian operator F from R n into R n , the generalized
Newton method is defined by
Qi and Sun [32, 35] established superlinear convergence of (1.3) by using a concept
of semismoothness. The concept of semismoothness was introduced by Mifflin for
real-valued functions [20]. In [35], F is said to be semismooth at x if the limit
lim
exists for any h 2 R n . Local behavior of the generalized Newton method is analyzed
in [13, 31, 32, 35].
The Rademacher theorem does not hold in function spaces. Hence the above
definitions of generalized Jacobian and semismoothness cannot be used in function
spaces.
In this paper, we introduce notions of slanting functions and slant differentiability
of operators in Banach spaces, and use these notions to formulate a concept of semi-
smoothness in infinite dimensional spaces, which coincides with the above notion of
semismoothness in the case of a locally Lipschitzian mapping on R n . These notions
will play a pivotal role in the formulation and convergence analysis of analogues of
Newton's method (smoothing and semismooth methods) for nondifferentiable operator
equations in function spaces.
The main feature of smoothing Newton methods in R n is to approximate F by
a parametric function f(x; ffl) : R n \Theta R++ ! R n which is continuously differentiable
with respect to x, and then to use the partial derivative f x at each step of the
Newton-like iteration. The error of f(\Delta; ffl k ) to F is bounded by
Smoothing Methods and Semismooth Methods 3
1. For complementarity problems, many smoothing
functions have the following properties [9]
exists for every x 2 R n
and
lim
khk
0:
The properties (1.4) and (1.5) suggest a superlinearly convergent Newton method
[6, 11]:
Note that f n\Thetan is a single valued function, and the superlinear convergence
of (1.6) is not based on the Rademacher theorem. This opens a way to study Newton
methods for nonsmooth problems in function spaces.
The organization of the paper is as follows. In Section 2 we introduce the notion
of a slanting function f o and slant differentiability for a general nonsmooth function
F in Banach spaces, and study some of their interesting properties. Using slanting
functions, we extend in Section 3 the semismooth Newton method and the smoothing
Newton method to Banach spaces. An application to a class of nonsmooth Dirichlet
problems is studied in Section 4.
We use ff; to denote scalars. The set of all positive real numbers is
denoted by R++ . Let L(X; Y ) denote the set of all bounded linear operators on X
into Y .
2. Slant differentiability. A function F : D ae X ! Y is said to be (one-
sided) directionally differentiable at x if the limit
exists, in which case is called the (one-sided) directional derivative of F at
x with respect to the direction h. For brevity we will drop "one-sided" in what follows
since this is the only notion of directional derivative that occurs in this paper.
A function F : D ae X ! Y is said to be B-differentiable at a point x if it is
directionally differentiable at x, and
lim
khk
0:
In this case, we call \Delta) the B-derivative of F at x. See [39] for the B-
differentiability (B for Bouligand).
In finite dimensional Euclidean spaces, Shapiro [41] showed that a locally Lipschitzian
function F is B-differentiable at x if and only if it is directionally differentiable
at x. Moreover, Qi and Sun [35] showed that F is semismooth at x if and only if F is
B-differentiable (hence directionally differentiable) at x and for each V 2 @F
4 X.Chen, Z. Nashed and L.Qi
However, these results do not hold in function spaces since the generalized Jacobian
is defined only in finite dimensional spaces. To circumvent this difficulty in infinite
dimensional spaces we introduce the following notion of slant differentiability.
Definition 2.1. A function F : D ae X ! Y is said to be slantly differentiable
at x 2 D if there exists a mapping f such that the family
of bounded linear operators is uniformly bounded in the operator norm
for sufficiently small and
lim
The function f o is called a slanting function for F at x.
Definition 2.2. A function F : D ae X ! Y is said to be slantly differentiable
in an open domain D 0 ae D if there exists a mapping f
f o is a slanting function for F at every point x 2 D 0 . In this case, f o is called a
slanting function for F in D 0 .
Definition 2.3. Suppose that f slanting function for F at
We call the set
xk!x
the slant derivative of F associated with f o at x 2 D. Here, lim xk!x f
the limit of f sequence fx k g ae D such that x k ! x and lim xk!x f
exists, and @S F (x) is the set of all such limits. (Note that f
is always nonempty.)
Slant differentiability captures a property that appears implicitly in some convergence
proofs of Newton-type methods for solving nonsmooth equations as well as
ill-posed smooth equations. For example, consider the parameterized Newton method
for solving ill-posed smooth equations. To overcome ill-posedness and singularity, we
use
is chosen such that F 0 I is nonsingular. Let f
and assume slanting function for F at x if F 0 (x) is
uniformly bounded in a neighborhood of x .
We now make a few comments on some unusual properties of slanting functions
which also explain the choice of the terms "slanting function" and "slant derivative".
Remarks
1. Unlike other notions of derivatives, the term "f does not appear in Definition
2.1, so for a slanting function f for F at x, f itself is not characterized
in general by a limit of a quotient or a sequence.
2. A function F may be slantly differentiable at every point of D, but there is
no common slanting function of F at all points of D. For example, if F is
Fr'echet differentiable at x, we take f
slanting function for F at x. But f o in general is not a slanting function of
F at other points of D. If F is continuously differentiable in D and we take
slanting function for F at every
point of D.
Smoothing Methods and Semismooth Methods 5
3. A slanting function f for F at x is a single valued function. A slantly differentiable
function F at x can have infinitely many slanting functions at x.
Even if F is continuously differentiable in D, F still can have infinitely many
slanting functions for all points of D. For example, we may let f o take the
same values of F 0 except at a finite number of points of D, and take arbitrary
values at these finite number of points. Then such f o is still a slanting function
of F for all points of D. One may conjecture that if F is continuously
differentiable in D and f o is a slanting function for F in D, then f
with F 0 except possibly on a set of measure zero.
4. If f are both slanting functions for F at x (in D), then
is also a slanting function for F at x (in D), where - 2 [0; 1]: Moreover,
lim
kf
On the other hand, if f are slanting functions for F and G at x (in
D), respectively, then h slanting function for ffF
at x (in D) where ff and fi are constants. Note that such a result for linear
combination does not hold for the generalized Jacobian [12].
5. f o is not continuous in general. For example, let
be a real number. Then the function
is a slanting function for F in X . The slant derivative of F for x 2 X is
In fact it is easy to see that if f o is continuous at x, then F is differentiable
at x and F 0 (x). The slant derivative of F associated with f o at x
reduces to a singleton @S F
6. For a locally Lipschitzian function F : R n is semismooth at x, then
any single-valued selection of the Clarke Jacobian or the B-subdifferential is
a slanting function of F at x. This may not be true if F is not semismooth
at x. For example, let
ae
The derivative F 0 (x) is discontinuous at 0. The function F is slantly differentiable
at Indeed let f o be any function for which lim h!0 f
Then
lim
0:
Hence F is slantly differentiable at 0 with infinitely many slanting functions
for F at 0. Note that such f o is not a slanting function for F at every point
f0g. If we let f slanting function for F at every
point but not a slanting function for F at 0.
6 X.Chen, Z. Nashed and L.Qi
7. For a slantly differentiable function F at x, the set @S F (x) is dependent on
the choice of a slanting function for F at x. Associated with any slanting
function, the set @S F (x) is bounded, since f uniformly bounded
for h sufficiently small. For example, let
ae
Let
ae
sin 1
slanting function for F at 0 and @S F
that F is neither directionally differentiable at 0 nor Lipschitzian in any
neighbourhood of 0. Note that the function f o in this example is not slantly
differentiable at 0.
8. A continuous function is not necessarily slantly differentiable. For example, let
ae p
jhj, and 1=
there is no
uniformly bounded function f o such that F
Definition 2.4. A function F is said to be Lipschitz continuous at
x if there is a positive constant L such that for all sufficiently small h,
We now present a necessary and sufficient condition for the slant differentiability,
for the proof we need the following lemma, which is a corollary of the Hahn-Banach
theorem.
Lemma 2.5. Let X be a normed space and h be a fixed element of X, h 6= 0.
Then there exists an element g 2 X , where X is the (norm) dual of X, such that
(Note by definition of X , g is a continuous linear functional on X, so it is bounded.)
Theorem 2.6. An operator slantly differentiable at x if and only
if F is Lipschitz continuous at x.
Proof. Suppose that F is slantly differentiable at x. By the definition of slant
differentiability, there are C ? 0 and ffi ? 0 such that for all khk - ffi, kf
and
khk
Hence, for all khk - ffi,
Smoothing Methods and Semismooth Methods 7
Conversely, suppose that F is Lipschitz continuous at x. We shall show that F
is slantly differentiable at x by constructing a slanting function for F at x. For each
fixed h 6= 0, by Lemma 2.5, there exists a continuous linear functional g h 2 X such
that g h fixed as above, define the following function
on an open domain containing x
khk
for h 6= 0, and define f to be any bounded linear operator on X into Y . Then f
maps D into L(X; Y ) since each g h is in X . For any z 2 X ,
khk
Thus
kf
khk
khk kzk:
Therefore
sup
z 6=0
kf
kzk
khk
that is
kf
Thus for sufficiently small h,
kf
Now using (2.2), and g h
khk
If X is a Hilbert space, then by the Riesz representation theorem every continuous
linear functional on X can be represented by an inner product. Thus the formula (2.2)
can be written in the form:
khk
Corollary 2.7. (Mean Value Theorem for Slantly Differentiable Functions)
Suppose that F : D ae X ! Y is slantly differentiable at x. Then for any h 6= 0 such
that x + h is in D, there exists a slanting function for F at x such that
8 X.Chen, Z. Nashed and L.Qi
Proof. This follows from the first part of Theorem 2.6 and the proof of the second
part of the same theorem.
Note that the above form of the mean value theorem is in equality form. It is a
selection theorem from the set of slanting functions for F at x. Mean value theorems
for smooth operators whose range is an infinite dimensional space are usually given in
the form of inequalities involving norms or majorants, or an inclusion form involving
the closed convex hull of the set of values of the derivative. For a comprehensive
overview of various types of mean value theorems for smooth operators, see pp. 171-
186 of [23].
Proposition 2.8. Suppose that F is slantly differentiable at x, and let f o be a
slanting function for F at x.
(a) F is directionally differentiable at x if and only if
lim
exists. If F is directionally differentiable at x, then
(b) F has a B-derivative at x if and only if
lim
exists uniformly with respect to h on each bounded set (say on
Proof. (a) Let h 2 X with
lim
is equivalent to
lim
Hence if F is directionally differentiable, then
The converse is also true.
(b) This follows from part (a) and the known (and easy to prove) fact that F has
a B-derivative at x if and only if
lim
exists uniformly with respect to h on each bounded set (see, for example, Nashed [23],
where a hierarchy of notions of differentiability are characterized by convergence of
"remainder" quotients R(th)=t as t approaches zero, uniformly with respect to h in
various classes of subsets).
Theorem 2.9. Suppose F is slantly differentiable at x and let f o be a slanting
function for F at x. Then the following statements are equivalent.
Smoothing Methods and Semismooth Methods 9
(a) For some function which is o(khk), f positively
homogeneous of degree 1 in h.
(b) lim t!0 exists for every h 2 X and
lim
khk
0:
(c) F is B-differentiable at x, and
Proof. (a)) (b): If f positively homogeneous of degree 1 in h,
then for any fixed t ? 0,
so
Note that only if for each fixed h, uniformly in
h on each bounded set. Hence for any h 2 X
lim
uniformly with respect to h on each bounded set. Moreover,
lim
khk
khk
0:
(b) ) (c): By part (b) of Proposition 2.1, statement (b) implies that F is B-
differentiable and lim t!0 statement (c) holds.
(c) positively
homogeneous of degree 1 in h, we have (a).
Proposition 2.10. Suppose that F is slantly differentiable in a neighborhood of
x and let f o be a slanting function for F in the neighborhood of x. Then the following
two statements are equivalent.
(a) There are a neighborhood N x of x and a positive constant C such that for any
(b) There are a neighborhood -
N x and a positive constant -
C such that for any u 2 -
every V 2 @S F (u) is nonsingular and kV
C:
Proof. Part (a) =) part (b): It is straightforward from the definition of @S F (u).
Part (b) =) part (a): It is due to the fact that f
Proposition 2.11. Suppose that F is slantly differentiable at x and let f o be
a slanting function for F at x. If there are a neighborhood N x of x and a positive
constant C such that for any u 2 N x , f o (u) is nonsingular and kf
then there is a positive constant -
C such that every V 2 @S F (x) is nonsingular and
C: Moreover, if Y is a finite dimensional space, the converse holds.
Proof. The first part follows the definition of @S F (x). The second part is due to
the fact that every bounded sequence has convergent subsequence in finite dimensional
spaces. Indeed, if the second part is not true, then there is a sequence fu k g such that
are singular or kf 1. By the definition of
@S F (x), there is a subsequence fu k j g ae fu k g such that f
is singular. This contradicts the assumption.
Nashed and L.Qi
3. Smoothing functions and semismooth functions. We generalize the definition
of smoothing functions for a nonsmooth function and the concept of semism-
mothness of a nonsmooth function in finite dimensional spaces to infinite dimensional
spaces.
Definition 3.1. We say that Y is a smoothing function of F
if f is continuously differentiable with respect to x and for any x 2 D and any ffl ? 0,
where - is a positive constant.
The smoothing function f is said to satisfy the slant derivative consistency
property at - x (in D) if
lim
exists for x in a neighborhood of -
x (in D) and f serves as a slanting function for F
at -
x (in D). Note that the limit in (3.2) is in the topology of the operator norm, so
the pointwise convergence of f x (x; ffl)h to f o (x)h for each fixed h is uniform on the set
Definition 3.2. We say that F is semismooth at x if there is a slanting
function f in a neighborhood N x of x, such that f o and the associated slant
derivative satisfy the following two conditions.
(a) lim t!0 exists for every h 2 X and
lim
(b)
Theorem 3.3. Suppose that F is slantly differentiable in a neighborhood N x of
x, and let f o be a slanting function for F in N x : Then F is semismooth at x if and
only if F is B-differentiable at x and
where @S F is the slant derivative associated with f o in N x .
Proof. Suppose that F is semismooth at x. Then from Theorem 2.9, part (a) of
Definition 3.2 implies that F is B-differentiable and
Thus part (b) of Definition 3.2 implies (3.3).
Now, we suppose that F is B-differentiable at x and (3.3) holds. Then for all
Smoothing Methods and Semismooth Methods 11
Hence part (b) of Definition 3.2 holds. Moreover, we have
By Theorem 2.9, part (a) of Definition 3.2 holds and so F is semismooth at x.
Theorem 3.3 implies that the definition of semismoothness used here coincides
with the definition of Qi-Sun [35] in finite dimensional spaces if we take a single-valued
selection of the Clarke Jacobian or the B-subdifferential as the slanting function.
To illustrate Theorem 3.3, we consider the system of "min" nonsmooth equations
in R n
where p and q are continuously differentiable functions from R n into itself. This
system is equivalent to the complementarity problem
Chen, Qi and Sun showed [9] that every smoothing function f(x; ffl) in the Chen-
Mangasarian smoothing function family [5] for the nonsmooth function F satisfies
(3.2). In particular, for
where ff 2 [0; 1] is dependent on the choice of a smoothing function. Such f
belongs to the set
at every point x 2 R n . (See [33] for @C F (x):) Hence, every smoothing function in the
Chen-Mangasarian smoothing function family satisfies the slant derivative consistency
property in R n . Moreover, the associated slant derivative is, for
which is bounded, nonempty and satisfies
Furthermore, the following fact is known [35]
On the other hand, by Proposition 2.8, we know
Hence, the nonsmooth function F is semismooth in the sense of Definition 3.2.
Now we consider superlinearly convergent Newton-type methods for nonsmooth
equations with slanting differentiable operators.
Nashed and L.Qi
Theorem 3.4. Suppose that F is slantly differentiable at a solution x of (1.1).
Let f o be a slanting function for F at x and kf in a neighborhood N
of x , where M is a positive constant. Then the iterative sequence fx k g generated by
the Newton-type method
superlinearly converges to x in a neighborhood N 0 of x . Here A(x) 2 L(X; Y ) and
Proof. By Definition 2.1 and the Banach Lemma [29], there is a neighborhood N 0
of x , N 0 ae N , and positive constants M
for any x 2 N 0 , A(x) is nonsingular and kA(x)
and
Therefore, starting from any x 0 2 N 0 the Newton method (3.4) is well defined and
the successive iterates satisfy the following inequalities:
Hence the sequence fx k g converges to x : Moreover, using Definition 2.1 and (3.5),
the inequalities above imply
Using Theorem 3.4 and Proposition 2.10, we can immediately obtain the following
theorem.
Theorem 3.5. Suppose that F is slantly differentiable at a solution x of (1.1).
Let f o be a slanting function for F at x and kf in a neighborhood N
of x , where M is a positive constant. Then the following statements hold.
(a) The Newton-type method (1.6) superlinearly converges to x in a neighborhood
N 0 of x .
(b) If f : D \Theta R++ ! Y is a smoothing function of F which satisfies the slant derivative
consistency property (3.2) in N , then the smoothing Newton method
superlinearly converges to x in a neighborhood N 0 of x .
(c) If F is semismooth at x , then the semismooth Newton method
superlinearly converges to x in a neighborhood N 0 of x .
Smoothing Methods and Semismooth Methods 13
4. An application to a class of nonsmooth elliptic partial differential
equations.
Let\Omega ae R 2 be a bounded region with piecewise smooth boundary
@\Omega
and let W be the class of functions
from\Omega to R satisfying
Z
R) be the space of functions in W endowed with the norm
Z
\Omega ju(x)jdx:
We consider the following nonsmooth Dirichlet problem
ae
in\Omega
R is a continuous function.
Let
Z
\Omega G(x; y)P (u(y))dy;
where G is the Green function for the boundary value problem (e.g. see [42])
ae
in\Omega
is the Dirac "generalized function" at y in \Omega\Gamma
The nonsmooth integral equation
is equivalent to the nonsmooth Dirichlet problem (4.1).
Theorem 4.1. Suppose that smoothing function of P
satisfying
where - is a positive constant. Then
Z\Omega
is a smoothing function of F and
where
x2\Omega Z\Omega
14 X.Chen, Z. Nashed and L.Qi
Proof. It is easy to see that f is continuously differentiable with respect to u and
Z
Moreover
Z
Z
Z
\Omega kG(x; y)kdy
-ffl:
Theorem 4.2. Suppose that P is slantly differentiable at u, and let p o be a
slanting function for P at u. Then F is slantly differentiable at u, and
Z
\Omega G(x; y)p
is a slanting function for F at u
Proof. Using the definition of f o as given in the statement of the theorem, we
have
'Z
Z\Omega
Z\Omega
'Z\Omega
slanting function for P at u, the above equality implies that f o is a
slanting function for F at u.
By properties of integrals, we have the following proposition.
Proposition 4.3. (1) If smoothing function of P which
satisfies the slant derivative consistency property (3.2) at u, then
is a smoothing function of F which satisfies the slant derivative consistency property
(3.2) at u
(2) If P is semismooth at u, then F is semismooth at u.
The above results demonstrate that superlinearly convergent smooth methods or
semismooth methods can be developed for the nonsmooth Dirichlet problem (4.1).
For example, we consider
that is
ae
where - and ff are constants. Let
Smoothing Methods and Semismooth Methods 15
Then p is a smoothing function of P in X , and satisfies the slant derivative consistency
property in X since
lim
ffl!0
ffl!0
otherwise:
We next show that p o is a slant function for P in X . In fact, for any u; h(6=
we have
Z
-f
Z
Z
u(x)+h(x)=ff
Note that
Letting khk ! 0, we have
khk
f
Z
Z
u(x)+h(x)=ff
dx
0:
Moreover, we can show that P is semismooth in X . Hence, by using
Z\Omega
we can obtain superlinearly convergent smoothing methods and semismooth methods
for the nonsmooth Dirichlet problem
ae
in\Omega
This nonsmooth Dirichlet problem is related to MHD (magnetohydrodynamics) equilibria
[18].
We report numerical results for the following example.
Example 4.1.
ae
where OE(0; problem
has an exact solution
ae
Z. Nashed and L.Qi
Table
Numerical result of Example 4.1: kFn
We use method (1.6) with the five-point finite difference method. The stopping
criterion is kFn Here Fn is the finite difference approximation function
with grids n. We report the value of kFn (x)k 1 at the last five iterations.
Nonsmooth optimization and operator equations involving nonsmooth operators
are becoming crucial in various areas of computational and applied mathematics,
for example in nonsmooth mechanics [21, 22, 30], optimal design of electromagnetic
devices [21], ill-posed problems involving nonsmooth operators and variational inequalities
[19, 24], bounded variation regularization and nondifferentiable optimization
problems in nonreflexive spaces [26, 27], inverse source problems [25], free boundary
problems [28], multi-body system identification [1], and nonlinear complementary
problems (see [8] and references cited therein). Various classes of these problems can
be reformulated as nonsmooth equations with locally Lipschitzian operators. Hence
the smoothing methods and semismooth methods studied in this paper can be applied
to these problems.
Acknowledgement
. This research was initiated while the first author worked in
the School of Applied Mathematics at the University of New South Wales where the
second author visited in 1993 and 1998. We would like to express our appreciation
for the support.
--R
Outer inverses and multi-body system identification
The global linear convergence of a non-interior path-following algorithm for linear complementarity problem
Smooth approximations to nonlinear complementarity problems
A class of smoothing functions for nonlinear and mixed complementarity problems
Superlinear convergence of smoothing quasi-Newton methods for nonsmooth equa- tions
Global and superlinear convergence of inexact Uzawa methods for saddle point problems with nondifferentiable mappings
Convergence of Newton's method for singular smooth and nonsmooth equations using adaptive outer inverses
Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities
Convergence domains of certain iterative methods for solving nonlinear equations
On homotopy-smoothing methods for variational inequalities
Optimization and Nonsmooth Analysis
Solution of monotone complementarity problems with locally Lipschitzian functions
Identification of the support of nonsmoothness
Multilevel algorithms for constrained compact fixed point prob- lems
Finite element analysis of a nondifferentiable nonlinear problem related to MHD equilibria
Regularization of nonlinear ill-posed variational inequalities and convergence rates
Semismooth and semiconvex functions in constrained optimization
Topics in Nonsmooth Mechanics
Mathematical Theory of Hemivariational Inequalities and Applications
Differentiability and related properties of nonlinear operators: some aspects of the role of differentials in nonlinear function analysis
On nonlinear ill-posed problems II: Monotone operator equations and monotone variational inequalities
Stable approximation of nondifferentiable optimization problems with variational inequalities
Stable approximation of a minimal surface problem with variational inequalities
Least squares and bounded variation regularization with non-differentiable functionals
Iterative Solution of Nonlinear Equations in Several Variables
Inequality Problems in Mechanics and Applications
motivation and algorithms
Convergence analysis of some algorithms for solving nonsmooth equations
A globally convergent successive approximation method for severely non-smooth equations
A nonsmooth version of Newton's method
Computational Solution of Nonlinear Operator Equations
Approximation of a nondifferentiable nonlinear problem related to MHD equilibria
A unified convergence theory for a class of iterative process
Local structure of feasible sets in nonlinear programming.
Newton's method for a class of nonsmooth functions
On concepts of directional differentiability
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B. Mermri , X. Chen, On characterizations and regularity of the solution of bilateral obstacle problems, Journal of Computational and Applied Mathematics, v.152 n.1-2, p.333-345, 1 March
Xiaojun Chen, Applications of smoothing methods in numerical analysis and optimization, Focus on computational neurobiology, Nova Science Publishers, Inc., Commack, NY, 2004 | nonsmooth elliptic partial differential equations;superlinear convergence;semismooth methods;nondifferentiable operator equation;smoothing methods |
588815 | Perfect Packing Theorems and the Average-Case Behavior of Optimal and Online Bin Packing. | We consider the one-dimensional bin packing problem under the discrete uniform distributions $U\{j,k\}$, $1 \leq j \leq k-1$, in which the bin capacity is $k$ and item sizes are chosen uniformly from the set $\{1,2,\ldots,j\}$. Note that for $0 < this is a discrete version of the previously studied continuous uniform distribution $U(0,u]$, where the bin capacity is 1 and item sizes are chosen uniformly from the interval $(0,u]$. We show that the average-case performance of heuristics can differ substantially between the two types of distributions. In particular, there is an online algorithm that has constant expected wasted space under $U\{j,k\}$ for any $j,k$ with $1 \leq j < k-1$, whereas no online algorithm can have $o(n^{1/2})$ expected waste under $U(0,u]$ for any $0 < u \leq 1$. Our $U\{j,k\}$ result is an application of a general theorem of Courcoubetis and Weber that covers all discrete distributions. Under each such distribution, the optimal expected waste for a random list of $n$ items must be either $\Theta (n)$, $\Theta (n^{1/2} )$, or $O(1)$, depending on whether certain ``perfect'' packings exist. The perfect packing theorem needed for the $U\{j,k\}$ distributions is an intriguing result of independent combinatorial interest, and its proof is a cornerstone of the paper. We also survey other recent results comparing the behavior of heuristics under discrete and continuous uniform distributions. | Introduction
. Suppose one is given items of sizes 1, 2, 3, . , j, one of each
size, and is asked to pack them into bins of capacity k with as little wasted space as
possible, i.e., one is asked to find a least cardinality partition (packing) of the set of
items such that the sizes of the items in each block (bin) sum to at most k. For what
Published electronically February 1, 2002. This paper originally appeared in SIAM Journal on
Discrete Mathematics, Volume 13, Number 3, 2000, pages 384-402.
http://www.siam.org/journals/sirev/44-1/39542.html
# Department of Computer Science, Athens University of Economics and Business, Athens, Greece
(courcou@csi.forth.gr).
- Bell Labs, Murray Hill, NJ 07974.
# Statistical Laboratory, University of Cambridge, Cambridge CB3 0WB, UK (R.R.Weber@
statslab.cam.ac.uk).
# Avaya Labs Research, Basking Ridge, NJ 07920 (mihalis@research.avayalabs.com).
values of j and k can the set be packed perfectly (i.e., so that the sizes of the items in
each block sum to exactly k)? Clearly the sum of the item sizes must be divisible by k,
but what other conditions must be satisfied? Surprisingly, the divisibility constraint
is not only necessary but su#cient. Readers might want to try their hand at proving
this. Relatively short proofs exist, as illustrated in the next section, but a certain
ingenuity is required to find one. The exercise serves as a warm-up for the following
more general and more di#cult theorem, also proved in the next section, in which
there are r copies of each size for some r # 1.
Theorem 1 (perfect packing theorem). For positive integers k, j, and r, with
perfectly pack a list L consisting of rj items, r each of sizes 1 through
into bins of size k if and only if the sum of the rj item sizes is a multiple of k.
In set-theoretic terms, the question answered by Theorem 1 is an intriguing puzzle
in pure combinatorics. But our motivation to work on it came from its relevance
to certain fundamental questions about the average-case analysis of algorithms. In
particular, consider the standard bin packing problem, in which one is given a bin
capacity b and a list of items each a i has a positive
size s i # b, and is asked to find a packing of these items into a minimum number
of bins.
In most real-world applications of bin packing, as in Theorem 1, the item sizes are
drawn from some finite set. However, the usual average-case analysis of bin packing
heuristics has assumed that item sizes are chosen according to continuous probability
distributions, which by their nature allow an uncountable number of possible item
sizes (see [3, 10], for example). The assumption of a continuous distribution has the
advantage of sometimes simplifying the analysis and has been justified on the grounds
that continuous distributions should serve as reasonable approximations for discrete
ones. But there are reasons to ask whether this is actually true. For example, consider
the continuous uniform distributions U(0, u], 0 < u # 1, where the bin capacity is 1
and item sizes are chosen uniformly from the interval (0, u], and the discrete uniform
distributions U{j, k}, 1 # j # k - 1, where the bin capacity is k and item sizes are
chosen uniformly from the set {1, 2, . , j}. The limit of the distributions U{mj,mk},
as m #, is equivalent to U(0, j/k] (after scaling by dividing the item sizes and
bin capacities by mk). However, in the limit, combinatorial questions such as those
addressed by Theorem 1 evaporate. This suggests that something important (and
may in fact be lost in the transition from discrete to continuous models.
The results in this paper show that this is indeed the case.
To describe the results, we need the following notation. If A is a bin packing
algorithm and L is a list of items, then A(L) is the number of bins used when A is
applied to L, and s(L) is the sum of the item sizes in L divided by the bin capacity.
Note that s(L) is a lower bound on the number of bins needed to pack L. The waste
in the packing of L by A is denoted by W A
an algorithm that always produces an optimal packing. In what follows, L n
denote a list of n items whose sizes are independent samples from a given distribution
F . The expected waste rate EW A
for an algorithm A and distribution F is defined
to be the expected value of W A (L n as a function of n. In what follows we typically
abbreviate this as simply the "expected waste." We say a distribution F is a bounded
waste distribution if EW OPT
As a consequence of Theorem 1 and a
classification theorem of Courcoubetis and Weber [11], we can prove the following.
Theorem 2. For any j, k with 1 < j < k - 1, EW OPT
This in itself does not represent a departure from the continuous model, since
U(0, u] is also a bounded waste distribution for all u, 0 < u < 1 [3, 16]. The distinction
comes when we consider "online" algorithms. In an online algorithm, items are
assigned to bins in the order in which they occur in the input list L. Each assignment
must be made without knowledge of the sizes or number of items later in the list, and
once an item is packed it cannot subsequently be moved. This mirrors many practical
situations but clearly is a substantial restriction on the power of an algorithm. In
particular, we can prove the following.
Theorem 3. If A is an online algorithm and u # (0, 1], then it cannot be the
case that E[W A
In contrast, in the discrete uniform case there is a single online algorithm that
has bounded expected waste for all the distributions U{j, k}, 1 # j < k - 1. This
is the recently discovered sum-of-squares algorithm (SS) of [12], defined as follows.
Suppose we are packing integer-sized items into bins of capacity b. When an online
algorithm packs an item x from such a list, the only thing relevant about the current
packing is the number N h of bins whose current contents total h, 1 # h # b - 1. SS
chooses the bin into which x is to be placed (either a new, previously empty bin or
one that is already partially filled but has enough room for x) so as to minimize the
resulting sum
h . Note that SS can be implemented to take O(b) time per
item [12], and so runs in linear time for any fixed bin size.
As shown in [12], algorithm SS performs well on average in a surprisingly general
sense. Let us say that a discrete distribution F is any triple (b, S, # p), where b is an
integral bin size, is a finite set of integral item sizes in the range
from 1 to b - 1, and #
rational probability vector, where p i > 0 is
the probability of item size s i and
ignore the possibility of items of
size b since such items always must start a new bin and completely fill it, leaving the
rest of the packing una#ected.) The following specialized version of the result of [12]
su#ces for our needs.
Theorem (see [12]). For any discrete distribution
Hence by Theorem 2, EW SS
The current paper is organized as follows. The proof of the perfect packing
theorem (Theorem 1) appears in section 2. Section 3 then presents the proof of
Theorem 2. We begin by describing the classification result of Courcoubetis and
Weber [11] upon which the proof depends. This result says that for any discrete
distribution F , EW OPT
must be one of #(n), #(n 1/2 ), or O(1). Which case
applies depends on the existence of certain perfect packings and is in general NP-
hard. However, Theorem 1 allows us avoid this complexity in the case of the discrete
uniform distributions U{j, k}. Theorem 3, this paper's contribution to the theory
of continuous distributions, is proved in section 4. We conclude in section 5 with a
survey of the results that have been proved about the average-case behavior of bin
packing algorithms under discrete and continuous distributions. As we shall see, there
are other significant di#erences between the discrete and continuous cases.
2. The Perfect Packing Theorem. We begin our proof of Theorem 1 with three
lemmas that list a number of special instances that lead to perfect packing. The first
lemma takes care of the special case, r = 1.
Lemma 4. Suppose m, j, and k are positive integers such that j # k and
1)/2. Then the set of j items, one each of sizes 1, . , j, perfectly packs into m
bins of size k.
Proof. The proof is by induction. Pick j and k and assume the theorem is true
for all pairs that are smaller in lexicographic order than (k, j). The theorem is clearly
true for k # 2 or j # 2, so assume k, j > 2.
then we can start by perfectly packing bins with pairs of items
(j after which the remaining items are those of sizes
plus the item of size k/2 if k is even. Since the sum of the sizes of
the items that have been packed at this point is a multiple of k, the sum of the sizes
of the remaining items is also a multiple of k. If k is odd, the unpacked items are an
instance of (k, and the induction hypothesis applies.
If k is even, then k/2 divides j(j + 1)/2 and all remaining items are no larger than
k/2. Thus the items 1, . , k - j - 1 form an instance of (k/2, k - j - 1) and by the
induction hypothesis can be perfectly packed into bins of size k/2. These half-bins
and the item of size k/2 can then be combined into bins of size k.
Now suppose j # k/2. If k is even, then we have an instance of (k/2, j) and the
induction hypothesis applies. If k is odd, first note that k/2 # j and j > 2 imply
which together with 2m. Thus we can
construct m pairs of items each of total size k
1. If we place one pair in each of our m bins, we now have
bins with gaps of size k - k items of sizes 1, . , j - 2m.
Because 1)/2, the sum of these item sizes must be m(k - k # ), and so an
application of the induction hypothesis to the instance completes the
proof.
Lemma 5. Consider r > 1 sets, the ith of which consists of j items of consecutive
(a) r is even or (b) j is odd.
Then these rj items perfectly pack into j bins of size equal to the sum of the average
item sizes in the r groups, i.e., r(j
Proof. The lemma will follow if we can show that for # bins
of size r(j + 1)/2 it is possible to pack perfectly the items into the j bins in such a
manner that each bin contains exactly one item from each of the r sets.
If r is even, then we simply take two of the sets and pack the ith largest item
in one set with the ith smallest item in the other set, i.e., as the pair (i,
bins to level j + 1. By repeating this r/2 times we fill j bins
of size r(j + 1)/2.
If r and j are both odd, then an extra step is required. The idea is first to pack
items in triples, one item from each of three sets, such that the sum of each triple is
the same. It is easiest to appreciate the construction by considering an example, say
9. The triples, which each sum to 15, are given in the columns below.
In general, the triples are (i, i
j. The result of packing these one per
bin is to fill all j bins to level 3(j 1)/2. The number of remaining sets is even and
the remaining spaces in the j bins are equal. Thus, the procedure for case (a) can be
applied to complete the packing in each bin.
The following lemma provides part of the induction step used in the proof of
Theorem 1.
Lemma 6. Consider a quadruple (k, j, r, m) of positive integers such that k # j
and 1)/2. Then there exists a perfect packing of r copies of 1, . , j
into m bins of size k if there exists a perfect packing for each lexicographically smaller
quadruple of this form, and if any one of the following holds:
(b) r does not divide k.
(c) k or r is even.
Proof. First, using the arguments of Lemma 4, we demonstrate how to reduce the
problem to a smaller instance if (a) holds. If j # k/2 and k is odd, then we can pack
bins with pairs (j - 1)/2. The remaining items, which
are of sizes 1, . , k - j - 1, define the smaller instance (k, k -
and k is even, then we can pack bins in the same
way, k/2. The remaining items, which are of sizes 1, . ,
and k/2, can be packed into bins of size k/2 by the induction hypothesis that there
exists a perfect packing for
If (b) holds, then r and m must have a common factor p > 1 and the problem
reduces to the instance (k, j, r/p, m/p).
Now suppose neither (a) nor (b) holds but (c) does. If k is even, then k/2 divides
k/2. Thus the problem reduces to a smaller
instance in which the bin size is k/2. If r is even, then since (b) does not hold, k is
divisible by r and so must be even too. Thus the same argument applies.
Finally, for case (d), assume that (a), (b), and (c) do not hold, i.e., j < k/2, r
divides k, and k and r are both odd. Let r
k. The fact that r is odd implies that
r 1 and r 2 are integers. The fact that r divides k implies that k 1 and k 2 are integers,
with 1)/2. Since by assumption j < k/2, we
have hence by hypothesis for the instance (k 1 , j, r 1 , m), we can pack r 1
copies of 1, . , bins of size k 1 . Similarly, if also then we can pack
copies of 1, . , bins of size k 2 . Since we can combine pairs
of bins of sizes k 1 and k 2 into bins of size k. Thus there is a reduction to smaller
instances holds.
Proof of the perfect packing theorem. Instances for which the theorem is to be
proved are described by the quadruples of Lemma 6. Notice that it would be enough
to specify the triple (k, j, r); however, it is helpful to mention m explicitly. The proof
of the theorem is by induction on (k, j, r) under lexicographical ordering. By Lemma 4
it is true for 1. Assume all quadruples that are smaller than (k, j, r, m) can be
perfectly packed and r > 1. We show there exists a perfect packing of r copies of
1, . , bins of size k. By Lemma 6, we need only consider the case when
and r are odd, r divides k, and (r - 1)k/2r < j < k/2. Note that in this case
(r - 1)k/2r is an integer and k/2r is 0.5 more than an integer. We show below that
we can perfectly pack all the items of sizes from
bins of size k. (Note that the lower bound on this range is greater than 1 because
of the above lower bound on j.) The theorem then follows because the remaining
items form a smaller quadruple, so by the induction hypothesis they can be perfectly
packed into bins of size k.
To follow the construction below, the reader may find it helpful to consider a
specific example. Consider the quadruple (k, j, r, m) = (165, 77, 5, 91). Note that k
and r are odd, r divides k, and j lies between (r -
show below how to perfectly pack all items of sizes 12, . , 77. The remaining items
form the smaller quadruple (165, 11, 5, 2).
To pack all items of sizes from j +1-(r-1)k/2r through j, we divide the range of
item sizes into intervals, i.e., sets of consecutive integers. Each interval is symmetric
about a multiple of k/2r and has one of two lengths depending on whether the interval
100 COFFMAN ET AL.
is symmetric about an odd or even multiple of k/2r. To form the intervals, we first
take the largest interval that is symmetric about (r - 1)k/2r; this is the interval
[(r -1)k/r-j, j]. Note that this interval does not include (r -2)k/2r since j <
rk/2r. Next we take the largest interval that can be formed from the remaining items
that is symmetric about (r-2)k/2r, obtaining the interval [j-k/r+1, (r-1)k/r-j-1].
Continuing in this fashion and taking intervals symmetric about further multiples of
k/2r, we end up with intervals of two kinds. First, there are (r - 1)/2 intervals
centered on even multiples of k/2r, with the interval centered on (r - 1 - 2i)k/2r
being ranges from 0 to (r - 3)/2. Second, there
is an equal number of intervals centered on odd multiples of k/2r, with the interval
centered on (r - 2i)k/2r being [j - ik/r ranges from 1
to (r - 1)/2. Note that the smallest endpoint is
claimed above.
For the numerical example above, there are two intervals of each type. Intervals
of the first type are [22, 44] and [55, 77]; they are of length 23 and symmetric about 33
and 66. Intervals of the second type are [12, 21] and [45, 54]; they are of length 10 and
symmetric about 16.5 and 49.5. In general, intervals of the first type have odd length
are symmetric about an even multiple of k/2r. Intervals of
the second type have even length k -2j -1 and are symmetric about an odd multiple
of k/2r. Our plan is to use Lemma 5 to perfectly pack into bins of size k those items
whose sizes lie in intervals of the same type.
We begin by considering all those intervals of the first type. These have odd
lengths and they are symmetric about points ik/r, 1)/2. There are
r items of each size in each of these intervals. Our strategy is to partition these
intervals into groups that satisfy the hypotheses of Lemma 5(b). That is, we arrange
for the midpoints of the intervals within each group to sum to k. Since the midpoints
correspond to the average item sizes for the corresponding intervals, and the number
of items in the intervals is odd, Lemma 5(b) implies that we can perfectly pack the
items in the intervals of each group. Constructing these groups is a bin packing
problem in which the midpoints of the intervals take on the role of item sizes. In
what follows we write "items" in quotes when speaking of the midpoints of intervals,
possibly normalized, and viewing them as items to be perfectly packed in bins of
some required size. In considering intervals of the first type, it is as though we had
r "items" of each of the sizes ik/r, and wished to pack them
into bins of size k. After a normalization that multiplies each item size by r/k, this is
equivalent to the problem of packing r "items" of each of the sizes 1, . , (r -1)/2 into
bins of size r. That is, we have a smaller version our packing problem,
with But by the induction
hypothesis this means that the desired packing can be achieved. In the example, it is
as though we had 5 "items" of sizes 33 and 66 that are to be packed in bins of size 165.
Normalizing by a factor of 1/33, this is equivalent to the problem instance (5, 2, 5, 3).
We must now pack items whose sizes lie in intervals of the second type. These
intervals are of even length, symmetric about the points ik/2r, for i odd and
1, . , r - 2. Again, there are r items of each size in these intervals. As above, we
exhibit a reduction to a smaller perfect packing problem. After we multiply item
sizes by 2r/k the problem is equivalent to perfectly packing r copies of "items" of
sizes 1, 3, 5, . , r - 2 into bins of size 2r. For the example, this is 5 copies of "items"
of sizes 1 and 3, to be perfectly packed into 2 bins of size 10. Unfortunately, if the
sum of the "item" sizes is an odd multiple of r the "items" cannot be perfectly packed
into bins of size 2r. For this reason, and also because it is convenient to do so even
when the sum of the "item" sizes is a multiple of 2r, we consider perfect packings
into bins of sizes r and 2r. Assume for the moment that r copies of "items" of sizes
can be perfectly packed into bins of sizes r and 2r. If they are packed
entirely into bins of size 2r, then the number of "items" in each bin must be even (as
all "item" sizes are odd), and so Lemma 5 applies and implies that the original items
can be perfectly packed into bins of size k. On the other hand, suppose a bin of size r
is required. The set of "items" that are packed into a bin of size r corresponds to a set
of intervals whose midpoints sum to k/2. Recall that the intervals are of even length.
We divide each such interval into its first half and its second half, obtaining twice as
many intervals, whose midpoints now sum to k. Now we can again use Lemma 5 to
construct the perfect packing.
The final step in the proof is to show that we can indeed perfectly pack r copies
of each of the item sizes 1, 3, 5, . , r - 2 into bins of sizes r and 2r. We shall use a
di#erent packing depending upon whether
For the case the "items" are perfectly packed by the following simple
procedure. We begin by packing one bin with (1, 1, r - 2), one bin with (2i
1, r-2i-2, r-2i) for each one bin with (2i-1, 2i+1, r-2i, r-2i)
for each (noting that in the final case, when #, we get three "items"
of size 2i This packs four "items" of each size larger than 1, and three
"items" of size 1. We can apply this packing # times, leaving us with one "item"
of each size larger than 1 and # "items" of size 1. Then we pack one bin with
This uses up all the remaining "items" (where
#+1 items of size 1 are used because there are two "items" of size 1 when 1). For
the numerical example, in which construction says that we should pack 5
copies of 1 and 3 into bins of size 5 and 10 by first packing one bin with (1, 1,
then one bin with (1, 3, 3, 3). This leaves one "item" of size 3 and two of size 1. These
perfectly pack into a bin of size 5.
When the procedure is very similar to that above. We begin by
packing one bin with (1, 1, r - 2), one bin with (2i +1, 2i+1, r -2i-2, r -2i) for each
As
before, this packs four "items" of each size larger than 1, and three "items" of size 1.
We apply this packing # times, leaving us with three "items" of each size larger than
1 and #+ 3 "items" of size 1. Then we pack one bin with (2i - 1, 2i
for each This leaves us with # "items" of size 1, two "items" of size
one "item" of each other size. Finally, as before, we pack one bin with
uses up all remaining items.
3. Proof of Theorem 2. Recall the theorem statement: For any distribution
U{j, k}, with j < k - 1, EW OPT
We rely on a general result of Courcoubetis and Weber [11]. Suppose
is a discrete distribution as defined in section 1. Note that a packing of items with
sizes from into a bin of size b can be viewed as a nonnegative integer
vector
interest are those vectors
that give rise to a sum of exactly b, which we shall call perfect packing configurations.
For instance, if one such configuration would be (1, 0, 2).
Let P S,b denote the set of all perfect packing configurations for a given S and b. Let
# S,b be the convex cone in R d spanned by all nonnegative linear combinations of
configurations in P S,b .
Theorem (Courcoubetis and Weber [11]). For any discrete distribution
(b, S, # p), the following hold.
(a) If # p lies in the interior of # S,b , then EW OPT
(b) If # p lies on the boundary of # S,b , then EW OPT
(c) If # p lies outside of # S,b , then EW OPT
In general it is NP-hard to determine which of the three cases applies to a given
distribution (as can be proved by a straightforward transformation from the PARTITION
problem [13]). However, for the distributions U{j, k}, j < k - 1, we can use
the following lemma, which we shall prove using the perfect packing theorem, to show
that (a) applies.
Lemma 7. For each i, j, k with exist positive integers
such that the set of r items consisting of r items of size i
together with r i items of each of the other j - 1 sizes can be packed perfectly into m i
bins of size k.
Note that this lemma implies that the j-dimensional vector -
is strictly inside the appropriate cone when 1. This is
because -
e is in the interior of the cone spanned by vectors of the form (r i , . , r
those vectors are sums of perfect packing configurations
by Lemma 7. The proof of Theorem 2 thus follows from case (a) of the above theorem.
Proof of Lemma 7. We make use of the perfect packing theorem. There are two
cases. If k we simply set r
that the total size of r i items each of the sizes 1, . , so by the
perfect packing theorem, we can perfectly pack them into j(j bins of size
i. The remaining s items of size i can then go one per bin to fill these
bins up to size precisely k.
On the other hand, suppose Now things are a bit more complicated.
We have r 1)/2. By
the perfect packing theorem r i items each of the sizes 1, . , perfectly pack
in s i bins of size k - i. (Such items exist because by assumption j < k - 1.) We
then add the additional s i items of size i to these bins, one per bin, to bring each
bin up to size k. There remain r i items each of sizes k - j through j, for a total of
items. These can be used to completely fill the remaining
bins with pairs of items of sizes (j, k - j), (j - 1, k
Note that if k is even, the last bin type contains two items of size k/2, but we have an
even number of such items by our choice of r presents no di#culty.
It is easy to verify that in both cases r i , s i , and m i are all less than 2k 2 .
4. Proof of Theorem 3. Recall the theorem statement: If L n has item sizes
generated according to U(0, u] for 0 < u # 1, and A is any online algorithm, then
there exists a constant c > 0 such that E[W A (L n )] > cn 1/2 for infinitely many n.
Proof. Let w(t) denote the amount of empty space in partially filled bins after t
items have been packed. We show that for any n > 0 the expected value of the average
of w(1), . , w(n) is # n 1/2 u 3 ). This implies that E[w(n)] must
be# (n 1/2 u 3 ), i.e.,
not o(n 1/2 ).
Consider packing item a t+1 . Let #(t) denote the number of nonempty bins that
have a gap of at least u 2 /8 after the first t items have been packed. There are at most
bins into which one can put an item larger than u 2 /8. Therefore, if a t+1 is to
leave a gap of less than # in its bin, either it must have size less than u 2 /8 or its size
must be within # of the empty space in one of these #(t) bins with gaps larger than
/8. The probability of this is at most [u 2 /8+#(t)]/u. By choosing
conditioning on whether #(t) is greater or less than n 1/2 , and noting that the size of
a t+1 is distributed as U(0, u] independent of #(t), we have
Now
(a s+1 is last in a bin and leaves gap #)
[P (a s+1 is last in a bin)
- P (a s+1 is last in a bin and leaves gap < #)]
[P (a s+1 is last in a bin) - P (a s+1 leaves gap < #)]
(a s+1 is last in a bin) -
u/4.
t be the sum of the first t item sizes, and note that S t is a lower bound on
the number of bins and hence on the number of items that are the last item in a bin.
We thus have
(a s+1 is last in a bin) # E[S t
Using the fact that we then have
If
we have for all t # n/2,
This implies
On the other hand, if
w(t) #n
These imply that E[w(n)]
is# (n 1/2 ).
It should be noted that the above proof relies heavily on the fact that the distribution
is continuous, since this is the reason why the union of n 1/2 intervals of size
cannot cover the full probability space. Our discrete distributions U{j, k} do not
have this failing, and for this reason we can obtain significantly better average-case
behavior for them.
104 COFFMAN ET AL.
Table
Expected waste in the symmetric case.
BF #(n 1/2 log 3/4 n) [26] #(n 1/2 log 3/4
Best online #(n 1/2 log 1/2 n) [26, 27] #(n 1/2 log 1/2
The upper bound is proved in the reference; the lower bound is conjectured based on experiments.
# The upper and lower bounds here appear to follow from the corresponding results for the continuous
case, but the details of the upper bound in particular still need to be worked out.
5. Concluding Remarks. The results in this paper were among the first to be
obtained about the average-case behavior of bin packing algorithms under discrete
distributions. Since they were announced in [4], many additional results have been
proved, illustrating further contrasts with (and similarities to) the case of continuous
distributions. In this concluding section we survey the literature and point out some
of the remaining open problems.
Let us begin by considering symmetric uniform distributions, as represented by
U(0, 1] and U{k - 1, k}, k # 1. (In general, a symmetric distribution is one that
satisfies p(s #) = p(s # b - #) for all #, 0 # b.) Table 1 summarizes what
is known about average-case behavior under these distributions. A horizontal line
separates the o#ine algorithms from the online ones. Except where noted, all results
in this table are theorems.
Four famous classical algorithms have been extensively studied. First fit (FF ) is
an online algorithm in which each item is placed in the first bin that has room for
it, where bins are sequenced according to the order in which they received their first
item. If no bin has room, a new bin is started. Best fit (BF ) is similar, except now
the item is placed in the bin with the smallest gap large enough to contain it (ties
broken in favor of the earlier bin). First fit decreasing (FFD) and best fit decreasing
(BFD) are the corresponding o#ine algorithms in which the list is first sorted so that
the items are in nonincreasing order by size, and then FF (BF ) is applied. From a
worst-case point of view, FF and BF are equivalent: in an asymptotic sense each can
produce packings that use 70% more bins than optimal, but neither can do any worse
[15]. The corresponding o#ine versions FFD and BFD each can use
more bins than optimal but can do no worse [14, 15].
The results in Table 1 show that these algorithms perform much better on average
than in the worst case, since they now have sublinear expected waste, a surprise when
it was first observed empirically in [2]. The o#ine versions continue to have an
advantage over their online counterparts, but it is of reduced practical significance.
And now there is a distinction in the behavior of FF and BF , with BF being the
better of the two.
The above remarks apply equally well to the discrete and continuous cases. As
to the comparison between these cases, we once again have a significant di#erence for
online algorithms. For any fixed value of k, the online algorithms in the table all have
expected waste, in contrast to the expected wastes in the continuous case of
#(n 2/3 ) for FF , #(n 1/2 log 3/4 n) for BF and #(n 1/2 log 1/2 n) for the best possible
online algorithm. (Here the notation #(f(n)) means that the lower bound is taken in
the Hardy and Littlewood sense of "not o(f(n))," i.e., "greater than cf(n) for some
Table
Possibilities for expected waste in the nonsymmetric case.
OPT #u (1) [3] # k (1) [.]
online# u (n 1/2 ), Ou (n 1/2 log 3/4 n) [.] [25] # k (1) [.]
. Results proved in this paper.
ruled out by theorems, but no occurrences are known either. For BFD and FFD,
it does not occur for any k # 10,000 [5].
# This is conjectured to hold for all u # (0, 1), based on experimental studies. To date it has been
proved only for u # [0.66, 2/3) and BF [18].
c > 0 and infinitely many n," rather than in the standard Knuthian sense of "greater
than cf(n) for some c > 0 and all su#ciently large n.")
In a sense, however, the online results for the discrete case are consistent with
those for the continuous one. Although technically an online algorithm is not allowed
to know the magnitude of n, if one formally sets in the formulas for expected
waste for the discrete case, one gets EW A
BF , and the best possible online algorithm. SS is not applicable to continuous
distributions, but note that in this asymptotic discrete sense it appears to be worse
than FF and BF . Indeed, experiments suggest that EW SS
Let us now turn to the nonsymmetric distributions U(0, u], u < 1, and U{j, k},
1. The known results for these distributions are summarized in Table 2.
Here for the first time we see di#erences between the continuous and discrete cases
for o#ine algorithms. In particular, for A # {FFD,BFD}, EW A
for all u # 1/2 [3, 16], but for many of the distributions U{j, k} with j # k/2 (the
corresponding discrete uniform distributions), we have EW A
Moreover, for u # (1/2, 1), EW A
growth rate never
occurs for U{j, k}. This follows from a theorem in [5] that says that for all discrete
distributions F , EW FFD
must be either O(1), #(n 1/2 ), or #(n).
The theorem also provides algorithms that determine the answers for a given
distribution (b, S, # p), find the constants of proportionality when the expected waste is
linear, and run in time polynomial in b and |S|. Unfortunately, although the answers
for the distributions U{j, k} with k # 10,000 have all been computed [5], these do
not suggest any simple rule as to how the choice among O(1), #(n 1/2 ), and #(n)
might depend on j and k. (As an example of the type of behavior that can occur,
for the U{j, 151} distributions the choice between linear and bounded expected waste
switches back and forth 10 times as j increases from 1 to 149.) The results for k #
10,000 do, however, exhibit several suggestive patterns. First (and this can be proved
to hold for arbitrarily large k), the expected waste is O(1) whenever j < # k or
expected waste #(n 1/2 ) does not occur for any U{j, k} with
suggesting that it may never occur. Third, for each U{j, k} with
(U{j, k}) are either both linear or both
bounded. If linear, the constant of proportionality for BFD is never larger than that
for FFD (but is occasionally smaller).
Here again there is a sense in which the discrete case is asymptotically consistent
with the continuous case, even though the expected waste for FFD and BFD is
always sublinear in the latter. As k increases, the maximum constants of proportionality
for the linear expected waste under U{j, k} appear to decrease. Indeed, it can be
106 COFFMAN ET AL.
shown that the constants for FFD are bounded by a function that declines at least
as fast as (log k)/k [5]. (This presumably holds for BFD as well.) The worst case is
the distribution U{6, 13}, for which the expected waste for both FFD and BFD is
n/624, which is less than 0.6% of the expected optimal number of bins. Moreover,
this is easily avoided, since not only does SS have bounded expected waste for this
distribution, but so do FF and BF (although this is the only case we have identified
where the online FF and BF algorithms outperform their o#ine cousins).
Turning now to the online algorithms FF and BF , we observe that their behavior
under discrete uniform distributions appears empirically to be similar to their behavior
under continuous ones. Based on extensive experiments, it is conjectured that FF
and BF both have linear expected waste under U(0, u] for 0 < u < 1, although to date
this has been proved only for u # [0.66, 2/3) and BF [18]. In the discrete case (U{j, k}
with experiments suggest that for su#ciently large k, the expected waste
for FF and BF is bounded when
linear. Some of this has been proved. In [19] it was shown that EW BF
0, and this result was extended to FF in [1]. In [4] it was
shown that EW FF
In practice, bounded expected waste is more common, at least for small k.
The growth rates for BF under U{j, k} with k # 11 were completely characterized
using multidimensional Markov chain arguments in [8], and linear expected waste only
occurs for U{8, 11}. The only general result proving linear expected waste mirrors
the result for the continuous case: EW BF
k is su#ciently large [18]. At this point we do not know if expected wastes other
than O(1) and #(n) are possible for FF or BF under any distributions U{j, k} with
classification theorem such as those for FFD, BFD, or OPT
has been proven, so the range of possibilities is not known to be limited to O(1),
#(n 1/2 ), and #(n), as it was for FFD and BFD.
There is also a gap between the lower bound proved in this paper on the best
possible online expected waste for continuous distributions U(0, u] and the best rate
known to be achievable. The former
is# (n 1/2 ) and the latter is O(n 1/2 log 3/4 n), as
proved in [25]. The algorithm of [25] works for any distribution, discrete or continuous,
but has drawbacks from a pragmatic point of view: the best current bound we have
on its running time is O(n 8 log 3 n) [12]. If one is willing to consider more specialized
algorithms, better running times are possible, at least theoretically. For any fixed
distribution F , there is an algorithm AF that runs in time O(n log n) and again
has expected waste of O(n 1/2 log 3/4 n) [24]. These algorithms have drawbacks too,
however, since the proof that they exist is nonconstructive. The question of whether
practical algorithms exist that attain these bounds, or indeed whether
lower bound is achievable, remains open.
Finally, in addition to the open problems mentioned above for the discrete and
continuous uniform distributions U(0, u] and U{j, k}, there is the question of what
happens for arbitrary discrete and continuous distributions. In the discrete case, the
above-mentioned classification theorems apply for BFD, FFD, and OPT , and say
that the corresponding expected waste must be O(1), #(n 1/2 ), or #(n). As also mentioned
above, the applicable cases for BFD and FFD and any specific distribution
can be determined in time polynomial in b and |S|. For OPT there is
also an algorithm for determining which case applies, as noted in [12]. This involves
solving up to |S| +1 linear programs with |S|b variables and |S| constraints. None
of these algorithms technically runs in polynomial time since b may be exponentially
larger than its contribution to instance size (log b). However, all are feasible for b in
PERFECT PACKING THEOREMS 107
excess of 1,000, which makes it possible to characterize the behavior of FFD, BFD,
and OPT for many interesting distributions on a case-by-case basis.
The theorem about SS presented in section 1 can be generalized to arbitrary
discrete distributions if one replaces SS by a simple variant SS # : As in SS, items
are packed so as to minimize
h , but now the choice must be made subject to
the following additional constraint: No item may be placed in a partially filled bin if
the resulting gap cannot be exactly filled with items whose sizes have already been
encountered in the list L. The resulting algorithm still runs in time O(nb) and satisfies
distributions F [12]. In addition, there
is a more complicated randomized variant that runs in time O(nb log b), satisfies the
above property, and also has the same constant of proportionality as OPT when the
expected waste is linear [12].
As to the case of arbitrary continuous distributions, we as yet have no general
classification theorems, although some partial results have been proved. Rhee
[22] provided a complicated measure-theoretic characterization of those F for which
is sublinear, but this does not appear to be computationally useful. A
result of Rhee and Talagrand [23] implies that if EW OPT
must be
O(n 1/2 ) or better. Rates strictly between O(1) and #(n 1/2 ) have not yet been ruled
out, however. Moreover, there is as yet no algorithm with the general e#ectiveness
of SS and its variants. The results of [24, 25] imply that there are online algorithms
whose expected waste is at most O(n 1/2 log 3/4
n) worse than the optimal expected
waste. For o#ine algorithms, Karmarkar and Karp have devised an algorithm which in
the worst case never uses more than the optimal number of bins plus O log 2 (OPT
O(log 2 n) [17]. This means that its expected waste is never more than the maximum of
O(log 2 n) and EW OPT
the algorithms of [24, 25], however, it is impractical,
having a running time for which our best current bound is O(n 8 log 2 n).
To conclude with an open problem that hearkens back to the main result of this
paper, note that our ability to determine the expected waste for FFD, BFD, and
OPT on a case-by-case basis can only take us so far, and more general results would be
desirable. Results for U(0, 1] and U{k - 1, k} typically continue to hold for arbitrary
continuous and discrete symmetric distributions, respectively, but the real world is
not dominated by symmetric distributions. It would be nice if we could identify
additional interesting classes of nonsymmetric distributions F for which general results
about EW OPT
can be proved, as we did in this paper for the discrete uniform
distributions. Are there interesting classes for which new perfect packing theorems
can provide us with similar general answers?
--R
An experimental study of bin packing
Some unexpected expected behavior results for bin packing
Stability of on-line bin packing with random arrivals and long-run average constraints
On the sum-of-squares algorithm for bin packing
Computers and Intractability: A Guide to the Theory of NP-completeness
Average Case Behavior of First Fit Decreasing and Optimal Packings for Continuous Uniform Distributions U(0
Linear waste of best fit bin packing on skewed distribu- tions
stochastic analysis of best fit bin packing
An Average-Case Analysis of Bin Packing with Uniformly Distributed Item Sizes
Optimal bin packing with items of random sizes
Optimal bin packing with items of random sizes.
The average case analysis of some on-line algorithms for bin packing
How to pack better than Best Fit: Tight bounds for average-case on-line bin pack- ing
--TR
--CTR
Janos Csirik , David S. Johnson , Claire Kenyon , James B. Orlin , Peter W. Shor , Richard R. Weber, On the Sum-of-Squares algorithm for bin packing, Journal of the ACM (JACM), v.53 n.1, p.1-65, January 2006 | approximation algorithms;bin packing;average-case analysis;online |
588918 | Convergence Properties of an Augmented Lagrangian Algorithm for Optimization with a Combination of General Equality and Linear Constraints. | We consider the global and local convergence properties of a class of augmented Lagrangian methods for solving nonlinear programming problems. In these methods, linear and more general constraints are handled in different ways. The general constraints are combined with the objective function in an augmented Lagrangian. The iteration consists of solving a sequence of subproblems; in each subproblem the augmented Lagrangian is approximately minimized in the region defined by the linear constraints. A subproblem is terminated as soon as a stopping condition is satisfied. The stopping rules that we consider here encompass practical tests used in several existing packages for linearly constrained optimization. Our algorithm also allows different penalty parameters to be associated with disjoint subsets of the general constraints. In this paper, we analyze the convergence of the sequence of iterates generated by such an algorithm and prove global and fast linear convergence as well as show that potentially troublesome penalty parameters remain bounded away from zero. | Introduction
. Introduction
In this paper, we consider the problem of calculating a local minimizer of the
smooth function
where x is required to satisfy the general equality constraints
and the linear inequality constraints
Here f and c i map ! n into !, A is a p-by-n matrix and b 2 ! p .
A classical technique for solving problem (1.1)-(1.3) is to minimize a suitable
sequence of augmented Lagrangian functions. If we only consider the problem (1.1)-
(1.2), these functions are defined by
where the components - i of the vector - are known as Lagrange multiplier estimates
and - is known as the penalty parameter (see, for instance, Hestenes [18], Powell [23]
and Bertsekas [3]). The question then arises how to deal with the additional linear
inequality constraints (1.3). The case where A is the identity matrix (that is when
specifies bounds on the variables) has been considered by Conn et al. in [5]
This research was supported in part by the Advanced Research Projects Agency of the Department
of Defense and was monitored by the Air Force Office of Scientific Research under Contract No
F49620-91-C-0079. The United States Government is authorized to reproduce and distribute reprints
for governmental purposes notwithstanding any copyright notation hereon.
This work was also supported by the Belgian national Fund for Scientific Research.
and [7]. They propose keeping these constraints explicitly outside the augmented Lagrangian
formulation, handling them directly at the level of the augmented Lagrangian
minimization. That is, a sequence of optimization problems, in which (1.4) is approximately
minimized within the region defined by the simple bounds, is attempted. In this
approach all linear inequalities other than bound constraints are converted to equations
by introducing slack variables and incorporated in the augmented Lagrangian
function. This strategy has been implemented and successfully applied within the
LANCELOT package for large-scale nonlinear optimization (see Conn et al. [6]). How-
ever, such a method may be inefficient when linear constraints are present as there
are a number of effective techniques specifically designed to handle such constraints
directly (see Arioli et al. [1], Forsgren and Murray [14], Toint and Tuyttens [24], or
and Carpenter [25], for instance). This is especially important for large-scale
problems. The purpose of the present paper is therefore to define and analyze
an algorithm where the constraints (1.3) are kept outside the augmented Lagrangian
and handled at the level of the subproblem minimization, thus allowing the use of
specialized packages to solve the subproblem.
Our proposal extends the method of Conn et al. [5] in that not only bounds but
general linear inequalities are treated separately. Fletcher [13, page 295] remarks on
the potential advantages of this strategy.
Furthermore, it is often worthwhile from the practical point of view to associate
different penalty parameters to subsets of the general constraints (1.2) to reflect different
degrees of nonlinearity. This possibility has been considered by many authors,
including Fletcher [13, page 292], Powell [23] and Bertsekas [3, page 124]. In this case,
the formulation of the augmented Lagrangian (1.4) can be refined: we partition the
set of constraints (1.2) into q disjoint subsets fQ j g q
redefine the augmented
Lagrangian as
where - is now a q-dimensional vector, whose j-th component is - j ? 0, the penalty
parameter associated with subset Q j . Because of its potential usefulness, and because
its analysis is difficult to directly infer from the single penalty parameter case, this
refined formulation will be adopted in the present paper.
The theory presented below handles the linear inequality constraints in a purely
geometric way. Hence the same theory applies without modifications if linear equality
constraints are also imposed and all the iterates are assumed to stay feasible with
respect to these new constraints. It is indeed enough to apply the theory in the affine
subspace corresponding to this feasible set. As a consequence, linear constraints need
not be included in the augmented Lagrangian and thus have the desirable property
that they have no impact on the structure of its Hessian matrix.
The paper is organized as follows. In Section 2, we introduce our basic assumptions
on the problem and the necessary terminology. Section 3 presents the proposed
algorithm and the definition of a suitable stopping criterion for the subproblem. The
global convergence analysis is developed in Section 4 while the rate of convergence
is analyzed in Section 5. Second order conditions are investigated in Section 6. Section
7 considers some possible extensions of the theory. Finally, some conclusions and
perspectives are outlined in Section 8.
2. The problem and related terminology. We consider the problem stated
in (1.1)-(1.3) and make the following assumptions.
AS1: The region nonempty.
AS2: The functions f(x) and c i (x), are twice continuously differentiable
for all x 2 B.
Assumption AS1 is clearly necessary for the problem to make sense. We note that
it does not prevent B from being unbounded.
We now introduce the notation that will be used throughout the paper.
Let g(x) denote the gradient r x f(x) of f(x) and H(x) denote its Hessian matrix
r xx f(x). We also define J(x) to be the m-by-n Jacobian of c(x), where
Hence
denote the Hessian matrix r xx c i (x) of c i (x). Finally, let g ' (x; -) and
denote the gradient, r x '(x; -), and the Hessian matrix, r xx '(x; -), of the
Lagrangian function
We note that '(x; -) is the Lagrangian solely with respect to the c i constraints. If we
define first-order Lagrange multiplier estimates componentwise as
where w [S] denotes the jSj-dimensional subvector of w whose entries are indexed by
the set S, we shall use the identity
r x \Phi(x;
Now suppose that fx k 2 Bg, f- k g and f- k g are infinite sequences of n-vectors,
m-vectors and positive q-vectors, respectively. For any function F , we shall use the
notation that F k denotes F evaluated with arguments x
So, for instance, using the identity (2.2), we have that
where we have written (2.1) in the compact form
We denote the vector w at iteration k by w k and its i-th component by w k;i . We also
use w k;[S] to denote the jSj-dimensional subvector of w k whose entries are indexed by
S.
Now let fx k subset K of the natural numbers N, be a convergent
subsequence with limit point x . Then we denote the matrix whose rows are those
of A corresponding to active constraints at x - that is the constraints which are
satisfied as equalities at x - by A . Furthermore, we choose Z to be a matrix
whose columns form an orthonormal basis of the null space of A , that is
A Z
We define the least-squares Lagrange multiplier estimates (corresponding to A )
at all points where the right generalized inverse
of J(x)Z is well defined. We note that, whenever J(x)Z has full rank, -(x) is
differentiable and its derivative is given in the following lemma
Lemma 2.1. Suppose that AS2 holds. If J(x)Z Z T
J(x) T is nonsingular, -(x) is
differentiable and its derivative is given by
where the i-th row of R(x) is (Z T
Proof. The result follows by observing that (2.5) may be rewritten as
g(x) and J(x)Z
for some vector r(x). Differentiating (2.7) and eliminating the derivative of r(x) from
the resulting equations gives the required result. 2
We stress that, as stated, the Lagrange multiplier estimate (2.5) is not directly calculable
as it requires a priori knowledge of x . It is merely introduced as an analytical
device.
Finally, the symbol k \Delta k will denote the ' 2 -norm or the induced matrix norm. We
are now in position to describe more precisely the algorithm that we propose to use.
3. Statement of the algorithm. We consider the algorithmic model we wish
to use in order to solve the problem (1.1)-(1.3). This model proceeds at iteration k by
computing an iterate x k which satisfies (1.3) and approximately solves the subproblem
min x2B
where the values of the Lagrange multipliers - k and penalty parameters - k are fixed
for the subproblem. Subsequently we update the Lagrange multipliers and/or decrease
the penalty parameters, depending on how much the constraint violation for (1.2) has
been reduced within each subset of the constraints. The motivation is simply to ensure
global convergence by driving, in the worst case, the penalty parameters to zero, in
which case the algorithms essentially reduce to the quadratic penalty function method
(see, for example, Gould [15]). The tests on the size of the general constraint violation
are designed to allow the multiplier updates to take over in the neighbourhood of a
stationary point.
The approximate minimization for problem (3.1) is performed in an inner iteration
which is stopped as soon as its current iterate is "sufficiently critical". We propose to
base this decision on the identification of the linear constraints that are "dominant" at
x (even though they might not be active) and on a measure of criticality for the part of
the problem where those constraints are irrelevant. Given ! - 0, a criticality tolerance
for the subproblem, we define, for a vector x 2 B, the set of dominant constraints at
x as the constraints whose indices are in the set
for some - 0 ? 0. Here a T
is the i-th row of the matrix A and b i is the
corresponding component of the right-hand side vector b. Denoting by A D(x;!) the
submatrix of A consisting of the row(s) whose index is in D(x; !), we also define
the cone spanned by the outward normals of the dominant constraints. The associated
polar cone is then
where cl(V ) denotes the closure of the set V . The cone T (x; !) is the tangent cone
with respect to the dominant constraints at x for the tolerance !. Note that D(x; !)
might be empty, in which case A D(x;!) is assumed to be zero, N(x; !) reduces to the
origin and T (x; !) is the full space.
We then formulate our "sufficient criticality" criterion for the subproblem as fol-
lows: we require that
is the projection onto the convex set V and ! k is a suitable tolerance at
iteration k. Once x k satisfying (3.3) has been determined by the inner iteration, we
denote
For future reference, we define Z k to be a matrix whose columns form an orthonormal
basis of V k , the null space of AD k
, and Y k to be a matrix whose columns form an
orthonormal basis of W
k . As above, we have that T k is the full space and N k
reduces to the origin when D k is empty. We note that, in this case, Z
I ,
the identity operator, and Y
We also note that V k ' T k , and hence that
since Z k Z T
k is the orthogonal projection onto V k .
It is important to note that the stopping rule (3.3) covers a number of more specific
choices, including the rule used in much existing software for linearly constrained
optimization (such as MINOS [21], LSNNO [24], or VE14 and VE19 from the Harwell
Subroutine Library [17]). The reader is referred to Section 7.2 for further details.
We are now in position to describe our algorithmic model more precisely. In this
model, we define ff k to be the maximum penalty parameter at iteration k (see (3.10)).
At this iteration, the parameters ! k and j k represent criticality and feasibility levels,
respectively.
partition of the set disjoint subsets
is given, as well as initial vectors of Lagrange multiplier estimates - 0
and positive penalty parameters - 0 such that
The strictly positive constants -
are specified. Set ff
approximately solves (3.1), i.e. such
that (3.3) holds.
[Test for convergence]. If kP T k
Step 3 [Disaggregated updates]. For
or Step 3b otherwise.
Step 3a [Update Lagrange multiplier estimates]. Set
Step 3b [Reduce the penalty parameter]. Set
where
Step 4 [Aggregated updates]. Define
If
then set
otherwise set
Increment k by one and go to Step 1.
Algorithm 3.1 is specifically designed for the first-order estimate (2.1), a formula
with potential advantages for large-scale computations. We refer the reader to Section
7.1 for a further discussion of a more flexible choice of the multipliers, covering,
among others, the choice of the least-squares estimates -(x) as defined in (2.5).
We immediately verify that our algorithm is coherent, in that
lim
Indeed, we obtain from (3.6) that ff k ! 1 for all k, and (3.14) then follows from (3.12)
and (3.13) if ff k tends to zero, or from (3.13) alone if ff k is bounded away from zero.
The restriction (3.6) is imposed in order to simplify the exposition. In a more
practical setting, it may be ignored provided the definition of ff 0 and (3.10) are replaced
by
and ff
respectively, for some constant fl s 2 (0; 1), and that (3.11) is replaced by
Algorithm 3.1 may be extended in other ways. For instance, one may replace the
definition of ! 0 , the first equation in (3.12) and the first equation of (3.13) by
. The definition of j 0 and the second equation
in (3.12) may then be replaced by
for some j s ? 0. None of these extensions alter the results of the convergence theory
developed below. The values used in the LANCELOT package in a similar context are
(relation (3.15) is also used with
ensuring that 0:01). The values
also seem suitable. The parameters ! and j specify the final accuracy requested by
the user.
Finally, the purpose of the update (3.9) is to put more emphasis on the feasibility of
the constraints whose violation is proportionally higher, in order to achieve a "balance"
amongst all constraint violations. This balance then allows the true asymptotic regime
of the algorithm to be reached. The advantage of (3.9) is that this balancing effect
is obtained gradually, and not enforced at every major iteration, as is the case in
Powell [23]. Furthermore Powell's approach increases the penalties corresponding to
the constraints that are becoming too slowly feasible, based on the ' 1 -norm. Thus it
is only when they have changed sufficiently so that they are all within the constraint
violation tolerance that the Lagrange multiplier update is performed. By contrast, we
update the multipliers of the well-behaved constraints (assuming they correspond to a
particular partition - which is likely since that is, partly at least, why the partitions
exist) independently of more badly behaved ones. In addition, by virtue of using the
-norm, we do not give quite the same emphasis to the most violated constraint.
4. Global convergence analysis. We now proceed to show that Algorithm 3.1
is globally convergent under the following assumptions.
AS3: The iterates fx k g considered lie within a closed, bounded
domain\Omega\Gamma
AS4: The matrix J(x )Z has column rank no smaller than m at any limit point,
x , of the sequence fx k g considered in this paper.
We notice that AS3 implies that there exists at least a convergent subsequence
of iterates, but does not, of course, guarantee that this subsequence converges to a
stationary point, i.e. that "the algorithm works". We also note that it is always
satisfied in practice because the linear constraints (1.3) includes lower and upper
bounds on the variables, either actual or implied by the finite precision of computer
arithmetic.
Assumption AS4 guarantees that the dimension of the null space of A is large
enough to provide the number of degrees of freedom that are necessary to satisfy the
nonlinear constraints and we require that the gradients of these constraints (projected
onto this null space) are linearly independent at every limit point of the sequence of
iterates. This assumption is the direct generalization of AS3 used by Conn et al. [5].
We shall analyse the convergence of our algorithm in the case where the convergence
tolerances ! and j are both zero. We first need the following lemma, proving
that (3.3) prevents both the reduced gradient of the augmented Lagrangian and its
orthogonal complement from being arbitrarily large when ! k is small.
Lemma 4.1. Let fx k g ae B; k 2 K, be a sequence which converges to the point x
and suppose that
where the ! k are positive scalar parameters which converge to zero as k 2 K increases.
Then
for some - 1 ? 0 and for all k 2 K sufficiently large.
Proof. Observe that, for k 2 K sufficiently large, ! k is sufficiently small and
sufficiently close to x to ensure that all the constraints in D k are active at x .
This implies that the subspace orthogonal to the normals of the dominant constraints
at x k , V k , contains the subspace orthogonal to the normals of the constraints active
at x . Hence, we deduce that
where we have used (3.5) to obtain the second inequality and (3.3) to deduce the third.
This proves the first part of (4.1).
We now turn to the second. If D k is empty, then Y k is the zero matrix and the
second part of (4.1) immediately follows. Assume therefore that D k 6= ;. We first
select a submatrix -
of AD k
that is of maximal full row-rank and note that the
orthogonal projection onto the subspace spanned by the fa i g i2D k
is nothing but
A T
A T
Hence we obtain from the orthogonality of Y k , the bound jD k j - p, (3.2) and (3.4)
and the fact that all constraints in D k are active at x for k sufficiently large, that
A T
A T
A T
A T
But there are only a finite number of nonempty sets D k for all possible choices of x k
and we may thus deduce the second part of (4.1) from (4.2) by defining
A T
A T
where the minimum is taken on all possible choices of D k and -
. 2
We now examine the behaviour of the sequence fr x \Phi k g. We first recall a result
extracted from the classical perturbation theory of convex optimization problems.
This result is well known and can be found, for instance, in [12, pp. 14-17].
Lemma 4.2. Assume that U is a continuous point-to-set mapping from S ' ! '
into the power set of ! n such that the set U(') is convex and non-empty for each
S. Assume that the real-valued function F (y; ') is defined and continuous on the
space and convex in y for each fixed '. Then, the real-valued function F
defined by
is continuous on S.
We now show that, if it converges, the sequence fr x \Phi k g tends to a vector which
is a linear combination of the rows of A with non-negative coefficients.
Lemma 4.3. Let fx k g ae B,k 2 K, be a sequence which converges to the point x
and suppose that the gradients r x \Phi k , k 2 K, converge to some limit r x \Phi . Assume
furthermore that (3.3) holds for k 2 K and that ! k tends to zero as k 2 K increases.
Then,
r x \Phi
for some vector - - 0, where A is the matrix whose rows are those of A corresponding
to active constraints at x .
Proof. We first define
with the aim to show that this quantity tends to zero when k 2 K increases. We
obtain from (4.3), the Moreau decomposition [20] of r x \Phi k and the Cauchy-Schwarz
inequality, that
1g. As, for
sufficiently close to x and ! k sufficiently small, all the constraints in D k must be
active at x , we have that N k is included in the normal cone N(x ; 0) and therefore the
vector PN k
belongs to this normal cone. Moreover, since the maximization
problem of the last right-hand side of (4.4) is a concave program, since x is feasible for
(1.3), and since kx large enough, we thus deduce that
is a global solution of this problem. Observing that
we obtain that
where we used the Cauchy-Schwarz inequality to deduce the last inequality. We may
now apply Lemma 4.1 and deduce from the second part of (4.1), (4.5) and the contractive
character of the projection onto a convex set containing the origin that
and thus, from (4.4) and our assumptions, that
Our assumption on the ! k sequence then implies that oe k converges to zero as k increases
in K.
Consider now the minimization problem
d;
subject to A(x
Since the sequences fr x \Phi k g and fx k g converge to r x \Phi and x respectively, we
deduce from Lemma 4.2 applied to the optimization problem (4.3) (with the choices
and the convergence of the sequence oe k to zero that the optimal value for problem
(4.6) is zero. The vector thus a solution for problem (4.6) and satisfies
r x \Phi
for some vector - - 0, which ends the proof. 2
The important part of our convergence analysis is the next lemma.
Lemma 4.4. Suppose that AS1 and AS2 hold. Let fx k g ae B; k 2 K, be a sequence
satisfying AS3 which converges to the point x for which AS4 holds and let -
where - satisfies (2.5). Assume that f- k g, k 2 K, is any sequence of vectors and that
nonincreasing sequence of q-dimensional vectors. Suppose further
that (3.3) holds where the ! k are positive scalar parameters which converge to zero as
increases. Then
(i) There are positive constants - 2 and - 3 such that
and
for all sufficiently large.
Suppose, in addition, that c(x
(ii) x is a Kuhn-Tucker point (first-order stationary point) for the problem (1.1)-
is the corresponding vector of Lagrange multipliers, and the sequences
converge to - for k 2 K;
(iii) The gradients r x \Phi k converge to g '
Proof. As a consequence of AS2-AS4, we have that for k 2 K sufficiently large,
exists, is bounded and converges to (J(x )Z . Thus, we may write
for some constant - 2 ? 0. Equations (2.3) and (2.4), the inner iteration termination
criterion (3.3) and Lemma 4.1 give that
for all k 2 K large enough. By assumptions AS2, AS3, AS4 and (2.5), -(x) is bounded
for all x in a neighbourhood of x . Thus we may deduce from (2.5), (4.10) and (4.11)
that
Moreover, from the integral mean value theorem and Lemma 2.1 we have that
Z 1r x -(x(s))ds \Delta
where r x -(x) is given by equation (2.6), and where Now the
terms within the integral sign are bounded for all x sufficiently close to x and hence
for all k 2 K sufficiently large and for some constant - 3 ? 0, which implies the
inequality (4.8). We then have that -(x k ) converges to - . Combining (4.12) and
(4.14) we obtain
which gives the required inequality (4.7). Then, since by assumption ! k tends to zero
as k increases, (4.15) implies that - k converges to - and therefore, from the identity
(2.3), r x \Phi k converges to g ' (x ; - ). Furthermore, multiplying (2.1) by - k;j , we obtain
Taking norms of (4.16) and using (4.15), we derive (4.9).
Now suppose that
c(x
Lemma 4.3 and the convergence of r x \Phi k to g ' (x ; - ) give that
for some vector - - 0. This last equation and (4.17) show that x is a Kuhn-Tucker
point and - is the corresponding set of Lagrange multipliers. Moreover (4.7) and
(4.8) ensure the convergence of the sequences f -(x
K. Hence the lemma is proved. 2
We finally require the following lemma in the proof of global convergence, which
shows that the Lagrange multiplier estimates cannot behave too badly.
Lemma 4.5. Suppose that, for some j (1 - j - q), - k;j converges to zero as k
increases when Algorithm 3.1 is executed. Then the product - k;j k- converges to
zero.
Proof. As - k;j converges to zero, Step 3b must be executed infinitely often
for the j-th subset. Let K be the set of indices of the iterations in
which Step 3b is executed.
We consider how the j-th subset of Lagrange multiplier estimates changes between
two successive iterations indexed in the set K j . Firstly note that - kv+1;[Q j
At iteration
where the summation is null if
Substituting (4.19) into (4.18), multiplying both sides by - kv+t;j , taking norms and
using (3.9), yields
and hence
Using the fact that (3.7) holds for we deduce that
Now defining
we obtain that
for all t such that k
Thus, from (4.22) and the inequality - ! 1, if ae v converges to zero, then ffi v and
hence, from (4.21), - kv+t;j k- both converge to zero. To complete the proof
it therefore suffices to show that ae v converges to zero as v tends to infinity.
Suppose first that ff k is bounded away from zero. Then we must have that (3.13)
is used for all k sufficiently large, with ff 1). This and
the definition of ae v in (4.20) imply that
min
for sufficiently large v. As (3.13) also guarantees that j k tends to zero, we deduce that
ae v converges to zero. This completes the proof for the first case.
Suppose now that ff k converges to zero. This implies that each of the q independent
penalty parameters is reduced an infinite number of times. Consider the progress
of ff k over the course of q successive decreases (3.11). As (3.11) only happens when
the currently largest penalty parameter, - k;j say, is reduced, as (3.9) requires that this
penalty parameter is reduced by - , and because there can only possibly be at most
parameters in the interval (- k;j ; - k;j ], it follows that ff k must be
reduced by at least - over q successive decreases (3.11). Thus, considering the possible
outcomes (3.12) and (3.13), each j kv+l must be bounded by a quantity of the form
t. Furthermore, at most q such terms can involve
any particular i and t. Therefore, since - ff kv ! 1, we obtain that
Thus we see that, as ff kv converges to zero, so does ae v , completing the proof for the
second case. 2
We can now derive the desired global convergence property of Algorithm 3.1,
which is analogous to Theorem 4.4 in Conn et al. [5].
Theorem 4.6. Assume that AS1 and AS2 hold. Let x be any limit point of the
sequence fx k g generated by Algorithm 3.1 of Section 3 for which AS3 and AS4 hold
and let K be the set of indices of an infinite subsequence of the x k whose limit is x .
Finally, let - conclusions (i), (ii) and (iii) of Lemma 4.4 hold.
Proof. Our assumptions are sufficient to reach the conclusions of part (i) of
Lemma 4.4. We now show that c(x and therefore that
c(x To see this, we consider a analyze two separate cases.
The first case is when - k;j is bounded away from zero. Hence Step 3a must be
executed every iteration for k sufficiently large, implying that (3.7) is always satisfied
for k large enough. We then deduce from (3.14) that c(x k ) converge to zero.
The second case is when - k;j converges to zero. Then Lemma 4.5 shows that
tends to zero. Using this limit and (3.14) in (4.9), we obtain that
tends to zero, as desired.
As a consequence, conclusions (ii) and (iii) of Lemma 4.4 hold. 2
We finally note that global convergence of Algorithm 3.1 can be proved under
much weaker assumptions on - k;j and ! k . The reader is again referred to Conn et al.
[9] for further details.
5. Asymptotic convergence analysis. The distinction between dominant and
non-dominant (floating) linear inequality constraints has some implications in terms of
the identification of those constraints that are active at a limit point of the sequence of
iterates generated by the algorithm. Given such a point x we know from Theorem 4.6
that it is critical, i.e. that \Gammag ' for the corresponding Lagrange
multipliers - . If we now consider a linear constraint with index
is active at x , we may define the normal cone N [i]
to be the cone spanned by the
outwards normals to all linear inequality constraints active at x , except the i-th
one. We then say that the i-th linear inequality constraint is strongly active at x if
\Gammag '
. In other words, the i-th constraint is strongly active at a critical
point if this point ceases to be critical when this constraint is ignored. Let us denote by
S(x ) the set of strongly active constraints at x . All non-strongly active constraints
at x are called weakly active at x . We now prove the reasonable result that all
strongly active constraints at a limit point x are dominant for k large enough.
Theorem 5.1. Assume that AS1-AS3 hold. Let fx k g, k 2 K, be a convergent
subsequence of iterates produced by Algorithm 3.1, whose limit point is x with corresponding
Lagrange multipliers - . Assume furthermore that AS4 holds at x . Then
for all k sufficiently large.
Proof. Consider a linear inequality constraint i 2 S(x ). Then, by definition
of this latter set, we have that \Gammag '
. Since Theorem 4.6 guarantees that
r x \Phi k converges to g ' (x ; - ) and as N [i]
is closed, we have that \Gammar x \Phi k 62 N [i]
for
large enough. Therefore, one obtains from the Moreau decomposition [20] of
\Gammar x \Phi k that
for some ffl ? 0 and for all sufficiently large k 2 K, where T [i]
. We have
also from (3.3) that kP T k
(\Gammar x \Phi k )k is arbitrarily small, because ! k tends to zero (see
(3.14)). Assume now that, for some arbitrarily large k 2 K, we have that i 62 D k . This
implies that T [i]
hence that (5.1) is impossible. We therefore deduce that i
must belong to D k , which proves the theorem. 2
This result is important and is the generalization of Theorem 5.4 by Conn et al.
[5]. It can also be interpreted as a means of active constraint identification, as is clear
from the following easy corollary.
Corollary 5.2. Suppose that the conditions of Theorem 5.1 hold. Assume
furthermore that all linear inequality constraints active at x have linearly independent
normals and are non-degenerate, in the sense that
where ri[V ] denotes the relative interior of a convex set V . Then D k is identical to
the set of active linear inequality constraints at x for all k 2 K sufficiently large.
Proof. The non-degeneracy assumption and the linear independence of the
active constraints normals imply that - is unique and only has strictly negative
components. Therefore each of the active linear inequality constraints at x is strongly
active at x , and the desired conclusion follows from Theorem 5.1. 2
We note here that the non-degeneracy assumption corresponds to strict complementarity
slackness in our context (see, for instance, Dunn [11], or Burke et al. [4]).
We now make some additional assumptions before pursuing our local convergence
analysis. We intend to show that all penalty parameters are bounded away from zero.
AS5: The second derivatives of the functions f(x) and c i (x) (1 - i - m) are Lipschitz
continuous at any limit point x of the sequence of iterates fx k g.
Suppose that (x ; - ) is a Kuhn-Tucker point for problem (1.1)-(1.3) and let
I be any subset of the linear inequality constraints which are active at x
that contains all strongly active constraints (S(x plus an arbitrary
subset of weakly active constraints at x . Then, if the columns of the matrix
Z form an orthonormal basis of the subspace orthogonal to the normals of
the constraints in I, we assume that the matrix
is nonsingular for all possible choices of the weakly active constraints in the
set I.
We note that AS6 implies AS4 and seems reasonable in that the definition of
strongly and weakly active constraints may vary with small perturbations in the prob-
lem, for instance when lies in one of the extreme faces of the cone N .
Our assumption might be seen as a safeguard against the possible effect of all such
perturbations.
We now make the distinction between the subsets for which the penalty parameter
converges to zero and those for which it stays bounded away from zero. We define
We also denote
and
ae k
We now prove an analog to Lemma 5.1 by Conn et al. [5] which is suitable for our
more general framework.
Lemma 5.3. Assume that AS1-AS3 hold. Let fx k g, k 2 K, be a convergent
subsequence of iterates produced by Algorithm 3.1, whose limit point is x with corresponding
Lagrange multipliers - . Assume that AS5 and AS6 hold at x . Assume
furthermore that Z 6= ;.
there are positive constants -
and an integer k 1
such that, if ff k 1 -
ff, then
and
(ii) If, on the other hand, P 6= ;, there are positive constants -
and an integer k 1 such that, if - k 1 ;Z -
ff, then
and
Proof. We will denote the gradient and Hessian of the Lagrangian function,
taken with respect to x, at the limit point
and H '
, respectively. Similarly,
J will denote J(x ). We also define We observe that the assumptions
of the lemma guarantee that Theorem 4.6 can be used.
We first note that there is only a finite number of possible D k , and we may thus
consider subsequences of K such that D k is constant in each subsequence. We also
note that each k 2 K belongs to a unique such subsequence. In order to prove our
result, it is thus sufficient to consider an arbitrary infinite subsequence -
K such that,
is independent of k. This "constant" index set will be denoted by D.
As a consequence, the cones N k and T k , the subspaces V k and W k and the orthogonal
matrices Z k and Y k are also independent of k; they are denoted by N , T , V , W , Z
and Y , respectively.
Using (2.3) and Taylor's expansion around x , we obtain that
where
ds
and
The boundedness and Lipschitz continuity of the Hessian matrices of f and c i in a
neighbourhood of x , together with the convergence of -
- k to - then imply that
and
for some positive constants - 8 and - 9 . Moreover, using Taylor's expansion again,
along with the fact that Theorem 4.6 ensures the equality c(x we obtain that
where
Z 1s
ds
(see Gruver and Sachs [16, page 11]). The boundedness of the Hessian matrices of the
c i in a neighbourhood of x then gives that
for some positive constant - 10 . Combining (5.9) and (5.12), we obtain
!/
where we have suppressed the arguments of the residuals r 1 , r 2 and r 3 for brevity.
Using the orthogonal decomposition of ! n into V \Phi W and defining
we may rewrite (5.14) as
r 4
where r 4
. Expanding this last equation gives thatB @
(5.15)B @
r
We now observe that (3.3), the inclusion V ' T and the fact that ! k tends to zero
imply that
0:
Substituting (5.16) in (5.15), removing the middle horizontal block and rearranging
the terms of this latter equation then yields that
!/
Roughly speaking, we now proceed by showing that the right-hand side of this relation
is of the order of ' k
We will then ensure that the vector on the left-hand side is of the same size, which is
essentially the result we aim to prove. We first observe that
from (4.1). We then obtain from (4.7) and (5.19) that
. Furthermore, from (5.10), (5.11), (5.13), (5.19) and (5.20),
. We now bound c(x k ) by distinguishing components from
Z and P . We first note that, since the penalty parameters for each subset in P are
bounded away from zero, the test (3.7) is satisfied for all k sufficiently large. Moreover,
the remaining components of c(x k ) satisfy the bound
for all j 2 Z and all k sufficiently large, using (4.9). Hence, using (5.3), (3.7) and
(5.22), we deduce that
Note that the first term of the last right-hand side only appears if P is not empty.
Since the algorithm ensures that
because may obtain from (4.1), (5.23) and (5.19) that
assumption AS6, the coefficient
matrix on the left-hand side of (5.17) is nonsingular. Let M be the norm of its
Multiplying both sides of the equation by this inverse and taking norms, we
obtain from (5.18), (5.21), (5.24) and (5.25) that
Suppose now that k is sufficiently large to ensure that
and let
Recall that ff 0 and hence -
ff, the relations (5.26)-(5.28) give
As converge to zero, we have that
for k large enough. Hence inequalities (5.29) and (5.30) yield that
If P is empty, we use (5.19), (5.31) and (5.18), the fact that - and the
inequality
to deduce (5.4), where - 4
deduce (5.5) from (4.7) and (5.4). Now, using (2.1),
and (5.6) then follows from (5.32) and (5.5).
If, on the other hand, P is not empty, (5.7) results from (4.7), (5.19), (5.31) with
Finally, (5.8)
results from (2.1) and (5.7). 2
For the remaining of this section, we will restrict our attention to the case where
the sequence of iterates converges to a single limit point. Obviously, this makes AS3
unnecessary. We briefly comment at the end of the section on why this additional
assumption cannot be relaxed.
We now show that, if the maximum penalty parameter ff k converges to zero, then
the Lagrange multiplier estimates - k converge to their true values - .
Lemma 5.4. Assume AS1 and AS2 hold. Assume that fx k g, the sequence of
iterates generated by Algorithm 3.1, converges to the single limit point x at which
AS6 holds, and with corresponding Lagrange multipliers - . Then, if ff k tends to zero,
the sequence - k converges to - .
Proof. Recall that AS6 implies AS4 and therefore that our assumptions are
sufficient to apply Theorem 4.6.
We observe that the desired convergence holds if - k;[Q j ] converges to -
It is thus sufficient to show this latter result for an arbitrary j between 1
and q. The result is obvious if Step 3a is executed infinitely often for the j-th subset.
Indeed, each time this step is executed, -
and the inequality (4.7)
guarantees that -
converges to - ;[Q j ] . Suppose therefore that Step 3a is not
executed infinitely often for this subset. Then k(- remain fixed for all
executed for each remaining iteration. But
then implies that kc(x k )
As ff k tends to zero and ff
k sufficiently large for which ff k strictly decreases. But then inequality (3.7) must be
satisfied for some k - k 3 , which is impossible, as this would imply that Step 3a is
again executed for the j-th subset. Hence Step 3a must be executed infinitely often.We now consider the behaviour of the maximum penalty parameter ff k and show
the important result that, under stated assumptions, it is bounded away from zero.
The proof of this result is inspired by the technique developed by Conn et al. [5]. When
the single penalty parameter definition of the augmented Lagrangian (1.4) is used (or,
equivalently, when one then avoids a steadily increasing ill-conditioning of the
Hessian of the augmented Lagrangian. Note that this ill-conditioning is also avoided
when q ? 1, as we show below in Theorem 5.6.
Theorem 5.5. Assume AS1 and AS2 hold and suppose that the sequence of
iterates fx k g of Algorithm 3.1 converges to a single limit point x with corresponding
Lagrange multipliers - , at which AS5 and AS6 hold. Then there is a constant ff min 2
(0; 1) such that ff
Proof. Suppose otherwise that ff k tends to zero (that is
that - k;j tends to zero for each j between 1 and q. Then Step 3b must be executed
infinitely often for each subset. We aim to obtain a contradiction to this statement by
showing that Step 3a is always executed for each subset for sufficiently large k. We
note that our assumptions are sufficient to apply Theorem 4.6. Furthermore, we may
apply Lemma 5.3 to the complete sequence of iterates.
First observe that
for all k - k 1 , where -
ff and k 1 are those of Lemma 5.3. Note that
for all k - k 1 . This follows by definition if (3.12) is executed. Otherwise it is a
consequence of the fact that ff k is unchanged while ! k is reduced, when (3.13) occurs.
Let k 4 be the smallest integer k such that
and
that (5.33) and (5.35) imply that
5 be such that
for all k - k 5 , which is possible because of Lemma 5.4. Now define k
let \Gamma be the set fk j (3.12) is executed at iteration k \Gamma 1 and k - k 6 g and let k 0 be the
smallest element of \Gamma. By the assumption that ff k tends to zero, \Gamma has an infinite number
of elements.
By definition of \Gamma, for iteration k 0 ,
. Then inequality
(5.6) gives that, for each j,
(from (5.36))
(from (5.33))
(from (5.34)).
As a consequence of this inequality, Step 3a will be executed for each j with - k 0
Inequality (5.5) together with (5.37) guarantee that
We shall now make use of an inductive proof. Assume that, for each j, Step 3a is
executed for iterations and that
Inequalities (5.38) and (5.39) show that this is true for We aim to show that
the same is true for Our assumption that Step 3a is executed gives that, for
iteration
. Then,
inequality (5.6) yields that, for each j,
(from (5.40))
(from (5.36))
Hence Step 3a will again be executed for each j with
Inequality (5.5) then implies that
(from (5.40))
(from (5.35))
which establishes (5.40) for executed for each
all iterations k - k 0 . But this implies that \Gamma is finite, which contradicts the assumption
that Step 3b is executed infinitely often for each subset. Hence the theorem is proved.This theorem was all that was needed in Conn et al. [5]. However, the situation
is more complex here because q may be larger than one. If the ill-conditioning of
the Hessian is to be avoided, we must now prove the stronger result that all penalty
parameters stay bounded away from zero.
Theorem 5.6. Assume AS1 and AS2 hold and suppose that the sequence of
iterates fx k g of Algorithm 3.1 converges to a single limit point x with corresponding
Lagrange multipliers - , at which AS5 and AS6 hold. Then there is a constant - ? 0
such that - k;j - for all k and all
Proof. Assume otherwise that Z is not empty, and hence that - k;Z converges
to zero. Then Step 3b must be executed infinitely often for j 2 Z . We aim to obtain
a contradiction to this statement by showing that, for any j 2 Z , Step 3a is always
executed for sufficiently large k. We may deduce from Theorem 5.5 that ff k attains
its minimum value ff min 2 (0; 1) at iteration k max , say. Hence, P 6= ;. Furthermore,
we may apply Lemma 5.3 to the complete sequence of iterates. Let k 7 - k max be the
smallest integer for which
ff and k 1 are those of Lemma 5.3, and where
Note that ff fi j +ffl
Consider the j-th subset, for some j 2 Z . At iteration k - k 7 , the algorithm
ensures that
if Step 3b is executed for the j-th subset, while (5.7) ensures that
if Step 3a is executed for the same subset. Summing on all j 2 Z , and defining
executed for the j-th subset at iteration kg
executed for the j-th subset at iteration kg;
we obtain that
ae
For the purpose of obtaining a contradiction, assume now that
ae k -2
for all k - k 7 . Then (5.42) gives that, for all k - k 7 ,
ae k+1
ae k
because of (5.41). Hence we obtain from (5.44) that
ae
Therefore, since ae k 7 ff (k\Gammak 7 +1)ffl
min tends to zero, we obtain that
ae k+1 !2 ff ff j+(k 7 \Gammak
min ff (k\Gammak 7 +1)fij
for all sufficiently large k, where the last equality results from the definition of k max
and (3.13). But this contradicts (5.43), which implies that (5.43) does not hold for all
sufficiently large. As a consequence, there exists a subsequence K such that
ae k !2
for all k 2 K. Consider such a k. Then, using (5.42) and (5.45), we deduce that
ae
where we have used (5.41) to obtain the second inequality. As a consequence, k+1 2 K
and (5.45) holds for all k sufficiently large. Returning to subset j 2 Z , we now obtain
from (5.8) and (5.45) that
for all k sufficiently large, because of (5.41). Hence Step 3a is executed for the subset
j and for all sufficiently large k, which implies that j does not belong to Z . Therefore
Z is empty and the proof of the theorem is completed. 2
As in Conn et al. [5], we examine the rate of convergence of our algorithms.
Theorem 5.7. Under the assumptions of Theorem 5.6, the iterates x k and the Lagrange
multipliers -
- k of Algorithm 3.1 are at least R-linearly convergent with R-factor
at most ff fi j
min , where ff min is the smallest value of the maximum penalty parameter
generated by the algorithm.
Proof. The proof parallels that of Lemma 5.3. First, Theorem 5.5 shows that
the maximum penalty parameter ff k stays bounded away from zero, and thus remains
fixed at some value ff min ? 0, for k - k max . For all subsequent iterations,
(5.
hold. Moreover, Theorem 5.6 implies that, for all hold for all
sufficiently large. Hence and because of (4.1), the bound on the right-hand
side of (5.25) may be replaced by -
Therefore, if k is sufficiently large that
and
inequalities (5.47)-(5.49) can be rearranged to yield
But then (5.19) gives that
(5.50) show that x k converges to x at least R-linearly, with R-factor ff fi j
min
. Inequalities
(4.7) and (5.50) then guarantee the same property for -
To conclude this section, we note that the conclusions of Theorems 5.5, 5.6 and 5.7
require that the complete sequence of iterates converges to a unique limit point. As
indicated above, this assumption cannot be relaxed. The counterexample presented
by Conn et al. [5] (where the linear inequality constraints are simple bound constraints
on the problem's variables) shows that the sequence of penalty parameters may indeed
converge to zero, if there is more than a single limit point.
6. Second order conditions. If we further strengthen the stopping test for the
inner iteration beyond (3.3) to include second-order conditions, we can then guarantee
that our algorithms converge to an isolated local solution. More specifically, we require
the following additional assumption.
AS7: Suppose that x k satisfies (3.3), converges to x for k 2 K, such that Z has
a rank strictly greater than m. Then, if Z is defined as in AS6, we assume
that Z T r xx \Phi k Z is uniformly positive definite (that is, its smallest eigenvalue
is uniformly bounded away from zero) for all k 2 K sufficiently large.
We can then prove the following result.
Theorem 6.1. Under assumptions AS1-AS3, AS5-AS7, the iterates x k , k 2 K,
generated by Algorithm 3.1 converge to an isolated local solution of (1.1)-(1.3).
Proof. By definition of \Phi,
r xx \Phi
(x) is the Jacobian of c(x) [Q j ] . Note that the rank of Z is at least that of
Z . AS7 then implies that there exists a nonzero vector s such that
and hence
for each j. For any such vector, AS7 further implies that
for some - 21 ? 0, which in turn gives that
because of (6.1) and (6.2). By continuity of H ' as x k and -
approach their limits,
this ensures that
for all nonzero s satisfying
which implies that x is an isolated local solution of (1.1)-(1.3) (see, for instance,
Avriel [2, Thm. 3.11]). 2
If we assume that the inner iteration stopping test is tightened so that r xx \Phi k is
required to be uniformly positive definite in the null space of the dominant constraints,
and if we assume that the non-degeneracy condition (5.2) holds, then Corollary 5.2
ensures that Z sufficiently large k and Theorem 6.1 holds. A weaker
version of this result also holds, where only positive semi-definiteness of the augmented
Lagrangian's Hessian is required, yielding then that x is a (possibly not isolated)
minimizer of the problem.
7. Extensions.
7.1. Flexible Lagrange multiplier updates. The formula (2.1) has definite
advantages for large-scale computations, but may otherwise appear unduly restrictive.
The purpose of the first extension we consider is to introduce more freedom in our
algorithmic framework, by replacing this formula by a more general condition, allowing
a much larger class of Lagrange multiplier updates to be used. More specifically, we
consider modifying Algorithm 3.1 as follows.
Algorithm 7.1
This algorithm is identical to Algorithm 3.1, except that Step 3 is replaced
by the following, where fl is a constant in (0; 1).
Step 3 [Disaggregated updates]. Compute a new vector of Lagrange multiplier
estimates - k+1 . For
or Step 3b otherwise.
Step 3a [Update Lagrange multiplier estimates]. Set
Step 3b [Reduce the penalty parameter]. Set
where - k;j is defined by (3.9) in Algorithm 3.1.
Algorithm 7.1 allows a more flexible choice of the multipliers than Algorithm 3.1,
but requires that some control is enforced to prevent their growth at an unacceptably
fast rate. It covers, among others, the choice of the least-squares estimates -(x) as
defined in (2.5).
The global convergence theory presented in Section 4 for Algorithm 3.1 can be
extended to cover Algorithm 7.1. This extension is detailed in Conn et al. [9]. Conn
et al. [10] extend the local convergence analysis of Section 5 to Algorithm 7.1, under
the additional condition that
holds for some positive constants - 22 and - 23 and all k 2 K sufficiently large, where K
is the index set of a subsequence of iterates (generated by Algorithm 7.1) converging
to the critical point x with corresponding Lagrange multipliers - . Both (2.1) and
(2.5) satisfy this condition because of Theorem 4.6.
We also note that Corollary 5.2 ensures that the least-squares multiplier estimates
are implementable when the non-degeneracy condition (5.2) holds. By this we
mean that the estimates
are identical to those defined in (2.5) for all k sufficiently large, and, unlike (2.5), are
well defined when x is unknown.
7.2. Alternative criticality measures. In Algorithms 3.1 and 7.1 we used the
criticality measure kP T k
in order to define the stopping criterion of the inner
iteration (see (3.3)), because it is general. However, this quantity might not be easily
computed in the course of the numerical method used to calculate x k , especially when
the dimension of the problem is high. It is therefore of interest to examine other
criticality measures that might be easier to calculate. It is the purpose of this section
to analyze such alternative proposals.
Given D k , N k , and AD k
as above, we first claim that (3.3) can be replaced by the
requirement that there exists a set of non-positive "dominant multipliers" f- ik g i2M k
is the jD k j-dimensional vector whose i-th component is - ik if or zero
otherwise. We prove this claim.
Lemma 7.1. Assume that there exists a non-positive - k such that (7.1) holds at
x k . Then (3.3) also holds at x k .
Proof. Since the vector A T
belongs, by construction, to the cone N k defined
in (3.4), we can immediately deduce from the definition of the othogonal projection
and (7.1) that
which is the desired inequality. 2
Condition (7.1) is appealing for two reasons. Firstly, a set of (possibly approx-
imate) multipliers is available in many numerical procedures that might be used to
perform the inner iteration and to compute a suitable x k ; one can then select those
multipliers which correspond to the dominant constraints, further restrict this choice
to the non-positive ones and finally check (7.1). Such a scheme is implicitly used by
both the Harwell [17] barrier-function quadratic programming codes VE14 and VE19
and the IMSL [19] general linearly constrained minimization package LCONG.
Alternatively, suitable multipliers can be computed, for instance by (approxi-
mately) solving the least-squares problem
min
-k
and selecting the non-positive components of the resulting vector -, or by (approxi-
mately) solving the constrained least-squares problem
min
-k:
Condition (7.1) is also appealing as it provides, in a single condition, both a stopping
condition on the inner iteration and a measure of the tolerated "inexactness" in solving
the associated least-squares problem, if this is the procedure chosen to obtain the
dominant multipliers.
We may therefore deduce from Lemma 7.1 that the convergence theory holds for
Algorithms 3.1 and 7.1 whenever (7.1) is used instead of (3.3).
Condition (7.1) can be further specialized. For instance, one might choose to
impose the familiar "reduced gradient" criterion
is an orthogonal matrix whose columns span the null space of the constraints
active at x k , provided that the multipliers associated with these linear constraints
are all non-positive. In this case, we have that
because T the tangent cone to the set determined by the linear inequality
constraints active at x k , contains T k . As a consequence, the convergence theory still
holds when this criterion, which has been implemented by several subroutines for
minimizing a general objective function subject to linear constraints (for example,
the NAG [22], quadratic programming code E04NFF and the more general package
E04UCF), is used as an inner-iteration stopping rule within Algorithms 3.1 and 7.1.
This is also true for reduced gradient methods (e.g. MINOS [21], or LSNNO [24])
which compute a full column rank matrix -
whose columns are generally non-
orthonormal but depend upon a subset of the (finite number) of coefficients for the
linear constraints. Indeed, the norm of -
bounded above and away from
zero, and a relationship that is a weighted form of (7.3) thus also holds in these cases.
In order to preserve coherence with the framework presented in Conn et al. [8],
we finally note that oe k as defined in (4.3) may also be viewed as a criticality measure.
Hence we might decide to stop the inner iteration when
The reader is referred to Conn et al. [9] for a proof that global convergence is still obtained
for this modification of Algorithms 3.1 and 7.1. However, the authors have not
been able to prove the desired local convergence properties with only (7.4). Instead,
the local convergence theory is covered for Algorithms 3.1 and 7.1 for the stronger
condition
(see Conn et al. [10] for details). This condition is theoretically interesting, but might
be practically too strong. Note, as we now show, that it implies a variant of (3.3).
Theorem 7.2. Assume that fx k g, k 2 K, is a convergent subsequence of vectors
of B such that (7.5) holds for each k 2 K, where the ! k converge to zero as k increases
in K. Then the inequality
also holds for each k 2 K sufficiently large and for some - 24 - 1.
Proof. We first consider the simple case where that is when no linear
inequality is present. In this case, it is easy to check from (4.3) that oe
But we must have that D
(\Gammar x \Phi k )k. We therefore obtain that
holds with - large enough to ensure that ! k - 1.
Assume now that p ? 0. The Moreau decomposition of \Gammar x \Phi k [20] is given by
obviously holds for any choice of - 24 . Assume
therefore that P T k
nonzero. We now show that x k B, where we
define
kAk1
Assume first that i 2 D k . Then \Gammaa i 2 N k and a T
because of the polarity of N k
and T k . Since x k 2 B, we obtain that
a T
On the other hand, if i 62 D k , we have that a T
(a T
Gathering (7.8) and (7.9), we obtain that x k B, as desired. Furthermore,
since kd k k - 1 by definition, we have verified that d k is feasible for the minimization
problem (4.3) associated with the definition of oe k . Hence,
where we have used successively the Moreau decomposition of \Gammar x \Phi k , the definition
of d k and the orthogonality of the terms in the Moreau decomposition. If ffl
(7.5) and (7.10) imply that
sufficiently large. Otherwise, we deduce from (7.10), (7.5) and (7.7) that
As a consequence of (7.11) and (7.12), we therefore obtain that (7.6) holds with
Combining all cases, we conclude that (7.6) holds with this last value of - 24 . 2
We finally note that Lemma 7.1 and Theorem 7.2 do not depend on the actual
form of the augmented Lagrangian (1.5), but are valid independently of the function
minimized in the inner iteration. This observation could be useful if alternative
techniques for augmenting the Lagrangian are considered for a merit function.
8. Conclusion. We have considered a class of augmented Lagrangian algorithms
for constrained nonlinear optimization, where the linear constraints present in the
problem are handled directly and where multiple penalty parameters are allowed.
The algorithms in this class have the advantage that efficient techniques for handling
linear constraints may be used at the inner iteration level, and also that the sparsity
pattern of the Hessian of the augmented Lagrangian is independent of that of the
linear constraints. The global and local convergence results available for the specific
case where linear constraints reduce to simple bounds have been extended to the more
general and useful context where any form of linear constraint is permitted.
We finally note that the theory presented is directly relevant to practical compu-
tation, as the inner iteration stopping rule (3.3) covers the type of optimality tests
used in available packages for linearly constrained problems. This means that these
packages can be applied to obtain an (approximate) solution of the subproblem, and
constitutes a realistic and attractive algorithmic development.
It is now the authors' intention to perform extensive numerical experiments on
large-scale problems. This development requires considerable care and sophistication
if an efficient solver for the subproblem is to be integrated with the class of algorithms
described here.
Acknowledgements
. The authors wish to acknowledge funding provided by a
NATO travel grant. They are also grateful to J. Nocedal and the anonymous referees
for their constructive comments.
--R
Computing a search direction for large-scale linearly constrained nonlinear optimization calculations
Nonlinear Programming: Analysis and Methods.
Constrained Optimization and Lagrange Multiplier Methods.
Convergence properties of trust region methods for linear and convex constraints.
A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds.
LANCELOT: a Fortran package for large-scale nonlinear optimization (Release
On the number of inner iterations per outer iteration of a globally convergent algorithm for optimization with general nonlinear equality constraints and simple bounds.
Global convergence of a class of trust region algorithms for optimization using inexact projections on convex constraints.
Global convergence of two augmented Lagrangian algorithms for optimization with a combination of general equality and linear constraints.
Local convergence properties of two augmented Lagrangian algorithms for optimization with a combination of general equality and linear constraints.
On the convergence of projected gradient processes to singular critical points.
Introduction to sensitivity and stability analysis in nonlinear programming.
Practical Methods of Optimization.
Newton methods for large-scale linear equality-constrained minimization
On the convergence of a sequential penalty function method for constrained minimization.
Algorithmic Methods in Optimal Control.
A catalogue of subroutines (release 11).
Multiplier and gradient methods.
D'ecomposition orthogonale d'un espace hilbertien selon deux c-ones mutuellement polaires
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A method for nonlinear constraints in minimization problems.
LSNNO: a Fortran subroutine for solving large scale nonlinear network optimization problems.
Indefinite systems for interior point methods.
--TR | constrained optimization;augmented Lagrangian methods;convergence theory;linear constraints |
588924 | The Effective Energy Transformation Scheme as a Special Continuation Approach to Global Optimization with Application to Molecular Conformation. | This paper discusses a generalization of the function transformation scheme used in Coleman, Shalloway, and Wu [Comput. Optim. Appl., 2 (1993), pp. 145--170; J. Global Optim., 4 (1994), pp. 171--185] and Shalloway [Global Optimization, C. Floudas and P. Pardalos, eds., Princeton University Press, 1992, pp. 433--477; Global Optim., 2 (1992), pp. 281--311] for global energy minimization applied to the molecular conformation problem. A mathematical theory for the method as a special continuation approach to global optimization is established. We show that the method can transform a nonlinear objective function into a class of gradually deformed, but ``smoother'' or ``easier'' functions. An optimization procedure can then be successively applied to the new functions to trace their solutions back to the original function. Two types of transformation are defined: isotropic and anisotropic. We show that both transformations can be applied to a large class of nonlinear partially separable functions, including energy functions for molecular conformation. Methods to compute the transformation for these functions are given. | Introduction
We are interested in solving the global minimization problem for molecular
conformation, especially protein folding.
How protein folds is one of the key biophysical problems of the decade.
Protein folding is fundamental for almost all theoretical studies of proteins
and protein-related life processes. It has many applications in the biotechnology
industry, notably, structure-based drug design for the treatment of
important diseases such as cancer and AIDS.
Optimization approaches to the protein folding problem are based on the
hypothesis that the protein native structure corresponds to the global minimum
of the protein energy. The problem can be attacked computationally
by minimizing the protein energy over all possible protein structures. The
structure with the lowest energy is presumed to be the most stable protein
structure.
Mathematically, for a protein molecule of n atoms, let
ng represent the molecular structure with each x i specifying the spatial
position of atom i. Then the energy minimization problem for protein folding
is to globally minimize a nonlinear function f(x) for all x 2 S, namely,
min x2S f(x); (1)
where S is the set of all possible molecular structures. The objective function
f(x) is the energy function for the protein. The usual form of f(x) is
is the pairwise energy function determined by
the distance between atoms i and j. A widely used pairwise energy
function is the Van der Waals energy function,
are all physical constants (see [2]).
Problem (1) is very difficult to solve in general. The reasons are as fol-
lows: First, in theory even simple versions of the problem have been proved
to be NP-complete [9]. Second, in practice the objective function often contains
exponentially many local minimizers; therefore, search for the global
minimizer can be computationally intractable. Third, the protein molecules
tend to be very large, typically containing O(10,000) atoms. For such large
problems, the required computation is unaffordable using general global optimization
methods.
However, because of its great practical importance, Problem (1) has been
studied intensively in many areas of computational science and optimiza-
tion. New algorithms on both sequential and parallel machines have been
developed; a variety of small to medium sizes of problems have been studied
[3, 4, 5, 6, 11, 12, 13, 14, 15, 17, 18, 19, 20]. In recent efforts smoothing
techniques are specifically designed for molecular conformation via global
minimization. Examples include the diffusion equation method [11, 14], the
packet annealing method [17, 18], as well as the effective energy simulated
annealing method [4, 5]. The basic idea behind these methods is to use special
techniques to smooth a given energy function so that search for a global
minimizer becomes more tractable. The methods usually use function transformation
schemes to transform a given energy function into a class of new
functions. A solution tracing procedure is then applied to the new functions
to locate a solution for the original function.
In this paper, we discuss an important generalization of the effective energy
transformation scheme introduced in [4, 5, 17, 18]. Instead of applying
the transformation to the probability distribution function, we now transform
the functions directly, generalizing the method to a broader class of
functions. More important, with this generalization, a mathematical theory
for the transformation as a special continuation approach to global optimization
is established. We show that the method can transform a nonlinear
objective function into a class of gradually deformed, but "smoother" or
"easier" functions. An optimization procedure can then be applied to the
new functions successively, to trace their solutions back to the original func-
tion. Two types of transformation are defined: isotropic and anisotropic.
We show that both transformation types can be applied to a large class of
nonlinear partially separable functions which includes typical energy functions
for molecular conformation. Methods to compute the transformation
for these functions are given.
The paper is organized as follows. Section 2 introduces the basic approach
and describes the function transformation method. Section 3 studies
the mathematical properties of the transformation as a special continuation
process. Section 4 characterizes the "smoothness" property and shows that
the transformed function becomes "smoother" in the sense that the small
high-frequency variations in the original function are averaged out after the
transformation. The numerical applicability of the transformation is discussed
in Section 5. The transformation is extended to the anisotropic type
in Section 6. The formulas to compute the transformation for molecular
conformation energy functions are derived. Finally, Section 7 contains concluding
remarks.
2 The Approach
In this section, we describe our function transformation idea which, in turn,
defines our basic approach to global optimization.
Suppose that we have a "poorly-behaved" nonlinear function with many
local minimizers. Because of "nonsmoothness," this type of function can be
very hard to minimize either locally or globally. To overcome this difficulty,
we suggest using a special technique to transform the objective function into
a class of gradually deformed, but "smoother" or "easier" functions. An optimization
procedure can then be applied to these new functions successively,
to trace their solutions back to the original function.
To deform the function, we define a parametrized integral transformation
as follows:
Given a nonlinear function f , the transformation
f is defined such that for all x,
Z
or equivalently;
Z
where - is a positive number and C - is a normalization constant such that
Z
Note that in contrast to the approaches in [4, 5, 17, 18], the transformation
here applies directly to the given function instead of its probability
distribution. This approach simplifies the transformation, and also makes it
much easier to compute and analyze.
To understand this transformation, consider that, given a random function
distribution function p(x 0 ) for the random variable
the expectation of the function g with respect to p is
Z
In light of (7), the transformation (4) yields a function value for
any x equal to the expectation for f sampled by a Gaussian distribution
function centered at x.
For example, consider the following nonlinear function:
which is a quadratic function augmented with a "noise" function. The transformation
for this function can be computed:
The function value fixed x is equal to the integration with
respect to the product of two functions, the original function f(x 0 ) and the
Gaussian distribution function p(x 0
Figure
(a)). The
parameter - determines the size of the dominant region of the Gaussian.
Since the most significant part of the integration is that within the dominant
region of the Gaussian, !f? - (x) can be viewed as the average value for the
original function f within a small -neighborhood around x. If - is equal to
zero, the transformed function is exactly the original function. For positive
-, the original function variations in small regions are averaged out, and the
transformed function will become "smoother" (Figure 1 (b)).
Figure
shows how the function behaves with increasing
-. Observe that when 0:0, the function is the original function; when
we increase - to 0.1, the function becomes "smoother;" when - is increased
further to 0.2, the function becomes entirely "smooth."
Figure
3 illustrates what the transformation implies for optimization. A
standard optimization procedure, the quasi-Newton method, is applied to
(a)
Figure
1: A one dimensional transformation example
Figure
2: A class of gradually deformed functions
the three functions in Figure 2. Figure 3 (a), (b), and (c) contain the corresponding
solutions x obtained with different choices of initial guesses x ffi .
Although globally convergent, the method may not find the right solution
if the "noise" is large. So for the function in Figure 2 (c), the method converged
to the right solution only when the initial guess was close enough to
the solution. When the initial guess was far from the solution, the method
failed to find the right solution (Figure 3 (c)). For the function in Figure 2
(b), although it is "smoother," the behavior of the method is essentially
unchanged. However, for the function in Figure 2 (a), the method always
converged to the right solution (Figure 3 (a)). If we apply the procedure to
the functions in Figure 2 (a) to (c) successively, and at each step take the
solution for the previous function as the starting point, the solutions for all
these functions can then be obtained.
The experiment above suggests a general global optimization method: to
optimize a difficult function, use the transformation technique to deform the
function into a class of "smoother" or "easier" functions, and then apply an
optimization procedure to the functions successively, to trace their solutions
back to the original function.
Continuation
What is the difference between the suggested approach and general homotopy
methods? The answer is that this approach is indeed a special type of
homotopy method. But the transformation is different from conventional ho-
motopies, and has the following three special features: First, the transformed
functions are not arbitrarily deformed functions. They all are approximations
to the original function in the sense that they are coarse estimates. Second,
the transformation is defined by a special parametrized integral transforma-
tion. If we increase the value of the parameter, the transformed function
will become "smoother" with small variations gradually removed, but maintaining
the overall function structure. Finally, if we apply an optimization
procedure to a transformed function, the obtained solution usually is close to
the solution for the original function. All these features are good for global
optimization (also for robust local optimization), but are not necessarily the
properties of conventional homotopies.
We show in the following that the proposed transformation is indeed
x*
(a)
x*
(b)
x*
(c)
Figure
3: The solutions for the functions in Figure 2 obtained by the quasi-Newton
method with different initial guesses
a well-defined homotopy and determines for any initial solution a unique
solution curve containing the stationary points for the transformed functions.
Assumption 1 The objective function f is twice continuously differentiable,
and the transformation (4) is well defined for the function as well as all its
derivatives.
Assumption 2 Let g be the gradient of f , and \Delta the Laplace operator
Then the operation \Delta can be applied to g, and the transformation (4) is well
defined for all derivatives involved. Also, \Deltag(x) is uniformly bounded and
satisfies a Lipschitz condition:
Assumption 3 The transformation !r 2 f ? - (x) satisfies a Lipschitz condition
and its inverse is uniformly bounded.
Note that to guarantee Assumptions 1 to 3, a sufficient condition on
f is that f and its derivatives are all integrable in terms of parametrized
integration (4).
We first state two sets of standard results for the proposed transformation
in the following lemmas without proof.
Assumption 1, 8-; x,
Assumption 1, 8x,
lim
lim
lim
For convenience, we define a function h(-; x), - 2 \Gamma, and x 2 S such that
is a vector space. With this
definition, the condition for x to be a stationary point of
x
Theorem 1 and h be defined as in (18). Then under
Assumptions 1 and 2, h 00
x-
exists and is uniformly bounded for all
and x 2 S, and also satisfies a Lipschitz condition in x:
In addition,
x-
Proof: Let p(-; x) be the Gaussian distribution function defined as follows:
-). Then by the definition of !f? - ,
Z
By Lemma 1,
x
Z
After differentiating (24) with respect to -, it follows that
Z
\Gamma-
Z
where
Z
Z
It is easy to verify that
Also note that
Z
Z
Therefore,
Replacing by (31), we see that
By Assumption 2, ! f 000 ? - (x) is well defined and uniformly bounded.
x-
exists and is uniformly bounded for all
f 000 (x) satisfies a Lipschitz condition by Assumption 2,
satisfies a Lipschitz condition:
immediately. So h 00
x-
(-; x) satisfies a Lipschitz
condition in x. 2
Theorem 2 Let f : R n ! R and h be defined as in (18). Then under
Assumptions 1 and 2, h 00
x-
exists and is uniformly bounded for all
and x 2 S, and also satisfies a Lipschitz condition in x:
In addition,
x-
\Deltag ? -
Proof: Let p n (-; x) be the Gaussian distribution function
where c n
-) n . Then by the definition of !f ? - ,
Z
By Lemma 1,
x
Z
Differentiate (38) with respect to - to obtain
Z
where
Z
Z
Z
Z
Z
where
Z
Z
From the proof of Theorem 1,
Substitute (44) back into (41) to obtain
Then
n- 2!g? - (x) (46)
and
Similar to the proof of Theorem 1, it follows immediately that h 00
x-
exists and is uniformly bounded for all - 2 \Gamma and x 2 S, and also satisfies a
Lipschitz condition in x. 2
Finally, we state and prove the main theorem in this section as follows:
Theorem 3 Let f be a function for which Assumptions 1, 2, and 3 all hold.
Then for any stationary point x 0 of there is a continuous and
differentiable curve x(- 2 \Gamma, such that x
stationary point of !f? - . The curve x(-) is also the unique solution of the
initial value problem
Proof: Since x 0 is a stationary point of !f? - 0
By Assumptions 1, 2, 3, Lemmas 1, 2, and Theorem 2, function h 0
x
is continuously
differentiable at all (-; S. So by the Implicit Function
Theorem, there is a continuously differentiable function x(-) at a neighborhood
of - 0 , such that x
x
for all - in the neighborhood.
We now show that x(-) also is defined uniquely in \Gamma.
By differentiating (51), we see that x(-) is a solution to the initial value
problem:
xx
x-
which, by Lemma 1 and Theorem 2, is equivalent to the problem (48)-(49).
Then it suffices to show that the right-hand side of (52) satisfies a Lipschitz
condition in x on \Gamma \Theta S, which guarantees a unique solution x(-) in \Gamma by
standard ordinary differential equation theory [10].
Under Assumption 3, for h 00
xx
xx
By Theorem 2, for h 00
x-
x-
Let
xx
x-
Then it is easy to verify that G(-; x) satisfies a Lipschitz condition in x on
with which completes the proof. 2
4 Smoothness
In Section 2 we illustrated that the transformed functions are "smoother"
than the original function in the sense that they vary slower and may even
have fewer local minimizers. In the following, we characterize more rigorously
the "smoothness" of the transformation.
f be the Fourier transformation for function f , and d
!f? - for function
Recall that the transformation !f ? - for f is just a convolution of
f and p, where p is the Gaussian distribution function
Therefore, the Fourier transformation of ! f ? - is equal to the product of
the Fourier transformations of f and p. The Fourier transformation of the
Gaussian distribution function is
where ! is the frequency. So, we have
d
We see from (62) that if - ! 0, then d
converges to -
f , and
converges to f . This is exactly the fact we stated in Lemma 2.
Also by (62), d
will be very small if ! is large and - is fixed.
This implies that high-frequency components of the original function become
very small after the transformation. This is why the transformed function is
"smoother." In addition, for larger - values, wider ranges of high-frequency
components of the original function practically vanish after the transforma-
tion, and therefore the transformed function becomes increasingly smooth as
increases. We state these properties formally in the following theorem.
Theorem 4 Let f , -
all be given and well defined.
fixed -, such that 8! with k!k ? ffi,
f (!)j
fixed -, let
(1=")=-. Then 8! with
": (64)From this theorem we learn that the relative size of d
can be
made arbitrarily small for all ! with k!k greater than a small value ffi. Since ffi
is inversely proportional to -, high-frequency components are removed when
- is large.
5 Numerical Applicability
The definition of the transformation (4) involves high-dimensional integration
which cannot be computed in general (except perhaps by the Monte Carlo
method, which is not appropriate for our purposes because it is too expen-
sive). So the transformation may not be applicable to arbitrary functions, at
least numerically. However, this transformation is computationally feasible
for a large class of nonlinear partially separable functions, and especially to
typical molecular conformation and protein-folding energy functions.
We state several useful properties of the transformation in the following:
First, for the sum of functions
the transformation of f is equal to the sum of the transformations of the f i 's:
Second, for the product of functions
Y
where the g i 's do not share common variables, the transformation of g is
equal to the product of the transformations of the g i 's:
Y
Finally, for a large subclass of nonlinear partially separable functions,
called the generalized multilinear functions,
Y
where the g i
j 's are one-dimensional nonlinear functions, we have
Y
involves only one-dimensional
integration, the transformation for a generalized multilinear function can be
computed using a standard quadrature rule.
In particular, let us consider a typical n-atom molecular conformation
energy function,
ng and h ij is the pairwise energy function
determined by the distance between atoms i and j. By (66), the
transformation of this energy function is equal to the sum of the transformations
of the pairwise energy functions. However, the computation for the
transformation still cannot be carried out directly, because there is
still more than one variable in each term. Nevertheless, the following theorem
provides a feasible way to compute the molecular energy transformation:
Theorem 5 Let f be defined as in (71). Then the transformation of f can
be computed using the formula
Z
Proof: We show the case when x 8i. The general case can be proved
similarly.
By the definition of !f? - , in form (5), for any x,
Z
where c - is such that
Z
Make the following variable transformation:
Then it is easy to verify that
Z
-Z
Z
(77)The integral for
involves only variable r ij and can be
computed with a standard numerical integration technique; therefore, the
transformation !f ? - (x) can be computed in this fashion.
Note that the integral for !f ? - (x) must exist, for otherwise the transformation
will have no definition. In practice, if the integral for a given f
does not exist, a modified function may need to be considered instead. For
example, the energy function given in (3) cannot be integrated directly because
the function goes to infinity when r ij becomes very small. Usually, this
can be cured by replacing the function for small r ij with finite interpolation
(see [4, 11, 17]).
Note also that the result in Theorem 5 applies only to energy functions
that can be formulated in form (71). Most popularly used energy functions
for molecular conformation and protein folding can be expressed as pairwise
forms, for example, the Lennard-Jones potential, the electrostatic potential,
the interaction potential for bonded atoms, etc. [2, 16]. However, some energy
functions do contain terms that are not pairwise distance functions; for
instance, the torsional potential usually is given as a function of the dihedral
angle. Special approximation techniques may be needed to transform this
type of function, We will not address this issue in this work.
6 Anisotropic vs. Isotropic
The transformation we have discussed so far is of the isotropic type in the
sense that it averages function variations equally along all directions in the
variable space. In practice, we might wish to average different sizes of function
variations along different directions (i.e., use different - values for different
variables) in order to obtain a more accurate overall structure of the
function. For this purpose, we can define a more general transformation,
called the anisotropic transformation.
Given a nonlinear function f , the anisotropic transformation
!f? for f is defined such that for all x,
Z
or equivalently;
Z
where is a diagonal matrix with positive diagonal elements:
and C
with c - i
determined such that
Z
Note also that in this definition,
From this definition, we see that the anisotropic transformation will be
reduced to the isotropic transformation when the diagonal elements of are
all identical.
Many of the important properties of the isotropic transformation carry
over to the anisotropic case. We state these properties in the following:
First, for the sum of the functions
we have
Second, for the product of the functions
Y
where the g i 's do not share common variables, we have
Y
where i 's are small diagonal matrices. If g i is a function of j variables
positive numbers - i
.
Third, for the generalized multilinear functions,
Y
where the g i
j 's are one dimensional nonlinear functions, we have
Y
We can also derive a simple formula to compute the anisotropic transformation
for the molecular conformation energy function:
Theorem 6 Let f be defined as in (71). Then the anisotropic transformation
of f can be computed using the formula
Z
k)e
Proof: We show only the case when x 8i. The general case can be
proved similarly.
By the definition of !f? , in form (79), for any x,
Z
is such that
Z
Make the following variable transformation:
Then we have
Using these relations we can verify that
Z
Z
which completes the proof. 2
The anisotropic transform determines for any initial solution a unique
solution function x(-) for the transformed functions, and therefore can also
be used as a continuation process for optimization, more general and powerful
than the isotropic transform. We state these results in Theorem 7 and 8. The
details for the proof are quite similar to those for Theorem 2 and 3, so we
will not present them.
Parallel to Assumptions 1, 2 and 3 for Theorem 2 and 3, we make the
following assumptions:
Assumption 4 The objective function f is twice continuously differentiable,
and transformation (78) is well defined for the function as well as its derivatives
Assumption 5 Let g be the gradient of f , and \Psi an operator,
Then the operation \Psi can be applied to g, and \Psig is a matrix with
Transformation (78) is well defined for all derivatives involved in \Psig. Also,
\Psig(x) is uniformly bounded and satisfies a Lipschitz condition:
Assumption 6 The transformation !r 2 f ? (x) satisfies a Lipschitz condition
and its inverse is uniformly bounded.
Let S be a vector space, and for a positive vector -
we define function h(-; x) such that 8(-;
where - is the diagonal vector of , that is,
Theorem 7 Let f be a given function and h be defined as in (101). Then
under Assumptions 4 and 5, h 00
x-
exists and is uniformly bounded for all
also satisfies a Lipschitz condition in x:
x-
x-
In addition,
x-
Theorem 8 Let f be a function for which Assumptions 4, 5, and 6 all hold.
Then for any stationary point x 0 of there is a
continuous and differentiable function x(- 2 \Gamma, such that x
is a stationary point of !f ? . The function x(-) is also the
unique solution of the initial value problem
7 Concluding Remarks
In this paper, we have discussed a generalization of the effective energy transformation
scheme used in [4, 5, 17, 18] for the global energy minimization
applied to molecular conformation. Instead of applying the transformation
to the probability distribution, here we transform the functions directly, generalizing
the scheme in [4, 5, 17, 18] to a broader class of functions. A
mathematical theory for the transformation as a special continuation approach
to global optimization is established. We have established that the
proposed method transforms a given nonlinear objective function into a class
of gradually deformed, but "smoother" or "easier" functions. A continuation
procedure can then be applied to these "smoother" or "easier" functions, to
trace their solutions back to the original function. Two types of transformation
are defined: isotropic and anisotropic. We have demonstrated that both
transformation types can be applied to a large sub-class of nonlinear partially
separable functions, and in particular, the energy functions for molecular con-
formation. Methods to compute the transformation for these functions are
given.
We believe that the proposed method provides a powerful and effective
tool for global or robust local optimization. We can see this partially from the
work in [4, 5], which can be viewed as a special application of the method.
In [4, 5], the transformation method, combined with simulated annealing,
was applied to the global energy minimization problem for molecular confor-
mation. Promising results were observed even if only simple algorithms and
approximated transformation were implemented.
More numerical work will be done in our future research. We will implement
a group of algorithms based on the theory presented in this paper.
While the transformation can now be computed with provided formulas, tracing
the solution curve can be carried out using advanced numerical methods.
There are at least three choices for the implementation of the tracing procedures
1. Use a general random search procedure to trace the changes of the
global solution when the transformed function is gradually changed
back to the original function.
2. Apply only local optimization procedures to each transformed function
to trace a set of solution curves, and choose the best among all solutions
obtained.
3. Solve the initial value problems for a set of solution curves, and choose
the best solution.
The first method is similar to the approach in [4, 5] where a simulated
annealing procedure was applied to the transformed functions. This method
converges to the global solution with certain probability, but a large number
of random trials usually are required to obtain the convergence. The
second approach is the most simple and efficient method, but the solution
curves to be traced must be selected cleverly, for otherwise the global solution
will not be guaranteed. The third method provides a more accurate and
reliable way to trace the solution curves. As we have shown in this paper,
the curves are solutions to well defined initial value problems. So standard
numerical IVP-methods can be used (e.g., predictor-corrector methods) [1].
The implementation of all three tracing procedures and the numerical comparison
among them will be of great interest for the further development of
the algorithms.
We are especially interested in applying these methods to the global energy
minimization problems for molecular conformation, especially protein
folding. A set of test problems will be considered including the Lennard-Jones
microcluster conformation problem, the distance geometry problem,
and several protein conformation problems.
While searching for native structures of protein molecules is certainly
very important, the proposed methods can also provide information about
the paths that solutions follow. Such information may contain insights about
how protein molecules change from arbitrary configurations to their native
structures.
Acknowledgments
This research was supported partially by the Cornell Theory Center, which
receives funding from members of its Corporate Research Institute, the National
Science Foundation (NSF), the Advanced Research Projects Agency
(ARPA), the National Institutes of Health (NIH), New York State, and IBM
Corporation.
The author thanks Lizhi Liao, Michael Todd, Lloyd Trefethen, and Wei
Yuan for constructive suggestions. He especially thanks Thomas Coleman
for many discussions relating to this work and for his helpful comments and
suggestions on the manuscript, and David Shalloway for many discussions on
the protein-folding problem as well as the original effective energy transformation
ideas.
--R
Allgower and Kurt Georg
Brooks III
David Shalloway and Zhijun Wu
David Shalloway and Zhijun Wu
David Kincaid and Ward Cheney
David Shalloway
David Shalloway
--TR
--CTR
Olivier Chapelle , Mingmin Chi , Alexander Zien, A continuation method for semi-supervised SVMs, Proceedings of the 23rd international conference on Machine learning, p.185-192, June 25-29, 2006, Pittsburgh, Pennsylvania
Jorge J. Mor , Zhijun Wu, Distance Geometry Optimization for Protein Structures, Journal of Global Optimization, v.15 n.3, p.219-234, October 1999
Mark S. Lau , C. P. Kwong, A Smoothing Method of Global Optimization that Preserves Global Minima, Journal of Global Optimization, v.34 n.3, p.369-398, March 2006
Jack Dongarra , Ian Foster , Geoffrey Fox , William Gropp , Ken Kennedy , Linda Torczon , Andy White, References, Sourcebook of parallel computing, Morgan Kaufmann Publishers Inc., San Francisco, CA, | integral transformation;global/local minimization;continuation methods;molecular conformation |
588931 | An Unconstrained Convex Programming Approach to Linear Semi-Infinite Programming. | In this paper, an unconstrained convex programming dual approach for solving a class of linear semi-infinite programming problems is proposed. Both primal and dual convergence results are established under some basic assumptions. Numerical examples are also included to illustrate this approach. | Introduction
. Many linear semi-infinite programming problems including
the L1 and Chebychev approximation problems [14, 15] appear in the following "dual
Program (D)
is a compact set in R n , a(t)
a are continuous functions defined on T .
A corresponding "primal form" linear semi-infinite programming problem can be
represented as follows.
Min
Z
c(t)x(t)d-(t)
s.t.
Z
a
where -(t) is the Lebesgue measure, a particular regular Borel measure, on T and
is measurable on T; x(t) - 0 a.e. with respect to -; and
R
To simplify our expressions, Eq. (1.2) will be denoted by
R
The duality theory relating Programs (P ) and (D) can be found in [17]. Under
certain conditions, a strong duality theorem holds. According to [17], there exist three
"basic" solution approaches, namely, the exchange methods, discretization methods,
and methods based on local reduction. All these methods usually replace Program (D)
by a sequence of finite linear programming problems for approximation and require a
Department of Industrial and Operations Engineering, The University of Michigan, Ann Arbor,
Michigan 48109, U.S.A.(cjlin@engin.umich.edu).
y Operations Research and Industrial Engineering, North Carolina State University, Raleigh, NC
27695-7913, U.S.A. The work of this author was supported in part by the 1994 NCSC-Cray Research
Grant (fang@eos.ncsu.edu).
z Institute of Applied Mathematics, National Cheng-Kung University, Tainan, Taiwan, R.O.C.
(soonyi@mail.ncku.edu.tw).
search procedure for global optimization on T . This could be very costly, especially
when T has higher dimensionality [13, 15, 16, 18].
In this paper, we propose a solution approach which identifies an optimal solution
of Program (D) as a limit point of the solutions of a sequence of finite dimensional
unconstrained convex programming problems. No global optimization is required
in the proposed approach but instead, a multidimensional integration is required.
The basic ideas are introduced in Section 2 and main results are given in Section 3.
Primal convergence results are derived in Section 4 and numerical examples are given
in Section 5. The last section includes some concluding remarks.
2. Basic Ideas. Following [11], given - ? 0, we perturb Program (P ) by adding
an entropic term -
R
log x(t)d-(t) to its objective to form a perturbed problem:
Min
Z
Z
log x(t)d-(t)
s.t.
Z
The conjugate dual [4, 21] of Program (P - ) is given as follows.
Program (D - )
Z
e
When its optimal solution exists, we denote x
- (t) as an optimal solution of Program
with an optimal objective value v(P - ). Similarly, w
- denotes an optimal
solution of Program (D - ) with an optimal objective value v(D - ).
Note that (D - ) is an unconstrained convex program which can be treated by
various numerical techniques [8]. This provides an alternative approach even though
the bottleneck becomes the numerical integration of a multidimensional integration
over a compact set T in R n . Compared to Program (D), solving (D - ) is like a penalty
function method [5] with an exponential penalty term -
R
penalty function methods used for semi-infinite programming problems can be referred
to [6, 20]. In this paper we focus on using the exponential penalty function for linear
semi-infinite programming as an extension of the methods developed for solving linear
programming problems [10, 12].
In order to show that the proposed approach works, we need to address three
basic issues [1]:
(i) The existence of an optimal solution w
- to Program (D -
(ii) The existence of a compact set and a positive constant \Theta such that w
lies
in the compact set, 8
(iii) If a sequence fw
g converges to w as - i ! 0, then w is an optimal solution
of (D).
In general, issues (i) and (ii) are much more difficult to deal with than the third
one. In this paper, we refer to Borwein and Lewis [2] for issue (i) and resolve the
other two in Section 3. Related results for the finite dimensional case can be found
in [9].
Throughout this paper, two commonly seen assumptions are made:
CONVEX PROGRAMMING APPROACH TO LSIP 3
Program (D) has an interior feasible solution, i.e., fw
(A2) The primal constraint qualification (PCQ) holds, i.e., there exists - x in L 1 (T;
is measurable on T and
R
with respect to the Lebesgue measure,
R
R
and
R
With the assumption (A2), Theorem 4.2 of [2] resolves issue (i) and assures that
both Programs (P - ) and (D - ) attain optimality with v(P -
w - is optimal to Program (D - ), then
x
is the unique optimal solution of Program (P - ). This provides a dual-to-primal
conversion formula. Also note that since the objective function of Program (D - ) is
strictly concave, under the assumption (A2), its optimal solution w
- can be obtained
by solving the following first order condition:
Z
a(t)e
3. Main Results. The objective of this section is to address issues (ii) and (iii).
Since issue (ii) is more difficult to handle, we start with issue (iii).
Theorem 3.1. If fw
is an optimal solution of
Program (D).
Proof.
Under the assumption (A1), Program (D) has a feasible solution -
with
Z
e
Z
e
If w is not feasible for Program (D), then there exists - t 2 T such that a( -
Since T is a compact set in R n and a j (t); c(t) are continuous on T , there exists a
neighborhood N and an ffl ? 0 such that a(t) T w \Gamma c(t) - ffl; for t 2 N
By using L'Hopital's rule, we have
Z
e
Hence the right-hand-side of Eq. since
Z
e
Z
Consequently, the left-hand-side of Eq. (3.1) approaches b T -
causes a contradiction. Therefore, w must be feasible for Program (D).
4 C.-J. LIN, S.-C. FANG AND S.-W. WU
(Optimality) If w is not an optimal solution of Program (D), then we can find
a feasible solution -
w with b T -
is optimal for Program (D - i ), we
know
Z
e
Z
e
\Gammac(t)
With the same reason as we had for Eq. (3.3), as
This
again causes a contradiction and hence completes the proof.
As a direct consequence, we have the next result.
Corollary 3.2. If fw
lim
Z
e
Proof. Since
Z
e
Z
e
as
Z
e
\Gammac(t)
Therefore,
lim
Z
e
To handle issue (ii), we make an additional assumption:
(Bounded level set assumption) There exists a constant L such that fw j
is nonempty and bounded.
Note that the assumption (A3) implies that Program (D) is solvable. We shall
keep using this constant L throughout the rest of the paper.
Given that ffl ? 0; l ? 0, - is the Lebesgue measure on T , and -
the following notations:
1. -
2.
3.
4. -
Consequently, the feasible region of the original problem F 4
becomes S(0) and the assumption (A3) becomes that
nonempty and bounded.
Now we prove that, under some conditions, there exist \Theta ? 0 and - l ? 0 such that
lies in a compact set S( - l) " B, 8 In this way, we have a convergent
subsequence which goes to an optimal solution w .
CONVEX PROGRAMMING APPROACH TO LSIP 5
The basic idea is to use -(w - ; ffl) as a measure of those t 0 s at which the constraint
lies outside S( - l) " B, we can prove that
-(w
bounded above by zero, which causes a violation of the first order necessary
Z
a(t)e
as
Theorem 3.3. Under the assumption (A3), there exists - l ? 0 such that S( -
is compact.
Proof. It is easy to see that S(l) " B is closed, for l ? 0. Hence we only have to
prove that there exists - l ? 0 such that S( -
If our claim is not true, there must exist a sequence fl i g, with lim i!1 l
that S(l i unbounded. Hence, we can select an unbounded sequence fw l i
with
for each l i . It is obvious that f w l i
is a bounded sequence, thus we
can find a subsequence fk i g of fl i g such that f wk i
converges to a point, say -
as
nonempty, we can find at least one w
being sufficiently small.
Equivalently, for any t 2 T ,
As
Therefore, for
any fi ? 0,
This contradicts the assumption
(A3), under which
By using the constant - l ? 0 obtained in Theorem 3.3, we can derive the following
results:
Lemma 3.4. Given ffl ? 0 and - l
Proof. Since -
is a closed subset of S( - l)"B, it is compact. With -
we have
there exists a sequence f -
is compact, there exists a subsequence which converges to a point -
with -
Hence
a.e. in T:
Remembering that a j (t) and c(t) are continuous on T , we know a(t) T -
. However, since -
B, from its definition, there exists - t 2 T such
6 C.-J. LIN, S.-C. FANG AND S.-W. WU
that a( -
contradicts Eq. (3.8). Thus we have and the
proof follows.
Lemma 3.5. Given any w 0 2 the line segment w
intersects -
at exactly one point.
Proof. Noting that w
there exists at least one - t 2 T such that
Note that for t 2 T with
Together with (3.11), we know such -
ff exists and -
Note that b T (w
We claim that w
Otherwise there are two possible cases:
(Case 1) There exists -
In this
case, we can find a small number fl ? 0 such that a( -
This contradicts the definition of -
ff.
(Case 2) For all
l. In this case, we
can find a small number fi ? 0 such that
This again contradicts the definition of -
ff.
Hence
We now prove that w
is the only
point where w
we know that there exists - t 2 T such that
By Eq. (3.9), we have a( -
Combining with Eq. (3.14), we know
there exists another intersection point, say w
then by the definition of -
ff, we see ff ? -
ff and a( - t) T (w
Hence
this causes a contradiction.
Lemma 3.6. Given ffl ? 0 and - l
Proof. Take
w be the intersection point of w
We can find ff ? 1 such that
(3.
CONVEX PROGRAMMING APPROACH TO LSIP 7
Note that Therefore for each t with
By Lemma 3.4, we further have
The next lemma shows that b T w
sufficiently small.
Lemma 3.7. There exists \Theta 1 ? 0 such that b T w
- L, for
Proof. By the assumption (A3), we can find a point -
w that is feasible for Program
(D) and b T -
Z
e
Z
e
Z
e
Z
e
w is feasible for (D), lim -!0 -
R
there exists a
sufficiently small \Theta 1 such that, for
Z
e
This completes the proof.
Combining the lemmas derived before, we now ready to prove the main result.
Theorem 3.8. Under the assumptions (A1)-(A3), if there exists -
that a(t) T -
, then there exists \Theta ? 0 such that w
- lies in a compact set
Proof. Suppose that the claim is not true. Then, Lemma 3.7 implies that there
exists an infinite sequence fw
whose elements are not in S( -
for each - i . Since
Z
a j (t)e
\Gammac(t)
we have
Z
w)e
By Lemma 3.6, for each i, we have -(w - i
8 C.-J. LIN, S.-C. FANG AND S.-W. WU
Z
w)e
Z
\Gammac(t)-fflg
Z
\Gammac(t)-fflg
R
\Gammac(t)-fflg (a(t) T -
d-(t) is bounded away from zero. Hence the right hand side of Eq. (3.19) approaches
1, as i !1. This contradicts Eq. (3.18) and completes the proof.
Consequently, the issues (ii) and (iii) have been taken care by the following corollary
Corollary 3.9. Under the assumptions (A1)-(A3), if there exists -
that a(t) T -
, then there exists \Theta ? 0 such that w
- lies in a compact set
sequence fw
with has at least
one convergent subsequence in S( - l) " B, whose limiting point is an optimal solution
of Program (D).
4. Dual Unboundedness and Primal Convergence in Optimal Value.
Note that (A3) is an assumption on bounded level sets. To detect the unboundedness
of the dual program (D), the following lemma may help.
Lemma 4.1. With the assumptions (A1) and (A2), suppose there exists -
such that a(t) T -
goes to infinity, as - i decreases to 0, then
either the program (D) is unbounded above or its optimal solution set is unbounded.
Proof. Assume our conclusion is false, then the program (D) is bounded above
and its optimal solution set is nonempty and bounded. In this case, we let L be the
optimal objective value which is sufficient for the assumption (A3) to hold.
With the assumptions (A1) and (A2) and the existence of -
Theorem 3.8 shows that the sequence fw
lies in a compact
set. This clearly causes a contradiction and the proof follows.
We now turn our attention to investigate the primal convergence of Program
in terms of the optimal objective value. It is important to point out that in our
formulation of program (P ), since the primal variable
where -(t) is the Lebesgue measure, a particular regular Borel measure, the program
may not achieve P -attainment [2, 3, 4]. In other words, program (P ) may not
have an optimal solution x (t) such that
x
x
Z
e
and x (t)
However, this does not affect the dual convergence results and
we shall show the convergence of program (P - i ) in terms of optimal objective value.
We start with the following two lemmas:
Lemma 4.2. If fw
such that, for each i ?
CONVEX PROGRAMMING APPROACH TO LSIP 9
Proof. If the claim is not true, then there exists a particular K such that, for any
lies in a compact set T , there is a subsequence which converges to - t
with
This contradicts the fact that
Hence the proof is complete.
Note that when w
is optimal to Program (D - i ), Eq. (2.4) holds as
Z
a j (t)e
m. Now, if there exists -
then Eq. (4.1) implies that
Z
w)e
\Gammac(t)
and
Z
e
This leads to the following result:
Lemma 4.3. If there exists -
R
Combining Lemmas 4.2 and 4.3, we have the following convergence result:
Theorem 4.4. If fw
0, and there exists -
such that a(t) T -
lim
Z
x
(t) log x
and
lim
Z
c(t)x
Proof. Given ffl ? 0, let
w . Lemma 4.2 provides an N ? 0 such that, for
Then from Eqs. (2.3), (4.2) and (4.3), for i ? N , we have
Z
x
(t) log x
Z
Z
e
\Gammac(t)
Therefore,
lim
Z
x
(t) log x
Since
Z
c(t)x
Z
x
(t) log x
Z
e
as
lim
Z
c(t)x
This clearly shows the primal convergence of program (P - i ) in terms of the optimal
objective value.
5. Numerical Examples. In this section, three numerical examples are re-
ported. Our purpose is not to claim any computational superiority of the proposed
method. Instead, we simply intend to illustrate the computational behavior the proposed
approach. Note that the proposed approach is flexible to use any commercial
or public unconstrained nonlinear optimizer, instead of developing special codes for
our own implementation. In our examples, the L-BFGS-B software [22] was applied.
5.1. L1 Problems. In this subsection, the following two commonly seen L1
problems [14, 15] were tested:
Problem 1
Problem 2
CONVEX PROGRAMMING APPROACH TO LSIP 11
Table
Problem 1
const. comp. slackness
It was reported in [14, 15] that 0.6931 and -1.78688 are approximate optimal
solutions to Problem 1 and Problem2, respectively.
We applied the L-BFGS-B subroutines [22] to solve Program (D - ) with the standard
default setting in the "driver1.f" of the L-BFGS-B solver. The stopping criteria
of L-BFGS-B was given by
where f is the function to be optimized and f k , f k+1 are the values of f in the k-th
and (k+1)-th iteration, respectively. Moreover, epsmch = 2:22 \Theta 10 \Gamma16 , which reflects
the machine accuracy for our Silicon Graphics workstation, and factr was set to be
A trivial initial solution with chosen for Problem 1.
The numerical results of the proposed approach is shown by Table 5.1.
In the table, "argmin
\Gammac(t)g, "min
\Gammac(t)g, and "comp.
Z
c(t)e
Z
c(t)x
Z
x
(t) log x
Z
x
The final w
we obtained for problem 1 is
Similarly, we chose a trivial initial solution with
2. The numerical results is shown in Table 5.2.
The final w
we obtained for problem 2 is
The first three columns of both tables clearly show that b T w
converges to the
reported solution up to the 4-th digit after the decimal point, as - i decreases. The
Table
Problem 2
const. comp. slackness
Table
Two dimensional problem
argmins argmin t min const. comp. slackness
last two columns of both tables also show that the dual feasibility and complementary
slackness conditions are satisfied up to the 4-th digit after the decimal point, when - i
is small enough.
The computational bottleneck of the proposed approach lies in the following in-
tegration
Z
e
Numerically, we need to discretize T for integration. When - i is getting smaller, a
finer discretization of T is needed. This could be very time-consuming, especially
when T has a high dimensionality. In our examples, we partition
400,000 intervals and use the Simpson's method for integration.
5.2. Two Dimensional Problem. In this subsection, the following two-dimensional
problem on page 112 of [15] was tested:
It was reported in [7] that 2.4356 is an approximate optimal objective value. We
followed all settings as described in the previous subsection with an initial solution
The numerical results are shown in Table 5.3.
CONVEX PROGRAMMING APPROACH TO LSIP 13
The final w
we obtained is
For this two dimensional problem, we partitioned [0; 1] \Theta [0; 1] into 1500 2 intervals
for integration. The computational behavior of this case is quite similar to
those reported in the previous subsection. The proposed approach indeed generates a
convergent solution which satisfies the dual feasibility and complementary slackness
conditions.
Because the numerical integration of
R
could be difficult for
a general problem, we do not claim any computational superiority of the proposed
approach. However, the proposed approach does provide a new angle to solve the
linear semi-infinite programming problems.
6. Concluding Remarks.
1. We have proposed an unconstrained convex programming approach to solving
linear semi-infinite programming problems. In this approach, solving Program
(D - ) with a sufficiently small - ? 0 provides an approximate solution
to Program (D).
2. A dual convergence result shows that, under the assumptions (A1)-(A3),
(D in terms of optimal solutions.
3. A primal convergence result shows that (P - in terms of
optimal objective values.
4. Compared to most known methods, the proposed approach does not require
any search procedures for finding a global optimizer over T . Instead, an
integration over T is required. We do not claim any computational superiority
of the proposed approach. However, it does provide an alternative. For
some problems, it might be easier to do integration than to perform a global
optimization over T .
5. The proposed approach is flexible to use any unconstrained nonlinear opti-
mizer. Three examples were included to illustrate the proposed approach.
Acknowledgment
:. The authors would like to thank Dr. A. R. Conn and referees
for their constructive comments.
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588960 | A Potential Reduction Newton Method for Constrained Equations. | Extending our previous work [T. Wang, R. D. C. Monteiro, and J.-S. Pang, Math. Programming, 74 (1996), pp. 159--195], this paper presents a general potential reduction Newton method for solving a constrained system of nonlinear equations. A major convergence result for the method is established. Specializations of the method to a convex semidefinite program and a monotone complementarity problem in symmetric matrices are discussed. Strengthened convergence results are established in the context of these specializations. | Introduction
In the paper [11], we have introduced the problem of solving a system of nonlinear equations
subject to additional constraints on the variables, i.e., a constrained system of equations. We
have demonstrated that constrained equations (CEs) provide a unifying framework for the study
of complementarity problems of various types, including the standard nonlinear complementarity
problem and the Karush-Kuhn-Tucker system of a variational inequality. Postulating a partitioning
property of the CE, we have introduced an interior point potential reduction algorithm for solving
the CE and applied this method to convex programs of different kinds. The goal of this paper is
to present a potential reduction Newton method for solving a CE, without assuming the existence
of the partitioning property that is key to the previous work.
The central problem studied in this paper is stated as follows. Let be a given
mapping from the real Euclidean space ! n into itself and
let\Omega be a given closed subset of ! n . The
constrained equation defined by the
find a vector x
We refer the reader to [11] for the initial motivation to study the CE. Later in this paper, we will
consider applications of our results to a semidefinite convex program and a monotone complementarity
problem on the cone of positive semidefinite matrices. These applications yield new interior
This work was based on research supported by the National Science Foundation under grants INT-9600343 and
CCR-970048 and the Office of Naval Research under grant N00014-94-1-0340.
y This work was based on research supported by the National Science Foundation under grant CCR-9213739 and
by the Office of Naval Research under grant N00014-93-1-0228.
point methods for solving these problems whose convergence can be established under some mild
assumptions.
The method proposed in this paper for solving the
combines ideas from the classical
damped Newton method for solving the unconstrained system of equations
and the family of interior point methods for solving constrained optimization and complementarity
problems. A general convergence theory for the proposed method will be presented and specializations
of the results to the aforementioned applications will be described. Unlike the previous study
[11] where we assume that the function H(x) has a certain partition conformal to the
set\Omega\Gamma we
make no such assumption herein. Instead, the present work is based on a set of broad hypotheses
on the
We explain some terminology and fix the notation used throughout the paper. For a given subset
S of ! n , we let int S, cl S, and bd S denote, respectively, the interior, closure, and boundary of S.
If the mapping H is (Fr'echet) differentiable at a point x in its domain, the Jacobian matrix of H
at x is denoted H 0 (x); thus the (i; j)-entry of H 0 (x) is equal to @H i (x)=@x j , for
is the Fr'echet derivative of H at x
along the direction v. If H(x; y) is a function of two arguments (x;
x denote the
partial Jacobian matrix of H with respect to the variable x. For a real-valued function
we write rOE(x) for the gradient vector of OE at the vector x . The p-norm of a vector x is
denoted by kxk p ; in particular, its 2-norm or Euclidean norm is denoted by kxk. For a nonnegative
vector a 2 ! n , we let [0; a] denote the line segment joining the origin and a.
The set of real matrices of order n is denoted M n ; the subset of symmetric matrices in M n
is denoted S n . The set M n forms a finite-dimensional inner-product vector space with the inner
product given by
where "tr" denotes the trace of a matrix. This inner product induces the Frobenius norm for
matrices given by
The subsets of S n consisting of the positive semidefinite and positive definite matrices are denoted
by S n
++ respectively. For two matrices A and B in S n , we write A
similarly, A OE B means
++ . For any matrix A 2 S n
denotes the square root of A;
i.e., A 1=2 is the unique matrix in S n
such that A.
Description and Analysis of the Algorithm
In this section, we describe the potential reduction Newton algorithm for solving the
where\Omega is a closed subset of ! n and H is a continuous mapping
This section is
divided into four subsections as follows: in the first subsection, we lay down the basic assumptions
satisfied by the
in the second subsection, we give some results which guarantee the
existence of a solution for the
in the third subsection, we present the detailed statement
of the algorithm; in the fourth subsection, we establish a convergence theorem for the algorithm.
2.1 Basic assumptions
We introduce several key assumptions on the
Subsequently, these assumptions will be
verified in the context of several applications of the CE.
(A1) The closed
set\Omega has a nonempty interior.
There exists a closed convex set S ' ! n such that
(a)
(b) the (open)
int\Omega is nonempty;
(c) the set H \Gamma1 (int
bd\Omega is empty.
H is continuously differentiable
on\Omega I , and H 0 (x) is nonsingular for all x
2\Omega I .
Assumption (A1) is needed for the applicability of an interior point method. The sets S and
\Omega I in assumption (A2) contain the key elements of the proposed algorithm. Notice that S pertains
to the range of H
and\Omega I to the domain. Initiated at a vector x 0
in\Omega I , the algorithm generates
a sequence of iterates fx k g
ae\Omega I so that the sequence fH(x k )g ae int S will eventually converge
to zero, thus accomplishing the goal of solving the
at least approximately. Assumption
facilitates the application of a Newton scheme for the generation of fx k g; this scheme relies
on a potential function for the
set\Omega I that is induced by such a function for int S. Specifically, we
postulate the existence of a potential function satisfying the following properties:
(A4) for every sequence fu k g ae int S such that
either lim
we have
lim
continuously differentiable on its domain and u T rp(u) ? 0 for all nonzero u 2 int S.
A condition equivalent to (A4) is stated in the following straightforward result.
Condition (A4) holds if and only if for all 0, the set
is compact.
The notion of the central path has played a fundamental role in all interior-point methods for
solving optimization and complementarity problems [2, 4, 5]. Inspired by this notion, we introduce
an important assumption on the potential function p that postulates the existence of a vector
satisfying a certain property; this vector will be used to define a modified Newton direction
that is key to the generation of the iterates for solving the
Although the vector a is
inspired by the central vector of all ones in the case where S is the nonnegative orthant, since our
present setting is very broad, the vector a should not be thought of as just a "central vector" for
int instead, a is closely linked with the potential function p which itself is fairly loosely restricted.
There exists a nonzero vector a 2 ! n and a scalar
oe
(a T u) (a T rp(u))
The basic role of the potential function p is to keep the sequence fH(x k )g away from the set
bd S n f0g while forcing it towards the zero vector. Hence, its role is slightly different from that of
a standard barrier function used in nonlinear programming, which in contrast penalizes an iterate
when it gets close to any boundary point of S. Later, we will identify this function for various sets
S in the applications to semidefinite problems. For now, we will consider the simple case where S
is the nonnegative orthant ! n
and establish the validity of conditions (A4)-(A6) for the function
log
is an arbitrary scalar. (Note: the ' 1 norm of u, instead of u T u, could also be used in
the first logarithmic term. The analysis remains the same with the constant i properly adjusted.)
Clearly, p is norm-coercive on ! n
lim
because for u ? 0,
log
log n
log
where the first and second inequalities follow from the fact that kuk 1 p nkuk and n log(
log u i n log n, respectively. Moreover, for any positive sequence fu k g converging to a
nonzero nonnegative vector with at least one zero component, the limit (1) clearly holds. Thus
holds. Moreover, with a taken to be the vector of all ones, (A6) also holds. Indeed, we have
for
a
thus
(a T rp(u)) (a T u)
where the last inequality follows from the fact that kuk 1 p n kuk and the arithmetic-geometric
mean inequality.
Other choices for the function p exist for
. The above choice will be generalized to the
case where S involves the cone of symmetric positive semidefinite matrices.
2.2 Existence of solutions
In this subsection, we study conditions that guarantee the existence of solutions of the
We start by giving a few definitions. Assume that M and N are two metric spaces and that
N is a map between these two spaces. For
Eg. The map G is said to be proper with respect to a set
is compact for every compact set K ' E. If G is proper with respect to N ,
we will simply say that G is proper. For D ' M , and E ' N such that G(D) ' E, the restricted
defined by ~
denoted by Gj (D;E) ; if
we write this ~
G simply as GjD . We will also refer to Gj (D;E) as "G restricted to the pair (D; E)",
and to GjD as "G restricted to D". We say that partition of the set V if
space M is said to be connected if there exists no
partition which both O 1 and O 2 are non-empty and open. A metric space M is said
to be path-connected if for any two points there exists a continuous
that
The following result and its proof can be found in Monteiro and Pang [7] (see Corollary 1 of
this reference).
Proposition 1 Let M and N be two metric spaces and N be a continuous map. Let M 0 '
M and N 0 ' N be given sets satisfying the following conditions:
is a local homeomorphism,
Assume that F is proper with respect to some set
E such that N 0 ' E ' N . Then F restricted to the pair (M local
homeomorphism. If, in addition, N 0 is connected, then F cl N 0 .
Using Proposition 1, we now derive two existence results for the
Theorem 1 Assume that conditions (A1)-(A3) hold and that there exists a convex set E ae S such
that
I ) is nonempty and H
proper with respect to E. Then,
.
In particular,
solution.
Proof. To apply Proposition 1, let
j\Omega I , N Using
(A2), we easily see that F (M by (A3) and the
inverse function theorem, it follows that F j M 0 is a local homeomorphism. Since
with respect to E by assumption, it follows from Proposition 1 that
cl cl
where the last equality follows from the fact that cl E) " cl (int
by elementary properties of convex sets (see subsection 2.1 in Chapter 3 of [1]). Moreover, it also
follows from Proposition 1 that restricted to the
int S) is a proper
local homeomorphism.
Theorem 2 Assume that conditions (A1)-(A3) hold and that F is proper with respect to S. Then
restricted
to\Omega I maps each path connected component
of\Omega I homeomorphically
onto int S. In particular,
solution.
Proof. Conclusion (i) follows immediately from Theorem 1 with S. Using the last conclusion
obtained in the proof of Theorem 1 and setting we conclude that F restricted to the pair
(\Omega I ; int S) is a proper local homeomorphism. If T
'\Omega I is a path connected component
of\Omega I
then F restricted to the pair (T ; int S) is a proper local homeomorphism since T is both open and
closed with respect
to\Omega I . Since every proper local homeomorphism from a path connected set into
a convex set is a homeomorphism (see for example Theorem 1 of [7]), (ii) follows.
2.3 The algorithm
The algorithm for solving the
damped Newton method applied to the
Referring the reader to [8] for the basic family of Newton methods for
solving this unconstrained equation, we highlight the modifications to deal with the presence of
the constraint
In essence, there are two major modifications. One, the Newton equation to
compute the search directions is modified using the (central) vector a in assumption (A6). Two,
the merit function for the line searches is based on the merit function:
This is different from the norm functions of H that are the common merit functions used in a
classical damped Newton method. Note that by (A3) and (A5) the function / is continuously
differentiable
on\Omega I .
With the above explanation, we now give the full details of the promised Newton method for
solving the
under the setting given in the last subsection.
Step 0. (Initialization) Let a vector x 0
2\Omega I and scalars ae 2 (0; 1) and ff 2 (0; 1) be given. Let a
sequence of scalars foe k g ae [0; oe) be also given. (The scalar
oe is as given in assumption (A6).) Set
the iteration counter
Step 1. (Computing the modified Newton direction) Solve the system of linear equations
a
a (3)
to obtain the search direction d k .
Step 2. (Armijo line search) Let m k be the smallest nonnegative integer m such that x k +ae m d k
2\Omega I
and
Step 3. (Termination test) If
prescribed tolerance;
stop; accept x k+1 as an approximate solution of the
Otherwise, return to Step 1 with
k replaced by k + 1.
By (A3) and the fact that x k
2\Omega I , the Newton equation (3) has a unique solution which
we have denoted by d k . The following lemma guarantees that d k is a descent direction for the
function / at x k . This property, along with the openness
of\Omega I , ensures that the integer m k can
be determined in a finite number of trials (starting with increasing it by one at each
thus guaranteeing the well-definedness of the next iterate x k+1 .
Lemma 2 Suppose that conditions (A5) and (A6) hold. Assume also that x
2\Omega I , d
are such that
a T H(x)
where a
are as in condition (A6). Then, r/(x) T d ! 0.
Proof. Let due to (4) and the assumption that x
2\Omega I . This
together with (2), (5), (4), (A5) and (A6) imply
a T u
a
oe
as claimed.
2.4 A convergence result
In what follows, we state and prove a limiting property of an infinite sequence of iterates fx k g
produced by the algorithm. Before stating the theorem, we observe that such a sequence necessarily
belongs to the
set\Omega I ; thus fH(x k )g ae int S. Since the sequence fx k g is infinite, we have
for all k.
Theorem 3 Assume conditions (A1)-(A6) hold and that lim sup k oe k ! oe. Let fx k g be any infinite
sequence produced by the potential reduction Newton algorithm. Then, the following statements
hold:
(a) the sequence fH(x k )g is bounded;
(b) any accumulation point of fx k g, if it exists, solves the
in particular, if fx k g is
bounded then the
solution.
Moreover, for any closed subset E of S containing the sequence fH(x k )g,
(c) if H is proper with respect to
(d) if H is proper with respect to E, then fx k g is bounded.
Proof. Let
all k. Hence, for any " ? 0 we have fu k g ae "g. Since by Lemma 1 the
set ("; fl) is compact, and hence bounded, we conclude that fu k g is bounded. Hence, (a) follows.
To show (b), let x 1 be an accumulation point of fx k g. Clearly x 1
2\Omega because\Omega is a closed
set. Assume for contradiction that be a subsequence converging
to x 1 and assume without loss of generality that foe converges to some scalar oe 1 . Since
oe k 0 for all k and lim sup k oe k ! oe, we must have oe 1 2 [0; oe). Since p(u k )
and
lim
there exists " ? 0 such that the subsequence fu ae ("; fl). Since by Lemma 1 the
set ("; fl) is compact, we conclude that u int S, and hence that x 1 2
assumption (A2), it follows that x 1
2\Omega I . Hence, by assumption (A3), H 0
exists. This implies that the sequence fd converges to a vector d 1 satisfying
a
a:
Hence, it follows from Lemma 2 that r/(x 1
converges to x 1
2\Omega I where / is continuous, it follows that f/(x k
converges. This implies that the whole sequence f/(x k )g converges due to the fact that it is
monotonically decreasing. Using the relation
for all k, we conclude that
lim
and hence that
lim
because
lim
Thus
lim
which implies that m k 2 for all k 2 sufficiently large. Consequently, by the definition of m k ,
we deduce that
for all k 2 sufficiently large. Letting k 2 tend to infinity in the above expression, we obtain
which contradicts the fact that ff ! 1 and r/(x 1 Consequently, we must have
0, and hence (b) follows.
Assume now that E is a closed subset of S containing the sequence fH(x k )g. To prove (c),
assume for contradiction that for an infinite subset ae f0;
lim inf
By an argument similar to that employed above, we conclude that for some " ? 0 we have
and the fact that E is closed, we conclude that (";
a compact subset of int S " E. Since H is proper with respect to int S " E, the inverse image of
is compact, and hence bounded. This implies that fx is bounded.
By (b), every accumulation point of the latter subsequence is a zero of H . This contradiction
establishes (c).
Finally, using (a) and the fact that E is closed, we conclude that fu k g is contained in a compact
subset E 1 of E. Since H is proper with respect to E, it follows that the set H is
bounded. Hence, (d) follows.
Statements (a), (b) and (c) of Theorem 3 do not claim the boundedness of the sequence fx k g.
In particular, existence of a solution to the
established only under the properness
condition of statement (d). A consequence of statement (c) is
consequently,
in the sense that for any such ", there
exists a vector x "
can be computed by the potential
reduction Newton method starting at the given vector x 0 .
The framework of the
that we have set forth so far is very broad. In addition to
not assuming any sign restriction on the components of H (like we did in [11]; see Assumption 1
therein), part of the generality of the present framework stems from the freedom in the choice of
the set S and the associated potential function p. Indeed, as we shall see in the special cases below,
the set S and the function p can often be constructed under very mild assumptions.
3 Monotone Complementarity Problems in Symmetric Matrices
We consider a mixed complementarity problem defined on the cone of symmetric positive semidefinite
matrices. The linear version of this problem was introduced by Kojima, Shindoh, and Hara
[3] and has received a great deal of research attention recently. In what follows, we consider a non-linear
version of this problem defined in [6]. This reference contains a fairly extensive bibliography
on interior point methods for solving optimization and complementarity problems defined on the
cone of semidefinite matrices; it will be the source for several results that will be used freely in the
subsequent development.
3.1 Implicit mixed complementarity problems
We recall the framework considered in [6]. Let F : S n
be a given mapping. The
mixed complementarity problem in symmetric matrices is to find a triple (X; Y; z) 2 S n \Theta S n \Theta ! m
satisfying
F (X; Y;
\Theta S n
As explained in [6] and the references therein, there are several equivalent ways of stating the
complementarity condition each leading to a different interior point method for solving
the above problem. In what follows, we consider the equivalent formulation of this problem as the
CE defined by the
where the
set\Omega and the
\Theta S n
are defined by
F (X; Y; z)
Similar treatment can be applied to other equivalent formulations and to generalizations of the
basic problem (6). Throughout the following discussion, F is assumed to be continuous on its
domain and continuously differentiable on S n
++ \Theta S n
Associated with the above mapping H , define the set
\Theta S n
g:
It has been shown in Lemma 1 of [6] that
\Theta S n
g:
The fundamental role of the set U in the study of the problem (6) is well explained in this reference.
This set continues to have an important role in the present algorithmic setting for solving the cone
complementarity problem.
We introduce an important assumption on the mapping F that will be used to verify the
nonsingularity of the Jacobian matrix H 0 (X; Y; z).
(B1) The mapping F is (X; Y )-differentiably-monotone at every triple (X; Y; z) 2 U \Theta ! m ; i.e., for
any such triple,
(B2) The mapping F is z-differentiably-injective at every triple (X; Y; z) 2 U \Theta ! m ; i.e., for any
such triple,
The following lemma asserts that the basic assumptions (A1)-(A3) in Subsection 2.1 are valid
under the above hypotheses.
Lemma 3 Consider the
with\Omega and H defined by (7) and (8), and let S j S n
If conditions (B1) and (B2) hold, then
\Omega I j
moreover, the
and the set S satisfy conditions (A1), (A2) and (A3).
Proof. Only the second assertion requires a proof. Conditions (A1) and (A2)(a) obviously hold.
Clearly U is an open set; since (I ; I) 2 U , (A2)(b) holds. Moreover, it is easy to see that the
alternative representation (10) implies (A2)(c). Next we establish that (A3) holds under (B1) and
(B2). This amounts to showing that for every (X; Y; z)
the following implication
holds:
Assume the left-hand condition holds. Then,
Condition (B1) and (14) imply that dX ffl dY 0. This together with (13) and the fact that
(see the proof of Theorem 3.1(iii) of [10]). In turn, this together
with imply
which yields due to (B2).
Next we deal with conditions (A4)-(A6). For this purpose, consider the potential function
++ \Theta S n \Theta defined by
++ \Theta S n
is an arbitrary constant.
Lemma 4 The potential function (15), the vector a j and the scalar
conditions (A4), (A5) and (A6).
Proof. Since for a matrix Z 2 S n , kZk 2
F is equal to the sum of the squares of the n eigenvalues of
Z, and det Z is equal to the product of these eigenvalues, the verification of (A4) for the function
p(M; N; v) is the same as in the previous case of a nonnegatively constrained equation (discussed
at the end of Subsection 2.1). Noting that
we have
and thus (A5) holds. We now show that (A6) is satisfied with the given a and oe. Indeed we have
which implies
Noting that (i) tr(M) equals the sum of the eigenvalues of M , (ii) tr(M the sum of the
inverses of the same eigenvalues, and (iii) kMk 2
the sum of these eigenvalues
squared, it follows from the same derivation as in the end of Subsection 2.1 that condition (A6)
holds.
According to (2), the potential function (15) induces the following merit function on the set
log
det
for any triple (X; Y; z) 2 U \Theta ! m . Here, k \Delta k F;2 denotes the norm on S n \Theta ! m defined by
We now give a detailed description of a specialized algorithm for solving the mixed complementarity
problem in symmetric matrices (6), based on the potential reduction Newton method for
solving the
oe defined as in (7), (8), Lemma 3, (15)
and Lemma 4, respectively.
Step 0. (Initialization) Let a pair of matrices (X
and ff 2 (0; 1) be given. Let a sequence of scalars foe k g be also given, where oe k 2 [0; 1) for all k.
Set the iteration counter
Step 1. (Computing the modified Newton direction) Solve the system of linear
A
to obtain the search triple (dX
Step 2. (Armijo line search) Let m k be the smallest nonnegative integer m such
and
Step 3. (Termination test) If
prescribed tolerance;
stop; accept the triple (X as an approximate solution of the problem (6). Otherwise,
return to Step 1 with k replaced by k + 1.
As an immediate consequence of Lemma 3, Lemma 4 and Theorem 3, we have the following
convergence result for the above algorithm.
Theorem 4 Assume that conditions (B1) and (B2) hold and lim sup k oe k ! 1. Let f(X
be any infinite sequence produced by the above algorithm for solving problem (6). Then, the following
statements hold:
(a) the sequence fH(X
(b) any accumulation point of exists, solves the problem (6); in particular, if
bounded then problem (6) has a solution.
We now make a few remarks. The above theorem guarantees neither that f(X k
bounded nor that it has an accumulation point. The conclusion that f(X k ; Y k ; z k )g is bounded
would follow from Theorem 3(d) with could prove that the map H is proper with respect
to the set S j S n
. Unfortunately, this requirement is rather strong. For monotone mixed
complementarity problems, we state in Proposition 2 below a result (from Monteiro and Pang [6,
Lemma 2]) asserting that the map H is proper with respect to S n \Theta F (U \Theta ! m ). Hence, if the latter
set contains the set
equivalently if the equality F (U \Theta ! m
then the sequence generated by the above algorithm f(X k ; Y k ; z k )g is bounded. Intuitively, the
equality F (U \Theta ! m hold for maps F satisfying some kind of strong monotonicity
condition. But since this type of condition is fairly restrictive, we do not pursue this issue any
further.
Another possible approach which would guarantee the boundedness of f(X is to
reduce the set S so as to have S ' S n \Theta F (U \Theta ! m ). This approach requires some knowledge of the
set F (U \Theta ! m ). We will see that for the complementarity problems studied in Subsection 3.2 and
Section 4, enough information about the set F (U \Theta ! m ) is available which allows us to choose a set
S together with a potential function satisfying the inclusion S ' S n \Theta F (U \Theta ! m )
and the conditions (A1)-(A6) of Subsection 2.1.
Before stating the properness result mentioned above, we give a few basic definitions.
mapping J(X; Y; z) defined on a subset dom(J) of M n \Theta M n \Theta ! m is said to be
Y )-equilevel-monotone on a subset V ' dom(J) if for any (X; Y; z) 2 V and (X
such that F (X; Y;
will simply say that J is (X; Y )-equilevel-monotone.
In the following two definitions, we assume that W , Z and N are three normed spaces and that
OE(w; z) is a function defined on a subset of W \Theta Z with values in N .
2 The function OE(w; z) is said to be z-bounded on a subset V ' dom(OE) if for every
sequence f(w k ; z k )g ae V such that fw k g and fOE(w k ; z k )g are bounded, the sequence fz k g is also
bounded. When dom(OE), we will simply say that OE is z-bounded.
Definition 3 The function OE(w; z) is said to be z-injective on a subset V ' dom(OE) if the following
implication holds: (w;
we will simply say that OE is z-injective.
The following is the promised result from Lemma 2 of Monteiro and Pang [6].
m be a continuous map and let H
m be the map defined by (8). Assume that the map F is (X; Y )-equilevel-monotone and
z-bounded on its domain. If the map H restricted to U \Theta ! m is a local homeomorphism, then H is
proper with respect to S n \Theta F (U \Theta ! m ).
3.2 Standard complementarity problem
In this section, we consider the standard nonlinear complementarity problem (NCP) in symmetric
matrices:
is a given continuous mapping which is continuously differentiable on S n
++ .
This problem is a special case of the implicit mixed complementarity problem of Subsection 3.1
z is not present) and F : S n
\Theta S n
\Theta S n
We make the following assumption on the mapping f .
is monotone on S n
Lemma 5 If condition (C1) holds then the
\Theta S n
defined by (17) satisfies
condition (B1) of Subsection 3.1.
Proof. By (C1), it follows that for every X 2 S n
, the linear map f 0 (X) is monotone in the sense
that
To verify (B1), assume that (dX; dY equivalently
by (18), we have
This shows that implication (11) holds for since implication (12) holds vacuously for
It is possible to solve the NCP (16) with the use of the potential reduction algorithm described
in Subsection 3.1. However, the sequence of iterates generated by this algorithm might
not be bounded. We now develop a different potential reduction algorithm in which the set S is
reduced so as to have S ' S n
\Theta F (U ), thus ensuring the boundedness of the sequence f(X
(see the discussion at the end of the previous subsection).
To describe the alternative algorithm, it is sufficient to identify the
the set S,
the potential function and the vector a and scalar oe in condition (A6). We let
\Theta S n
\Theta S n
where F is given by (17). Moreover, we let S j S n
\Theta S n
be defined by
F
\Theta S n
is an arbitrary constant. Finally, we let a j (I ; I) and oe j 1. Clearly, the
set\Omega I and
the merit function /
and
log
det
Lemma 6 The
the set S, the potential function !, the vector a and the
scalar oe defined above satisfy conditions (A1)-(A6) of Subsection 2.1.
Proof. Condition (A2)(b) follows from the fact that
2\Omega I for all sufficiently large. The
other conditions are either straightforward or are shown using Lemma 5 and the same arguments
used in the proofs of Lemma 3 and Lemma 4.
Before giving the convergence result for the potential reduction Newton method in the above
framework, we state the following result which will be used to establish boundedness of the iterates
generated by this method.
Lemma 7 Suppose that f : S n
is a continuous map which is continuously differentiable
on S n
++ and satisfies condition (C1). Then, for the maps F and H defined by (17) and (19)
respectively, we have:
(a) F
\Theta S n
proper with respect to S n \Theta S n
(c) if 0 2 F (S n
\Theta S n
proper with respect to S n \Theta S n
Proof. By Proposition 4(a) and Corollary 3 of [6] with
\Theta S n
\Theta S n
). Using this inclusion, we easily see that statement (a) holds.
We next show (b). By Lemma 6, H 0 (X; Y ) is invertible for all (X; Y restricted
to U is a local homeomorphism. Thus it follows from Lemma 2 that H is proper with respect to
once we prove that S n
++ \Theta S n
++ be
arbitrary.
\Theta S n
\Theta S n
such that ~
X). For ffl ? 0,
let
X). Clearly, X ffl 0 for every ffl ? 0. By
the continuity of f and the fact that U
Y 0, we have Y ffl 0 for ffl ? 0 sufficiently small. Since
that U belongs to F (S n
\Theta S n
We omit the proof of (c) which is similar to that of (b).
We will skip the straightforward formulation of the potential reduction Newton method specialized
to the above choices of the
potential function
and scalar oe; instead we directly give the convergence properties of the method.
Theorem 5
be a continuous function which is continuously differentiable on
++ and satisfies condition (C1). Suppose that f(X k ; Y k )g is a sequence generated by the potential
reduction Newton method with the
potential function
and scalar
oe as specified above. Then, the following statements hold:
(a) every accumulation point of f(X k ; Y k )g is a solution of the NCP (16);
(b) if there exists ~
such that f( ~
(c) if there exists "
++ such that f( "
, then the sequence f(X k ; Y k )g is bounded.
Proof. Statement (a) follows from Theorem 3(b). To prove statement (b), note first that the
assumption implies that 0 2 F (S n
\Theta S n
Hence, by Lemma 7(b), we conclude that H is proper
with respect to S n \Theta S n
++ . It follows from Theorem 3(c) with
to zero. The proof of (c) follows similarly from Lemma 7(c) and Theorem 3(d) with
Statement (a) is within expectation; statement (b) is interesting because its assumption is
the feasibility of the NCP in symmetric matrices (16). A consequence of of statement (b) is that
feasibility of this problem (which is also monotone by assumption (C1)) is sufficient for the sequence
to converge to zero although no boundedness of the sequence f(X k ; Y k )g is asserted.
The latter assertion is established under the strict feasibility of the problem (16); this is statement
(c).
4 Convex Semidefinite Programs
In this section we consider the convex semidefinite program studied in [6, 9], namely:
minimize '(x)
subject to G(x) 0
are given smooth mappings. Under a suitable
constraint qualification, if x is a locally optimal solution of the semidefinite program, then there
must exist (j ; U
such that
is the Lagrangian function defined by
Clearly, the first-order optimality condition (21) and the feasibility of x is equivalent to the implicitly
mixed complementarity problem (6) in which the map F : S n
\Theta S n
is defined by
\Theta S n
and the following correspondence of variables are made: (U; V ) Hence, as
in Subsection 3.1, the feasibility of x and the first-order optimality condition (21) can be formulated
as the
set\Omega and the
\Theta S n
are defined by
\Theta S n
\Theta S n
Our goal is to solve the
by the potential reduction Newton method. For this purpose,
we make the following blanket assumptions on problem (20):
(D1) the objective function continuously differentiable and convex;
(D2) the map G continuously differentiable and positive semidefinite convex
(psd-convex), that is
(D3) the map affine, and the (constant) gradients frh j (x)g p
are linearly
independent;
(D4) for every (x; U;
the following implication holds:
(D5) the feasible set
is nonempty and bounded.
We propose below a new interior point method for solving the convex semidefinite program
(20) based on the potential reduction Newton algorithm of Subsection 2.3. This method not
only generalizes the algorithm developed in Section 4.2 of [11] to the context of the nonlinear
semidefinite programming problem but it also allows for more general choice of starting points.
The new algorithm uses a novel potential function / which depends on the starting point. A
key advantage of the new algorithm is that strong convergence properties can be established for
arbitrary starting points. This differs from the results in [11] which either require the starting point
to satisfy the linear equality constraint (Theorem 5 in the reference) or do not guarantee
the boundedness of the sequence of multipliers (Theorem 4 in the reference).
p+m denote an arbitrary starting point and let c 0 j h(x 0 ) and
any matrix such that
\Theta S n
Note that S depends on the starting point when
The following technical lemma is a partial restatement of Lemma 6 of [6] and is used in the
subsequent Lemma 9 to establish that the
and the set S defined above satisfy conditions
(A1)-(A3) of Subsection 2.1.
Lemma 8 Assume that G is an affine function. Then
the following statements hold:
(a) for every U 2 S n
, the function x
(b) if condition (D5) holds then, for every
!, the set
is bounded.
Lemma 9 Assume that problem (20) satisfies conditions (D1)-(D4). The following three statements
hold:
(a) the map F defined by (23) satisfies (B1) and (B2) of Subsection 3.1;
(b) the
with\Omega and H defined by (25) and (24), respectively, and the set S defined by
(26), satisfy conditions (A1), (A2) and (A3) of Subsection 2.1; and
(c) the map H restricted to the set U \Theta ! p+m is a local homeomorphism.
Proof. Since the case where c is easy to deal with, the proof below focuses on the case where
Conditions (A1) and (A2)(a) are obvious. Clearly, we have
which is nonempty because it contains the tuple (U using (10) we easily see
that the set H \Gamma1 (int
bd\Omega is empty. We have thus proved that condition (A2) holds. Using
the same arguments as in the proof of Lemma 3, we can show that if statement (a) holds, then
is nonsingular for every (U; V; j; x) 2 U \Theta ! p+m ; in particular, we can conclude that
holds due to (27), and that H restricted to the set U \Theta ! p+m is a local homeomorphism by
the inverse function theorem. Thus the remaining proof is to show that F satisfies (B1) and (B2).
For this purpose, assume that (U; V; x;
for some (dU; dV; dx; dj) 2 S n \Theta S n \Theta ! p+m , or equivalently
Lemma 8(a) together with conditions (D1), (D2) and (D3) and the fact that U 0 imply that
L(x; U; j) is a convex function of x. Hence, we have that dx T L 00
xx (x; U; j)dx 0. Multiplying (29)
on the left by dx T and using this last observation together with (28) and (30), we obtain
xx (x; U; j)dx 0: (31)
Thus F satisfies (B1). Assume now that
Then all the relations above hold with (dU; dV In particular, (28), (30) and (31) imply
that h 0 (x;
Hence, we conclude that
to (D4). Using this and the fact that relation (29) hold with
which in turn implies that due to (D3). We have thus shown that F satisfies (B2).
Associated with the set S, we now introduce the following potential function
defined for any tuple (A; B; c; d) 2 int S by
det
where i is a suitable constant.
We establish in the next result that if i 3n=2, then the above potential function satisfies
conditions (A4), (A5) and (A6) of Subsection 2.1.
then the potential function (32), the tuple a
and the constant
conditions (A4), (A5) and (A6) of Subsection 2.1.
Proof. The verification of (A4) is similar to the one of Lemma 4. Define
~
It is easy to see that
~
~
~
The definition of and ~
together with a simple algebraic manipulation reveals that
and hence that (A5) holds. Moreover, using the fact that
F and (trP
for every P 2 S n and i 3n=2, we obtain for every (A; B; c; d) 2 int S,
2i
F
Hence (A6) holds with
The next two results will be used in Theorem 3 to establish the boundedness of the sequence of
iterates generated by the potential reduction Newton method under the framework of this section.
Lemma 11 Assume that problem (20) satisfies conditions (D1)-(D5). Then the
\Theta
defined in (25) is proper with respect to the set S n \Theta F (U \Theta ! p+m ).
Proof. Using Proposition 4(a) and Lemma 7 of [6], we conclude that the map F defined in (23) is
)-equilevel monotone on S n
\Theta S n
\Theta ! p+m . Moreover, by Proposition 4(c) and Lemma 9 of [6],
it follows that F is (j; x)-bounded on S n
p+m . Since by Lemma 9 the map H restricted
to U \Theta ! m+p is a local homeomorphism, we conclude from Proposition 2 that H is proper with
respect to S n \Theta F (U \Theta ! p+m ).
In the next result we describe in more detail the set F (U \Theta ! p+m ) for the map F given by (23).
Lemma 12 Assume that problem (20) satisfies conditions (D1)-(D5). Then F (U \Theta ! p+m
is the map given by (23) and
Moreover, F is a convex set.
Proof. The inclusion F (U \Theta ! p+m straightforwardly from the definition of the
map F and the set U . Assume now that (B; c; d) 2 F \Theta ! m . We have proved in Lemma 10 of [6]
that if conditions (D1)-(D5) holds and (0; Consider now the
problem
minimize ~
subject to ~
where ~
. It is easy to
see that the functions ~
G and ~
h also satisfies conditions (D1)-(D5). Hence, applying Lemma
of [6] to this new problem, we conclude that (0; 0;
F is defined like the
function F in (23) with ', G and h replaced by ~
G and ~ h, respectively. A simple verification
shows that (0; 0;
F (U \Theta ! p+m ) is equivalent to (B; c; d) 2 F (U \Theta ! p+m ). We have thus shown
that F (U \Theta ! p+m Using conditions (D2) and (D3), and some standard arguments, we
can easily show that F is a convex set.
We establish one technical lemma which will be used to prove an important conclusion of the
main result of this section, Theorem 6.
Lemma 13 Let fU k g and fV k g be two sequences in S n
++ such that
lim
Then
lim
Proof. Since its eigenvalues are all real. Since
it follows that all the eigenvalues of U k V k are real too. This implies that the eigenvalues of (U k
are all positive. Therefore,
Since the right-hand norm converges to zero as k ! 1, the same holds for the left-hand norm.
Thus the spectrum of U k V k converges to the single element f0g. Since this spectrum is the same
as that of (U k the desired limit (33) follows.
The following is the main convergence result of the potential reduction Newton method specialized
to the convex semidefinite program (20).
Theorem 6 Suppose that problem (20) satisfies conditions (D1)-(D5), and that f(U
is a sequence generated by the potential reduction Newton method of Subsection 2.3 initialized at
an arbitrary tuple (U
(25), (26) and (32), respectively, a j Assume also that
1=2. Then, the following statements hold:
(a) every accumulation point of f(U is a solution of the
(b) the sequence f(V k ; x k )g is bounded; thus fx k g has at least one accumulation point;
(c) lim k!1 H(U
(d) every accumulation point of the sequence fx k g is an optimal solution of problem (20);
(e) if there exists
that is problem (20) has a Slater
point, then the whole sequence f(U
Proof. By Lemmas 9 and 10, the assumptions of the theorem guarantee
conditions (A1)-(A6) of Subsection 2.1. Hence,
by Theorem 3, we conclude that statement (a) holds and that the sequence fH(U
bounded. By the definition of H , this implies that fV k are bounded, and
hence fx k g ae fx
8(b) the latter set is bounded, we conclude that fx k g is bounded. Clearly, this and the fact that
bounded imply that fV k g is also bounded. Hence, statement (b) follows.
The proofs of statements (c) and (e) are based on statements (c) and (d) of Theorem 3. For
simplicity, we assume in the remaining proof that c 0 j h(x 0 ) 6= 0; the proof when c
analogous. Define
\Theta
Note that E is a closed subset of S. Moreover, using (D3) and the fact that the third component
of a is zero, we easily see that fh(x k )g ae [0; c 0 ]. Clearly, this implies that fH(U
In view of (c) and (d) Theorem 3, statement (c) and (e) follow once we establish that the map H
is proper with respect to
\Theta
and also proper with respect to E under the assumption that (0; We prove first the
properness assertion with respect to int S " E. By Lemmas 11 and 12, we know that H is proper
with respect to S n \Theta F (U \Theta ! p+m Hence, it suffices to show that int S " E is
contained in S n \Theta F \Theta ! m , or equivalently that
Using the definition of F and Lemma 8(b), it is easy to see that
cl
Moreover, it follows immediately from the definition of F and (35) that
cl F
Let (B; c) be an arbitrary element of the left-hand set in (34). Since c 2 [0; c 0 ], we have
some t 2 [0; 1]. Hence,
Since (0; cl F by (D5), by (26), and cl F is a convex set due to Lemma 12 and
Proposition III.1.2.7 of [1], we conclude that (tG cl F . Hence, by
(37) and (38), we have (B; c) = (B; holds.
Assume now that (0; To prove the properness assertion with respect to E, it suffices to
show that E ' S n \Theta F \Theta ! m , or equivalently
If (B; c) is in the left-hand set then we have
by (26) and F is convex by Lemma 12, we conclude that
Hence, by (36) and the fact that B tG 0 , we have (B; c) = (B;
holds.
Finally, we prove statement (d). For each k, let
It follows that x k is an optimal solution of the convex program
due to the fact that x k together with the multiplier pair (U the optimality condition
for this problem. Now let x 1 be an arbitrary accumulation point of fx k g. Clearly, x 1 is a feasible
solution of (20) due to Theorem 6(c). To show the global optimality of x 1 , assume that ~
x is
an arbitrary feasible solution of (20). Let t k 2 [0; 1] be such that h(x k
~
is feasible to (40). Since ft k g converges to zero, it follows that
converges to ~ x. Moreover, since H(U by the definition of S (26), we have
for each k (cf. (38)),
Hence, it follows that
log det
for all k. Rearranging this inequality, we obtain
log det
log det
I
log det
log det
where the last inequality follows from the fact that
Hence, as k goes to 1, we may invoke Lemma 13 to conclude that We have
thus proved that x 1 is an optimal solution of (20).
Remark. The significance of part (d) of Theorem 6 is that it does not require the sequence of
multipliers to be unbounded.
Assuming that G 0 0, it is possible to show that the potential function (32), a j
and
the inequality in condition (A6) for every (A; B; c; d) in the set
E is defined in the proof of Theorem 6. Using this fact, it is possible to establish a convergence
result similar to Theorem 6 for a j 1. The crucial point to note is that Theorem
3 still holds if we assume the inequality in condition (A6) to be valid only for points in the sequence
)g. Details are omitted.
--R
Convex Analysis and Minimization Algorithms I
A unified approach to interior point algorithms for linear complementarity problems
The complementarity problem for maximal monotone multifunctions
Pathways to the optimal set in linear programming
On two interior-point mappings for nonlinear semidefinite complementarity problems
Iterative Solution of Nonlinear Equations in Several Variables
First and second order analysis of nonlinear semidefinite programs
Existence of search directions in interior-point algorithms for the SDP and monotone SDLCP programs
An interior point potential reduction method for constrained equations
--TR
--CTR
Samuel Burer , Renato D. C. Monteiro , Yin Zhang, Interior-Point Algorithms for Semidefinite Programming Based on a Nonlinear Formulation, Computational Optimization and Applications, v.22 n.1, p.49-79, April 2002
Tong , Liqun Qi , Yu-Fei Yang, The Lagrangian Globalization Method for Nonsmooth Constrained Equations, Computational Optimization and Applications, v.33 n.1, p.89-109, January 2006
Christian Kanzow , Nobuo Yamashita , Masao Fukushima, Levenberg-Marquardt methods with strong local convergence properties for solving nonlinear equations with convex constraints, Journal of Computational and Applied Mathematics, v.172 n.2, p.375-397, 1 December 2004
Christian Kanzow , Nobuo Yamashita , Masao Fukushima, Levenberg-Marquardt methods with strong local convergence properties for solving nonlinear equations with convex constraints, Journal of Computational and Applied Mathematics, v.173 n.2, p.321-343, 15 January 2005
Stefania Bellavia , Maria Macconi , Benedetta Morini, STRSCNE: A Scaled Trust-Region Solver for Constrained Nonlinear Equations, Computational Optimization and Applications, v.28 n.1, p.31-50, April 2004
Jong-Shi Pang , Defeng Sun , Jie Sun, Semismooth homeomorphisms and strong stability of semidefinite and Lorentz complementarity problems, Mathematics of Operations Research, v.28 n.1, p.39-63, February | potential function;global convergence;interior point methods;primal-dual methods;complementarity problems;semidefinite programming;potential reduction algorithm;newton method;constrained equation;variational inequality |
588961 | A Practical Algorithm for General Large Scale Nonlinear Optimization Problems. | We provide an effective and efficient implementation of a sequential quadratic programming (SQP) algorithm for the general large scale nonlinear programming problem. In this algorithm the quadratic programming subproblems are solved by an interior point method that can be prematurely halted by a trust region constraint. Numerous computational enhancements to improve the numerical performance are presented. These include a dynamic procedure for adjusting the merit function parameter and procedures for adjusting the trust region radius. Numerical results and comparisons are presented. | Introduction
. In a series of recent papers, [3], [6], and [8], the authors have
developed a new algorithmic approach for solving large, nonlinear, constrained optimization
problems. This proposed procedure is, in essence, a sequential quadratic
programming (SQP) method that uses an interior point algorithm for solving the
quadratic subproblems and achieves global convergence through the application of a
special merit function and a trust region strategy. Over the past several years the
theory supporting this approach has been analyzed and strengthened. This theory is
presented in a companion paper [4]. In addition, implementations of the algorithm
have been extensively tested on a variety of large problems, including standard test
problems and problems of engineering and scientific origin, ranging in size from several
hundred to several thousand variables with up to several thousand constraints.
Specific strategies have been developed for handling the parameters utilized by the
algorithm and for dealing with nontrivial pathologies (e. g. , linearly dependent active
constraint gradients or inconsistent linearized constraints in the quadratic subprob-
lem) that often occur in large scale problems. In this paper we present the results of
these efforts.
Based on its theoretical foundation and on our numerical experience we are confident
that this algorithm provides an efficient means for attacking a large, sparse,
nonlinear program with equality and/or inequality constraints. Rigorous comparisons
of algorithms for large nonlinear problems is notoriously difficult, especially given the
extensive set of options typically available in codes for such problems. Nevertheless,
our algorithm, with the (conservative) default parameter settings, has been successful
on problems that have caused difficulties for other algorithms and, consequently, we
are encouraged to believe that it is competitive at the current stage in the development
of methods for solving these large problems.
Below we give an outline of our basic procedure and in the succeeding sections we
provide more specific detail on the component parts of the implemented algorithm,
including the strategies and safeguards that we have used. We also exhibit and comment
on the results of some of our numerical tests. This paper relies heavily on the
results from the paper on the theory for motivation of the basic ideas.
Applied and Computational Mathematics Division, National Institute of Standards and Tech-
nology, Gaithersburg, MD 20899
y Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213-3890
z Mathematics Department, University of North Carolina, Chapel Hill, NC 27599
We assume the general nonlinear programming problem to be of the form
subject to: g(x) 0
are smooth functions. Nonlinear equality
constraints are not included in our description here in order to avoid distracting
technicalities. The modifications necessary for their insertion can be inferred from [6].
Nonlinear equality constraints are included in our code and in some of the problems
we tested. The sequential quadratic programming method is the backbone of our
algorithm. (See [7] for a review of these techniques.) At the kth step we have an
iterate, x k , denoting the current approximation to the solution of (NLP ). In addition
to the x-iterate we also maintain a non-negative iterate, z k 2 R m , which measures
the infeasibility at x k . At this stage (NLP ) is modeled by a quadratic program of
the form
min
subject to: rg(x k
is taken to be an appropriate approximation to the Hessian of the Lagrangian
for (NLP ), i.e.,
where
and H xx represents the Hessian with respect x of the function to which it is applied.
(See Section 4.5 for a discussion of the choice of B k used in our numerical experiments.)
In this form (QP ) generates a step that provides a search direction for improving the
current iterate.
There are two significant points to be made concerning this phase of our algorithm.
First, we apply an interior point quadratic program solver to (QP ); more specifically,
we use the method found in [1] where solutions are calculated by solving a sequence of
low dimensional quadratic programs. Pertinent details of this solver and its properties
relative to its use in our SQP method can be found in Section 2. Second, we do not
try to solve (QP ) with complete accuracy at each iteration; rather, we often terminate
the interior point method prematurely. In particular, we halt the quadratic program
solver when the steplength exceeds a "trust region radius" that is modified at each
iteration according to how well the improvement in our merit function is predicted.
Thus our algorithm can be said to be a "truncated Newton method" in the sense
of [18] (see also [15]). This particular merit function and a more useful "working
version" are discussed in Section 3 and our strategy for updating the trust region
radius is given in Section 4.2.
The output of the (QP ) solver is a vector that determines the direction of the
step in the x-variable, which in turn yields a step direction for the "slack" variable
z as explained in Section 3. The combined step direction of these two variables is a
descent direction for the working version of the merit function and also for constraint
thus we can choose the steplength in this direction to decrease the merit
function and/or the infeasibility of the iterate. The choice of steplength determines
the new iterate x k+1 and also the new value z k+1 . The strategy for choosing the
steplength and other algorithmic details, including the modifications and safeguards
necessary to make an implementation robust, are given in Section 4.
The results of our numerical tests are contained in Section 5. These results
demonstrate the overall effectiveness of the procedure and highlight the beneficial
effect of our trust region strategy and other procedures. Finally, in Section 6 we
briefly consider weaknesses in the current version of the algorithm and suggest possible
avenues of research to improve its efficiency.
For a discussion of the theoretical and practical questions related to large scale
nonlinear programming see the recent surveys [12], [14] and [21].
2. An Interior Point QP Solver. Interior point methods for linear programming
have been demonstrated to be very successful, especially on large problems,
and recent research has lead to their extension to quadratic programs. A particular
method, the method of optimizing over low-dimensional subspaces, has performed well
on linear programs and has been extended to the quadratic programming case (see
[1], and [2] and the references contained therein). This method, for which good numerical
results for quadratic programs have been reported, has properties that make
it particularly compatible with the SQP algorithm we are describing in this paper. A
brief description of the essential features of this method and their importance for our
purposes follow. The many details of the actual algorithm that are not reported here
may be found in the above references.
The quadratic program that we solve, (QP ), has the form
subject to: A T s
n\Thetan , A 2 R n\Thetam , and b 2 R m . The assumptions on (2.1) that
are necessary to apply the interior point algorithm are that the problem be bounded,
that A have full column rank, and that there exist feasible points (i.e., that the
constraints be consistent). Note that Q can be indefinite and that no assumption of
a full-dimensional interior is required. If equality constraints are present, they are
handled by writing them as two inequalities.
An important prerequisite for solving (2.1) by an interior point method is a feasible
initial point. Our algorithm uses a "Big M " method to construct the Phase I
problem
min
subject to: A T s
where e is a vector of ones and ' is the "artificial" variable. Clearly for ' large
enough the point (s; is feasible for (2.2) and if M is sufficiently large the
algorithm applied to (2.2) will reduce ' until the artificial variable is nonpositive, at
which point the current value of s is feasible and the M' and e' terms are dropped.
If no such value of the artificial variable can be found, then (2.2) is not consistent and
the algorithm stops. As discussed below, we make use of the step obtained from (2.2)
even if it is not feasible for (QP ). Note that when equality constraints are present,
the entire solution procedure takes place in Phase I and ' will always be present.
The defining characteristic of the algorithm is that it proceeds by solving a sequence
of low-dimensional subspace approximations to (2.1). In our application we
follow the reported results in which the dimension of the subspace is taken as three.
The following is an outline of the O3D (for Optimizing over 3-Dimensional subspaces)
version of the algorithm. As the variable ' is treated essentially the same as the
components of s in the O3D algorithm (see, however, Step 6 below) the dependence
on ' is incorporated into the formulation given in (2.1).
O3D Algorithm for Quadratic Programming
1. Given a feasible point, s 0
2. Generate 3 independent search directions be the
matrix whose columns are p i .
3. Form and solve the restricted quadratic program
subject to: A T ~ s
where ~
Call the solution i .
4. Set s j+1 := s for an appropriate value of the steplength ae 2 (0; 1):
5. If stopping criteria are met, exit.
6. Go to 2. (At this step, if the component of the vector s corresponding to
the artificial variable ' has become nonpositive, it is eliminated from the
problem.)
The search directions in Step 2 are solutions to
\Theta AD 2 A T
where fi is a scalar depending on the current iterate,
with , and the t i are particular values chosen such that one of these
directions is always a descent direction with respect to the objective function. The
steplength ae is set to the lesser of 99% of the distance to the boundary and the distance
to the minimum of the objective function.
The form of the matrix in (2.3) allows for efficient exploitation of the sparsity.
Note that if Q is positive semi-definite, then the matrix in (2.3) is positive definite for
all interior points; otherwise, it may not be. In the latter case, a modification similar
to that in [20] is used. In our application of this algorithm, using this procedure
obviates the need for the matrix B k to be positive definite, which in turn allows us
to use the Hessian of the Lagrangian or a finite difference approximation thereof.
The standard stopping criterion for the algorithm is that at least one of the following
holds: (a) the relative change in two successive values of the objective function
is small; (b) the relative difference between the primal and the dual objective function
values is small; or (c) the difference between two successive iterates is small. For use
in our SQP algorithm we have added: (d) the length of the solution vector exceeds
a specified value. This additional condition has been implemented to allow for trust
region strategies; in particular, this criterion will cause the algorithm to halt if (QP )
is unbounded. In any case, the terminal vector will be a useful direction in the context
of our purposes; this point will be discussed in the next section.
The most recent version of O3D described in [1] contains an option to perform
a special "recentering step" after each subspace optimizing step (Step 4) that has
generally improved the efficiency. This option is not used in the results reported here.
(See Section 6 for a further comment.)
3. Updating the Iterates: the Merit Functions. In this section we review
the definitions and properties of our merit functions and provide formulas for updating
the iterates. The reader is referred to the companion paper for proofs and motivations
of these concepts.
As stated in Section 1, at each iteration our algorithm yields a pair
x k is an approximation to the solution of (NLP ) and z k is the corresponding approximate
slack vector. The step directions for the updated values of these approximations
are based on the (approximate) solution, to the quadratic program
min ffi;'
subject to: rg(x k
obtained as described in the preceding section. The vector ffi k gives the step direction
for x k and we determine the step direction, q k , for the slack vector z k by the formula
\Theta rg(x k
Note that if ffi k is feasible for (QP ) then
\Theta rg(x k
In this case z k is the slack vector for (QP ) corresponding to ffi k and thus is the
slack variable for the linear approximation of g(x k+1 ). Given the step direction we
then update the iterate by means of the formulas
z
for some value of the steplength parameter ff: Observe that if z k 0 then the fact
that means that z k+1 will be non-negative if ff 2 [0; 1]. In
our algorithm the non-negativity of the slack vector iterates is preserved and, in fact,
it sometimes turns out to be useful to maintain the z k at a positive level (see Section
4.8).
It is important to emphasize that the ffi k are determined by (QP ), the quadratic
approximation to (NLP ), and are not dependent on the choice of z k . The z k are
generated solely for use with the merit function described below. That is, we do not
solve the slack variable problem. A comment on the notation is also in order at this
point: We denote the iterate by and the step by (ffi k ; q k ), whereas conventional
notation would be to use ' x k
z k
and
It should be clear from the context what is meant.
In optimization algorithms the value of a steplength parameter is generally chosen
so as to reduce the value of a suitably chosen merit function. Typically, a merit function
for (NLP ) is a scalar-valued function that has an unconstrained minimum at x ,
a solution to (NLP ). Because a reduction in this function implies that progress is being
made towards the solution, it can be used to determine an appropriate steplength
in a given search direction.
In [5] and [6] a merit function for equality-constrained problems was derived that
has important properties vis-a-vis the steps generated by the SQP algorithm. Using
a slack-variable formulation of (NLP ) a merit function for the inequality constrained
problem can be constructed having the form
d c(x; z) T
where z is nonnegative, d is a scalar,
and
We use this merit function (and its approximations defined below) for choosing the
value of the steplength parameter ff. As noted above, the approximate slack vectors
generated by our algorithm, z k , always remain non-negative; thus the non-negativity
constraint on the z for / d imposes no theoretical difficulty.
The function c(x; z) defined above plays an important role in our algorithm as it
is used to measure the feasibility of the pair (x; z). That is, if we define the function
where k\Deltak denotes the standard Euclidean norm and set
then C 0 corresponds to the feasible set of (NLP ) and hence close to feasible
if it is in C j for small j.
For d sufficiently small the merit function / d has the desirable property that a
solution of (NLP ) corresponds to a (constrained) minimum of / d . In addition, if d is
small and ffi k is the exact solution to (QP ) (which implies that ' then the step
descent direction for / d when sufficiently close to feasibility.
Despite these useful properties, / d has two deficiencies that limit its use in an efficient
algorithm. First, (ffi k ; q k ) is a descent direction of / d only near feasibility, and, second,
the evaluation of rf and rg and additional nontrivial computational algebra are
required to assess a prospective point. In order to overcome these difficulties, the
approximate merit function
d
d
where
is developed as a "working" version of / d at As the values of
k and
are fixed, / k
d can be more easily evaluated than / d in a line search algorithm for
choosing an appropriate value of ff. This approximate merit function, / k
d , not only
has essentially the same properties as / d with respect to the step
has the stronger property that the step is a descent direction for / k
d everywhere.
Moreover, for sufficiently small and outside of a ball around the solution
a "sufficient" reduction in / k
d implies a "sufficient" reduction in / d . (We mean by
"sufficient" reduction that a Wolfe condition is satisfied.) Thus we are able to use / k
d
as a surrogate for / d for testing the progress of our iterates towards a minimum.
A further important property of the step ffi k , under the assumption that it is the
exact solution to (QP ), is that it is a descent direction for the function r defined by
(3.4). Thus a basic algorithm for the case where the (QP ) can be solved exactly is as
follows: Given an initial value of j use the steps (ffi k ; q k ) to reduce r until the iterates
are in C j . Once the iterates are contained in C j if a sufficient reduction in / k
d does
not yield a sufficient reduction in / d then reduce j. If, in the course of the algorithm,
remains bounded away from zero, then convergence follows from the fact that the
Wolfe condition is satisfied for / d . If j goes to zero, then convergence follows from
the observation that the radius of the ball in which the Wolfe condition is not satisfied
also goes to zero. This is essentially the algorithm for which global convergence is
proved in the paper on the theory.
In this paper we are primarily interested in enhancements that convert the theoretical
algorithm into one that is practical and efficient. This requires that we make
provisions for situations when the assumptions under which we performed the convergence
analysis are not valid and that we adopt numerical procedures to reduce the
computational effort. As we note below, not all of these modifications have been (or
even can be) theoretically justified, but we believe that the firm foundation of the
underlying algorithm and the evidence accumulated in extensive numerical testing
validate their use.
In the implementation of our algorithm a trust region constraint is used that
possibly truncates the quadratic programming algorithm before an exact solution
is achieved. In this case the theory described above does not apply for the step
obtained from the approximate solution, Although a general
convergence theory based on this step is not yet available, it is shown in the theory
paper that if the approximate solution is obtained from the O3D algorithm and if ' k
is not too large then the resulting step has the appropriate descent properties for the
functions r, / d , and / k
d at In particular, convergence can be achieved if ' k
goes to zero in a suitable manner. These properties justify our use of the truncation
procedure to speed up the algorithm. It is important to note that this approximation
procedure also allows us to handle the difficulty that arises in sequential quadratic
programming methods when the quadratic subproblem is inconsistent.
4. The Truncated SQP Algorithm. In this section we give a somewhat detailed
description of our algorithm. Initially we assume that the Hessian approxima-
are positive definite, the matrices
A k , are nonsingular, and the linearized
constraints in (QP ) are consistent. In real-world applications these assumptions are
not always valid so we have tried to make our algorithm flexible enough to perform
well in situations where these assumptions fail to hold. We describe some of these
adaptations at the end of this section.
The implementation of the algorithm depends upon four important parameters
that need to be either computed or modified throughout the course of the algorithm.
The globalization parameter, j, was introduced in (3.5). It is a measure of the size
of the domain about the feasible region in which the direction (ffi k ; q k ) is a descent
direction for the true merit function / d . A current estimate of j is maintained in the
algorithm. The trust region parameter, , is an upper bound on the (weighted) norm
of our approximate solution to (QP ),
where D is a positive definite diagonal matrix. The trust region radius is updated
at every iteration. The parameter, ff, is the steplength parameter. It determines
the length of the step in the variables (x; z) in the direction (ffi k ; q k ). It is chosen
to guarantee progress towards the solution in decreasing either the merit function
or infeasibility. Finally, d, the merit function parameter, must be small enough to
guarantee that the theoretical properties described in the preceding section are valid.
Although the theory allows arbitrarily small values of d, the algorithm becomes very
slow if d is too small, thus it is monitored throughout the algorithm and either increased
or decreased as appropriate.
The outline of the algorithm is followed by specific comments on the procedures
and their justifications. This version contains some of the practical modifications
described above. To simplify the notation we define
Recall that r is given by (3.4).
Basic Truncated SQP Algorithm
1. Initialization: Given x
a. Initialize the slack variable z 0 0;
b. Set k := 0:
2. Calculation of the basic trust region step:
a. While kffik ! , iterate (using O3D) on
min
subject to: rg(x k
to obtain ffi k and ' k .
b. Set
\Theta rg(x k
\Theta rg(x k
otherwise
c. Decrease d if necessary.
3. Computation of the steplength parameter:
a. Choose ff 2 (0; 1] such that / k
d is sufficiently reduced.
b. If
reduce ff if necessary until r is sufficiently reduced.
c. If reduce ff if necessary so that
4. Update of the estimate of the globalization parameter:
a. If
set
5. Update of the variables and check for termination:
a. Set
z
b. If convergence criteria are met, quit.
c. Update B k to B k+1 .
6. Adjustment of the merit function and trust region parameters:
a. Update d if necessary.
b. Adjust the trust region radius .
7. Return:
a.
b. Go to Step 2.
4.1. The Globalization Parameter. The globalization step is based on work
in [6] and [4]. In Step 3 we require that the approximate merit function be reduced
and, in addition, if the current iterate lies outside the set C j we require that the
constraint infeasibilities also be reduced. This is possible as a result of the descent
properties described in Section 3. If we have a good estimate of j and
then the true merit function can also be reduced; if this is not the case, then our
estimate of j is too large and we reduce its value in Step 4. This procedure will
eventually lead to a sufficiently small value of j. Note that this arrangement allows
steps that may increase the merit function, but only in a controlled way. It also allows
steps that may increase the constraint infeasibilities, but only when inside of C j .
4.2. Updating . Our procedure for updating , the trust region radius, in
Step 6b is similar to the standard strategy used in trust region algorithms (see [17] or
[31]) in that we base the decision on how to change on a comparison of a predicted
relative reduction, pred k ; and an actual relative reduction, ared k , in a function used
to measure the progress toward the solution. (Various formulas for the predicted relative
reduction, pred k , have been suggested for different merit functions, especially for
equality constrained programming problems; see, for example, [19]). What is distinctive
about our procedure is that we use different functions for computing pred k and
ared k depending on the current status of the algorithm. When the linearized constraints
are satisfied we use the approximate merit function to compute the predicted
and actual reductions. When the trust region constraint causes O3D to terminate in
Phase I, i.e., when the linearized constraints are not satisfied, predicted and actual
reductions in infeasibility are used.
In the case when a feasible solution to (QP ) is obtained then / k
d is used to
compute the predicted and actual reductions. Our method for defining pred k differs
from the standard methods used in unconstrained optimization because the step-
finding subproblem is not based solely on the merit function and, moreover, the
trust region constraint does not appear explicitly in the subproblem. Nevertheless
in updating we want to assess how well an approximation to / k
d agrees with / k
d in
the direction uses a quadratic approximation of the Lagrangian
for the objective function with linearized constraints, we form our approximation to
d based on a quadratic approximation to the function / k
1 given by
and a linear approximation to
Note that / k
d
z). Based on these considerations and the
results of [16] we define the predicted relative reduction by
pred
ae
d
oe
where the derivatives are with respect to x and z and the steplength parameter ff k is
the size of the most recently accepted step. The value of the actual relative reduction,
ared k , is taken to be the difference in the values of / k
d at the points
and divided by the value of / k
valid criticism of the formula for
pred k is its dependence on higher order derivatives. Therefore we use the available
approximation of the Hessian of the Lagrangian for r 2 / k
1 . For example, cell-centered
finite difference approximations to the Hessian of the Lagrangian function were used
in the numerical results presented here, unless analytic second derivative formulas
were readily available.
The above choice for pred k is not used when the step returned by O3D is not
feasible. In these situations the resulting step is dominated by a feasibility improving
component and it makes little sense for the adjustment to to be determined by / k
rather, a comparison of the predicted and actual improvement in constraint infeasibility
seems more appropriate. Therefore, in this case the function r(x; z) is used for
comparison purposes. The values of pred k and ared k are given as follows for the case
when the O3D algorithm terminates in Phase I:
pred
and
ared
These heuristics for choosing pred k and ared k appear to work well. Specifically,
they allow the trust region radius, , to be increased even in the event that the step
returned by O3D does not satisfy linearized constraints or it results in an increase in
the true merit function. In our experience, the alternative formulas based solely on
constraint violations never are employed close to the solution. Indeed, the iterates
preceding convergence have always been observed to be well inside C j where satisfying
the linearized constraints and decreasing the merit functions usually pose no problem.
4.3. The steplength ff. The steplength ff is determined in Step 3 of the al-
gorithm. The "sufficient decrease" referred to in 3a and 3b requires that the Wolfe
condition be satisfied. For a given function OE and potential step w from point v this
condition requires that ff satisfy
for some fixed oe 2 (0; 1). In the numerical experiments reported in Section 5 we employed
a simple backtracking procedure (with factor one-half) to find ff to satisfy this
condition for both / k
d and for r. We have also experimented with more sophisticated
line search methods motivated by unconstrained optimization techniques as in [18],
but the observations to date suggest that the more complicated line searches result in
very little improvement of our algorithm, except when the iterates are quite far from
the solution.
4.4. Adjusting d. Choosing an effective value for the merit function parameter
d is essential in our algorithm. While it is clear that (in a compact set) a sufficiently
small value of d will assure that the results given in [4] are valid, there are three
very important practical reasons why the parameter must be adjusted rather than
fixed. First, if the angle between the direction generated by O3D and the gradient of
the approximate merit function becomes nearly orthogonal the steps might become
too small. We adjust d to avoid this possibility. Second, the approximate merit
d , is changing at each iteration and it is possible a previous iterate might
be acceptable to the current / k
d , i.e., cycling might occur. This worry can also be
alleviated by adjusting d. A third reason for changing d is to allow for larger steps.
It is seen from the theory and has been verified by numerical experience that if d
is too small then the form of the merit function forces the path of the iterates to
follow the "nearly active" constraints closely. This causes the algorithm to take very
small steps and, in particular, to be slow in moving away from a nonoptimal active
set. By making it possible to increase d we can significantly improve the algorithm's
performance.
In the implementation of our algorithm there are two opportunities to adjust d:
in Step 2, after solving the quadratic subproblem, and in Step 6, after the step has
been taken. In the first of these adjustments d can only be decreased; in the second,
the parameter may be increased or decreased.
In Step 2, the angle between the gradient of the approximate merit function
d and the step direction (ffi k ; q k ) is computed. If these two vectors become nearly
orthogonal, we conclude that d is not small enough to ensure a good decrease in / k
d ,
and we decrease the parameter. To be more specific, we compute
If w(d) \Gamma:1 we calculate a value "
d so that w( "
d) \Gamma:5: We safeguard the procedure
by not allowing more than a certain percentage decrease in d. In the current version
we use 50%.
If d was not decreased in Step 2 we consider modifying it after a step has been
taken (Step 6). Here the primary concern is to avoid cycling. To do so we compute
an interval for the penalty parameter as follows. For a fixed integer we seek a value
of the parameter,
d, such that
d
Inequality (4.2) implies that none of the past iterates will be acceptable to the
approximate merit function with the new value of
d. (Thus if cycling would
be possible). To accomplish this, we use the decomposition
1 and / k
are defined in Section 4.2. We then compute the values of / k
and / k
consider the inequalities
d
d
We define d u
i and d l
i to be the upper and lower values of d that ensure that inequality
(4.4) is satisfied. Then letting
d
and
we obtain an interval (d l ; d u ). Assuming that this interval exists it is the case that
if the value of d for the next step is chosen in this interval, the next iterate will
not return to one of the previous iterates. In practice a value of 5 is usually
more than sufficient to prevent cycling. If the interval doesn't exist, then we make no
change.
Given that we can choose d to avoid cycling, our second objective at this juncture
is to increase d to allow bigger steps. If the d u is larger than the current d then we
can safely increase d without worrying about possible cycling. However, we safeguard
this increase in two ways. First, we require that the predicted reduction based on the
approximate merit function must be greater than the predicted reduction of infeasibility
in the linearized constraints. This restriction prevents d from being increased
prematurely due primarily to a large decrease in constraint infeasibilities. Specifically,
writing the predicted reduction in / k
d (see (4.1)) as
we insist that for a new value of d
Second, we use a maximum allowable change (currently a factor of 2) to limit the
growth of d. Computationally, these simple procedures for updating d appear to be
effective, especially in the presence of highly nonlinear constraints and poorly scaled
problems.
4.5. The Hessian Approximation. In the numerical experimentation reported
here, we have used a finite difference approximation to the Hessian of the Lagrangian
. Although the Hessian of the Lagrangian at a strong solution is positive definite
on the appropriate subspace, it may be indefinite in general. Even if it is positive
definite the finite difference approximation may not be. We experimented with two
approaches for handling this possibility. First, we simply modified the approximate
Hessian matrix by adding non-negative elements to the diagonal ensuring that the
Cholesky factorization of the matrix had positive elements along its diagonal (see
[20]). This modification was easy to implement, but it was observed to slow convergence
on some problems. While this modification guarantees that a positive definite
matrix will be delivered to the (QP ) solver, if it takes place when the iterates get
close to the solution, it generally precludes local q-superlinear convergence.
An alternative to modifying the approximate Hessian of the Lagrangian is simply
to allow O3D to iterate on the indefinite QP subproblem, halting the iterations when
the solution exceeds the trust region radius. We implemented this approach and
it seemed to yield superior results to those obtained by making the approximate
Hessian positive definite (especially when the iterates were close to a solution) even
though, theoretically, we can only prove that we obtain a descent direction when the
approximate Hessian is positive definite.
4.6. Convergence Criteria. The convergence criteria used are standard, and
similar to those in [3]. We first insist that the constraints be satisfied to a close
tolerance; specifically we require
We also require that either
or
The criterion (4.9) is a stronger indication that a KKT point has been reached. The
weaker criterion (4.10) suggests that progress slowed drastically and that iterates may
or may not have drawn close to a solution. For this reason criterion (4.9) is usually
preferable to criterion (4.10). The Lagrange multipliers returned by the quadratic
program are used in (4.9) unless the trust region constraint determines the approximate
solution of the (QP ). In that case, we use the least squares approximation to
the multipliers, replacing all negative multipliers with machine zeros. In all of the
problems solved to date, the trust region never comes into play when the iterates get
close to the solution; therefore the (QP ) multipliers are used for the convergence test
at the solution.
4.7. Inconsistent Quadratic Subproblems. One difficulty that can occur
when making linear approximations to nonlinear constraints is that (QP ) may be
inconsistent. In this case O3D will, even if it runs to completion, not exit Phase
I and will return a positive value of the artificial variable. (Note that this always
occurs if equality constraints are present.) For small ' the resulting direction is a
descent direction for / k
d and for r. As a result, the step taken in this direction
will generally decrease infeasibility, making it less likely that an inconsistent set of
linearized constraints will be encountered during subsequent iterations.
More recent versions of our algorithm include a constraint relaxation procedure
that appears to yield an acceptable step, even in the event that inconsistent
linearizations of constraints are encountered. Because this situation did not surface
during the numerical experiments presented in this paper, we do not include a description
of our perturbation procedure. We do note, however, that we have encountered
important application problems where this procedure was crucial to the performance
of our algorithm (see for example [24]).
4.8. Updating slack variables. One difficulty in our algorithm is the updating
of slacks in the event that the SQP step does not satisfy the linearized constraints
well enough, i.e., ' k is not small enough. This can occur when (QP ) is inconsistent
or when a trust region bound is encountered during the solution of (QP ). In this case
our slack variable updating scheme would ensure that non-negative slacks remain non-
negative, but the direction may not be one of descent. We resolve this dilemma by
opting for descent, i.e., computing q k with ' replacing any negative slacks
using the following rule:
If z k+1
z k+1
ae ffl Mach g i
\Gammag i
where ffl Mach is machine epsilon. This is sometimes referred to as 'closing' the constraints
(see for example [33]).
4.9. Linearly Dependent Constraint Gradients. Linearly dependent constraint
gradients cause many theoretical and computational difficulties in constrained
optimization. In our theoretical algorithm we obtain convergence even when there are
linearly dependent constraint gradients provided the approximate multipliers do not
become unbounded. In practice, even though O3D has no difficulty in dealing with
this problem, evaluating the merit function and computing the least squares approximation
to the Lagrange multipliers become problematical. Computational experience
shows we solve many problems with degeneracy in the constraints. Simply maintaining
slacks to to be positive as described above allows us to factor the crucial matrices
and continue with the algorithm. However, the algorithm failed to solve some problems
that had a large amount of degeneracy in the linearized constraint matrix. This
was, of course, problem dependent but it was observed that the current implementation
can usually solve problems where up to 25 percent of the constraint gradients are
linearly dependent. This degeneracy causes the performance of the merit functions
to deteriorate. In particular, the least squares approximation to Lagrange multipliers
seems to be especially poor, resulting in only very small steps being allowed, even
close to the solution.
5. Numerical Results. The modified algorithm was coded in Fortran and is
installed on a SPARCstation 10 using IEEE floating point arithmetic (64 bit). The
current implementation is being used to solve a wide variety of medium to large
scale problems. In this section we report the results of a set of performance tests
designed specifically to answer questions about the trust region strategy and the
procedure to update the penalty parameter, d. We conclude the section with the
results of our algorithm applied to some test problems that are publicly available. We
emphasize that all of the problems were solved with the same default settings of the
parameters, (see Table 1), i.e., no attempt was made to pick parameter settings to
optimize performance on individual problems.
Although in many of the applications some analytic derivatives were available,
no use of analytic derivative information was used in these numerical experiments.
When possible, first and second derivatives were computed using forward and central
finite differences respectively. A costly one-time calculation provided a zero/non-zero
stencil of the Hessian of the Lagrangian and the Jacobian matrix of the constraint
function. These stencils were then used for the duration of the solution process.
For some problems, these finite difference approximations are not convenient to use.
This can be the case with control problems governed by partial differential equations
(see [29] or [30]). If the partial differential equation is solved using a finite element
method, with piecewise linear elements, then evaluating the derivative of the objective
Parameter Value
z Mach )
Table
Numerical values of default parameters
function with respect to the control variables can be quite cumbersome. In such
cases, which occurred in the control problems in our test suite, one can approximate
the first derivatives of the objective function by solving an adjoint problem with
a computational cost comparable to one function evaluation. (For examples, see
[22].) The objective function portion of the Hessian of the Lagrangian can then be
approximated with forward finite differences.
A set of eight problems was chosen as the first test suite. These problems ranged
in size from 500-1000 variables and from 1000-2000 constraints. The first four are
relatively straightforward nonlinear programming test examples, while the last four
are from actual applications: two discretized control problems, a density estimation
problem from statistics, and a "molecular distance" problem. A more complete description
of these problems is found in the Appendix. The problems all have nonlinear
inequality constraints and exploitable sparsity. Problem 4 (NLP4) was designed to
have a controllable percentage of linear dependency in the constraint gradients to
demonstrate any weaknesses in the algorithm associated with this difficulty. We ran
three versions of our algorithm on each problem; using a positive definite modification
of the Hessian matrix, as discussed in Section 4, with and without the trust region
strategy and using the unmodified Hessian with the trust region. (Using the unmodified
Hessian results in failure in most cases if no trust region strategy is employed.)
In addition, each problem was run from two starting points; one, labeled "c", which
was close to the solution in the sense that each of the variables was of the same order
of magnitude as in the solution and a distant start, labeled "f".
The results of the numerical tests on these problems are summarized in Tables
1-3. The first two columns of each table gives the number of SQP iterations ("nl-i")
and the total number of O3D iterations ("qp-i"). The next two columns contain the
stopping criterion that was met and the value of the gradient of the Lagrangian at the
solution. Unless the algorithm failed, (which is denoted by "Failure" in the tables)
feasibility condition (4.8) was satisfied for all solutions. The stopping criterion is
denoted by either a "1" or a "2" depending on whether (4.9) or (4.10) was satisfied.
If both were conditions were satisfied, a "3" appears in the column. The remaining
columns give information about the values of the parameter d for each run; columns
five through eight giving the initial, maximum, minimum, and final values of this
parameter and the final column giving the last iteration at which d was changed.
The results of the tests illustrate that using the unmodified Hessian with the trust
region was most effective in reducing the number of O3D iterations and the number
of SQP iterations. The trust region strategy prevented long, unprofitable steps from
being generated when far from the solution and the use of the unmodified Hessian
allowed the trust region to become inactive near the solution thus allowing rapid
local convergence. Requiring the Hessian to be positive definite often precluded rapid
local (q-superlinear) convergence and, when used in conjunction with the trust region
strategy, resulted in the trust region's being active close to the solution.
The results also show that the value of the parameter d varied over several orders
of magnitude. The procedures discussed in Section 4 that allowed the value of d to
increase or decrease greatly enhanced the algorithm; earlier tests using either a fixed
value of d or only allowing a reduction in d yielded inferior results.
Another modification in our algorithm, not reflected in the table or included in
the description in the preceding section, was made to force the O3D algorithm to take
a minimum number of steps. We found that when the trust region radius became
small the algorithm would sometimes exit O3D after only one iteration, resulting in
a poor step direction. This poor step would result in a further decrease in , and
eventually the algorithm would fail. When we required a minimum number of steps
to be taken in O3D (our choice was 7) this problem disappeared.
Recently a collection of test problems has become available for the testing and
comparing of optimization algorithms, (see [13]). The Constrained and Unconstrained
Testing Enviroment (CUTE), are quickly becoming standards with which researchers
can establish the viability and effectiveness of their numerical algorithms. These problems
are replacing the smaller and well scaled test problems of Hock and Schittkowski
[25] and Schittkowski [32] which were not intended to be used to test large scale al-
gorithms. Our results on the CUTE test problems are summarized in Tables 6, 7
and 8. These problems were solved to the same stopping conditions as the problems
above. Likewise, the same table format was used to present these numerical results.
For detailed description of these problems, structure, motivation, and sources see [9].
While it appears that the CUTE test problem set is rich in both large and small
scale unconstrained and equality constrained test problems, at present there are not
many large scale problems that include inequality constraints (and particularly non-linear
inequality constraints). We chose problems that reflected the class of problems
our algorithm was designed to solve. At least one inequality constraint was present
in each problem. The number of variables and/or constraints was large enough so
that the exploitation of special sparsity structure was important. The problems we
selected from CUTE to report on were "CORKSCREW, MANNE, SVANBERG" and
"ZIGZAG". The associated problem sizes are recorded in Table 5.
It is worth commenting that much of the machinery developed in this paper deals
with effectively handling nonlinear inequality constraints. The performance of our
algorithm on the CUTE test problem set is, therefore, slightly deceiving since many
of the constraints in these problems are simple bounds on the primal variables or
purely linear. (For instance, approximately 83% of the constraints in CORKSCREW,
50% of the constraints in MANNE, and 66% of the constraints in ZIGZAG were linear
and many of them were equality constraints). Although these caused no problem for
our algorithm, the structure of these constraints was not completely exploited and
the extra machinery of our code resulted in an overhead with no performance benefit.
Clearly an algorithm designed specifically to deal with linear equality constraints
should outperform our algorithm on these problems. The problem on which our
algorithm appeared to perform best was SVANBERG, a problem with only inequality
constraints (and a substantial number of them are nonlinear).
We succeeded in solving all four problems with a reasonable number of inner and
outer iterations. However, many of our algorithmic enhancements contributed little
to the solution process. The measure of distance to feasibility (the j-tube strategy),
the nonmonotone updating of penalty parameter d, and the trust region strategy were
essentially dormant during the solution process regardless of the iterates' proximity
to the solution or to feasibility. In fact, the only evidence of our enhancements on the
small number of CUTE test problems that we solved occurred when d was decreased
slightly while solving the problem MANNE employing modified Hessians with a trust
region strategy (see the third and fourth rows of Table 7). It is noteworthy that the
iterates that resulted from solving this problem with the penalty parameter artificially
held fixed at were identical to iterates that resulted for the adjusted d solution.
This appears to illustrate that in this case the adjustment of d was purely superficial.
Truss - c 103 3561 1 4.4e-8 9.87e-1 1.01e00 9.57e-2 9.57e-1 100
Molec-c 37 1376 1 9.8e-9 9.88e-1 1.21e1 1.17e-2 9.81e-1 34
Molec-f
Table
Modified Hessians with no trust region
BCHeat -c 257 2898 1 8.1e-8 9.18e-1 4.15e00 1.38e-1 4.10e00 249
BCHeat -f 289 3071 1 9.9e-8 1.00e00 3.74e00 2.44e-1 3.89e00 281
Molec-c
Molec-f 44 621 1 6.6e-8 1.00e00 1.48e00 7.39e-2 9.52e-2 38
Table
Modified Hessians with trust region
Truss-c 94 2242 1 3.9e-8 9.87e-1 1.03e00 1.26e-1 9.11e-1 87
Molec-c
Molec-f
Table
Un-modified Hessians with trust region
Problem Variables Constraints
CORKSCREW 96 159
ZIGZAG 304 1206
Table
Minimization parameters
Problem nl-i qp-i conv krx lk1 d0 maxd mind final d last d-cha
Table
Modified Hessians with no trust region
Problem nl-i qp-i conv krx lk1 d0 maxd mind final d last d-cha
Table
Modified Hessians with trust region
Problem nl-i qp-i conv krx lk1 d0 maxd mind final d last d-cha
Table
Un-modified Hessians with trust region
6. Future Directions. In this paper we have discussed in some detail an SQP
algorithm for solving large scale nonlinear problems. The numerical results with
default parameter settings indicate that the procedures that we have implemented
are robust, effective, and efficient; the convergence theory in [4] provides a sound
theoretical basis for the procedure. Nevertheless, there are several areas in which the
techniques used here can be improved to allow the solution of larger and more difficult
problems.
Algorithmically, we observe that the current implementation requires the factorization
of both (rg T rg+Z) and (rgrg T ), the latter in O3D. While the sparse matrix
package makes this reasonable for the problems that we have currently considered, it
is clearly expensive to maintain both.
The results reported here use analytic or finite difference Hessian approximations.
An examination of the details of O3D reveals that a limited memory BFGS or limited
memory SR1 could be readily incorporated into the code. We have done some
experimentation with such techniques; the results will be reported elsewhere [26].
Many of the problems that we have seen have been degenerate and this significantly
slows the convergence of the method. The primary culprit is the extremely
poor multiplier estimates provided by the least squares procedure. Improvements in
this area are certainly required.
In some problems (not reported here) that have nonlinear equality constraints,
we have occasionally observed significant difficulty in trying to satisfy the linearized
equality constraints, i.e., in completing Phase I. In these cases we have had some success
in relaxing the constraints [26]. In the context of O3D, this can be accomplished
by simply fixing the artificial variable at some positive value and continuing the O3D
iterations. In this approach, we often find that O3D converges and the "recentering"
procedure mentioned in Section 2 has led to further improvements. The theory in [4]
supports these ideas. The details will, again, be reported elsewhere.
--R
An interior-point method for general large scale quadratic programming problems
An interior point method for linear and quadratic programming problems
A merit function for inequality constrained nonlinear programming problems
A family of descent functions for constrained optimization
A truncated SQP algorithm for large scale nonlinear programming problems
Cute: Constrained and unconstrained testing environment
Functional and numerical solution of a control problem originating from heat transfer
On exact and approximate boundary controlla- bilities for the heat equation
Large scale numerical optimization: Introduction and overview
Lancelot: A Fortran Package for Large-Scale Nonlinear Optimization
SIAM Journal on Numerical Analysis
Globally convergent inexact Newton methods
A robust trust region algorithm with nonmonotonic penalty parameter scheme for constrained optimization
New York
Constrained nonlinear pro- gramming
Numerical Methods for Nonlinear Variational Problems
Molecular conformations from distance matrices
Optimal signal sets for non-gaussian detectors
Test Examples for Nonlinear Programming Codes
The Use of Optimization Techniques in the Solution of Partial Differential Equations from Science and Engineering
The solution of the metric stress and sstress problems in multidimensional scaling using Newtons method
Numerical solution of a nonlinear parabolic control problem by a reduced SQP method
Optimal Control of Systems Governed by Partial Differential Equations
Computing a trust region step
More Test Examples for Nonlinear Programming Codes
On the role of slack variables in quasi-Newton methods for constrained optimiza- tion
Nonparametric Probability Density Estimation
Nonparametric Function Estimation
where it was used to illustrate separability in nonlinear programming.
"distance matrices"
--TR
--CTR
Thomas F. Coleman , Jianguo Liu , Wei Yuan, A New Trust-Region Algorithm for Equality Constrained Optimization, Computational Optimization and Applications, v.21 n.2, p.177-199, February 2002
E. Bradley , A. OGallagher , J. Rogers, Global solutions for nonlinear systems using qualitative reasoning, Annals of Mathematics and Artificial Intelligence, v.23 n.3-4, p.211-228, 1998
Rubin Gong , Gang Xu, Quadratic surface reconstruction from multiple views using SQP, Integrated image and graphics technologies, Kluwer Academic Publishers, Norwell, MA, 2004 | interior point;large scale;trust region;nonlinear programming;SQP;merit function |
589001 | A Multiple-Cut Analytic Center Cutting Plane Method for Semidefinite Feasibility Problems. | We consider the problem of finding a point in a nonempty bounded convex body $\Gamma$ in the cone of symmetric positive semidefinite matrices ${\cal S}^m_+$. Assume that $\Gamma$ is defined by a separating oracle, which, for any given $m\ti m$ symmetric matrix $\hat{Y}$, either confirms that $\hat Y \in \Gamma$ or returns several selected cuts, i.e., a number of symmetric matrices Ai, i=1,. . .,p, p\le p_{\max}$, such that $\Gamma$ is in the polyhedron $ \{ Y \in {\cal S}^m_+ \mid A_i \bullet Y \le A_i \bullet \hat{Y}, i=1,\ldots,p \}.$ We present a multiple-cut analytic center cutting plane algorithm. Starting from a trivial initial point, the algorithm generates a sequence of positive definite matrices which are approximate analytic centers of a shrinking polytope in ${\cal S}^m_+$. The algorithm terminates with a point in $\Gamma$ within $O(m^3p_{\max}/\epsilon^2)$ Newton steps (to leading order), where $\epsilon$ is the maximum radius of a ball contained in $\Gamma$. | Introduction
be the set of mm symmetric matrices and let S m
be its subset of symmetric positive
semidenite matrices. We consider the problem of nding a point in a convex subset of
We assume that contains a full-dimensional closed ball with radius > 0: The set
is implicitly dened by a separating oracle, which, for any given mm symmetric matrix ^
Y ,
either conrms that ^
Y 2 or returns several cuts, i.e., a number of symmetric matrices A
such that is in the polyhedron fY
Here p max is the maximum number of cuts admitted in each iteration.
In a recent paper [8], we presented an analytic center cutting plane method for the case p
1, in which a single cut is added in each iteration. The method was shown to have a worst-case
complexity of O(m leading order). However, to make a cutting plane algorithm
practically e-cient, adding multiple cuts is often necessary. The purpose of this paper is to
propose and analyze an analytic cutting plane method that uses multiple cuts for solving the
convex semidenite feasibility problem mentioned above. In admitting multiple cuts in an
analytic center cutting plane method, we face some new theoretical problems that are dierent
from the single-cut situation, these include (a) the problem of nding a feasible starting point
for the Newton iteration after several new cuts have been added; (b) the estimation of the
number of Newton steps needed to obtain a new approximate center through estimating the
changes in the primal-dual potential function.
Our paper extends the multiple-cut schemes of Go-n and Vial, Luo, and Ye [2, 5, 10] from
Such extensions not only broaden the applications of cutting plane methods, but
also extend several classical theoretical results for non-negative vectors to positive semidenite
matrices. We note that for our multiple-cut analytic center cutting plane algorithm, the
complexity analysis on the number of Newton iterations per oracle call follows the approach
in [3].For the complexity analysis on the number of oracle calls, we follow the approach in [10],
but we simplify the proofs of some results analogous to those in [10] by considering all the
added cuts simultaneously instead of inductively.
In this paper we will show that, starting from a trivial initial point, the multiple-cut algorithm
generates a sequence of positive denite matrices which are approximate analytic centers of a
shrinking polytope in S m
. The algorithm will stop with a solution in at most O(m 3
(to leading order) Newton steps. Our analysis show that when the the problem is specialized
to the space of positive semidenite diagonal matrices (which is equivalent to the non-negative
the complexity bound is reduced to O(m 2 p This complexity bound is
lower than the existing bound of O(m
obtained in [2] and [10], where the same cuts
are considered. Our bound appears to be better than that obtained in [5]. (Note that the
proof for the bound appeared in [5] is incomplete, and to our best knowledge, a provable
bound should be O(m
Furthermore, the analysis in [5] is carried out only for the
so-called shallow cuts that are placed at some distances away from the current testing point
and hence may not be as e-cient as our proposed algorithm where the cuts pass through the
testing point.
We are able to obtain better complexity results than existing ones even when the problem is
specialized to IR m
because in each oracle call, we only admit cuts that are su-ciently good.
We shall not give the precise denition of \goodness" here but refer the reader to section 4.
Roughly speaking, base on our criteria, the admitted cuts A in each oracle call are
eective in reducing the size of polytope in the sense that each should be able to delete a sizable
portion of the current polytope that can not be otherwise deleted by the other admitted cuts.
One obvious advantage of having such a selection criterion is that the number of cuts added
in each iteration will be reduced since only eective cuts are admitted, and this translates into
saving in the computational cost in each Newton step.
We will now introduce some notations. For matrices
A Y := tr(AY
We write Y 0 and Y 0 if Y is positive denite and positive semidenite, respectively. For
Y 0, we denote its symmetric square root by Y 1=2 . The 2-norm of a vector x is denoted by
kxk, and the matrix 2-norm of a matrix A is denoted by kAk. For A
are the eigenvalues of A. Note that
k(A)k1 . We will use these facts in the paper without explicitly mentioning them. For a
positive vector x 2 IR n , we write
Generally, we use capital letters for matrices, lower case ones for vectors, and Greek letters for
scalars. For convenience, we let
Let svec be an isometry identifying S m with IR
m so that K
smat be the inverse of svec. Given any G 2 S m , we let G
m to be the unique
symmetric matrix such that
It is easy to see that if G is positive denite, then G
G is positive denite, and (G
G 1=2
G 1=2 . If G is nonsingular, then (G
Throughout, we make the following assumptions:
A1. is a convex subset of S m
.
A2.
where
A3. contains a full dimensional ball of radius > 0. That is, there exists Y c
that fY
Note that Assumption A2 is made for convenience. It can be satised by scaling if the original
set ^ is bounded. That is, suppose there exists a constant
> 0 such that for all
. Then the scaled set = fY=
2 A multiple-cut analytic center cutting plane method
We rst dene the analytic center and then propose a multiple-cut analytic center cutting
plane method at the end of this section.
, be all the cuts dening the kth working
set
k . Dene
Then the
set
k can be represented
as
We dene the following potential function on the
set
and denote
The unique minimizer of k (Y )
over
k is known as the analytic center
of
k .
It is easy to see that the analytic center of the initial working
set
0 is I=2; where I is the
identity matrix. As a matter of fact,
Y
Y
The minimum of 0 (Y ) must satisfy 1 (Y
It is known [7, Proposition 5.4.5] that k is a strongly 1-self-concordant function
on
and
diag (s), and should be the
mm matrix within the round brackets. However, we have identied that mm matrix with
a vector in IR
m through the linear isometry svec.
The optimality conditions for minimizing k are:
denotes the vector of ones)
I Y 0; Z; V 0; s; x > 0:
With a slight abuse of language, we also call the solution (
V ) of (2.1) the analytic
center
of
k .
Denition 2.1 Given a point (Y; s;
We call (Y; s; x; Z; V ) an -approximate (analytic) center
of
all the
linear equalities in (2.1) are satised, and x; s > 0, Z; V 0. Obviously, a 0-approximate
center is exactly the analytic center
of
.
Denition 2.2 Given Y
It was shown [8] that the following lemma holds.
Lemma 2.3 Given Y We have
Remark. Given Y
the minimizer For such a Y , we
will call Y an -approximate center
of
k in the sense that the point (Y; s; x Y is an
-approximate center.
We will now describe our algorithm.
A multiple-cut analytic center cutting plane algorithm.
3=2), and pick - 2 (; 1). Set
Let
0 be the initial working
set and let Y be the initial point.
Step 1. At the k-th iteration, call the oracle to either conrm that Y k is a feasible point of
or return p k matrices A n k
Otherwise, construct the new working
set
Step 2. Find a point ~
Y in the interior
of
(discussed in section 3).
Step 3. (Recentering) Starting with the point
Y in Step 2, perform the dual Newton
method:
3.1. If - k+1 (Y ) < , set Y to Step 1.
3.2. Otherwise, Set
smat
where
is determined as follows: if - k+1 (Y )
-,
. Go to Step 3.1.
3 Restoration of centrality
In our cutting plane algorithm, approximate analytic centers are found by using the dual
Newton method. Our aim in this section is to estimate the number of Newton steps required
to nd an approximate analytic center for a newly constructed working set. We do so by
estimating the amount of potential value we should reduce for the new set. The mechanics
are as follows. Since the potential function is 1-self-concordant, each Newton step can reduce
the potential function by a constant amount. Thus to estimate the number of Newton steps
needed to nd an approximate analytic center for a new working set, all we need is to estimate
the amount of potential value we should reduce for the new set.
To nd an approximate analytic center for a new working set, ideally, we would want the
Newton method to start with the preceding approximate analytic center Y k . However, Y k is
not in the interior of the new working
set
k+1 since the new cuts pass through this point.
Thus our immediate task is to nd an interior point
in
k+1 , and then use this point as the
starting point for the Newton method.
Let n k be the number of cuts dening the
set
k . Suppose that p k new cuts are added to form
the new
set
. Recall that
Then the
sets
and
k+1 can be written as
Suppose is an -approximate center with < 1
3=2. (Note that by lemma
We will now construct a point ( ~
s; ~
that is in
the interior
of
. To this end, consider the following convex minimization problem:
Evidently, the above problem has a unique solution that is also the unique solution to the
KKT-conditions:
Let (~!; ~
) be an approximate solution of the above KKT conditions where (3.1a) is satised
exactly and maxfj2p k ~
~
1=2. Note that in this case,
~
Note that to nd such a pair (~!; ~
), we can apply Newton method to (3.1a) and (3.1b), where
the computational work for each Newton iteration is O(p 3
In general, this constitutes only a
very small fraction of the total computational work involved in nding an approximate analytic
center
for
k+1 . In order not to lengthen the paper unnecessarily, we shall not establish the
complexity of the Newton method for nding (~!; ~
) in this paper. Interested reader can refer
to [3] for such results.
~
~
~
~
We refer the reader to [3] for an illuminating discussion on the motivation for considering the
optimization problem (3.1a){(3.1b) in constructing the strictly interior point
of
above.
It is readily shown that the following result holds:
F
Lemma 3.1 For any vector the following inequality holds:
Proof. Refer to [11].
Lemma 3.2 Suppose (Y is an -approximate center with < 1. Then the
following inequalities hold:
Proof. We shall omit the proof of the rst equality as it is easy. Now we proceed with the
proof of the second one. We have
where we have used a theorem of Ostrowski [4, p. 225] in the second equality above, and i 's
are scalars such that
min (Z 1=2
Noting that max (Z 1=2
proved the required inequality. The last
inequality in the lemma can be proven similarly.
Theorem 3.3 The point ( ~
s; ~
constructed in (3.3){(3.4) satises the last three conditions
in (2.1).
Proof. First, we show that ~
Y I. We have
since k(S k
3=2 < 1 from (3.7). On the other hand, we also have
~
since kY 1=2
3=2 < 1. The fact that ~
Y I can be shown similarly. Furthermore
where we used the fact that from (3.1a), B T
.
Next we show that ~ x > 0 and ~
We have
~
since by lemma 3.2,
Furthermore,
Up to this point, we have succeeded in nding an interior point
of
k+1 that is derived from
Y k . Our next task is to estimate the potential value of the new point
in
.
Lemma 3.4 Suppose - k (Y k ) . Then the potential value k+1 ( ~
Y ) satises the following
inequality
Proof. Let ~
Y and U We have
Note that we used the fact that d B T
. Now
e (U 1=2
where
Note that e T
By applying lemma 3.1 to (3.10), we have
~
Note that in the last second inequality above, we used the Cauchy inequality to derive the
~
!.
Substituting the result in (3.12) into (3.9), we prove the lemma.
>From lemma 3.4, we see that the upper bound for the dual potential value k+1 ( ~
the term ln ~
. If we were to consider the dual potential value alone, then nding an upper
bound for ln ~
is necessary. But we have found that nding a tight upper bound for this
term is di-cult. As a result, we have decided to consider the primal-dual potential value for
which nding an upper bound for ln ~
is not necessary. To this end, let us dene the primal
potential function associated
with
k . For any k (x; Z; V
++ that satises
the primal potential of (x; Z; V ) is dened by
The primal-dual potential function associated
with
k is
We should emphasis that the primal-dual potential function is introduced solely for the purpose
of estimating the potential value of ( ~
V ). It is not needed in our cutting plane
algorithm described in section 2.
Now we shall proceed to establish an analog of lemma 3.4 for the primal potential function.
Before doing that, we need the following lemma.
Lemma 3.5 For the directions (x; Z;V ) given in (3.4), the following inequality holds:
Proof. Noting that
!, we have
d T ~
Thus
(Y 1=2
(I U 1=2
F
F
Note that in the last inequality above, we used (3.7) and the fact that
Lemma 3.6 For the point (~x; ~
constructed in (3.6), the following inequality holds:
Proof. We have
where
Note that e T
by lemma 3.2,
F
F
By lemma 3.1 and (3.17), we get from (3.16),
By applying lemma 3.5 and (3.7), we prove the lemma.
The next lemma is an analog of lemma 3.4 for the primal-dual potential function.
Lemma is an -approximate center with < 1
3=2. Then
where
Proof. Combining the results in lemmas 3.4 and 3.6, we have
Note that
~
~
~
~
By substituting (3.20) into (3.19), the lemma is proven.
With lemma 3.7, we can nally establish an explicitly known upper bounded for the primal-dual
potential value k+1 ( ~
Theorem 3.8 Suppose (Y is an -approximate center with < 1
3=2.
Then
where () is the constant given in (3.18).
Proof. We have
It is readily shown that
Next we need to get an upper bound for the term k (Y k
(3.22). By following the proof of lemma 2.1 in [1] and using the quadratic convergence result
in [8], it is readily shown that
Similarly, it can be shown that
Combining (3.24) and (3.25), we get
By putting the results in lemma 3.7, (3.23) and (3.26) into (3.22), the theorem is proven.
With the estimate of k+1 ( ~
theorem 3.8, we are now ready to estimate the number
of dual Newton steps required to nd an approximate analytic center
for
k+1 by using the
point ~
Y as the initial point.
Theorem 3.9 Given an -approximate center Y k
of
k with < 1
3=2. The total number
of dual Newton steps required to nd an -approximate center Y k+1
of
k+1 is
O (p k
where the constant O(1) is independent of k.
Proof. By theorem 2.2.3 in [7], each dual Newton step reduces k+1 by a positive constant
long as a point ^
Y with -
not yet found, while keeping
the primal iterate xed. Now, starting at ( ~
V ), the total value of k+1 needed to be
reduced is not more than k+1 ( ~
theorem 3.8 implies that
at most"
Newton steps are required to reach a point ^
Y with -
Y onwards, by Lemma
4.3 in [8], quadratic convergence can be achieved, so it needs at most ln(ln(
additional
full Newton steps to nd a point Y k+1 satisfying - k+1 (Y k+1 ) . (We can choose for example,
4 Potential changes and Complexity
Recall
that
is an -approximate analytic center
of
k with < 1
3=2. Let
Then
Let
Y k and
Y k+1 be the analytic centers
of
and
where
In this section, we estimate the amount that the dual potential will increase when the working
set change
from
to
. To this end, we rst establish a lemma that is an extension of a
result in [10].
Lemma 4.1 Suppose n; p are positive integers, and is a positive n-vector with e T
Then for any positive constant , the following inequality holds:
Y
where is a positive constant no greater than 1:3
Proof. We need only to consider the case where n 2 as the inequality holds trivially when
1. Consider the maximization problem:
Y
It is shown in [10] that the maximizer has the form
and
p=2
Thus
Y
Y
1=p
Y
1=p
Y
1=p
where
1+1=p e 1=(p+1)
e
Lemma 4.2 Suppose Y k is an approximate analytic center
of
3=2.
Then
where is a constant depending only on .
Proof. For simplicity, we will drop the subscripts k and k our notations in this proof,
and denote for
example,
and
by
and
Let
Y ,
Y+ , and
Y
Let
U 1=2
U 1=2 ];
Note that
G
G T .
First, we establish an upper bound for ln
We have
G
Thus
By part (iii) of theorem 2.2.2 in [7], we have
[1 3-(Y )] 1=3
3:
Thus
Hence
Y
pln
and the desired upper bound is established.
Now observe that
Y
Y
det
det
Y
det
det
U
Using the bound in (4.3), we have
Y
det
det
Y
det
det
U
The inequality (4.2) follows once we have shown that
Y
det
det
Y
det
det
U
Note that
A
U
U
e
U 1=2
U 1=2 )C A
and by using (2.1), we have
U 1=2
Z
Z
U
Z
U
By Lemma 4.1, (4.5) is proved.
The complexity analysis is based on the following idea. For the sequence of working
set
k , we
can establish upper and lower bounds on
). The upper bound is approximately n k
which is a consequence of the assumption that contains a ball of radius and the fact that
k is dened by n k cuts. The lower bound is obtained by estimating
which is
a consequence of Lemma 4.2. A sophisticated estimation of r k gives rise to a lower bound that
is proportional to n k ln(n k =m 3 ). The algorithm must terminate before the lower and upper
bounds con
ict each other.
We rst establish an upper bound for
Lemma 4.3
Let
k be dened by n k linear inequalities and the positive semidenite
constraint. Suppose Assumptions A1-A3 hold. Then
Proof. Assumptions A1-A3 imply that there exists a point Y c 2 , such that
(i) All eigenvalues of Y c and I Y c are greater than or equal to ;
(ii) For any A 2 S m with
We will brie
y describe how to prove (Y c ) e before continuing with the proof of the lemma.
Suppose j is an eigenvalue of Y c and v j is a corresponding unit eigenvector. Consider the
. Since this matrix has a zero eigenvalue, it lies on the boundary of
0 and by Assumption A3, we have
The fact that (Y c ) (1 )e can be proven similarly.
Now we continue with the proof of the lemma. Since
Noting that
Y
Y
we have the desired inequality.
Now we turn to nding a lower bound for
r i2
Obviously, we need to estimate r i for each i. We rst seek to bound
i by D 1
dened as follows. Let I is the identity matrix of order
m. For
let
Lemma 4.4 Let A n i +j (with be the cuts generated from the
approximate analytic center Y ii ,
k. For any point Yk , let
Then
In particular,
Proof. We rst estimate s n i +j . We have
The last inequality holds because by Assumption A2,
I
implying that e (Y
Next, let
Note that in deriving (4.8), we used the fact that S i
for each i, and that
In our complexity analysis, we will make the following assumptions.
Assumption A4. p max m, where p
Assumption A5. Let
There exists a xed constant 1 such that for each
Assumption A4 is made for technical reason. It is used in proof of lemma 4.5. Such an
assumption also appeared in the papers [3] and [10]. Note that Assumption A4 can be relaxed
to p max O(m). But for simplicity, we xed the constant at 1.
Note that Assumption A5 holds trivially with . For the special case where a single
cut is used in each iteration, it holds with 1. Thus by xing at an intermediate value
between 1 and p max , we admit only cuts that are su-ciently good in the sense that the matrix
have too many small eigenvalues. Of course, one may not want to x at the
extreme value 1 since then the criterion is likely to reject most of the cuts unless there are
many mutually orthogonal (with respect to
The main advantage of having Assumption A5 is that in each oracle call, we have an objective
criterion to select only cuts that are useful among possibly a large number of ineective cuts.
In this way, the number of cuts added in each iteration will not be unnecessarily large, and
hence the computational time in each iteration will not grow as rapidly compared to the case
where the cuts are admitted unchecked. The choice of in practice would depend on the
problem at hand. It should dynamically be adjusted as information on the quality of the cuts
are obtained as the cutting algorithm progresses. If the choice of is too stringent and many
good cuts are rejected, then we can progressively increase its value so that more good cuts are
selected.
However, without a priori information on the quality of the cuts, we propose to choose to be
a small constant, say 5, based on the following empirical observation we made from numerical
experiments. We conducted numerical experiments on random matrices of the form V T V
mp , for 260. The elements of V are drawn
independently from the standard normal distribution. We computed the ratio between the
largest eigenvalue of V T V and Tr(V T V )=p for each V , and found that these ratios are less
than 2 for all the 3510 cases we tested.
Now let us continue with our complexity analysis. Let
Since
we have
Next, we establish an upper bound for the right hand side of the above inequality. Its proof is
modeled after that of [10, lemma 3.5]. However, we simplied the proof by considering all the
cuts simultaneously instead of handling them one by one as in [10].
Lemma 4.5
9m
8m
Proof. From the equation
Y
we have
9m
where we used the fact that max (B T
and the
inequality ln(1 +x) 8x=9 for 0 x 1=8. We also made use of Assumption A4 that p i m.
>From (4.10), it follows immediately that
But
m+m
implying that
8m
Combining (4.11) and (4.12), the lemma is proved.
With the above lemma, we can now formally state a lower bound for
Lemma 4.6 Suppose Assumptions A1{A5 hold. Then
8m
where is the constant appeared in (4.2).
Proof. The proof is similar to that of theorem 10 in [10] by making use of (4.9) and Lemma
4.5.
We will next estimate the number oracle calls required to nd a feasible point of .
Lemma 4.7 Suppose the Assumptions A1{A4 hold, and p max m. Then the analytic center
cutting plane method stops with a feasible before k violates the following inequality
8m
exp
Proof. From Lemmas 4.3 and 4.6, we have
8m
Thus, the algorithm must terminate before k violates the above inequality, i.e., the algorithm
must terminate before k violates the following inequality:
8m
Since
the algorithm must terminate before k
violates the inequality in the lemma.
Theorem 4.8 Suppose the Assumptions A1-A4 hold, and p max m. Then the analytic center
cutting plane method terminates in at most O (m 3 p max steps, where the
notation O means that lower order terms are ignored. The total number of cuts added is not
more than O (m 3
Proof. Ignoring lower order terms (assuming k m) and by the assumption that is a
constant independent of p max , the above lemma implies that the algorithm stops as soon as k
For large k, ln n k is negligible compared to n k , hence the algorithm requires at most
cuts. By Theorem 3.9, the total number of Newton steps is
O
The theorem is proved.
For feasibility problems in IR m
m should be replaced by m in Lemma 4.7. Thus the complexity
bound is O(m 2 p max for the number of required Newton steps. This bound is better
than the bounds obtained in [2], [5], and [10].
Acknowledgement
We thank the referees for their constructive comments that greatly help to improve the paper.
--R
Complexity analysis of an interior cutting plane method for convex feasibility problems
Multiple cuts in the analytic center cutting plane method
Convex nondi
Matrix Analysis
Analysis of a cutting plane method that uses weighted analytic center and multiple cuts
Cutting plane algorithms from analytic centers: e-ciency estimates
An analytic center cutting plane method for semide
A potential reduction algorithm allowing column generation
Complexity analysis of the analytic center cutting plane method that uses multiple cuts
Interior Point Algorithms: Theory and Analysis
--TR | multiple cuts;analytic center;cutting plane methods;semidefinite programming |
589002 | A Superlinearly Convergent Sequential Quadratically Constrained Quadratic Programming Algorithm for Degenerate Nonlinear Programming. | We present an algorithm that achieves superlinear convergence for nonlinear programs satisfying the Mangasarian--Fromovitz constraint qualification and the quadratic growth condition. This convergence result is obtained despite the potential lack of a locally convex augmented Lagrangian. The algorithm solves a succession of subproblems that have quadratic objectives and quadratic constraints, both possibly nonconvex. By the use of a trust-region constraint we guarantee that any stationary point of the subproblem induces superlinear convergence, which avoids the problem of computing a global minimum. We compare this algorithm with sequential quadratic programming algorithms on several degenerate nonlinear programs. | Introduction
. Recently, there has been renewed interest in analyzing and
modifying the algorithms for constrained nonlinear optimization for cases where the
traditional regularity conditions do not hold [5, 12, 11, 20, 24, 23]. This research
has been motivated by the fact that large-scale nonlinear programming problems
tend to be almost degenerate (have large condition numbers for the Jacobian of the
active constraints). It is therefore important to define algorithms that are as little
dependent as possible of the ill-conditioning of the constraints. In this work, we term
as degenerate those nonlinear programs (NLPs) for which the gradients of the active
constraints are linearly dependent. In this case there may be several feasible Lagrange
multipliers.
Many of the previous analysis and rate of convergence results for degenerate NLP
[5, 12, 11, 20, 24, 23] are based on the validity of some second-order conditions. These
are essentially equivalent to the condition in unconstrained optimization that, for
a critical point of a function f(x) to be a local minimum, f xx - 0 is a necessary
condition and f xx - 0 is a sufficient condition. Here - is the positive semidefinite
ordering. The place of f xx in constrained optimization is taken for these conditions
by L xx , the Hessian of the Lagrangian, which is now required to be positive definite
on the critical cone for one or all of the Lagrange multipliers [7, 21].
This work differs from previous approaches in that we assume only that
1. At a local solution x of the constrained nonlinear program, the first-order
Mangasarian-Fromovitz [18, 17] constraint qualification holds.
2. The quadratic growth condition (QG) [6, 15] is satisfied:
for some oe ? 0 and all x feasible in a neighborhood of x .
3. The data of the problem are twice continuously differentiable.
These assumptions are equivalent to a weaker form of the second-order sufficient
conditions [14, 6] which does not require the positive semidefiniteness of the Hessian
of the Lagrangian on the entire critical cone. In a recent a paper [2] it has been shown
Thackeray 301, Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15213
(anitescu@math.pitt.edu). Part of this work was completed while the author was the Wilkinson
Fellow at the Mathematics and Computer Science Division, Argonne National Laboratory. This
work was supported by the Mathematical, Information, and Computational Sciences Division sub-program
of the Office of Advanced Scientific Computing, U.S. Department of Energy, under Contract
W-31-109-Eng-38. This work was also supported by award DMS-9973071 of the National Science
Foundation.
that these conditions guarantee that x is an isolated stationary point and that a
steepest-descent like algorithm induces linear convergence to x . The framework used
here accommodates even problems for which no locally convex augmented Lagrangian
exists [2], which do not satisfy the assumptions of most other convergence results
[5, 12, 11, 20, 24].
In this paper we define an algorithm that is superlinearly convergent even in the
very general conditions outlined above. The trade-off is that the subproblems to be
solved are more complex than a quadratic program. The algorithm can be justified
by a particular perspective on Newton's method for unconstrained optimization. If
f(x) is the function to be minimized without constraints then sufficiently close to a
solution x Newton's direction, d, is a solution of the quadratic minimization problem.
min
xx f(x)d:
The term f(x) is constant for this minimization problem, but we include it to emphasize
that we can regard d as a solution of the second-order approximation to the
problem. If we have an inequality constrained nonlinear program,
subject to g i
its second-order approximation at x is the following problem
min
xx f(x)d
subject to g(x) +r x
We call such a problem a quadratically constrained quadratic problem (QCQP). To
ensure that the problem is bounded even for x far from the solution x , we add to
the problem a trust-region constraint, which is also quadratic:
The problem is generally not convex and thus finding the global optimum may be a
difficult problem. Also, the trust-region constraint may interfere with the order of
convergence. However, we show that for x close to x and for sufficiently small but
fixed:
1. The trust region constraint is inactive at any stationary point of the QCQP.
2. Any stationary point d of the QCQP used as a progress direction induces
superlinear convergence.
Therefore, finding a local solution to the QCQP is sufficient to induce superlinear
convergence of the iterates, which considerably reduces the conceptual complexity of
a sequential QCQP (SQCQP) algorithm. Note that the QCQP subproblem is identical
to the one used in [16], although the analysis conditions in this work are more general.
The paper is structured as follows. In Subsection 1.1 we discuss the different conditions
defining a stationary point of a nonlinear program and the quadratic growth
condition. Section 2 characterizes stationary points of the second-order approximation
(QCQP) of the nonlinear program at x . We show that, if the trust-region constraint
defines a sufficiently small region then the Mangasarian-Fromovitz constraint qualification
is satisfied at any feasible point and is the unique stationary point of
the QCQP. As a result, in Section 3 we prove that, for x sufficiently close to x , the
trust-region constraint is inactive at any stationary point of QCQP and we prove the
superlinear convergence of the SQCQP algorithm. We conclude with Section 4, where
we briefly discuss possible approaches to solving the QCQP subproblem.
1.1. Previous Work, Framework, and Notations. We deal with the NLP
problem
f(x) subject to g(x) - 0;
are twice continuously differentiable.
We call x a stationary point if the Fritz John conditions conditions hold: There
exist
Here L is the Lagrangian function
A local solution x of (1.2) is a stationary point [19]. If certain regularity conditions
hold at x (discussed below), then there exists - 0 such that x with -
In that case (1.3) are referred to as the KKT (Karush-
Kuhn-Tucker) conditions [3, 4, 8] and - are referred to as the Lagrange multipliers.
For that case, which is the one that most oftenly appears in this work, we define the
Lagrangian as
and the Karush-Kuhn-Tucker conditions become
r x L(x;
Since our analysis is limited to a neighborhood of a point x that is a strict local
minimum, we assume that all constraints are active at x , or g(x Such a
situation can be obtained by choosing a sufficiently small trust-region and simply
dropping the constraints i for which g i (x since this relationship holds in an
entire neighborhood of x . This does not reduce the generality of our results, but it
simplifies the notation because now we do not have to refer separately to the active
set.
The regularity condition, or constraint qualification, ensures that a linear approximation
of the feasible set in the neighborhood of x captures the geometry of
the feasible set. Often in local convergence analysis of constrained optimization algo-
rithms, it is assumed that the constraint gradients r x are linearly
independent, so that the Lagrange multiplier in (1.6) is unique. We assume instead
the Mangasarian-Fromovitz constraint qualification (MFCQ) [18, 17]:
It is well known [9] that MFCQ is equivalent to boundedness of the set M(x ) of
Lagrange multipliers that satisfy (1.6), that is,
Note that M(x ) is certainly polyhedral in any case. Another condition equivalent
to MFCQ (1.7) is [10]
such that
The critical cone at x is [7, 22]
\Phi
We briefly review some of the second-order conditions in the literature. In the
framework of [7], the second-order sufficient conditions for x to be an isolated local
solution of (1.2) are [7, 8]:
If these conditions hold at x for some - , then the quadratic growth condition is
satisfied, irrespective of the validity of the first-order constraint qualification [7, 8].
An important consequence of the condition (1.11) is that x is a local minimum of
the augmented Lagrangian
for a sufficiently large constant c.
A refinement of the second-order conditions was introduced in [14]. In the presence
of MFCQ, those conditions require that
Further analysis shows that, in presence of MFCQ, these conditions are necessary and
sufficient for the quadratic growth condition to hold [6, 14, 15, 22].
If the condition (1.12) holds, but (1.11) does not, then there may be no augmented
Lagrangian with a positive semidefinite Hessian, as it is shown with an example in
[2]. This is an interesting aspect since it invalidates the usual working assumption
of Lagrange multiplier methods [4]. It also shows that the analysis in this paper is
done without assuming the existence of an augmented Lagrangian that has x as an
unconstrained minimum.
In our analysis we use the L1 nondifferentiable exact penalty function:
If the MFCQ (1.7) conditions hold at x , then the quadratic growth condition
(1.1) and the second order conditions (1.12) are each equivalent to the following
condition [6]
for some oe ? 0 and all x in a neighborhood of x .
For some function h : we denote by c 1h , c 2h bounds depending on the
first and second derivatives of h. The positive and negative parts of h(x) are h
With this notation Also, in our notation, r x g i (x), - and
r x g(x)- are column vectors.
In this work we need to estimate distances to sets described by linear constraints:
where M eq and M in are n eq \Theta n and, respectively, n in \Theta n matrices and q eq and q in are
respectively, n in dimensional vectors. By Hoffman's Lemma [13], if P 6= ;,
there exists c P ? 0 such that
where by D( ~
d; P) we denoted the distance from ~
d to the set P. This result allows us
to relate the distance from a point ~
d to a polyhedral set in terms of the infeasibility
of ~
d in the representation (1.15).
2. Stationary Points of Quadratically Constrained Quadratic Pro-
grams. In this section we investigate the stationary points of the quadratically constrained
quadratic program
min d2IR n a T d
subject to b T
are n \Theta n
symmetric matrices and a 2 m. We denote this program by
TRQCQP(fl). Our assumptions concerning (2.1) are:
1. At
2. The quadratic growth condition is satisfied near 0: There exists fl 0
and oe 1 ? 0 such that
a
A local solution of (2.1) is clearly
The aim of this section is to show that under assumptions (2.3) and (2.2), there
exists is the only stationary point of TRQCQP(fl) (2.1), for
any 0 - As a consequence any algorithm that reaches a stationary point of
finds its global optimum. The results from [2] ensure that
an isolated stationary point of TRQCQP(fl) (2.1). However, the developments of this
section are necessary to ensure that additional stationary points are not introduced
by the trust region constraint.
The proof has the following steps, each stated for sufficiently small fl.
Lemma 2.4 proves that MFCQ (1.7) is satisfied for all stationary points ~
d
of (2.1). Therefore, at any stationary point there exist Lagrange multipliers
that satisfy (1.6);
ultimately implies that for any Lagrange multiplier - at a stationary
point ~
d of (2.1) there exists a sufficiently close Lagrange multiplier
at active subset is included in the active subset of -. This
leads to the identity (-
~
~
which helps bound above
the variations in the objective function of (2.1) in the proof of Theorem 2.7.
proves that the multiplier of the trust-region constraint is bounded
above. This in turn implies Lemma 2.6: the Lagrange multipliers of all
potential stationary points are uniformly bounded.
ffl Theorem 2.7, the main result of this section, proves that ~
is the unique
stationary point of (2.1).
Subsection 2.1 contains additional results implied by Hoffman's Lemma (1.16),
which are used in Section 3.
2.1. Sensitivity results for Lagrange Multipliers. An immediate consequence
of MFCQ (2.2) is that the set of Lagrange Multipliers of TRQCQP(fl) (2.1) at
is nonempty and bounded.
Lemma 2.1. There exists c M ? 0 such that, for any w 2 IR
n , and for any - 2 IR
satisfying
a
there exists a - 2 M such that
Proof Follows by direct application of Hoffman's Lemma (1.16), after using
that jjwjj 1 - jjwjj. \Pi
Lemma 2.2. There exists j ? 0 such that for all w 2 IR n with jjwjj - j and any
satisfying
a
there exists - 2 M such that -
Proof Assume the contrary: For any k 2 IN , there exists w k 2 IR
n , such that
k and there exists - k satisfying
a
and an index set I k ae f1; mg, such that - k
I
From Lemma 2.1, D(-
is a
compact set, and the set of subsets of f1; is finite, there exists a subsequence
such that I
From our assumptions - I 6= 0, 8- 2 M . On the other hand, since - kq
I
I
and - kq ! - we must have -
I which is a contradiction. The proof is complete.
Lemma 2.3. There exists c M ? 0 and j ? 0 such that for any w 2 IR
n with
any - satisfying
a
there exists - 2 M with and such that -
Proof Let j be the quantity defined by Lemma 2.2. Let I ae f1;
that there exists a - satisfying (2.5) and - I = 0. Lemma 2.2 implies that there exists
such that - I = 0. Let M I be the set of - 2 IR
m such that
From Lemma 2.2, M I is not empty. From Hoffman's Lemma (1.16), there exists
c M I ? 0 such that, for all - 2 IR
m , we have
From Lemma 2.1 choose - 2 M such that
From the definition of M (2.4) we have that
Thus, from (2.7) we must have
I
I
We also have from our choice of - (2.8) that
we thus have
conjunction with the
preceding inequality and (2.9) implies that
Hence from (2.8) and the preceding inequality we have that
The conclusion now follows after taking
I
2.2. Stationary Points of Quadratically Constrained Quadratic Pro-
grams. In this section we analyze the stationary points of TRQCQP(fl) (2.1) for
sufficiently small values of the parameter fl. We choose fl 00
1 such that
are the quantities appearing in MFCQ (2.2) with choose
which guarantees that whenever jjdjj - fl 1 , both (2.10) and the quadratic growth
condition
Lemma 2.4. There exists
(1.7) at all its stationary points d with fl such that
The important consequence of this lemma is that Lagrange multipliers exist at
any stationary point of TRQCQP(fl) (2.1).
Proof Take the quadratically constrained quadratic program
min d2IR n d T d
subject to b T
with global solution as well as the
quadratic growth condition (1.1). From [2], is an isolated stationary point of
(2.12). Therefore there exists a fl 0
such that the only stationary point d of (2.12)
that satisfies d T d - (fl 0
Take now
g. Assume that there exists fl,
MFCQ (1.7) is not satisfied at some stationary point -
d of TRQCQP(fl) (2.1). From
and (1.3) it follows that there exists -
not both equal to 0, such
that
If -
would imply
or, after multiplying with p from (2.10) we get
d)
which implies -
contradiction with the assumption that not both - 0 and -
are
and from (2.13) we get -
dividing with -
(b
(b T
But this means that -
d 6= 0 is a stationary point of (2.12) with a Lagrange multiplier
, which contradicts the properties of our choice of fl 2 . The proof is complete. \Pi
Lemma 2.5. Consider the following quadratically constrained quadratic program
subject to \Gamma i
Then there exists the only stationary
point of (2.15) that satisfies jjdjj - fl 3 is
Proof Choose fl 0
g. From (2.11) this implies that for all d with
3 the quadratic growth condition (2.3) and (2.10) holds. Also, from Lemma
2.4, MFCQ (1.7) holds at any stationary point of (2.15).
Take ~
d 6= 0 a feasible point of (2.15). We now estimate the variation of the
constraints and objective function in a specific direction from ~
d, in order to decide
under what conditions ~
can be a stationary point of (2.15). Let the active set
at ~
d be
We estimate the first-order behavior of \Gamma i (d) in the direction \Gamma ~
is the
vector from (2.10) and fi - 0. For
d we get
~
d) T (\Gamma ~
\Gammab T
~
~
~
d)
\Gammab T
~
d)
d
where we used (2.10) and that, from (2.16), if
d then \Gammab T
~
~
For the objective function we have that
(r d \Psi( ~
~
d) T (\Gamma ~
\Gammaa T ~
d T A ~
~
d T ~
~
~
d T ~
~
d T ~
d T A ~
~
where we used the quadratic growth condition (2.3). Choose now
d
2:
Assume that ~
d 6= 0,
d
. Using that
d T A ~
~
d
d
d
d
d
d
d
d
d
where we used that from our choice of fl 00
3 (2.21) and since
d
3 we have c fi
d
and c fi
d
We also used the definition of c ffi (2.22) and that c 1 - c ffi .
Using (2.23) in (2.18) we get
r d \Psi( ~
d) T (\Gamma ~
d
d
d
Using (2.19) and (2.20) in (2.17) we get for all
d
d) T (\Gamma ~
d
d
d
d
0:
From (2.25) and (2.24) we get that if ~
and ~
3 (2.21) then there exists a direction
~
that produces strict decreases in the objective function and the active constraints.
Therefore ~
d cannot be a stationary point of (2.15). Otherwise (1.3) implies that there
exist the multipliers -
m , not all of them of 0 such that
d)
d
From (2.25) and (2.24) we get, after multiplying with ~
that
d) T ~
d
d) T ~
which is a contradiction that proves the lemma with c ffi defined in (2.22) and
Lemma 2.6. There exists
d with
d
stationary point of TRQCQP(fl) (2.1) with Lagrange multipliers - 2 IR
Proof We take
defined in (2.11), fl 2 is the quantity from Lemma 2.4 and fl 3 is the quantity
from Lemma 2.5. Lemma 2.4 ensures that the Lagrange multipliers exist at any
stationary point of TRQCQP(fl) (2.1).
Assume the contrary of the conclusion of the Lemma: For any k 2 IN , there exists
~
d k a stationary point of TRQCQP(fl) (2.1) with
multipliers
and (1.6). In particular,
a +A ~
~
By Lemma 2.5 since
must have c k
can choose - such that for a subsequence k q , q !1, we have lim q!1
with jj- jj
d , where
d
We can now divide through (2.27) with
take the limit as
and
d
~
multiply with p and use (2.10) and the fact that jj- jj
to get
~
d
which is a contradiction. This proves the lemma. \Pi
Theorem 2.7. There exists fl 5 ? 0, such that, for any fl such that
TRQCQP(fl) (2.1) has the unique stationary point
Proof Choose
ae
oe
where j is the quantity from Lemma 2.3, c ffi is the quantity from Lemma 2.5, -1
is the quantity from Lemma 2.6 and 4, are the bounds on the trust
regions that ensure that all preceding results hold.
Let ~
d 6= 0 be a stationary point of TRQCQP(fl) (2.1) with
Lemma 2.4 TRQCQP(fl) (2.1) satisfies MFCQ (1.7) at ~
d. Therefore there exist the
Lagrange multipliers - 0, c 1 - 0 which, together with ~
d satisfy (1.6), or
a
d)
~
d)
~
d T ~
~
~
d T ~
Since
d
applies to give that c 1 - c ffi . Since
d
4 we have that jj-jj 1 -1 from Lemma 2.6. We define
~
~
d:
After applying the triangle inequality and using (2.28) we have that
d
~
d
~
d
d
d
For the last inequality,
c- and
d
results in jjwjj - j. From (2.30)
and (2.31) we have that
a
This implies, from Lemma 2.3, that there exists - 2 M ( a Lagrange multiplier for
TRQCQP(fl) (2.1) at
(2.
a
Adding the last equality to the first equation in (2.30), dividing by 2 and multiplying
with ~
d T we obtain
a T ~
dA ~
~
~
d)
d
0:
We now use the identity u
well as the fact that (-
~
~
from (2.33) and (2.30) to obtain
d +2
~
dA ~
d +2
~
~
~
d)
d
a T ~
dA ~
(-
~
d)
d
which results in
a T ~
dA ~
d
(-
~
Since ~
d is feasible for TRQCQP(fl) (2.1) and since
d
the quadratic
growth condition (2.3) holds to give that a T ~
dA ~
d
. Define
From (2.33), (2.31) and (2.32) we have jj-
d
Using all these bounds
in (2.34), together with the arithmetic-quadratic mean inequality we get
d
- a T ~
dA ~
d
(-
d) -4
d
d
d
Since
d
our assumption, we obtain, after dividing through the previous
inequality with
d
that
d
Choose now
ae
oe
From (2.35) it follows that the unique stationary point of TRQCQP(fl) (2.1) with
The proof is complete. \Pi
3. Sequential Quadratically Constrained Quadratic Programming. In
this section, we introduce the sequential quadratically constrained quadratic programming
algorithm. We prove that under the conditions set forth in the introduction,
the algorithm induces superlinear convergence. Since our main interest is the rate of
convergence of the algorithm, we do not address global convergence issues.
We consider the following form of the algorithm:
1. Choose a starting point x k ,
2. Let stationary point of
xx f(x)d
subject to g i (x) +r x
3. Take x restart.
At every step, the algorithm solves a problem with quadratic constraints and a
quadratic objective, none of which are assumed to be convex. We name the above
algorithm sequential quadratically constrained quadratic programming or SQCQP.
As outlined in Subsection 1.1, we assume without loss of generality that g i (x
eventually considering a sufficiently small trust-region, and
that the quadratic growth condition (1.1) and MFCQ (1.7) hold at a local solution
x of the nonlinear program (1.2). From [14, 6] these conditions are equivalent to
MFCQ (1.7) and (1.12), which are expressed only in terms of the derivatives of the
data up to the second order. We show that (3.1) is feasible for fl fixed and x in some
neighborhood of x . Since it is also bounded, a stationary point must exist.
Due to the fact that it captures the entire information up to second order for
(1.2) at x , the quadratically constrained quadratic program
xx f(x )d
subject to r x g(x )
satisfies MFCQ (1.7) and (1.12) at As a result of [14, 6] it follows that (3.2)
satisfies MFCQ (2.2) and the quadratic growth condition (2.3). Therefore, all the
results from Section 2 apply for (3.2). We follow a line of proof similar to the one in
Section 2.
ffl Theorem 3.1 proves that MFCQ (1.7) is satisfied by (3.1) in a neighborhood
of x and that the trust-region constraint is inactive at any stationary point
d of (3.1). Corollary 3.2 further insures that in a neighborhood of x , the
Lagrange multipliers of (3.1) are uniformly bounded.
ultimately implies that for any Lagrange multiplier - at a stationary
point d of (3.1) at there exists a sufficiently close Lagrange
multiplier - at whose active subset is included in the active subset of
-. This in turn leads to the conclusions of Lemma 3.4 that (- i +-
is a stationary point of (3.1).
This helps bound above the variations in the objective function of (3.1) in
the proof of Theorem 3.5.
Theorems 3.5 and 3.6 prove the superlinear convergence of a sequence x
initiated sufficiently close to x , where d k is any stationary point of
(3.
Theorem 3.1. There exists fl 6 ? 0 and a neighborhood N fl 6
any fl with there exists a neighborhood N fl (x ) of x such that
(i) The QCQP (3.1) is feasible for any x 2 N fl (x ).
(ii) For any x 2 N fl 6
any d with jjdjj - fl 6 we have
are the quantities entering the definition of MFCQ (1.7).
(iii) For any sequence x k 2 N fl (x
and with ~
d k a stationary point of (3.1) at must have ~
(iv) The constraint d T d - fl 2 is inactive for any x 2 N fl (x ) and d stationary
point of (3.1).
Proof Since (3.2) satisfies MFCQ (2.2) and the quadratic growth condition
(2.3) at 0, from Theorem 2.7 there exists
6 , such that, for any 0 !
6 , ~
is the only stationary point of (3.2). Choose now fl such that
6 . Since (3.2)
satisfies MFCQ (2.2), then, from [21], for any sufficiently small perturbation of (3.2)
we still obtain a feasible nonlinear program. We regard (3.1) as a perturbation of (3.2)
and we therefore have, from the fact that f; g are twice continuously differentiable, that
there exist a neighborhood N 2
such that (3.1) is feasible for any x 2 N 2
which proves part (i) as long as N fl
which will be established later.
We also have that, for all
since from MFCQ (1.7) is a bound on the second derivatives
of m. If we chose fl 00
4c2g
, d with jjdjj - fl 00
6 and N fl 6
4c2g
we get from the previous bound that, since now c 2g jjx \Gamma x jj - i0,
c 2g jjdjj - i0,
0which shows part (ii), after defining
00g. We now choose N 3
both the conclusions of (i) and (ii) hold. In
particular, for any fl 2 (0; fl 6
must have a stationary point since it
is feasible and bounded.
Assume now that the conclusion (iii) does not hold: There exists fl ? 0, with
and a sequence x k ! x , x k 2 N 3
and the corresponding
stationary points d k of (3.1) are bounded bellow
sufficiently
large. Since d k is a stationary point of (3.1) at must satisfy the first-order
necessary conditions (1.3) for some multipliers - k
Since the multipliers -
1, and the direction
we can extract a subsequence k q such that x kq ! x ,
1. Taking the limit as q ! 1 in (3.3) we obtain
from the continuity of all data involved in terms of (x; d), that d is a stationary point
of (3.2). Since d 6= 0 this contradicts the outcome of Theorem 2.7 that is valid due
to our choice of fl 6 . This proves (iii).
Assume now that (iv) does not hold. It then follows that there exists a sequence
stationary point and such that
fl. But this contradicts
the conclusion of (iii) and thus there exists a neighborhood N fl
that for x 2 N fl (x ) any stationary point of (3.1) satisfies d T d ! fl 2 and for which the
conclusions of parts (i),(ii) and (iii) hold. The proof is complete. \Pi
Corollary 3.2. Any stationary point of (3.1) satisfies the Kuhn-Tucker conditions
1.6, for any 0 - There exists
-1 such that, for any x 2 N fl (x ), any stationary point d of (3.1) and any Lagrange
multipliers - satisfying the Kuhn-Tucker conditions we have jj-jj 1 -1 .
Proof Theorem 3.1(iv), we have that for any
stationary point d, we must have jjdjj ! fl. Therefore only the
can be active at a stationary point d. Then by
Theorem 3.1(ii), MFCQ (1.7) is satisfied at d and thus there exist multipliers - 0
satisfying the Kuhn-Tucker conditions and in particular:
Multiplying through with p we get
after using the usual norm inequalities we get
Since on N fl (x ) the expression from the right hand side is bounded above, there
exists -1 for which the conclusion of this corollary holds. \Pi
Lemma 3.3. There exists fl 7 ? 0 and a constant c ? 0 such that for any fl with
there exists a neighborhood N 1
for any d a stationary point of (3.1) with the Lagrange multipliers - there exist the
Lagrange multipliers at
and
Proof Take fl such that d be a stationary
point of (3.1) with the Lagrange multipliers - 0 (which exist from Corollary (3.2)).
From the Kuhn-Tucker conditions we obtain
and thus
Using that jjr x and that jjr x g i
c 2g jjx \Gamma x jj, where c 2f and c 2g are bounds on the second derivatives of f and g, we
get from (3.5) and Corollary 3.2 that
We choose
, where j is the quantity from Lemma 2.2. From (3.6)
it follows that, for any fl -
that, since jjdjj -
We can therefore apply Lemma 2.2 and (3.6) to get that there exists - 2 M(x )
with the properties required, after taking
is the constant from Lemma 2.3. \Pi
Lemma 3.4. Let x 2 N 1
is the neighborhood
obtained in Lemma 3.3. Let - be a Lagrange multiplier associated with a stationary
point d at x of (3.1). Let - 2 M(x ) such that -
and such that
where \Theta P (d) is a continuous function that satisfies \Theta P
Proof Using the first-order Taylor remainder formula [1] for g i (y) around
for the fact that g i (x) is twice continuously differentiable for
we obtain that
is a continuous function, it follows that
is a continuous function on jjwjj - fl 7 with the property that \Theta i We have
that d is a stationary point of (3.1) and as a result satisfies g i
Replacing w with d in (3.7) we obtain
now
\Theta P
\Theta i (d):
From the definition of \Theta i (d) we have that \Theta P is continuous and that \Theta P
From the definition of P (x) (1.13), we get that,
This proves point (i). Since - is such that - our hypothesis, and
since d is a stationary point of (3.1) and thus satisfies the complementarity condition
xx g(x)d
this implies that, for
xx g(x)d
or, by using (3.7),
and thus
which completes the proof of (ii) and of the Lemma. \Pi
From here on we use extensively that, for h twice continuously differentiable, we
have
is a continuous function with / 3h Indeed by
Taylor's theorem we have that there exist continuous functions / 1
3h
3h
3h
which satisfy / 1
3h
3h
3h
and
which in turn implies, after choosing / 2
3h
3h
and using the Cauchy-Schwarz
inequality, that
The relation (3.9) now follows by comparing (3.9), (3.10) and (3.12) and taking
3h (z). If h were three times continuously differentiable, then
would be related to the third derivative of h, from the error formula of trapezoidal
integration [1], which is the origin of our subscript notation.
Theorem 3.5. Let be a sequence such that x k ! x , x k 6= x . Let d k
be a stationary point of (3.1) for is the quantity
from Lemma 3.3. Then
lim
Proof Since x k ! x , the sequence x k eventually reaches N 1
means that Lemmas 3.4 and 3.3, as well as all preceding results apply for
sufficiently large k. Using (3.9) we get that
is a bound obtained by using (3.11) for f(x) between x k
and x k . / 0
3f is a continuous function satisfying / 0
From Corollary 3.2, there exist the Lagrange multiplier - k , which, together with
d k satisfies the Kuhn-Tucker conditions (1.6) for (3.1) at
there exists a - \Lambdak 2 M(x ) such that
Using the Kuhn-Tucker conditions (1.6) to replace r x
r x f(x ) in terms of g and the Lagrange multipliers, and using the bounds
that follow from Corollary 3.2, we get from (3.13)
3g
is a bound obtained from applying (3.11) to g i (x),
between the points x k taking the maximum among
the resulting bounds. / 0
3g a continuous function satisfying / 0
We now
make use of the identity ab for the terms
Continuing the bounding in (3.15) we get
3f
(3.
We now bound all terms involving - and - . Using that
and that g is twice continuously differentiable and thus
we get
mc c 2g (
Using that jj-jj 1 -1 from Corollary 3.2, (3.9) for g i (x) and that g i (x
as well as Lemma 3.4 (ii) we obtain that
Putting together the bounds from (3.16), (3.17) and (3.18) we obtain
Since the bound on the right hand side is nonnegative, we can use Lemma 3.4 (i)
and the quadratic growth condition (1.14) to get that
oe
where
are continuous functions of their arguments that satisfy \Phi 1 (0;
We now use that ab - to get from (3.20) that
oe
0, from Theorem 3.1, there exists K 1 such that 8k - K 1
we have
Taking the corresponding term to the right-hand side, we get that, 8k - K 1 ,
Now using the continuity of \Phi 2 and \Theta P , and that, from Theorem 3.1 (iii), d k ! 0,
we get that
lim
oe
lim
0:
or that
lim
Using now the consequence of the triangle inequality
and dividing the relation with
and taking the limit, this implies that
lim
and thus
lim
Dividing (3.22) by the last limit we get that
lim
which proves the claim of the Theorem. \Pi
Theorem 3.6. Let fl be such that is the quantity from
Lemma 3.3. There exists a radius r such that for any x 2 B(x
a stationary point of (3.1), then
Whenever started inside B(x ; r ), the SQCQP algorithm produces a sequence x k !
x that is superlinearly convergent,
lim
Proof Assume the contrary: For any q 2 IN , there exists x q 6= x such that
q and d q a stationary point of (3.1) such that
Therefore x q ! x , and by Theorem 3.5
lim
which contradicts (3.23). As a result there exists r with the properties required by
the Theorem. When started with x 0 2 B(x ; r ), the SQCQP algorithm produces a
sequence x , such that
We can now use Theorem 3.5 to claim that
lim
which proves the superlinear convergence of x k to x . The proof is complete. \Pi
Note If the data of the problems are three times continuously differentiable,
then the functions /
are Lipschitzian in their respective arguments, which
considerably simplifies the notation for the proof of Theorem 3.5. For instance,
case. Using essentially the same proof, the conclusion
can then be strengthened to show that the order of convergence is at least 34. Conclusions. We present an algorithm that achieves superlinear convergence
of the iterates to a local minimum of the nonlinear program (1.2) at which MFCQ
(1.7) and the quadratic growth condition (1.1) are satisfied. The conditions we
impose allow even situations for which no locally convex augmented Lagrangian exists,
a case not accommodated by most previous results in the literature.
At each step we solve a subproblem generated by approximating the function and
the constraints by the second-order Taylor series at the current iterate. We also add
a trust-region constraint, which insures that the problem is bounded. The algorithm
therefore solves at each step a quadratically constrained quadratic program (QCQP)
and we thus call it sequential quadratically constrained quadratic program (SQCQP).
The subproblem to be solved is not necessarily convex. However we prove that
for a suitable, fixed size of the trust region, the associated constraint is inactive at
any stationary point of QCQP. As a result, any stationary point of the QCQP induces
superlinear convergence of the iterates, which obviates the need for finding the global
optimum of the subproblem.
A subproblem that has quadratic constraints is more difficult to solve than a
subproblem with linear constraints, the latter being the case of Sequential Quadratic
Programming algorithms [19]. One could of course solve the QCQP with a nonlinear
programming technique. The algorithm in [2] achieves at least linear convergence
on the subproblem under the conditions considered here. Since in this work a more
accurate model of the constraints is considered, compared to SQP, it would be expected
that a smaller number of exterior iterations and thus of function evaluations is
needed before completion. However, given the complexity of the subproblem, this will
not necessarily results in superior runtime. Nevertheless, algorithms can be derived
to deal directly with quadratically constrained problem via semidefinite relaxation
[16]. Devising methods that specifically accommodate quadratic constraints will be
the subject of future research.
--R
An Introduction to Numerical Analysis
Degenerate
New York
Local Analysis of Newton-Type Methods for Variational Inequalities and Non-linear Programming
Introduction to Sensitivity and Stability Analysis in
Practical Methods of Optimization
A necessary and sufficient regularity condition to have bounded multipliers in nonconvex programming
Differential stability in nonlinear programming
Stabilized sequential quadratic programming
Stability in the presence of degeneracy and error estimation
On approximate solutions of systems of linear inequalities
Necessary and sufficient conditions for a local minimum.
On Sensitivity analysis of nonlinear programs in Banach spaces: the approach via composite unconstrained optimization
The Fritz John necessary optimality conditions in the presence of equality constraints
Superlinear convergence of an interior-point method despite dependent constraints
Applications to nonlinear programming Mathematical Programming Study 19
Sensitivity analysis of nonlinear programs and differentiability properties of metric projections
Superlinear convergence of a stabilized SQP method to a degenerate solution
Modifying SQP for degenerate problems
--TR | quadratic constraints;sequential quadratic programming;degenerate constraints;superlinear convergence |
589013 | A Computationally Efficient Feasible Sequential Quadratic Programming Algorithm. | A sequential quadratic programming (SQP) algorithm generating feasible iterates is described and analyzed. What distinguishes this algorithm from previous feasible SQP algorithms proposed by various authors is a reduction in the amount of computation required to generate a new iterate while the proposed scheme still enjoys the same global and fast local convergence properties. A preliminary implementation has been tested and some promising numerical results are reported. | Introduction
Consider the inequality-constrained nonlinear programming problem
min f(x)
s.t.
are continuously
differentiable. Sequential Quadratic Programming (SQP) algorithms are
widely acknowledged to be among the most successful algorithms available
for solving (P ). For an excellent recent survey of SQP algorithms, and the
theory behind them, see [2].
Denote the feasible set for (P ) by
In [17, 8, 14, 15, 1], variations on the standard SQP iteration for solving
are proposed which generate iterates lying within X. Such methods are
sometimes referred to as "Feasible SQP" (or FSQP) algorithms. It was observed
that requiring feasible iterates has both algorithmic and application-oriented
advantages. Algorithmically, feasible iterates are desirable because
ffl The QP subproblems are always consistent, i.e. a feasible solution
always exists, and
ffl The objective function may be used directly as a merit function in the
line search.
In an engineering context, feasible iterates are important because
ffl Often f(x) is undefined outside of the feasible region X,
ffl Trade-offs between design alternatives (all requiring that "hard" constraints
be satisfied) may be meaningfully explored, and
ffl The optimization process may be stopped after a few iterations, yielding
a feasible point.
The last feature is critical for real-time applications, where a feasible point
may be required before the algorithm has had time to "converge" to a solution
An important function associated with the problem (P ) is the Lagrangian
which is defined by
Given a feasible estimate x of the solution of (P ) and a symmetric matrix
H that approximates the Hessian of the Lagrangian L(x; -), where - is
a vector of non-negative Lagrange multiplier estimates, the standard SQP
search direction, denoted d 0 (x; H) or d 0 for short, solves of the Quadratic
Program (QP)
s.t.
Positive definiteness of H is often assumes as it ensures existence and uniqueness
of such solution. With an appropriate merit function, line search pro-
cedure, Hessian approximation rule, and (if necessary) Maratos effect [13]
avoidance scheme, the SQP iteration is known to be globally and locally
superlinearly convergent (see, e.g., [2]).
A feasible direction at a point x 2 X is defined as any vector d in R n such
that x+ td belongs to X for all t in [0; - t ], for some positive - t. Note that the
SQP direction d 0 , a direction of descent for f , may not be a feasible direction
at x, though it is at worst tangent to the active constraint surface. Thus, in
order to generate feasible iterates in the SQP framework, it is necessary to
"tilt" d 0 into the feasible set. A number of approaches have been considered
in the literature for generating feasible directions and, specifically, tilting the
SQP direction.
Early feasible direction algorithms (see, e.g., [27, 17]) were first-order
methods, i.e. only first derivatives were used and no attempt was made to
accumulate and use second-order information. Furthermore, search directions
were often computed via linear programs instead of QPs. As a conse-
quence, such algorithms converged linearly at best. Polak proposed several
extensions to these algorithms (see [17], Section 4.4) which took second-order
information into account when computing the search direction. A few
of the search directions proposed by Polak could be viewed as tilted SQP directions
(with proper choice of the matrices encapsulating the second-order
information in the defining equations). Even with second-order information,
though, it is not possible to guarantee superlinear convergence of these algorithms
because no mechanism was included for controlling the amount of
tilting.
A straightforward way to tilt the SQP direction is, of course, to perturb
the right-hand side of the constraints in QP 0 (x; H). Building on this obser-
vation, Herskovits and Carvalho [8] and Panier and Tits [14] independently
developed similar feasible SQP algorithms in which the size of the perturbation
was a function of the norm of d 0 (x; H) at the current feasible point
x. Thus, their algorithms required the solution of QP 0 (x; H) in order to
define the perturbed QP. Both algorithms were shown to be superlinearly
convergent. On the other hand, as a by-product of the tilting scheme, global
convergence proved to be more elusive. In fact, the algorithm in [8] is not
globally convergent, while the algorithm in [14] has to resort to a first-order
search direction far from a solution in order to guarantee global convergence.
Such a hybrid scheme could give slow convergence if a poor initial point is
chosen.
The algorithm developed by Panier and Tits in [15], and analyzed under
weaker assumptions by Qi and Wei in [20], has enjoyed a great deal of
success in practice as implemented in the FFSQP/CFSQP [26, 12] software
packages. We will refer to their algorithm throughout this paper as FSQP.
In [15], instead of directly perturbing QP 0 (x; H), tilting is accomplished
by replacing d 0 with the convex combination
an (essentially) arbitrary feasible descent direction. To preserve the local
convergence properties of the SQP iteration, ae is selected as a function
ae(d 0 ) of d 0 in such a way that d approaches d 0 fast enough (in particular,
as the solution is approached. Finally, in order to avoid
the Maratos effect and guarantee a superlinear rate of convergence, a second
order correction d C (denoted ~
d in [15]) is used to "bend" the search direction.
That is, an Armijo-type search is performed along the arc x+td+t 2 d C , where
d is the tilted direction. In [15], the directions d 1 and d C are both computed
via QPs but is is pointed out that d C could instead be taken as the solution of
a linear least squares problem without affecting the asymptotic convergence
properties.
From the point of view of computational cost, the main drawback of
algorithm FSQP is the need to solve three QPs (or two QPs and a linear
least squares problem) at each iteration. Clearly, for many problems it
would be desirable to reduce the number of QPs at each iteration while
preserving the generation of feasible iterates as well as the global and local
convergence properties. This is especially critical in the context of those
large-scale nonlinear programs for which the time spent solving the QPs
dominates that used to evaluate the functions.
With that goal in mind, consider the following perturbation of QP 0 (x; H).
Given a point x in X, a symmetric positive definite matrix H, and a non-negative
scalar j, let (d(x; H; j); fl(x; H; j)) solve the QP
s.t.
where fl is an additional, scalar variable.
The idea is that, away from KKT points of (P), fl(x; H; j) will be negative
and thus d(x; H; j) will be a descent direction for f (due to the first
constraint) as well as, if j is strictly positive, a feasible direction (due to
the m other constraints). Note that when j is set to one the search direction
is a special case of those computed in Polak's second-order feasible
direction algorithms (again, see Section 4.4 in the book [17]). Further, it is
not difficult to show that when j is set to zero, we recover the SQP direc-
tion, i.e. d(x; H; values of the parameter j, which we
will call the tilting parameter, emphasize feasibility, while small values of j
emphasize descent.
In [1], Birge, Qi, and Wei propose a feasible SQP algorithm based on
QP (x; H; j). Their motivation for introducing the right-hand-side constraint
perturbation and the tilting parameters (they use a vector of parameters,
one for each constraint) is, like ours, to obtain a feasible search direction.
Specifically, motivated by the high cost of function evaluations in the application
problems they are targeting, their goal is to ensure that a full step of
one is accepted in the line search as early on as is possible (so that costly
line searches are avoided for most iterations). To this end, their tilting parameters
start out positive and, if anything, increase when a step of one is
not accepted. A side-effect of such an updating scheme is that the algorithm
cannot achieve a superlinear rate of convergence, as the authors point out
in Remark 5.1 of [1].
In the present paper, our goal is to compute a feasible descent direction
which approaches the true SQP direction fast enough so as to ensure superlinear
convergence. Furthermore, we would like to do this with as little
computation per iteration as possible. While computationally rather expen-
sive, algorithm FSQP of [15] has the convergence properties and practical
performance we seek. We thus start with reviewing its key features. For x
in X, define
the index set of active constraints at x. In FSQP, in order for the line-search
(with the objective function f used directly as the merit function) to
be well-defined, and in order to preserve global and fast local convergence,
the sequence of search directions fd k g generated by algorithm FSQP is
constructed so that the following properties hold:
is a KKT point for (P ),
is not a KKT point,
is not a KKT point, and
We will show in Section 3 that given any symmetric positive definite matrix
H k and non-negative scalar automatically satisfies P1 and
P2. Furthermore, it satisfies P3 if j k is strictly positive. Ensuring that P4
holds requires a bit more care.
In the algorithm proposed in this paper, at iteration k, the search direction
is computed via solving QP and the tilting parameter j k
is iteratively adjusted to ensure that the four properties are satisfied. The
resultant algorithm will be shown to be locally superlinearly convergent and
globally convergent without resorting to a first-order direction far from the
solution. Further, the generation of a new iterate only requires the solution
of one QP and two closely related linear least squares problems. In contrast
with the algorithm presented in [1], our tilting parameter starts out positive
and asymptotically approaches zero.
Recently there has been a great deal of interest in interior point algorithms
for nonconvex nonlinear programming (see, e.g., [5, 6, 24, 4, 16, 23]).
Such algorithms generate feasible iterates and typically only require the
solution of linear systems of equations in order to generate new iterates.
SQP-type algorithms, however, are often at an advantage over such methods
in the context of applications where the number of variables is not too
large but evaluations of objectives/constraint functions and of their gradients
are highly time-consuming. Indeed, because these algorithms use
quadratic programs as successive models, away from a solution, progress
between (expensive) function evaluations is often significantly better than
that achieved by algorithms making use of mere linear systems of equations
as models.
In Section 2, we present the details of our new FSQP algorithm. In
Section 3, we show that under mild assumptions our iteration is globally
convergent, as well as locally superlinearly convergent. The algorithm has
been implemented and tested and we show in Section 4 that the numerical
results are quite promising. Some related issues are discussed in Section 5.
Finally, in Section 6, we offer some concluding remarks and discuss some
extensions to the algorithm which are currently being explored.
Algorithm
We begin by making a few assumptions that will be in force throughout.
Assumption 1: The set X is non-empty.
Assumption 2: The functions f
are continuously differentiable.
Assumption 3: For all x 2 X with I(x) 6= ;, the set frg j
is linearly independent.
A point x 2 R n is said to be a Karush-Kuhn-Tucker (KKT) point for
the problem (P ) if there exist scalars (KKT multipliers) - ;j ,
such that
(1)
It is well known that, under our assumptions, a necessary condition for
optimality of a point x 2 X is that it be a KKT point.
Note that, with x 2 X, QP (x; H; j) is always consistent: (0;
the constraints. Indeed, QP (x; H; j) always has a unique solution (d; fl) (see
by convexity, is its unique KKT point; i.e. there
exist multipliers - and - j , together with (d; fl), satisfy
\Gammaj
(2)
A simple consequence of the first equation in (2), which will be used through-out
our analysis, is an affine relationship amongst the multipliers, namely
Parameter j will be assigned a new value at each iteration, j k at iteration
k, to ensure that d(x k has the necessary properties. Strict positivity
of j k is sufficient to guarantee that Properties P1 to P3 are satisfied. As it
turns out however, this is not enough to ensure that, away from a solution,
there is adequate tilting into the feasible set. For this, we will force j k to be
bounded away from zero away from KKT points of (P ). Finally, P4 requires
that j k tend to zero sufficiently fast as d 0 tends to zero, i.e., as a
solution is approached. In [14], a similar effect is achieved by first computing
of course, we want to avoid that here.
Given an estimate I E
k of the active set I(x k ), we can compute an estimate
k ) of d 0 by solving the equality constrained QP
s.t.
which is equivalent (after a change of variables) to solving a linear least
squares problem. Let I k be the set of active constraints, not including the
"objective descent" constraint hrf(x k
I k
We will show in Section 3 that d E sufficiently
large. Furthermore, we will prove that, when d k is small, choosing
is sufficient to guarantee global and local superlinear convergence. Proper
choice of the proportionality constant (C k in the algorithm statement below),
while not important in the convergence analysis, is critical for satisfactory
numerical performance. This will be discussed in Section 4.
In [15], given x, H, and a feasible descent direction d, the Maratos
correction d C (denoted ~
d in [15]) is taken as the solution of the QP
s.t.
if it exists and has norm less than minfkdk; Cg, where - is a given scalar
satisfying 2 - 3 and C a given large scalar. Otherwise, d C is set to zero.
(Indeed, a large d C is meangingless and may jeopardize global convergence.)
In Section 1, it was mentioned that a linear least squares problem could be
used instead of a QP to compute a version of the Maratos correction d C
with the same asymptotic convergence properties. Given that our goal is
to reduce the computational cost per iteration, it makes sense to use such
an approach here. Thus, at iteration k, we take the correction d C
k to be
the solution d C exists and is not too large (specifically,
if its norm is no larger than that of d k ), of the equality-constrained QP
(equivalent to a least squares problem after a change of variables)
s.t.
direct extension of an alternative considered in [14]. In
making use of the best available metric, such an objective, as compared
to the pure least squares objective kd C k 2 , should yield a somewhat better
iterate without significantly increasing computational requirements (or affecting
the convergence analysis). Another advantage of using metric H k is
that, asymptotically, the matrix underlying LS C will be the
same as that underlying LS E resulting in computational sav-
ings. In the case that LS C inconsistent, or the computed
solution d C
k is too large, we will simply set d C
k to zero.
The proposed algorithm is as follows. Parameters ff, fi are used in the
Armijo-like search, - is the "bending" exponent in LS C , and ffl ' , C, C, and
D are used in the update rule for j k .
Algorithm FSQP 0
Parameters:
positive definite,
Computation of search arc.
(i). compute (d k ; the active
set I k , and associated multipliers
(ii). compute d C
exists and satisfies
kd C
k. Otherwise, set d C
the first value of t in the sequence
that satisfies
Updates.
(i). set x k+1 / x
k .
(ii). compute H k+1 , a new symmetric positive definite estimate
to the Hessian of the Lagrangian.
(iii). select C k+1 2 [C; C].
has a unique solution
and unique associated multipiers, compute d E
and the associated multipliers
In such case,
D and - E
else set j k+1 / C k+1
' .
(iv). set k
3 Convergence Analysis
Much of our analysis, especially the local analysis, will be devoted to establishing
the relationship between d(x; H; j) and the SQP direction d 0 (x; H).
Given x in X and H symmetric positive definite, d 0 is a KKT point for
solution d 0 (x; H)) if and only if there exists a
multiplier vector - 0 such that
Further, given I ae mg, an estimate d E is a KKT point for LS E (x; H; I)
(thus its unique solution d E (x; H; I)) if and only if there exists a multiplier
vector - E such that
Note that the components of - E for j 62 I play no role in the optimality
conditions.
3.1 Global Convergence
In this section we establish that, under mild assumptions, FSQP 0 generates
a sequence of iterates fx k g with the property that all accumulation points
are KKT points for (P ). We begin by establishing some properties of the
tilted SQP search direction d(x; H; j).
Lemma 1. Suppose Assumptions 1 through 3 hold. Then, given H symmetric
positive definite, x 2 X, and j - 0, d(x; H; j) is well-defined and
is the unique KKT point of QP (x; H; j). Further-
more, d(x; H; j) is bounded over compact subsets of X \Theta P \Theta R + , where P
is the set of symmetric positive definite n \Theta n matrices and R + the set of
nonnegative real numbers.
Proof. First note that the feasible set for QP (x; H; j) is non-empty, since
Now consider the cases
separately. From (2) and (4), it is clear that, if is a solution to
only if d is a solution of QP 0 It
is well known that, under our assumptions, d 0 (x; H) is well-defined, unique,
and continuous. The claims follow. Suppose now that j ? 0. In that case,
(d; fl) is a solution of QP (x; H; j) if and only if d solves the unconstrained
problem
ae
oe
and
ae
oe
Since the function being minimized in (6) is strictly convex and radially
unbounded, it follows that (d(x; H; j); fl(x; H; j)) is well-defined and unique
as a global minimizer for the convex problem QP (x; H; j), and thus unique
as a KKT point for that problem. Boundedness of d(x; H;
subsets of X \Theta P \Theta R + follows from the first equation in (2), our regularity
assumptions, and (3), which shows (since j ? 0) that the multipliers are
bounded.
Lemma 2. Suppose Assumptions 1 through 3 hold. Then, given H symmetric
positive definite and j - 0
(i). fl(x; H; only if
(ii). d(x; H; only if x is a KKT point for (P ). Moreover, if
either (thus both) of these conditions holds, then the multipliers - and
- for QP (x; H; are related by
and - .
Proof. To prove (i), note that since (d;
QP (x; H; j), the optimal value of the QP is non-positive. Further, since
H ? 0, the quadratic term in the objective is non-negative, which implies
Now suppose that d(x; H; feasibility of the first
QP constraint implies that fl(x; H; Finally, suppose that fl(x; H;
it is clear that
and achieves the minimum value of the objective. Thus, uniqueness gives
Suppose now that d(x; H; by (2) there
exist a multiplier vector - and a scalar multiplier - 0 such that
We begin by showing that - ? 0. Proceeding by contradiction, suppose
by (3) we have
Note that,
I
Thus, by the complementary slackness condition of (2) and the optimality
conditions (7),
By Assumption 3, this sum vanishes only if - contradicting
(8). Thus - ? 0. It is now immediate that x is a KKT point for (P )
with multipliers -
Finally, to prove the necessity portion of part (ii) note that if x is a
KKT point for (P ), then (1) shows that (d; is a KKT point for
Uniqueness of such points (Lemma 1) yields the result.
The next two lemmas establish that the line search in Step 2 of Algorithm
FSQP 0 is well defined.
Lemma 3. Suppose Assumptions 1 through 3 hold. Suppose x 2 X is not
a KKT point for (P ), H is symmetric positive definite and j ? 0. Then
(i).
(ii).
Proof. Both follow immediately from Lemma 2 and the fact that d(x; H;
and fl(x; H; must satisfy the constraints in QP (x; H; j).
Lemma 4. Suppose Assumptions 1 through 3 hold. Then, if
a KKT point for (P ) and the algorithm will stop in Step 1(i) at iteration k.
On the other hand, whenever the algorithm does not stop in Step 1(i), the
line search is well defined, i.e. Step 2 yields a step t k equal to fi j k for some
Proof. Suppose that j
with
The latter case cannot hold, as the stopping
criterion in Step 1(i) would have stopped the algorithm at iteration k \Gamma 1.
On the other hand, if
then in view of the optimality
conditions (5), and the fact that x k is always feasible for (P ), we see that
x k is a KKT point for (P ) with multipliers
0; otherwise:
Thus, by Lemma 2, d and the algorithm will stop in Step 1(i). The
first claim is thus proved. Also, we have established that
Step 2 is reached. The second claim now follows immediately from Lemma 3
and Assumption 2.
The previous lemmas imply that the algorithm is well-defined. In addi-
shows that if Algorithm FSQP 0 generates a finite sequence
terminating at the point xN , then xN is a KKT point for the problem (P ).
We now concentrate on the case in which an infinite sequence fx k g is gen-
erated, i.e. the algorithm never satisfies the termination condition in Step
1(i). Note that, in view of Lemma 4, we may assume throughout that
Before proceeding, we make an assumption concerning the estimates H k
of the Hessian of the Lagrangian.
Assumption 4: There exist positive constants oe 1 and oe 2 such that, for
all k,
Lemma 5. Suppose Assumptions 1 through 4 hold. Then the sequence fj k g
generated by Algorithm FSQP 0 is bounded. Further, the sequence fd k g is
bounded on subsequences on which fx k g is bounded.
Proof. The first claim follows from the update rule in Step 3(iii) of Algorithm
. The second claim then follows from Lemma 1 and Assumption
4.
Given an infinite index set K, we will use the notation
\Gamma! x
to mean
Lemma 6. Suppose Assumptions 1 through 3 hold. Suppose K is an infinite
index set such that x k
is bounded on K, and d k
\Gamma! 0:
sufficiently large and the QP multiplier sequences
are bounded on K. Further, given any accumulation
point is the unique solution of QP
Proof. In view of Assumption 2 frf(x k )g k2K must be bounded. Lemma 2(i)
and the first constraint in QP
Thus,
\Gamma! 0. To prove the first claim, let j 0 62 I(x ). There exists
such that g j 0 sufficiently large. In view of
Assumption 2, and since d k
\Gamma! 0, fl k
\Gamma! 0, and fj k g is bounded on K, it
is clear that
sufficiently large, proving the first claim.
Boundedness of f- k g k2K follows from non-negativity and (3). To prove
that of f- k g k2K , using complementary slackness and the first equation in
(2), write
Proceeding by contradiction, suppose that f- k g k2K is unbounded. Without
loss of generality, assume that k- k k 1 ? 0, for all k 2 K and define for all
Note that, for all k 2 K, k- k k Dividing (10) by k- k k 1 and taking
limits on an appropriate subsequence of K, it follows from Assumptions 2
and 4 and boundedness of f- k g that
for some - ;j , 1. As this contradicts Assumption
3, it is established that f- k g k2K is bounded.
To complete the proof, let K 0 ' K be an infinite index set such that
\Gamma! j and assume without loss of generality that H k
and - k
\Gamma! - . Taking limits in the optimality conditions (2) shows that,
indeed, (d;
and - . Finally, uniqueness of such points (Lemma 1) proves the result.
Lemma 7. Suppose Assumptions 1 through 4 hold. Then, if K is an infinite
index set such that d k
\Gamma! 0, all accumulation points of fx k g k2K are KKT
points for (P ).
Proof. Suppose K 0 ' K is an infinite index set on which x k
\Gamma! x 2 X.
In view of Assumption 4 and Lemma 5, assume, without loss of generality
that H k
\Gamma! H , a positive definite matrix, and j k
In view of
Lemma 6, (0; 0) is the unique solution of QP It follows from
Lemma 2 that x is a KKT point for (P ).
We now state and prove the main result of this subsection.
Theorem 1. Under Assumptions 1 through 4, Algorithm FSQP 0 generates
a sequence fx k g for which all accumulation points are KKT points for (P ).
Proof. Suppose K is an infinite index set such that x k
\Gamma! x . In view of
Lemma 5 and Assumption 4, we may assume without loss of generality that
\Gamma! d , j k
are considered separately.
Suppose first that j there exists an infinite
index set K 0 ' K such that either d E
d
\Gamma! 0. If the latter case holds, it is then clear that x
\Gamma! x , since
\Gamma! 0. Thus, by Lemma 7, x is a KKT point for
suppose instead that d E
From
the second set of equations in (5), one can easily see that I
sufficiently large, and using an argument very similar to that
used in Lemma 6, one can show that f- E
k g k2K 0 is a bounded sequence. Thus,
taking limits in (5) on an appropriate subsequence of K 0 shows that x is a
KKT point for (P ).
Now consider the case j ? 0. We show that d k
\Gamma! 0. Proceeding
by contradiction, without loss of generality suppose there exists d ? 0 such
that kd k k - d for all k 2 K. From non-positivity of the optimal value of
the objective function in QP
Assumption 4, we see that
Further, in view of (9) and since j ? 0, there exists j ? 0 such that
From the constraints of QP
and
using Assumption 2, it is easily shown that there exists
such that for all k 2 K, k large enough,
The rest of the contradiction argument establishing d k
exactly
the proof of Proposition 3.2 in [14]. Finally, it then follows from Lemma 7
that x is a KKT point for (P ).
3.2 Local Convergence
While the details are often quite different, overall the analysis in this section
is inspired by and occasionally follows that of Panier and Tits in [14, 15]. The
key result is Proposition 1 which states that, under appropriate assumptions,
the arc search eventually accepts the full step of one. With this and the
fact, to be established along the way, that titled direction d k approaches
the standard SQP direction sufficiently fast, superlinear convergence follows
from a classical analysis of M.J.D. Powell's. As a first step, we strengthen
the regularity assumptions.
Assumption
are three times continuously differentiable.
A point x is said to satisfy the second order sufficiency conditions with
strict complementary slackness for (P ) if there exists a multiplier vector
ffl The pair (x ; - ) satisfies (1), i.e. x is a KKT point for (P ),
positive definite on the subspace
ffl and - ;j ? 0 for all j 2 I(x ) (strict complementary slackness).
In order to guarantee that the entire sequence fx k g converges to a KKT
point x , we make the following assumption. (Recall that we have already
established, under weaker assumptions, that every accumulation point of
is a KKT point for (P ).)
Assumption 5: The sequence fx k g has an accumulation point x which
satisfies the second order sufficiency conditions with strict complementary
slackness.
It is well known that Assumption 5 guarantees that the entire sequence
converges. For a proof see, e.g., Proposition 4.1 in [14].
Lemma 8. Suppose Assumptions 1, 2', and 3 through 5 hold. Then the
sequence generated by Algorithm FSQP 0 converges to a point x satisfying
the second order sufficiency conditions with strict complementary
slackness.
From this point forward, - will denote the (unique) multiplier vector
associated with KKT point x for (P ). It is readily checked that, for any
symmetric positive definite H, (0; - ) is the KKT pair for QP 0
As announced, as a first main step, we show that our sequence of tilted
SQP directions approaches the true SQP direction sufficiently fast. (This is
achieved in Lemmas 9 through 18.) In order to do so, define d 0
k to be equal
to d 0 are as computed by Algorithm FSQP 0 .
Further, for each k, define - 0
k as a multiplier vector such that (d 0
(4) and let I 0
g: The following lemma
is proved in [15] (with reference to [14]) under identical assumptions.
Lemma 9. Suppose Assumptions 1, 2', and 3 through 5 hold. Then
(iii) For all k sufficiently large, the following two equalities hold
I 0
We next establish that the entire tilted SQP direction sequence converges
to 0. In order to do so, we establish that d(x; H; j) is continuous in a
neighborhood of positive
definite. Complicating the analysis is the fact that we have yet to establish
that the sequence fj k g does, in fact, converge. Given j - 0, define the set
ae' rf(x )
\Gammaj
oe
Lemma 10. Suppose Assumptions 1, 2', and 3 through 5 hold. Then, given
any j - 0, the set N (j ) is linearly independent.
Proof. Let H be symmetric positive definite. Note that, in view of Lemma 2,
Now suppose the claim does not hold, i.e. suppose there
exists scalars - j , j 2 f0g [ I(x ), not all zero, such that
\Gammaj
0: (11)
In view of Assumption 3, - 0 6= 0 and the scalars - j are unique modulo a
scaling factor. This uniqueness, the fact that d(x and the first
scalar equations in the optimality conditions (2) imply that -
are KKT multipliers for QP Thus, in view of (3),
0:
But this contradicts (11), which gives
hence N (j ) is linearly independent.
Lemma 11. Suppose Assumptions 1, 2', and 3 through 5 hold. Let j - 0
be an accumulation point of fj k g. Then, given any symmetric positive definite
is the unique solution of QP (x ; H; j ) and the
second order sufficiency conditions hold, with strict complementary slackness
Proof. In view of Lemma 2, QP its unique
solution. Define the Lagrangian function L : R n \Theta R \Theta R \Theta R m ! R for
Suppose -
are KKT multipliers such that (2) holds with
- and -. Let be the index for the first constraint
in QP (x ; H; j ), i.e. hrf(x ); di - fl. Note that since (d ; fl the
active constraint index set I for QP
(Note that we define I as including 0, while I k was defined as a subset of
Thus the set of active constraint gradients for QP
is N (j ).
Now consider the Hessian of the Lagrangian for QP (x ; H; j ), i.e. the
second derivative with respect to the first two variables (d; fl),
and given an arbitrary h 2 R n+1 , decompose it as
clearly,
and for h 6= 0, h T r 2 L(0; 0; -
Hy is zero if and only if
ff 6= 0. Since for such h
ff
it then follows that r 2 L(0; 0; -
-) is positive definite on N (j ) ? , the tangent
space to the active constraints for QP (x ; H; j ) at (0; 0). Thus, it is
established that the second order sufficiency conditions hold.
Finally it follows from Lemma 2(ii) that -
- which, together
with Assumption 5, implies strict complementarity for QP
at (0; 0).
Lemma 12. Suppose Assumptions 1, 2', and 3 through 5 hold. Then, if K
is a subsequence on which fj k g converges, say to
and - k
Finally,
Proof. First, proceed by contradiction to show that the first two claims hold
and that, in addition,
\Gamma! (0; 0); (12)
i.e., suppose that on some infinite index set K 0 ' K either - k is bounded
away from -
-, or - k is bounded away from -
from zero. In view of Assumption 4, there is no loss of generality is assuming
that H k
\Gamma! H for some symmetric positive definite H . In view of
Lemmas 10 and 11, we may thus invoke a result due to Robinson (Theorem
2.1 in [21]) to conclude that, in view of Lemma 2(ii),
\Gamma! (0; 0); - k
a contradiction. Hence the first two claims hold, as does (12). Next, proceeding
again by contradiction, suppose that d k 6! 0. Then, since fH k g and fj k g
are bounded, there exists an infinite index set K on which fH k g and fj k g
converge and d k is bounded away from zero. This contradicts (12). Thus
It immediately follows from the first constraint in QP
that
Lemma 13. Suppose Assumptions 1, 2', and 3 through 5 hold. Then, for
all k sufficiently large, I
Proof. Since is bounded and, in view of Lemma 12, (d k ;
Lemma 6 implies that I k ' I(x ), for all k sufficiently large. Now suppose it
does not hold that I sufficiently large. Thus, there exists
and an infinite index set K such that j 0 62 I k , for all k 2 K. Now,
in view of Lemma 5, there exists an infinite index set K 0 ' K and j - 0
such that j k
Further, Lemma 12 shows that - j 0
all k sufficiently large, k 2 K 0 , which, by complementary slackness, implies
this is a contradiction,
and the claim is proved.
Now define
and, given a vector - define the notation
Note that, in view of Lemma 9(iii), for k large enough, the optimality
conditions (4), yield
R T
The following well-known result will be used.
Lemma 14. Suppose Assumptions 1, 2', and 3 through 5 hold. Then the
R T
is invertible for all k large enough and its inverse remains bounded as k !
1.
Lemma 15. Suppose Assumptions 1, 2', and 3 through 5 hold. For all k
sufficiently large, d E
are uniquely defined, and d E
k .
Proof. In view of Lemma 13, the optimality conditions (5), and Lemma 14,
for all k large enough, the estimate d E
k and its corresponding multiplier
vector
are well defined as the unique solution of
R T
The claim then follows from (13).
Lemma 16. Suppose Assumptions 1, 2', and 3 through 5 hold. Then
(iii) For all k sufficiently large, I
Proof. Claim (i) follows from Step 3(iii) of Algorithm FSQP 0 , since in view
of Lemma 12, Lemma 15, and Lemma 9, fd k g and fd E
both converge to 0.
In view of (i), Lemma 12 establishes that
(ii) is proved. Finally, claim (iii) follows from claim (ii), Lemma 13, and
Assumption 5.
We now focus our attention on establishing relationships between d k , d C
and the true SQP direction d 0
k .
Lemma 17. Suppose Assumptions 1, 2', and 3 through 5 hold. Then
Proof. In view of Lemma 15, for all k sufficiently large, d E
k exist and
are uniquely defined, and d E
k . Lemmas 12 and 9 ensure that Step 3(iii)
of Algorithm FSQP 0 chooses
sufficiently large, thus
(i) follows. It is clear from Lemma 13 and the optimality conditions (2) that
d k and - k satisfy
R T
for all k sufficiently large, where 1 jI(x )j is a vector of jI(x )j ones. It thus
follows from (13), Assumption 2, and Lemmas 12, 14 and 16 that
and in view of claim (i), claim (ii) follows. Finally, since (from the QP
constraint and Lemma is clear that
O(kd k
Lemma 18. Suppose Assumptions 1, 2', and 3 through 5 hold. Then d C
O(kd 0
Proof. Let
Expanding we see that, for some - j 2 (0; 1),
z -
\Gammag
Assumption 2' we conclude that c
O(kd 0
sufficiently large, in view of Lemma 13, d C
k is well-defined
and satisfies
thus
R T
Now, the first order KKT conditions for LS C us there
exists a multiplier - C k 2 R jI(x )j such that
R T
Also, from the optimality conditions (15) we have
where
In view of Lemma 17, q
k and - C
R T
d C
or equivalently, with - 0
R T
The result then follows from Lemma 14.
In order to prove the key result that the full step of one is eventually
accepted by the line search, we now assume that the matrices fH k g suitably
approximate the Hessian of the Lagrangian at the solution. Define the
projection
(R T
Assumption
lim
0:
The following technical lemma will be used.
Lemma 19. Suppose Assumptions 1, 2', and 3 through 5 hold. Then there
exist constants
(ii) for all k sufficiently large
sufficiently large,
Proof. To show part (i), note that in view of the first QP constraint, negativity
of the optimal value of the QP objective, and Assumption 4,
The proof of part (ii) is identical to that of Lemma 4.4 in [14]. To show
(iii), note that from (15) for all k sufficiently large, d k satisfies
R T
Thus, we can write d
(R T
The result follows from Assumption 3 and Lemma 17(i,iii).
Proposition 1. Suppose Assumptions 1, 2', and 3 through 6 hold. Then,
sufficiently large.
Proof. Following [14], consider an expansion of g j (\Delta) about x k
I(x ), for all k sufficiently large,
where we have used Assumption 2', Lemmas 17 and 18, boundedness of all
sequences, and (16). As - ! 3, it follows that g j
for all k sufficiently large. The same result trivially holds for j 62 I(x
for k large enough, the full step of one satisfies the feasibility condition in the
arc search test. It remains to show that the "sufficient decrease" condition
is satisfied as well.
First, in view of Assumption 2 0 and Lemmas 17 and 18,
From the top equation in optimality conditions (2), equation (3), Lemma 17(i),
and boundedness of all sequences, we obtain
The last line in (2) and Lemma 17(i,iii) yield
Taking the inner product of (19) with d k , then adding and subtracting the
quantity
using (20), and finally multiplying the result
by 1gives2 hrf(x k ); d k
Further, Lemmas 17 and
Combining (18), (21), and (22), and using the fact that, for k large enough,
With this in hand, arguments identical to those used following equation (4.9)
in [14] show that
for all k sufficiently large. Thus the "sufficient decrease" condition is satisfied
A consequence of Lemmas 17, 18, and Proposition 1 is that the algorithm
generates a convergent sequence of iterates satisfying
Two-step superlinear convergence follows.
Theorem 2. Suppose Assumptions 1, 2', and 3 through 6 hold. Then Algorithm
generates a sequence fx k g which converges 2-step superlinearly
to x , i.e.
lim
0:
The proof is not given as it follows step by step, with minor modifications,
that of [18, Sections 2-3].
Finally, note that Q-superlinear convergence would follows if Assumption
6 were replaced with the stronger assumption
lim
0:
(See, e.g., [2].)
4 Implementation and Numerical Results
Our implementation of FSQP 0 (in C) differs in a number of ways from
the algorithm stated in Section 2. (It is readily checked that none of the
differences significantly affect the convergence analysis of Section 3.) Just
like in existing C implementation of FSQP (CFSQP: see [12]) the distinctive
character of linear (affine) constraints and of simple bounds is exploited
(provided the nature of these constraints is made explicit). Thus the general
form of the problem description tackled by our implementation is
min f(x)
s.t.
where a
(componentwise). The details of the implementation are spelled out below.
Many of them, including the update rule for H k , are exactly as in CFSQP.
In the implementation of QP no "tilting" is effected in connection
with the linear constraints and simple bound, since clearly the un-
tilted SQP direction is feasible for these constraints. In addition, each non-linear
constraint is assigned its own tilting parameter j j
Thus QP replaced with
The
k 's are updated independently, based on independently adjusted C j
's.
In the algorithm description and in the analysis all that was required of
was that it remain bounded and bounded away from zero. In practice,
though, performance of the algorithm is critically dependent upon the choice
of C k . In the implementation, an adaptive scheme was chosen in which the
new values C j
are selected in Step 3 based on their previous values C j
on the outcome of the arc search in Step 2, and on a preselected parameter
if the full step of one was accepted
are left unchanged; (ii) if the step of one was not accepted even though
all trial points were feasible, then, for all j, C j
k is decreased to minfffi c C j
(iii) if some infeasibility was encountered in the arc search, then, for all j
such that g j caused a step reduction at some trial point, C j
k is increased to
k is kept constant. Here, g j is said to
cause a step reduction if, for some trial point x, g j is violated (i.e., g j (x) ?
but all constraints checked at x before g j were found to be satisfied at that
point. (See below for the order in which constraints are checked in the arc
search.)
It was stressed in Section 2 that the Maratos correction can be computed
using an inequality-constrained QP such as QP C , instead of a LS C . This was
done in our numerical experiments, in order to more meaningfully compare
the new algorithm with CFSQP, in which an in an inequality-constrained
QP is indeed used. The implementation of QP C and LS E involves index sets
of "almost active" constraints and of binding constraints. First we define
I n
I a
is the machine precision. Next, the binding sets are defined as
I b;n
I b;l
mn is now the QP multiplier corresponding to the nonlinear
constraints and where - a
are the QP multipliers
corresponding to the affine constraints, the upper bounds, and the
lower bounds, respectively. Of course, no bending is required from d C
k in
connection with affine constraints and simple bounds, hence if I n
simply set d C
Otherwise the following modification of QP C is used:
s.t.
I a
Since not all simple bounds are included in the computation of d C
k , it is
possible that x k
k will not satisfy all bounds. To take care of this,
we simply "clip" d C
k so that the bounds are satisfied. Specifically, for the
upper bounds, we perform the following:
for j 62 I b;u
do
if (d C;j
d C;j
The same procedure, mutatis mutandis, is executed for the lower bounds.
We note that such a procedure has no effect on the convergence analysis of
Section 3 since, locally, the active set is correctly identified and a full step
along
k is always accepted. The least squares problem LS E used to
compute d E
k is modified similarly. Specifically, in the implementation, d E
k is
only computed if m n ? 0, in which case we use
s.t.
The implementation of the arc search (Step 2) is as in CFSQP. Specif-
ically, feasibility is checked before sufficient decrease, and testing at a trial
point is aborted as soon as infeasibility is detected. Like in CFSQP, all linear
and bound constraints are checked first, then nonlinear constraints in an
order maintained as follows: (i) at the start of the arc search from a given
iterate x k the order is reset to be the natural numerical order; (ii) within
an arc search, as a constraint is found to be violated at a trial point, its
index is moved to the beginning of the list, with the order of the others left
unchanged.
An aspect of the algorithm which was intentionally left vague in Sections
2 and 3 was the updating scheme for the Hessian estimates H k . In the
implementation, we use the BFGS update with Powell's modification [19].
Specifically, define
where, in an attempt to better approximate the true multipliers, if - k ?
we normalize as follows
A scalar ' k+1 2 (0; 1] is then defined by
0:8
the rank two Hessian update is
Note that while it is not clear whether the resultant sequence fH k g will,
in fact, satisfy Assumption 6, this update scheme is known to perform very
well in practice.
All QPs and linear least squares subproblems were solved using QPOPT [7].
For comparison sake, QPOPT was also used to solve the QP subproblems
in CFSQP. While the default QP solver for CFSQP is the public domain
code QLD (see [22]), we opted for QPOPT because it allows "warm starts"
and thus is fairer to CFSQP in the comparison with the implementation of
more QPs are solved with the former). For alls QPs in both
codes, the active set in the solution at a given iteration was used as initial
guess for the active set for the same QP at the next iteration.
In order to guarantee that the algorithm terminates after a finite number
of iterations with an approximate solution, the stopping criterion of Step 1
is changed to
small. Finally, the following parameter values were selected:
Further, we always set H
experiments were run on a Sun Microsystems Ultra 5 workstation
For the first set of numerical tests, we selected a number of problems
from [9] which provided feasible initial points and contained no equality con-
straints. The results are reported in Table 1, where the performance of our
implementation of FSQP 0 is compared with that of CFSQP (with QPOPT
as QP solver). The column labeled # lists the problem number as given
in [9], the column labeled ALGO is self-explanatory. The next three columns
give the size of the problem following the conventions of this section. The
columns labeled NF, NG, and IT give the number of objective function eval-
uations, nonlinear constraint function evaluations, and iterations required
to solve the problem, respectively. Finally, f(x ) is the objective function
value at the final iterate and ffl is as above. The value of ffl was chosen in
order to obtain approximately the same precision as reported in [9] for each
problem.
The results reported in Table 1 are encouraging. The performance of our
implementation of Algorithm FSQP 0 in terms of number of iterations and
function evaluations is essentially identical to that of CFSQP (Algorithm
FSQP). The expected payoff of using FSQP 0 instead of FSQP however is
that, on large problems the CPU time expended in linear algebra, specifically
in solving the QP and linear least squares subproblems, should be much
less. To assess this, we carried out comparative tests on the COPS suite of
problems [3].
The first five problems from the COPS set [3] were considered, as these
problems either do not involve nonlinear equality constraints or are readily
reformulated without such constraints. (Specifically, in problem "Sphere"
the equality constraint was changed to a "-" constraint; and in "Chain" the
equality constraint (with replaced with two inequalities, with
the left-hand side constrained to be between the values
the solution was always at 5.) "Sawpath" was discarded because it involves
few variables and many constraints, which is not the situation at which the
new algorithm is targeted. The results obtained with various instances of
the other four problems are presented in Table 2. The format of that table
is identical to that of Table 1 except for the additional column labeled NQP.
In that column we list the total number of QP iterations in the solution of
the two major QPs, as reported by QPOPT. (Note that QPOPT reports
CFSQP
CFSQP
CFSQP 9 19 7 6.0000000E+00
43
CFSQP
Table
1: Numerical results on Hock-Schittkowski problems.
zero iteration when the result of the first step onto the working set of linear
constraints happens to be optimal. To be "fair" to FSQP 0 , we thus do not
count the work involved in solving LS E either. We also do not count the
QP iterations in solving QP C , the "correction" QP, because it is invoked
identically in both algorithms.) The reason for the smaller ffl on Cam is this
allowed CFSQP to reach the globally optimal objective function values (as
per [3]).
The results show a typical significantly lower number of QP iterations
with the new algorithm and, as in the case of the Hock-Schittkowski prob-
lems, a roughly comparable behavior of the two algorithms in terms of number
of function evaluations. Note that in the two instances where the NQP
count is less for CFSQP than for FSQP 0 , different local minima are reached,
which makes the comparison meaningless. Finally, the abnormal terminations
on Sphere-50 and Sphere-100 are both due to QPOPT's failure to
solve a QP-the "tilting" QP in the case of CFSQP.
One issue of interest is whether our convergence results still hold under
weaker assumptions. To wit, Qi and Wei showed in [20] that the algorithm
of [15] still enjoys global convergence (all limit points are KKT) and local
(two-step) superlinear convergence when Assumption 3 (LICQ) is replaced
with the Mangasarian-Fromovitz constraint qualification (MFCQ)-or even
with a condition slightly weaker than MFCQ. They further showed that, if
that algorithm is slightly modified, local superlinear convergence is preserved
without strict complementarity assumption, provided the strong second-order
sufficiency condition (SSOSC) is assumed. For algorithm FSQP 0 as
stated however, LICQ and strict complementarity are essential, in connection
with Step 3 (iii). First, Assumption 3 is needed in order for LS E
to be well-defined close to a solution x . Second, strict positivity of the multipliers
associated with active constraints at the solution is needed in order
for the components of - E
k to be nonnegative when a solution is approached.
Barring this, the condition - E
may never hold and the
update rule j k+1 / C k+1 \Delta kd k k 2 may not be used close to the solution, in
which case superlinear convergence would not take place. Careful modifications
of Algorithm FSQP 0 might (at least in theory) accommodate weaker
assumptions though. First, in the absence of Assumption 3, LS E
could possibly be replaced with a QP with jI E
inequality constraints (with
essentially no penalty in CPU cost). Second, by replacing the nonnegativity
condition by a requirement of the type - E;j - \Gammaffl k , where - E;j is now a "reg-
ular" multiplier (see [20]) and where ffl k ? 0 would be made to go to zero
at an appropriate (slow) rate, it may be possible to preserve convergence
to KKT points while insuring that, even in the absence of strict comple-
mentarity, the test would be satisfied close to the solution, thus allowing
superlinear convergence to take place. This would require detailed analysis
though, and may be as likely to hurt as to help in practice.
A second issue worth discussing is that of possible low cost solution for
the QP and two linear least-squares problems. The indexes of constraints
appearing in LS E and LS C at iteration k are generally different, I E
28 142 .776859 1.E-4
CFSQP 42 8177 44 350 .776859
CFSQP 591 345458 154 2771 .783873
CFSQP 795 28328 246 587 660.675
CFSQP failure
CFSQP 977 3784 575 1259 4.81189
CFSQP
Table
2: Numerical results on COPS problems.
for the former, I k for the latter. However, it was proved that, for k large
enough, both of these sets are equal to I(x ). When that is the case, it
is readily checked that LS E
involve the
same matrix, and thus that the latter can be solved at low cost once the
former has been solved. Unfortunately, the linear systems arising in the
solution of QP are different. In particular, they involve j k .
6 Conclusions
We have presented here a new SQP-type algorithm generating feasible it-
erates. The main advantage of the algorithm presented here is a reduction
in the amount of computation required in order to generate a new iterate.
While this may not be very important for applications where function evaluations
dominate the actual amount of work to compute a new iterate, it
is very useful in many contexts. In any case, we saw in the previous section
that preliminary results seem to indicate that decreasing the amount of
computation per iteration did not come at the cost of increasing the number
of function evaluations required to find a solution.
A number of significant extensions of Algorithm FSQP 0 are being ex-
amined. It is not too difficult to extend the algorithm to handle mini-max
problems. The only real issue that arises is how to handle the mini-max
objectives in the least squares sub-problems. Several possibilities, each with
the desired global and local convergence properties, are being examined.
Another extension that is important for engineering design is the incorporation
of a scheme to efficiently handle very large sets of constraints and/or
objectives. We will examine schemes along the lines of those developed in
[11, 25]. Further, work remains to be done to exploit the close relationship
between the two least squares problems and the quadratic program.
A careful implementation should be able to use these relationships to great
advantage computationally. For starters, updating the Cholesky factors of
H k instead of H k itself at each iteration would save a factorization in each
of the sub-problems. Finally, it is possible to extend the class of problems
(P ) which are handled by the algorithm to include nonlinear equality con-
straints. Of course, we will not be able to generate feasible iterates for such
constraints, but a scheme such as that studied in [10] could be used in order
to guarantee asymptotic feasibility while maintaining feasibility for all
inequality constraints.
--R
A variant of the Topkis-Veinott method for solving inequality constrained optimization problems
Sequential quadratic programming.
An interior point algorithm for large scale nonlinear programming.
On the formulation and theory of the Newton interior-point method for nonlinear programming
A primal-dual interior method for nonconvex nonlinear programming
User's guide for qpopt 1.0: A fortran package for quadratic programming.
A successive quadratic programming based feasible directions algorithm.
Test Examples For Nonlinear Programming Codes
Nonlinear equality constraints in feasible sequential quadratic programming.
Feasible sequential quadratic programming for finely discretized problems from SIP.
User's Guide for CFSQP Version 2.5: A C Code for Solving (Large Scale) Constrained Nonlinear (Minimax) Optimization Problems
Exact Penalty Functions for Finite Dimensional and Control Optimization Problems.
A superlinearly convergent feasible method for the solution of inequality constrained optimization problems.
On combining feasibility
Computational Methods in Optimization.
Convergence of variable metric methods for nonlinearly constrained optimization calculations.
A fast algorithm for nonlinearly constrained optimization calculations.
On the constant positive linear dependence condition and its application to SQP methods.
Perturbed Kuhn-Tucker points and rates of convergence for a class of nonlinear-programming algorithms
QLD: A Fortran Code for Quadratic Programming
A primal-dual interior-point method for nonconvex optimization with multiple logarithmic barrier parameters and with strong convergence properties
An interior point algorithm for non-convex nonlinear programming
An SQP algorithm for finely discretized continuous minimax problems and other minimax problems with many objective functions.
User's Guide for FSQP Version 3.7: A FORTRAN Code for Solving Nonlinear (Minimax) Optimization Problems
Methods of Feasible Directions.
--TR
--CTR
Dudy Lim , Yew-Soon Ong , Bu-Sung Lee, Inverse multi-objective robust evolutionary design optimization in the presence of uncertainty, Proceedings of the 2005 workshops on Genetic and evolutionary computation, June 25-26, 2005, Washington, D.C.
Zhibin Zhu, An efficient sequential quadratic programming algorithm for nonlinear programming, Journal of Computational and Applied Mathematics, v.175 n.2, p.447-464, 15 March 2005
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Borys Shchokin , Farrokh Janabi-Sharifi, Design and kinematic analysis of a rotary positioner, Robotica, v.25 n.1, p.75-85, January 2007 | feasible SQP;feasible iterates;sequential quadratic programming;FSQP;SQP |
589022 | Monotonicity of Fixed Point and Normal Mappings Associated with Variational Inequality and Its Application. | We prove sufficient conditions for the monotonicity and the strong monotonicity of fixed point and normal maps associated with variational inequality problems over a general closed convex set. Sufficient conditions for the strong monotonicity of their perturbed versions are also shown. These results include some well known in the literature as particular instances. Inspired by these results, we propose a modified Solodov and Svaiter iterative algorithm for the variational inequality problem whose fixed point map or normal map is monotone. | Introduction
. Given a continuous function f : R n ! R n and a closed convex
set K in R n ; the well-known finite-dimensional variational inequality, denoted by
f ), is to find an element x 2 K such that
It is well-known that the above problem can be reformulated as nonsmooth equations
such as the fixed point and normal equations (see e.g. [9, 18]). The fixed point
equation is defined by
and the normal equation is defined by
positive scalar, and \Pi K (\Delta) denotes the projection operator on the
convex set K; i.e.,
Throughout the paper, k \Delta k denotes the 2-norm (Euclidean norm) of the vector in R
It turns out that x solves VI(K; f) if and only if - ff (x and that if x solves
ff f(x ) is a solution to \Phi ff conversely, if \Phi ff (u
\Pi K (u ) is a solution to VI(K; f):
Recently, several authors studied the P 0 property of fixed point and normal maps
when K is a rectangular box in R n , i.e., the Cartesian product of n one-dimensional
This work was partially supported by Research Grants Council of Hong Kong under grant
CUHK4392/99E.
y Department of Systems Engineering and Engineering Management, The Chinese University
of Hong Kong, Shatin, New Territory, Hong Kong, and Institute of Computational Mathematics
and Scientific/Engineering Computing, Chinese Academy of Sciences, Beijing, China (Email:
ybzhao@se.cuhk.edu.hk).
z Corresponding author. Department of Systems Engineering and Engineering Management, The
Chinese University of Hong Kong, Shatin, New Territory, Hong Kong (Fax: (852) 26035505; Tel:
Y. B. ZHAO AND D. LI
intervals. For such a K, Ravindran and Gowda [17] (respectively, Gowda and Tawhid
[8]) showed that - ff (x) (respectively, \Phi ff (x)) is a P 0 -function if f is. Notice that the
monotone maps are very important special cases of the class of P 0 -functions. It is
worth considering the problem:
(P) When are the mappings - ff (x) and \Phi ff (x) monotone if K is a general closed
convex set?
Intuitively, we may conjecture that the fixed point map and the normal map
are monotone if f is. However, this conjecture is not true. The following example
shows that for a given ff ? 0 the monotonicity of f , in general, does not imply the
monotonicity of the fixed point map - ff (x) and the normal map \Phi ff (x):
Example 1.1. Let K be a closed convex set given by
and
For any x; y we have that 0: Hence the function f is
monotone on R 2 . We now show that for an arbitrary scalar ff ? 0 the fixed point
mapping - ff not monotone in R 2 . Indeed, let
It is easy to verify that - ff
Thus, we have
which implies that - ff (\Delta) is not monotone on R
Example 1.2. Let K be a closed convex set given by
given as in Example 1.1. We now show that for an arbitrary
the normal mapping \Phi ff not monotone in
R 2 . Indeed, let We have that \Phi ff
Thus, we have
which implies that \Phi ff (\Delta) is not monotone on R
From the above examples, we conclude that certain condition stronger than the
monotonicity of f is required to guarantee the monotonicity of - ff (x) and \Phi ff (x). One
such condition is so-called co-coercivity condition. We recall that f is said to be co-
coercive with modulus fi ? 0 on a set S ae R n if there exists a constant fi ? 0 such
that
The co-coercivity condition was used in several works, such as Bruck [1], Gabay [7] (in
which this condition is used implicitly), Tseng [25], Marcotte and Wu [15], Magnanti
and Perakis [13, 14], and Zhu and Marcotte [29, 30]. It is also used to study the strict
feasibility of complementarity problems [27]. It is interesting to note that in an affine
MONOTONICITY OF FIXED POINT AND NORMAL MAPPINGS 3
case the co-coercivity has a close relation to the property of psd-plus matrices [12, 30].
A special case of the co-coercive map is the strongly monotone and Lipschitzian map.
We recall that a mapping f is said to be strongly monotone with modulus c ? 0 on
the set S if there is a scalar c ? 0 such that
It is evident that any co-coercive map on the set S must be monotone and Lipschitz
continuous (with constant not necessarily strongly monotone (for
instance, the constant mapping) on the same set.
In fact, the aforementioned problem (P) is not completely unknown. By using the
co-coercivity condition implicitly and using properties of nonexpansive maps, Gabay
actually showed (but did not explicitly state) that - ff (x) and \Phi 1=ff (x) are monotone
if the scalar ff is chosen such that the map I \Gamma fff is nonexpansive. Furthermore, for
strongly monotone and Lipschitzian map f , Gabay [7] and Sibony [20] actually showed
that - ff (x) and \Phi 1=ff (x) are strongly monotone if the scalar ff is chosen such that the
contractive. Throughout this paper, we use the standard concept
"nonexpansive" map and "contractive" map in the literature to mean a Lipschitzian
map with constant
However, it is easy to give an example to show that - ff (x) and \Phi ff (x) are still
monotone (strongly monotone) even when ff is chosen such that I \Gamma fff is not nonexpansive
(contractive). For instance, let
We see that the
function f is co-coercive with modulus fff is not nonexpansive for
remains monotone. As a result, the main purpose of this paper
is to expand the results of Sibony [20] and Gabay [7]. We show that if f is co-coercive
(strongly monotone and Lipschitz continuous, respectively), the monotonicity (strong
monotonicity, respectively) of the maps - ff (x) and \Phi ff (x) can be ensured when ff lies
in a larger interval in which the map I \Gamma fff may not be nonexpansive (contractive,
respectively). The results derived in this paper are not obtainable by the proof based
on the nonexpansiveness and contractiveness of maps.
The other purpose of the paper is to introduce an application of the monotonicity
of - ff (x) and \Phi ff (x): This application (see Section 3) is motivated by the globally
convergent inexact Newton method for the system of monotone equations proposed
by Solodov and Svaiter [21]. See also [22, 23, 24]). We propose a modified Solodov
and Svaiter method to solve the monotone equations - ff 0: This
modified algorithm requires no projection operations in the line-search step.
2. Monotonicity of - ff (x) and \Phi ff (x). It is known (see Sibony [20] and Gabay
[7]) that if f is strongly monotone with modulus c ? 0 and Lipschitz continuous
with constant L ? 0, then I \Gamma fff is contractive when
nonexpansive, this in turn implies that - ff (x) and \Phi 1=ff (x) are both strongly monotone
Similarly, it follows from Gabay [7] (see Theorem 6.1 therein) that
if f is co-coercive with modulus fi ? 0; then I \Gamma fff is nonexpansive for
and thus we can easily verify that - ff (x) and \Phi 1=ff (x) are monotone for
In this section, we prove an improved version of the above-mentioned results. We
prove that i) when ff lies outside of the interval (0; 2c=L 2 ), for instance, 2c=L 2 - ff -
are still strongly monotone although I \Gamma fff , in this case, is
not contractive, and ii) when ff lies outside of the interval (0; 2fi], for instance, 2fi !
remain monotone although I \Gamma fff is not nonexpansive.
This new result on monotonicity (strongly monotonicity) of - ff (x) and \Phi 1=ff (x) for ff ?
not obtainable by using the nonexpansive (contractive) property
4 Y. B. ZHAO AND D. LI
of I \Gamma fff: The reason goes as follows: Let f be co-coercive with modulus fi ? 0 on
the set S ' R n ; where
Clearly, such a scalar fi is unique and is not a constant
mapping. We now verify that I \Gamma fff is nonexpansive on S if and only if
It is sufficient to show that if ff ? 0 is chosen such that I \Gamma fff is nonexpansive on S,
then we must have ff - 2fi: In fact, if I \Gamma fff is nonexpansive, then for any x; y in S
we have
which implies that
By the definition of fi, we deduce that ff=2 - fi; the desired consequence. Similarly, let
f be strongly monotone with modulus c ? 0 and Lipschitz continuous with constant
on the set S, where
and
We can easily see that is not a single point set.
It is also easy to show that I \Gamma fff is contractive if and only if
Since the map I \Gammafff is not contractive (nonexpansive, respectively) for ff - 2c=L 2
our result established in this section cannot follow directly
from the proof of Sibony [20] and Gabay [7].
We also study the strong monotonicity of the perturbed fixed point and normal
maps defined by
and
respectively. This is motivated by the well-known Tikhonov regularization method for
complementarity problems and variational inequalities. See for example, Isac [10, 11],
Venkateswaran [26], Facchinei [3], Facchinei and Kanzow [4], Facchinei and Pang [5],
Gowda and Tawhid [8], Qi [16], Ravindran and Gowda [17], Zhao and Li [28], etc. It is
worth mentioning that Gowda and Tawhid [8] showed that when perturbed
mapping \Phi 1;" (x) is a P-function if f is a P 0 -function and K is a rectangular set. We
show in this paper a sufficient condition for the strong monotonicity of - ff;" (x) and
\Phi ff;" (x): The following lemma is helpful.
Lemma 2.1. (i) Denote
(2.
MONOTONICITY OF FIXED POINT AND NORMAL MAPPINGS 5
Then
(ii) For any ff ? 0 and vector b 2 R n ; the following inequality holds for all v 2 R n ;
Proof. By the property of projection operator, we have
Adding the above two inequalities leads to
i.e.,
This proves the result (i).
Given ff ? 0 and b 2 R n ; it is easy to check that the minimum value of ffkvk 2 +v T b
is \Gammakbk 2 =(4ff): This proves the result (ii).
We are ready to prove the main result in this section.
Theorem 2.1. Let K be an arbitrary closed convex set in R n and K ' S ' R
(i) If f is co-coercive with modulus fi ? 0 on the set S; then for any fixed scalar
the fixed point map - ff (x) defined by (1.1) is monotone on
the set S.
(ii) If f is strongly monotone with modulus c ? 0 on the set S; and f is Lipschitz
continuous with constant L ? 0 on S; then for any fixed scalar ff satisfying
the fixed point map - ff (x) is strongly monotone on the set S:
(iii) If f is co-coercive with modulus fi ? 0 on the set S; then for any 0 ! ff ! 4fi
the perturbed map - ff;" (x) is strongly monotone in x on the
set S.
Proof. Let ff ? 0 and 0 - 2=ff be two scalars. For any vector x; y in
By using the notation of (2.1) and Lemma 2.1, we have
6 Y. B. ZHAO AND D. LI
If f is co-coercive with modulus fi ? 0, using " - 2=ff we see from the above that
'ff
Setting in the above inequality, we see that for 0 ! ff - 4fi the right-hand side
is nonnegative, showing that - ff is monotone on the set S: This proves the result (i).
Also, if ff ! 4fi and
4fi ); the right-hand side of the above inequality is
greater than or equal to rkx \Gamma showing that - ff;" is
strongly monotone on the set S: The proof of the result (iii) is complete.
Assume that f is strongly monotone with modulus c ? 0 and Lipschitz continuous
with constant L ? 0: We now prove the result (ii). For this case, setting
(2.2), we have that
For it is evident that the scalar
4c
0:
Result (ii) is proved.
Similarly, we have the following result for \Phi ff (x):
Theorem 2.2. Let f be a function from R n into itself and K be a closed convex
set and K ' S ' R
(i) If f is co-coercive with modulus fi ? 0 on the set S; then for any constant ff
such that ff ? 1=(4fi); the normal map \Phi ff (x) given by (1.2) is monotone on the set
S.
(ii) If f is strongly monotone with modulus c ? 0 and Lipschitz continuous with
constant L ? 0 on the set S; then for any ff satisfying ff ? L 2 =(4c); the normal map
\Phi ff (x) given by (1.2) is strongly monotone on the set S.
(iii) If f is co-coercive with modulus fi ? 0 on the set S; then for any constant
ff ? 1=(4fi); the perturbed normal map \Phi ff;" (x), where strongly monotone
in x on the set S.
MONOTONICITY OF FIXED POINT AND NORMAL MAPPINGS 7
Proof. Let ff; "; r be given such that ff 0: For any vector x; y in
u x and u y be defined by (2.1) with
2.1 we have
kf (\Pi K
and
that further implies
By using the above three inequalities, we have
\Gammaf (\Pi K
\Gammaf (\Pi K
kf (\Pi K
Let f be co-coercive with modulus fi ? 0 on the set S. Setting in the above
inequality, and using the co-coercivity of f , we have
4ff kf (\Pi K
4ff
kf (\Pi K
For ff ? 1=(4fi); the right-hand side is nonnegative, and hence the map \Phi ff is monotone
on the set S: This proves the result (i).
By the co-coercivity
of f , the inequality (2.6) can be further written as
kf (\Pi K
8 Y. B. ZHAO AND D. LI
the right-hand side of the above is nonnegative, and thus
the map \Phi ff;" is strongly monotone on the set S: Result (iii) is proved.
Finally, we prove the result (ii). Assume that f is strongly monotone with modulus
continuous with constant L ? 0. For any vector x; y in
we note that the equation (2.5) holds for any ff ?
the equation (2.5) reduces to
Given ff ? L 2 =(4c): Let r be a scalar such that
Notice that
Substituting the above into (2.7) and using inequalities (2.3) and (2.4), we have
where the last inequality follows from the Lipschitz continuity and strong monotonicity
of f: The right-hand side of the above is nonnegative. Thus, the map \Phi ff is strongly
monotone on the set S: This proves the result (ii).
The following result is an immediate consequence of Theorems 2.1 and 2.2.
Corollary 2.1. Assume that f is monotone and Lipschitz continuous with
constant L ? 0 on a set S ' K:
; then the perturbed map - ff;" (x) is strongly
monotone in x on the set S.
then the perturbed normal map \Phi ff;" (x) is
strongly monotone in x on the set S.
Proof. Let " 2 (0; 1) be a fixed scalar. It is evident that under the condition
of the corollary, the function F strongly monotone with modulus
continuous with constant L+ ": Therefore, from Theorem 2.1(ii)
we deduce that if 0 the map - ff;" (x) is strongly monotone on S:
Similarly, the strong monotonicity of \Phi ff;" (x) follows from Theorem 2.2(ii).
The Item (iii) of both Theorem 2.1 and Theorem 2.2 shows that for any sufficiently
small parameter ", the perturbed fixed point and normal maps are strongly monotone.
This result is quite different from Corollary 2.1. When ff is a fixed constant, Corollary
MONOTONICITY OF FIXED POINT AND NORMAL MAPPINGS 9
2.1 does not cover the case where " can be sufficiently small. Indeed, for a fixed ff ? 0,
the inequalities
4" fail to hold when " ! 0:
Up to now, we have shown that the fixed point map - ff (x) (respectively, the
normal map \Phi ff (x)) is monotone if f is co-coercive with modulus fi ? 0 and ff 2 (0; 4fi]
(respectively, ff 2 (1=(4fi); 1)): This result includes the known ones from Sibony [20]
and Gabay [7] as special cases. Under the same assumption on f and ff; we deduce
from the Item (iii) of Theorems 2.1 and 2.2 that the perturbed forms - ff;" and \Phi ff;" are
strongly monotone provided that the scalar " is sufficiently small. In the succeeding
sections, we will introduce an application of the above results on globally convergent
iterative algorithms for VI(K; f) whose fixed point map or normal map is monotone.
3. Application: Iterative algorithm for VI(K; f) . Since - ff (x) and \Phi ff (x)
are monotone if the function f is co-coercive and ff lies in certain interval, we can
solve the co-coercive variational inequity problems via solving the system of monotone
equation - ff Recently, Solodov and Svaiter [21] (see also,
[22, 23, 24]) proposed a class of inexact Newton methods for monotone equations. Let
F(x) be a monotone mapping from R n into R n : The Solodov and Svaiter's algorithm
for the equation proceeds as follows:
Algorithm SS. [21] Choose any x 0
Inexact Newton step. Choose a positive semidefinite matrix G k . Choose
and
where
Line-search step. Find y being the
smallest nonnegative integer m such that
\GammaF
Projection step. Compute
repeat.
As pointed out in [21], the above inexact Newton step is motivated by the idea of
proximal point algorithm [2, 6, 19]. Algorithm SS has an advantage over other Newton
methods that the whole iteration sequence is globally convergent to a solution of the
system of equations, provided a solution exists, under no assumption on F other than
continuity and monotonicity. Setting or \Phi ff (x), from Theorems 2.1 and
2.2 in this paper and Theorem 2.1 of [21], we have the following result.
Theorem 3.1. Let f be a co-coercive map with constant fi ? 0: Substitute F(x)
in Algorithm SS by - ff (x) (respectively, \Phi ff (x)) where
ff ? 1=4fi). If - k is chosen such that C 2 - k - C 1 kF(x k )k, where C 1 and C 2
are two constants, then Algorithm SS converges to a solution of variational inequality
provided that a solution exists.
While Algorithm SS can be used to solve the monotone equations - ff
and \Phi ff each line-search step needs to compute the values of - ff
and \Phi ff that represents a major cost of the algorithm in calculating
projection operations. Hence, in general cases, Algorithm SS has high computational
Y. B. ZHAO AND D. LI
cost per iteration when applied to solve \Phi ff To reduce this major
computational burden, we propose the following algorithm which needs no projection
operations other than the evaluation of the function f in line-search steps.
Algorithm 3.1. Choose x 0
Inexact Newton Step: Choose a positive semidefinite matrix G k . Choose
Compute
where
Line-search step. Find y being the smallest
nonnegative integer m such that
Projection step. Compute
The above algorithm has the following property.
Lemma 3.1. Let - ff (x) be given as (1.1). At kth iteration, if m k is the smallest
nonnegative integer such that (3.2) holds, then y satisfies the following
estimation:
Proof. By the definition of - ff (x), the nonexpansiveness of projection operator
and (3.2), we have
\Gamma\Pi K
Also,
\Gamma- ff
MONOTONICITY OF FIXED POINT AND NORMAL MAPPINGS 11
By (3.1) and positive semi-definiteness of G k , we have
\Gamma- ff
Combining (3.3), (3.4) and (3.5) yields
\Gamma- ff
The proof is complete.
Using Lemma 3.1 and following the line of the proof of Theorem 2.1 in [21], it is
not difficult to prove the following convergence result.
Theorem 3.2. be a continuous function such that there exists
a constant ff ? 0 such that - ff (x) defined by (1.1) is monotone. Choose G k and - k
such that kG k k - C 0 and - are three fixed positive
numbers and p 2 (0; 1]: Then the sequence fx k g generated by Algorithm 3.1 converges
to a solution of the variational inequality provided that a solution exists.
Algorithm 3.1 can solve the variational inequality whose fixed point mapping
- ff (x) is monotone for some ff ? 0. Since the co-coercivity of f implies the monotonicity
of the functions - ff (x) and \Phi ff (x) for suitable choices of the value of ff; Algorithm
3.1 can locate a solution of any solvable co-coercive variational inequality problem.
This algorithm has an advantage over Algorithm SS in that it does not carry out
any projection operation in the line-search step, and hence the computational cost is
significantly reduced.
4. Conclusions . In this paper, we show some sufficient conditions for the monotonicity
(strong monotonicity) of the fixed point and normal maps associated with
the variational inequality problem. The results proved in the paper encompass some
known results as particular cases. Based on these results, an iterative algorithm for
a class of variational inequalities is proposed. This algorithm can be viewed as a
modified Solodov and Svaiter's method but has lower computational cost than the
latter.
Acknowlegements. The authors would like to thank two anonymous referees
for their incisive comments and helpful suggestions which help us to improve many
aspects of the paper. They also thank Professor O. L. Mangasarian for encouragement
and one referee for pointing out refs. [7, 20].
--R
On the Douglas-Rachford splitting method and the
Beyond monotonicity in regularization methods for
Total Stability of Variational Inequalities
Finite termination of the
Applications of the method of multipliers to variational inequalities
Existence and limiting behavior of trajectories associated with P0 equations
A survey of theory
Tikhonov's regularization and the complementarity problem in Hilbert spaces
A generalization of Karamardian's condition in complementarity theory
A decomposition property for a class of square matrices
A unifying geometric solution framework and complexity analysis for variational inequalities
The orthogonality theorem and the strong-f-monotonicity condition for variational inequality algorithms
On the convergence of projection methods: application to the decomposition of affine variational inequalities
regularization methods for variational inequality problems
Regularization of P 0
Normal maps induced by linear transformations
Control Optim.
M'ethods it'eratives pour les 'equations et in'equations aus d'eriv'ees partielles non- lin'earies de type monotone
A globally convergence inexact Newton method for systems of monotone equations
A truly globally convergent Newton-type method for the monotone nonlinear complementarity problem
A new projection method for variational inequality problems
Further applications of a matrix splitting algorithm to decomposition in variational inequalities and convex programming
An algorithm for the linear complementarity problem with a P 0
On condition for strictly feasible condition of
On a new homotopy continuation trajectory for
New classes of generalized monotonicity
--TR
--CTR
Zhang , Weijun Zhou, Spectral gradient projection method for solving nonlinear monotone equations, Journal of Computational and Applied Mathematics, v.196 n.2, p.478-484, 15 November 2006 | iterative algorithm;fixed point and normal maps;strongly monotone maps;variational inequalities;cocoercive maps |
589026 | Second-Order Algorithms for Generalized Finite and Semi-Infinite Min-Max Problems. | We present two second-order algorithms, one for solving a class of finite generalized min-max problems and one for solving semi-infinite generalized min-max problems. Our algorithms make use of optimality functions based on second-order approximations to the cost function and of corresponding search direction functions. Under reasonable assumptions we prove that both of these algorithms converge Q-superlinearly, with rate at least 3/2.This paper is a continuation of [E. Polak, L. Qi, and D. Sun, Comput. Optim. Appl., 13 (1999), pp. 137--161]. | Introduction
As is also the case with ordinary min-max problems, generalized min-max problems can
be either finite or semi-infinite. Both are of the form
where
# is a smooth function and # n
# m is a nonsmooth, vector-valued
function. In the case of finite min-max problems, the components of #(-) are of the form 2
where the functions f
are continuously di#erentiable and
the sets q j := {1, 2, ., q j } are of finite cardinality 3 .
In semi-infinite generalized min-max problems the components of #(-) are of the form
where the functions
Finite generalized min-max problems are obviously a special case of semi-infinite generalized
min-max problems, since when the sets
we can define the functions f j,k (x) by
The best known generalized minimax problem occurs when an optimization problem
with a max function cost and equality and inequality constraints is set up for solution
using exact penalty functions, which results in an unconstrained optimization problem
with f 0 (x) in (1.1) of the form:
r
where # e and # i are two positive penalty parameters.
Another simple example occurs in a least squares problem involving max functions, in
which case
We denote components of a vector by superscripts and elements of a sequence or a set by subscripts.
3 Given any positive integer q, we use the notation q := {1, 2, ., q}.
where each # j (x) is as in (1.3).
As a last example, in trying to approximate a structural optimization problem the aim
of which was to minimize the sum of the probability of failure 4 plus the cost of the steel
in the structure, using linearizations of a state-limit function, we obtained a cost function
of the form
u#B#
# is a ball of radius #, centered at the origin in the space of the random variables u, and
g(x, u) is a smooth state-limit function which defined the boundary between outcomes
that result in structural failure from those that do not.
Functions of the form f 0 with #(-) as in (1.4), are the best known example
of quasidi#erentiable functions and are treated in depth in [4]. Hence generalized min-max
problems can be solved using algorithms developed for quasi-di#erentiable functions,
see, e.g., [2, 3, 6, 7, 8, 9, 11, 21]. Under the additional assumption that #F (y)/#y j > 0
for all y # m and generalized min-max problems can be solved using
transformations 5 into a smooth, constrained nonlinear programming problem (see e.g.,
[1, 5, 12]). Direct methods that depend on the assumption that #F (y)/#y j > 0 for all
can be found, for example, in [6, 9] and in [20].
We will consider semi-infinite generalized min-max problems under the following hypotheses
Assumption 1.1 We will assume that
(a) The functions F (-) and # j (-, y), j # m, y # Y j , are at least once continuously di#er-
(b) There exists a positive number c F > 0 such that #F (y)/#y j
(c) The sets Y j are either compact sets of infinite cardinality, or sets of finite cardinality,
of the form given in (1.5). 2
Parts (a) and (b) of Assumption 1.1 ensure that when both the F (-) and the # j (-)
are convex, the function f 0 (-) is also convex. In addition, as we will see, when all parts
of Assumption 1.1 hold, the function f 0 (-) has a subgradient. In [20], this fact was used
in defining an optimality function and an associated descent direction for the problem
4 The probability of failure was given by # g(x,u)#0
#(u)du, with #(-) the normal probability density
function.
5 These transformations result in a smooth problem with more variables than in the nonsmooth prob-
lem. There is a fair bit of anecdotal evidence that they can induce considerable ill-conditioning in the
smooth problem because they introduce arbitrary scaling. In general, solving nonsmooth problems using
transformation techniques appears to be less e#cient than using algorithms that exploit problem
structure.
P, and in extending the Pshenichnyi-Pironneau-Polak (PPP) Algorithm 4.1 in [17] (see
also [22, 13, 14]) to finite generalized min-max problems, and the Polak-He PPP Rate-
Preserving Algorithm 3.4.9 in [17] (see also [15]) to semi-infinite generalized min-max
problems.
In this paper we show that techniques used in [18] and [19] for constructing Q-
superlinearly converging algorithms for solving finite and semi-infinite min-max problems,
of the form (1.1) and (1.2), can be extended to construct Q-superlinearly converging algorithms
for the solution of both finite and semi-infinite generalized min-max problems.
In Section 2, we present a continuous optimality function and its associated search
direction function which, together with a backstepping rule, constitute the backbone of
our algorithms. In Section 3, we extend the Polak-Mayne-Higgins Newton's method [18],
for solving finite min-max problems, to generalized finite min-max problems. We prove
the Q-superlinear convergence of this extention in Section 4. In Section 5, we make use of
the theory of consistent approximations developed in [17] and the algorithm presented in
Section 3 to develop an algorithm for solving generalized semi-infinite min-max problems
and prove its convergence and Q-superlinear convergence. Section 6 is devoted to some
numerical results to demonstrate the behavior of the proposed algorithms. We sum up in
the concluding Section 7.
2. Optimality Conditions
We will now present optimality conditions for the semi-infinite generalized min-max problem
problem, defined in (1.1), (1.2), (1.4), both in "classical" form and in terms of an
optimality function which leads to a superlinearly converging second-order algorithm.
Lemma 2.1 [20] Suppose that F
# is continuously di#erentiable and that # :
# m is a locally Lipschitz continuous function that has directional derivatives at
every x # n . Let f
# be defined by
Then, given any x # n , and any direction vector h # n , the function f 0 (-) has a
directional derivative df 0 (x; h) which is given by
Suppose that Assumption 1.1 is satisfied. Then it follows from Lemma 2.1 that the
directional derivative of f 0 (-), at a point x # n in the direction h, is given by
#F
#F
where
When all the sets Y j are as in (1.5), (2.3) assumes the form
#F
where the functions f j,k (-) are defined by
and the sets - q j (x) by
Hence the following result is obvious.
Theorem 2.2 Suppose that -
x is a local minimizer for the problem (1.1), (1.2), (1.4).
Then for all h # n ,
#F
#F
Furthermore, (2.8) holds if and only if 0 #f 0 (-x), where the subgradient #f 0 (-x) is given
by
#F
Since (2.8) is a necessary condition of optimality, any point - x # n that satisfies (2.8)
will be called stationary.
When all the sets Y j are of the form (1.5), the expressions (2.8) and (2.9) assume the
following
#F
#F
(#(-x))#f j,k (-x) # . (2.11)
Definition 2.3 We will say that
# is an optimality function for problem (1.1),
(1.2),
(a) #(-) is upper semi-continuous,
(c) for any -
holds if and only if
Assumption 2.4 We will assume that
(a) the functions # in (1.1), (1.2), (1.4), are twice
Lipschitz continuously di#erentiable on bounded sets,
(b) the functions #
are locally Lipschitz continuous,
(c) there exist constants 0 < c # C < #, such that for all j # m, y
and
. (2.13)For the sake of convenience, for any x, h # n and w # m , we define
u(x, h, w) := #F (#(x)), -
and
v(x, h, w) := 1
The reason for the introduction of the artificial variable w is as follows. The function
is a perfectly good second-order approximation to F (#(x + h)), but unfortunately, it is
not always convex and hence leads to problems in developing an algorithm for solving
semi-infinite generalized min-max problems. By introducing the artificial variable w, we
can define the function
which, as we will later see, is a convex second-order approximation to F (#(x
hence much more useful in algorithm construction.
We define the function # n
# and the associated search direction function
# n by
{ min
and
{ min
h, w))} . (2.20)
Note that
We will shortly see that the function #(-) is an optimality function for the problem
(1.1), (1.2), (1.4). For any y, #y # m , let
Lemma 2.5 Suppose that Assumptions 1.1 and 2.4 are satisfied. For any y, #y # m , let
(y, #y) be the solution set of (2.22).
Then# (y, #y) is non-empty and compact and for
any w #
(y, #y), we have
Proof. Since #F (y) > 0 and # 2 F (y) is positive semi-definite, for any w # 0 and
#w# we have
(y, #y) is nonempty and compact.
Suppose that w #
# (y, #y). Then w # satisfies the following first-order optimality
conditions which follow directly from (and are equivalent to) the KKT conditions:
i.e.,
#w #F (y) +# 2 F (y)(#y
Clearly, (2.26) implies that for any w #
(y, #y), we have
#F (y) +# 2 F (y)#y +# 2 F (y)w # 0. (2.27)Lemma 2.6 Suppose that Assumptions 1.1 and 2.4 are satisfied. Then for any z # n
there exists an # > 0 such that for all h # n with #h# and for all x # n with
#x - z# we have
i.e.,
defined by (2.17).
Proof. Since h, -) is a convex quadratic function, any w # m
satisfying the following first-order conditions
#w, #F (#(x)) +# 2 F (#(x))( -
is a solution of (2.18). Then, because #F (y)/#y j # c F , for every
#(-) is uniformly continuous on any compact set and -
we see that for
any z # n there exists an # > 0 such that for all h # n with #h# and for all x # n
with #x - z#, This implies that for all those h and x, we have
Hence our proof is complete. 2
The above lemma shows that -
is identical to -
su#ciently small.
This fact will be used in proving our superlinear convergence results.
In general, -
not convex in h. We will now show that -
is convex in h.
Lemma 2.7 Suppose that Assumptions 1.1 and 2.4 are satisfied. Then for any fixed
is a convex function. Moreover, -
f 0 (-) is continuous.
Proof. First we will show that -
is a convex function. For any y # m and #y # m ,
we have
where
It is easy to verify that S(#y) is a concave function and that its subgradient is given by
# (y, #y)}, (2.34)
(y, #y) is the solution set of (2.33). It now follows from (2.32) that -
locally Lipschitz continuous in #y and that its subgradient gradient at #y is given by
# (y, #y)}. (2.35)
Since, by Lemma 2.5, for any w #
# (y, #y)
we conclude that for any s # -
every
is convex in
(because it is the composition of a convex function with positive elements in the
subgradient and a vector function whose components are convex).
Next, we will prove that -
continuous. First, since #F (y)/#y j # c F > 0 and
positive semi-definite for all j # {1, - , m} and y # m , it follows from (2.22)
(y, #y) is uniformly bounded in a neighborhood of given point (z,
It now follows from Corollary 5.4.2 in Polak [17] that -
F (-) is continuous. Hence
which implies that -
is continuous on # n
with y := #(x) and #y := -
The following theorem shows that #(-) is indeed an optimality function for the problem
(1.1), (1.2), (1.4) and that the set-valued function H(-) is a descent direction function for
Theorem 2.8 Suppose that Assumptions 1.1 and 2.4 are satisfied. Consider the functions
#(-) and H(-) defined by (2.19) and (2.20), respectively. Then
(i) For all x # n ,
(ii) For all x # n ,
where df 0 (x; h) is the directional derivative of f 0 at x in the direction h and # =2
(iii) For any x # n , 0 #f 0 (x) if and only if is the subgradient
of f 0 (-) at x, defined in (2.9). Moreover, for any x # n such that
have
(iv) The set valued map H(-) is (a) bounded on bounded sets, (b) compact valued, and (c)
outer-semicontinuous, i.e., for any x # n , H(x) is closed and, for every compact
set S such that H(x) # there exists a # > 0 such that H(z) #
z # B(x, #) := {y # n
|#y - x#}.
(v) The function #(-) is continuous.
Proof. (i) admissible in (2.19) that #(x) # 0 for all x # n .
directly from the definition of #(x) in (2.19) that for
any h # H(x),
#F
Thus we have shown that (2.40) holds.
(iii) For any x # n , let
min
u(x, h,
u(x, h, 0) . (2.42)
We will first prove that
It is easy to see that Hence we
only need to show that
Suppose that there exists an h # n such that
For any j # {1, - , m}, we have
and
Thus, there exists a constant C 0 such that
which further implies that there exists a constant C 1 such that
Since u(x, -, 0) is a convex function and u(x, 0, su#ciently small we have
which contradicts that
Next, with #f 0 (x) the subgradient of f 0 (-) at x, defined in (2.9), by emulating the
proof of Lemma 2.5.5 in [17], we can prove that for any x # n , 0 #f 0 (x) if and only if
therefore if and only if
Finally we will show that for any x # n such that
For the sake of contradiction, suppose that there exists an x # n such that
{0}. Then there exist 0
such that
u(x, h, w)
which, together with the fact that v(x, h, w) # 0 implies that u(x, h, w) # 0. Hence
we conclude that both u(x, h,
u(x, h, w) < 0, which contradicts (2.43). However, implies that
h, 0) is strongly convex in h and u(x, 0,
(iv) According to our definition, for each h # n there exists a w(h) # m
such that
which, together with the facts that #F (y) > 0, y # m and v(x, h, w(h)) # 0, implies
that
Since for each j # {1, - , m} and h # n ,
it follows from (2.51) that for all y in any bounded neighborhood of x,
Consequently, for any x # n , H(x) is nonempty and bounded and H(-) is bounded on
bounded sets. Since -
continuous (Lemma 2.7), it follows that H(x) is closed.
Next we will prove that for every x # n and every compact set S such that
there exists a # > 0 such that H(z) # not, then there
exists an x # n and a compact set S such that H(x) # and a sequence {x i }
converging to x such that H(x i ) # S #. Hence there exists a sequence {h i } such that
is a compact set, without loss of generality, we can assume that
By definition of H(x i ),
f 0 (-) is continuous, it follows from (2.55), that
which implies that - h # H(x). This contradicts that H(x) # Thus, we have shown
that H(-) is outer-semicontinuous.
(v) Finally, it follows from Corollary 5.4.2 in Polak [17] that # is continuous. 2
By introducing an additional variable, we can rewrite the expression for #(x), defined
in (2.19), as follows
Problem (2.57) is a quadratic problem with quadratic constraints. Under suitable assump-
tions, (2.57) is actually a convex quadratic problem with convex quadratic constraints,
and hence provides a convenient means for computing the optimality function value #(x)
and an associated search direction h # H(x).
Theorem 2.9 Suppose that Assumptions 1.1 and 2.4 are satisfied and the sets Y j are as
in (1.5). For any x # n , let #(x) be the solution set of (2.57), i.e., any (p, h) #(x)
solves (2.57). Then
(i) Problem (2.57) is a convex quadratic problem with convex quadratic constraints.
(ii) For x # n , #(x) is nonempty and compact and #(-) is outer-semicontinuous and
bounded on bounded sets.
(iii) If z # n is such that there exist a neighborhood
N(z) of z and an # > 0 such that for any (p, h) #(x), x # N(z), we have
Proof. (i) Under the conditions of Assumptions 1.1 and 2.4, # 2 F (#(x)) is positive
semidefinite and for each j # {1, 2, - , m}, -
strongly convex. Hence (2.57) is a
convex quadratic problem with convex quadratic constraints.
(ii) Since for all y in a bounded neighborhood N(x) of x and j # {1, 2, - , m},
it follows that for all y # N(x) and (p,
#(y,
we have
# as #(p, h)#. (2.61)
Hence, for all x # n , #(x) is nonempty and compact, and #(-) is bounded on bounded
sets.
The outer-semicontinuity of #(-) follows from the fact that #(-) is continuous and the
constraint set in (2.57) is outer-semicontinuous.
(iii) Since z # n is such that #(z). For any x # n , the KKT
conditions for (2.57) are
is the subgradient of -
with respect to h.
Suppose that (p, h) #(z). By (iii) of Theorem 2.8, we have Hence it follows
from (2.62) and the fact that -
which implies that positive
semidefinite. Thus, we have proved that since #(-) is outer-
semicontinuous, it follows that if x # z and (p, h) #(x), then
It now follows from (2.62), (2.64), and the fact that for any y # m , #F (y)/#y j
m}, that there exists a neighborhood N(z) of z such that for all x # N(z),
the multiplier # in the KKT (2.62) must have all components positive and hence for all
x # N(z), the KKT conditions for (2.57) become
Thus, for any x # N(z) and j # {1, 2, - , m}, there exist nonnegative numbers - j,k
satisfying # k#q j
such that for any (p, h) #(x),
where
and for any k # q j such that
we have
We conclude from (2.65), (2.66) and (2.67) that for all x # N(z) and (p, h) #(x),
#, p#
where the last inequality follows from the fact that f j,k (x) # j (x) for all k # q j and
m. By shrinking N(z) if necessary, we conclude from (2.67), (2.70) and Assumptions
1.1 and 2.4 that there exists a positive number # > 0 such that for all x # N(z) and
3. An Algorithm for Solving Generalized
Finite Min-Max Problems
An algorithm for solving generalized finite min-max problems is obviously of interest in
its own right. However, we will also need it as a subroutine for our algorithms for solving
generalized semi-infinite min-max problems. Hence, for the time being, we will assume
that the sets Y j are of the form (1.5) and that the functions f j,k (-) are as in (2.6). As a
result, our generalized finite min-max problem assumes the form (1.1), (1.2), (1.4), with
min
where, in view of Assumption 1.1, the functions F (-) and f j,k (-), j # m, k # q j are all
continuously di#erentiable, where f j,k (-) are defined by (2.6).
We are now ready to state an algorithm for solving generalized finite min-max prob-
lems. This algorithm is a generalization of the Polak-Mayne-Higgins Newton's algorithm
for solving finite min-max problems [18].
Algorithm 3.1 (Solves Problem (3.1))
Parameters. # (0, 1), # (0, 1), and # > 0.
Step
Step 1. Compute the optimality function value # i := #(x i ) and a search direction h i #
according to the formulae (2.19) and (2.20).
Step 2. If # Else, compute the step-size
where N := {0, 1, 2, .
Step 3. Set
replace i by to Step 1. 2
Lemma 3.2 [20] Suppose that Assumption 1.1 holds. Then for any y, y # m such that
y,
Lemma 3.3 [20] Suppose that Assumptions 1.1 and 2.4 are satisfied. Then there exists
a constant # > 0 such that for all x, x # n and # [0, 1],
Theorem 3.4 Suppose that Assumptions 1.1 and 2.4 are satisfied and that all the Y j ,
are of the form (1.5), so that problem (1.1), (1.2), (1.4) reduces to problem (3.1).
If {x i } # i=0 is an infinite sequence generated by Algorithm 3.1 and x # is the unique solution
of (3.1), then {x i } # i=0 converges to x # .
Proof. Suppose that {x i } # i=0 is an infinite sequence generated by Algorithm 3.1. Since
f(-) is strongly convex by Lemma 3.3, the sequence {x i } # i=0 is bounded. Suppose that -
x is
an accumulation point of this sequence. Since the cost function f 0 (-) is continuous, f 0 (-x)
is an accumulation point of the cost sequence. Hence, since, by construction, the cost
sequence {f 0
Now, for the sake of contradiction, suppose that #(-x) < 0. Since for any x # n ,
H(x) is compact, and H(-) is bounded on bounded sets and is outer-semicontinuous ((iv)
of Theorem 2.8), it follows from Theorem 5.3.7 (b) in Polak [17] that there exists a
subsequence {j i } # i=0 of the integers such that x
x and h j i # - h # H(-x), as i #. It
follows from (ii) and (iii) of Theorem 2.8 that -
be such that 0 < # < 1. Then it follows from the definition of the directional
derivative of f 0 (-) that there exists a k # N such that
Hence,
for all # > 0 such that
. (3.10)
# := 1# . Then, since f 0 (-) and #(-) are continuous and h j i # - h, as i #, there
exists a # > 0 such that for all x
which shows that for all x
. Next, since #(-) is continuous, there
exists -
# (0, #) such that for all x
It therefore follows from
the step-size rule (3.2) that for all x
#),
#(-x) . (3.12)
implies that f 0
contradicting the fact that f 0 Hence we conclude that
and therefore that -
strongly convex, the whole
sequence {x i } converges to x # . 2
4. Rate of Convergence of Algorithm 3.1
Proposition 4.1 Suppose that Assumptions 1.1 and 2.4 are satisfied and that - x is the
unique solution of f 0 (-). Then for all x # n ,
Proof. By Lemma 3.3, f 0 (-) is a strongly convex function. Hence, for any x # n we
have
#F
#F
c
#F
c
where - q j (x) is defined by (2.7). It now follows from (2.5) and (4.2) that
Proposition 4.2 Suppose that Assumptions 1.1 and 2.4 are satisfied. Then for any
compact convex set S there exists a # > 0 such that for any x, z # S,
defined in (2.17).
Proof. First, it follows from Polak [17, Lemma 2.5.4] or [18], that there exists a constant
such that for any x, z # n ,
be a compact set, and let L 2 < # be a Lipschitz constant for # 2 F (-) on S,
such that for any z # S,
#F (#(z))# L 2 . (4.6)
Then for all x, z # S,
where L 3 := mL 2 L 1 /6. Since, by assumption, S is compact, it follows that there exists a
positive number L 4 such that for all x, z # S,
and
Hence for all x, z # S,
with
4 . (4.12)
Similarly, we can prove that for all x, z # S,
Thus we have shown that (4.4) holds. 2
Theorem 4.3 Suppose that Assumptions 1.1 and 2.4 are satisfied, that all the Y j ,
are of the form(1.5), so that problem (1.1), (1.2), (1.4) reduces to problem (3.1). If
is a sequence constructed by Algorithm 3.1, in solving problem (3.1), then, {x i } # i=0
converges superlinearly with Q-order at least 3/2.
Proof. First we will prove that after a finite number of iterations, the step-size # i
stabilizes to 1, so that eventually x holds for the sequence {x i } # i=0 . We will
then complete our proof by making use of results in [17, Corollary 2.5.8].
It follows from Theorem 3.4 that the sequence {x i } # i=0 converges to the unique minimizer
x of f 0 (-). Hence we conclude from Theorem 2.8 that
In view of this, we conclude from Lemma 2.6 that there exist a positive number # > 0
and a nonnegative integer i 0 such that for all
Suppose that i 0 is su#ciently large to ensure that for all
x# . (4.16)
Then, making use of (4.1), we find that, for
because -
by (4.15). It now follows from Proposition 4.2 that
there exists a # > 0 such that for all
Now, by Theorem 2.9, there exist a positive integer and an # 1 > 0 such that
for all
Next, Proposition 4.2 and (4.15), imply that for all
Hence, from (4.20) and (4.19), we have
It now follows from (4.21) and the fact that h i # 0 as i # that for all i su#ciently
large,
We therefore conclude from [17, Corollary 2.5.8] or [18], (4.18) and (4.19) that {x i } # i=0
converges to -
x superlinearly with Q-order at least 3/2. 2
5. An Algorithm for Solving Generalized
Semi-Infinite Min-Max Problems
We are now ready to tackle the generalized semi-infinite min-max problems defined in
(1.1), (1.2), (1.4). Such problems can be solved only by discretization techniques. We
will use discretizations that result in consistent approximations (as defined in Section 3.3
of [17]) and use them in conjunction with a master algorithm that calls Algorithm 3.1 as
a subroutine. We will see that under a reasonable assumption, the resulting algorithm
retains the rate of convergence of Algorithm 3.1.
5.1. Consistent Approximations
Let N 0 be a strictly positive integer, and, for N # N 0 := {N 0 ,
be finite cardinality subsets of Y j , j # m, such that Y j,N # Y j,N+1 for all N and the closure
of the set lim Y j,N is equal to Y j , j # m. Then we define the family of approximating
problems PN , N # N 0 , as follows:
PN min
where
N (x)), and for j # m,
It should be clear that the approximating problems PN are of the form (3.1) and that
one can define optimality functions # N (-) for them of the form (1.5). We will refer to the
original problem (1.1), (1.2), (1.4) as P.
Definition 5.1 [17] We will say that the pairs (PN , # N ), in the sequence {(PN , # N )} N#N0
are consistent approximations to the pair (P, #), if the problems PN epi-converge to P,
(i.e., the epigraphs of the f 0
N (-) converge to the epigraph of f 0 (-) in the sense defined in
Definition 5.3.6 in [17]), and for any infinite sequence {xN } N#K , K # N 0 , such that
Assumption 5.2 We will assume as follows:
(a) For every N # N 0 , the problem (5.1) has a solution.
(b) There exists a strictly positive valued, strictly monotone decreasing function # :
N #, such that #(N) # 0, as N #, and a K < #, such that for every
there exists a y # Y j,N such that
#y - y # K#(N). (5.4)For example, if for all j # m, Y j is the unit cube in # m j , i.e., Y
then we can define Y
I
with a(N) := 2 N-N 0 . In this case it is easy to see that
constructions can be obtained for other polyhedral sets.
For any x, h # n and w # m , we define
uN (x, h, w) := #F (#N (x)), -
and
where
and
We infer from (2.19) that the optimality functions # N (-), for the problems PN have
the following form:
{ min
h, w))}. (5.9)
Since the cardinality of the sets Y j,N is finite, it is obvious that the # N (x) can be evaluated.
As was also done in the Polak-Mayne-Higgins Rate-Preserving method [19] (see also
[20]), we use an alternative optimality function for the problems PN for precision adjustment
in our algorithm. This optimality function is defined by
#F
where
#F
with # > 0, a constant.
Similarly (as in [20]), we define an alternative optimality function for the problem P
by
#F
where
#F
with # > 0 the same constant as in (5.11).
Proposition 5.3 [20] Suppose that Assumptions 1.1 and 5.2 are satisfied, and that for
N (-) is defined by (5.2) and -
# N (-) by (5.10). Let S # n be a bounded subset
and let L < # be a Lipschitz constant valid for the functions # j (-) and # x # j (-) on
q. Then there exists a constant C S < # such that for all x # S, N # N 0 ,
and
| -
5.2. The Superlinear Rate Preserving Algorithm
Algorithm 5.4 (Solves Problem (1.1), (1.2), (1.4))
Parameters. # (0, 1), # > 0, D > 0, # > 3.
Step
Step 1. Compute the optimality function value -
according to (5.10) and (5.11), i.e.,
#F
where
#F
(#N
Step 2. If
go to Step 3. Else, replace N by N + 1, and go to Step 1.
Step 3. Compute the second optimality function value # N according to (5.9), i.e.,
{ min
and the corresponding search direction h i according to
{ min
Step 4. Compute the step-size
and go to Step 5.
Step 5. Set
to Step 1. 2
Remark.
(a) It follows from Proposition 5.3 that -
the loop consisting of Step 1 and Step 2 of Algorithm 5.4 yields a finite
discretization parameter N i . For simplicity, we will assume that Algorithm 5.4 does
not produce an iterate x i such that -
(b) Note that the work needed to compute x i by Algorithm 5.4 increases with the iteration
number i. 2
Lemma 5.5 Suppose that Assumptions 1.1, 2.4, and 5.2 are satisfied, and that Algorithm
5.4 has constructed a sequence {x i } # i=0 together with the corresponding sequence of
discretization parameters {N i } # i=0 . If the sequence {x i } # i=0 has at least one accumulation
point, then N i #, as i #.
Proof. For the sake of contradiction, suppose that the sequence {x i } # i=0 has an accumulation
point -
x and that the sequence {N i } # i=0 is bounded. Then, because {N i } # i=0 is a
monotonically increasing sequence of integers, there exists an i 0 # N , such that for all
. Hence for the construction of the sequence {x i } # i=0 is
carried out by Algorithm 3.1 applied to problem (5.1) with Furthermore, it
follows from (5.18) that there exists an # > 0, such that -
However, it follows from Theorem 3.4 that # N # and from the continuity of -
that - # N #(x i
where the infinite subsequence {x i } i#K ,
converges to -
which contradicts the previous finding, and hence completes our
proof. 2
Theorem 5.6 Suppose that Assumptions 1.1, 2.4, and 5.2 are satisfied, and that Algorithm
5.4 has constructed a bounded sequence {x i } # i=0 . Then every accumulation point -
x
of {x i } # i=0 satisfies -
Proof. By applying Theorem 3.3.23 of [17] or Theorems in Section 5 of [16] and Lemma
5.5 to Algorithm 5.4, we obtain the desired result. 2
Theorem 5.7 Suppose that Assumptions 1.1, 2.4, and 5.2 are satisfied, and that Algorithm
5.4 has constructed a bounded sequence {x i } # i=0 . Then {x i } converges to the unique
x of f 0 (-) with Q-order 3/2.
Proof. First, by Theorem 5.6 and the fact that f 0 (-) has a unique minimizer -
x, the whole
sequence {x i } converges to -
x. Hence, one can deduce from Theorem 4.3 and the proof of
[17, Theorem 3.4.20], that {x i } converges to - x with Q-order 3/2. Since the derivation is
straightforward, we omit the details here. 2
6. Some Numerical Results
We now present some numerical results that illustrate the behavior of the algorithm
proposed in Section 5 for generalized semi-infinite programming problems. The algorithm
was implemented in Matlab. Throughout the computational experiments, the parameters
used in the algorithm were 3.1. For both
examples, we used the starting point (1, 1). The iteration of the algorithm is stopped at x i
if for some N the meshsize #(N) < 0.005 and |# N developed
in [24], which was based on a smoothing Newton method [23] for variational inequalities,
was used to solve our search direction finding subproblem (2.57).
Example 1. In this case, f 0
with
and
Example 2. In this case, the functions f are also defined as in
Example 1, but F (-) is defined by
The numerical results are summarized in Table 1 and Table 2. In these two tables
the first column gives the residue ||x i -
x|| (we used the last iterate as a substitute for
x) and the discretization level (the meshsize at the present level is decreased to half of
the previous one) refined by the master algorithm at the i-th step. It is clear from the
numerical results that the rate of convergence is superlinear.
Iteration
Discretization level
Table
1: Numerical results for Example 1
Iteration
Discretization level
Table
2: Numerical results for Example 2
7. Conclusion
We have presented two superlinearly converging algorithms, one for solving finite generalized
min-max problems of the form (1.1), (1.2), (1.3) and one for solving generalized
semi-infinite min-max problems of the form (1.1), (1.2), (1.4). These algorithms were
obtained by making use of the concepts underlying the construction of the Polak-Mayne-
Higgins Newton's method [18] and the Polak-Mayne-Higgins Rate-Preserving method [19],
respectively. The construction of the algorithms depends on the cost unction having a
subgradient and their rate of convergence depends on convexity and second order smooth-
ness, and hence Assumption 2.4 is essential.
Our numerical results are consistent with our theoretical prediction that the algorithms
converge Q-superlineary.
Acknowledgement
. The authors wish to thank Prof. R. T. Rockafellar for suggesting
the function -
as a way to get around the possible non-convexity of the function
in h, as well as for the formula (2.57) which shows that our optimality function
is defined by a quadratically constrained quadratic programming problem.
--R
"Nondi#erentiable optimization via approximation,"
"Quasidi#erentiable functions: necessary conditions and descent directions,"
"An algorithm for minimizing a certain class of quasidi#erentiable functions,"
"A smooth transformation of the generalized minimax problem,"
"A quadratic approximation method for minimizing a class of qua- sidi#erentiable functions,"
"A linearization method for minimizing certain quasidi#erentiable functions,"
"Randomized search directions in descent methods for minimizing certain quasidi#erentiable functions,"
"Descent methods for quasidi#erentiable minimization,"
"Proximal control in bundle methods for convex nondi#erentiable minimization,"
"The method of common descent for a certain class of quasidi#erentiable functions,"
"Algorithms for a class of nondi#erentiable problems,"
"On the rate of convergence of certain methods of centers,"
"Basics of minimax algorithms,"
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"On the use of consistent approximations in the solution of semi-infinite optimization and optimal control problems,"
Optimization: Algorithms and Consistent Approximations
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--TR
--CTR
Huang , Defeng Sun , Gongyun Zhao, A Smoothing Newton-Type Algorithm of Stronger Convergence for the Quadratically Constrained Convex Quadratic Programming, Computational Optimization and Applications, v.35 n.2, p.199-237, October 2006 | optimality functions;superlinear convergence;consistent approximations;generalized min-max problems;second-order methods |
589032 | A Primal-Dual Method for Large-Scale Image Reconstruction in Emission Tomography. | In emission tomography, images can be reconstructed from a set of measured projections using a maximum likelihood (ML) criterion. In this paper, we present a primal-dual algorithm for large-scale three-dimensional image reconstruction. The primal-dual method is specialized to the ML reconstruction problem. The reconstruction problem is extremely large; in several of our data sets the Hessian of the objective function is the product of a 1.4 million by 63 million matrix and its scaled transpose. As such, we consider only approaches that are suitable for large-scale parallel computation. We apply a stabilization technique to the system of equations for computing the primal direction and demonstrate the need for stabilization when approximately solving the system using an early-terminated conjugate gradient iteration.We demonstrate that the primal-dual method for this problem converges faster than the logarithmic barrier method and considerably faster than the expectation maximization algorithm. The use of extrapolation in conjunction with the primal-dual method further reduces the overall computation required to achieve convergence. | Introduction
. In this paper we consider the image reconstruction problem
in emission tomography. This problem is encountered in the eld of nuclear medicine,
which is concerned with the study of organ function through radioactively labeled
\tracer" compounds. The quantity of interest in this problem is the spatial concentration
of radioactive emissions within the object under study. The quality of the
reconstructed image can depend upon a number of factors including the number of
emission events (i.e., counts) collected by the scanner and the method used to reconstruct
the image. In studies that are characterized by poor counting statistics (that
is, few counts), statistical reconstruction methods that model the Poisson nature
of the emission process have been shown to improve image quality over traditional,
non-statistical reconstruction methods [35, 57]. The low-count problem has generated
considerable interest in the medical imaging community because low radiotracer doses
and short scanning durations are highly desirable.
The estimation of emission density in an organ is an inherently three-dimensional
(3-D) process. Volume, or 3-D acquisition improves the counting statistics compared
with 2-D acquisition (in which axially oblique coincidences are either physically or
electronically blocked from detection) but increases the problem size considerably.
Since the 3-D problem may involve image and measurement vectors with millions
of elements, the amount of computation required to perform 3-D statistical reconstructions
can be quite substantial. In our computational studies for example, the
larger reconstructions consist of 1.4 million image variables which are reconstructed
from a measurement vector with 63 million elements. As such, it is important to use
reconstruction methods that converge rapidly. The statistical image reconstruction
Ariela Sofer is partly supported by National Science Foundation grants DMI-9414355 and DMI
9800544.
y Center for Information Technology, National Institutes of Health, Bethesda, Maryland, 20892-
5624 (johnson@mail.nih.gov).
z Department of Systems Engineering and Operations Research, George Mason University, Fairfax,
Virginia 22030-4444 (asofer@gmu.edu).
C.A. JOHNSON AND A. SOFER
problem can be posed as a constrained nonlinear optimization problem. In this paper
we present a primal-dual method for performing statistical 3-D reconstructions in
emission tomography that has been specialized to the intricacies of the application.
We demonstrate the rapid convergence of our primal-dual method in computational
studies on low-count, 3-D positron emission tomography (PET) data.
This paper is organized as follows. In Section 2 we present the statistical model
and develop the objective function. Section 3 reviews the EM method for ML recon-
struction. In Section 4 we develop a primal-dual method for ML reconstruction and
discuss initialization, stabilization, and extrapolation enhancements. Computational
tests comparing the primal-dual results to a logarithmic barrier approach and the EM
method on small animal data are presented in Section 5. Some concluding remarks
are made in Section 6.
2. Statistical model and objective function. We begin our discussion by
forming a nite parameter space for the image estimates, as is customary [20]. Consider
the situation depicted in Figure 2.1 where a grid of boxes or voxels has been
imposed over the emitting object (for simplicity, the Figure is depicted in 2-D; the
concept is readily extended to 3-D). Given a set of measurements along lines of coin-
cidence, we seek to estimate x expected number of counts
emitted from voxel i. Let X i be the number of radioactive events emitted from voxel
are assumed to be independent Poisson-distributed random variables with mean
system matrix C 2 < nN is used to model a number of physical eects
including spatially dependent resolution and attenuation. The elements C i;j of the
system matrix represent the probability that an event emitted from voxel i will be
detected by detector pair (coincidence line). The number of events emitted from voxel
and detected at coincidence line j is therefore are also independent
Poisson variables. The measurements y j are thus realizations of sums of
independent Poisson variables y
The above is a considerably simplied model of the actual measurement process; for
further discussion on its validity to the present situation, see [24].
Given our simplied Poisson model, the likelihood may be written as
Y
Y
e
The ML objective function is formed by taking the log likelihood
log
Ignoring the constant term, we dene our objective function fML (x) as
is a vector of 1's, so that q is the sum of the
columns of C (which need not necessarily be 1). Dening
Fig. 2.1. Relationship between estimate x i and measurement y j . Shown here is the case of
PET, where emission-count measurements are taken along coincidence lines from pairs of detectors.
A nite parameter space is formed by imposing a grid of voxels over the emitting region. The
estimate of the expected emission intensity within voxel i is x i .
to be a forward transformation, we can write the gradient and Hessian of the objective
function, respectively, as
The Hessian is negative semidenite (since
so the objective function (2.1) is concave. Thus, any local maximum
will also be a global maximum.
Equation (2.2) sheds some insight into the computational costs associated with
maximizing the objective function. Given a current solution estimate x k ; computing
the gradient requires rst computing a forward transformation ^
then computing a backward transformation
k y from the forward transformation.
The costs of performing the forward transformation and backward transformation are
similar and together dominate the computation associated with iterative reconstruction
methods, especially in large scale. We shall revisit this computational structure,
which is common to all iterative reconstruction methods.
Since the underlying activity distribution is non-negative, the ML reconstruction
problem is a constrained optimization problem with lower-bound constraints:
maximize fML (x)
subject to x 0:
The ML objective function has a nite maximum and compact level sets on x 0
[36].
2.1. Maximum a posteriori reconstruction. Without regularity conditions
on x, estimating the spatial emission distribution is a statistically ill-posed problem
[7, 33]. The fully converged ML reconstruction, being dominated by noise and edge
artifact, is not generally of biomedical interest [55]. Regularization can be included
4 C.A. JOHNSON AND A. SOFER
in the objective function by introducing a Bayesian formulation [20, 37]. Given prior
probabilities P fxg and P fyg for the image and measurements, respectively, we dene
the posterior probability
The estimate of x is then obtained by maximizing the posterior probability P fxjyg.
A common choice for the image prior is the Gibbs distribution P
although other priors (e.g., Gaussian, Gamma) have been investigated [37, 39]. The
popularity of Gibbs priors stems in part from their ability to capture the local correlation
property of images [19]. The energy function R is dened as a sum of "potential"
functions designed to discourage non-smoothness in a neighborhood
denotes the neighborhood of voxel i. In order to maintain concavity and
twice continuous dierentiability in the objective function, the potential function V i;l
is chosen to be convex with continuous rst and second derivatives. In our studies we
have used the potential function V i;l
z
log
z
and - is a shaping constant that we typically set to 1 [38].
For maximum a posteriori (MAP) reconstructions, the objective function is the
log-posterior likelihood log P fxjyg. Ignoring a constant, our objective function become
The MAP reconstruction problem can also be posed as a constrained optimization
problem
maximize fMAP (x)
subject to x 0:
We note for future reference the following:
Although the function R (with the potential function (2.5)) is concave it is not strictly
concave. only for vectors v that are a scalar multiple of
the unit vector e N , and since e T
negative denite and that fMAP is strictly concave [38]. In addition, fMAP has a
nite maximum and bounded level sets on x 0 [37].
2.2. The optimization problem. For convenience of notation, let us pose the
reconstruction problem as a constrained minimization problem:
subject to x 0;
(2.
0. The case
corresponds to the unregularized
ML; in general we shall be more interested in the fully converged MAP where
The Karush-Kuhn-Tucker (KKT) rst-order necessary conditions for optimality
of (2.8) at a point x are existence of Lagrange multipliers so that
x is the Lagrangian function. Due to the strict convexity of
f , the second-order su-ciency conditions are satised, and x is the unique minimizer
of f .
3. The EM algorithm. The expectation maximization (EM) method, as presented
by Dempster, Laird, and Rubin [8] for ML estimation, is an iterative algorithm
for computing ML estimates when the measurements are viewed as incomplete data.
Shepp and Vardi [53] and Lange and Carson [36] applied the EM method to emission
and transmission tomography problems, respectively. The EM algorithm has been
proven to converge to an optimal solution of (2.4) [36, 56].
The EM algorithm for emission tomography can be derived [56, 27] from the
optimality conditions for the reconstruction problem. For the unregularized problem
can be written as
diag (x), and Premultiplication by X , and utilizing the complementary
slackness condition yields
or since
Applying a xed-point algorithm x
to the above equation yields the
EM update equation
where x k is the current image estimate,
diag
. Given a positive initial solution x the algorithm maintains non-negativity
at every iteration and converges to a xed point x which is
an optimal solution of (2.4). The asymptotic rate of convergence is governed by the
spectral radius of rM which is typically very close to unity. In one example
using reasonable assumptions about the scanner geometry, the lower bound of the
spectral radius was calculated to be .99938 [17]. Indeed, EM has been observed to
converge very slowly, especially close to the optimal solution. The slow convergence of
the EM algorithm has limited its clinical applicability. The cost of one EM iteration
is equivalent to the cost of one gradient calculation.
In MAP-EM, the presence of the regularizing term in (2.6) precludes a closed-form
update equation such as (3.1) for ML-EM. We mention two algorithms that are commonly
used for MAP-EM reconstructions: the \one step late" (OSL) algorithm and
6 C.A. JOHNSON AND A. SOFER
DePierro's algorithm. Green's OSL algorithm approximates R (x) with the constant
, thereby permitting a closed-form approximated update [16, 17]
. OSL converges to the MAP solution provided that
, where
is an upper threshold for the prior strength. DePierro's algorithm is
a \true" MAP-EM implementation that substitutes the convex function R (x) with a
separable, convex, and twice continuously dierentiable function R x; x k
R (x), so
that separable maximizations can be performed on the variables [9, 10]. Regularization
improves the convergence rate of EM, with larger prior strengths resulting in lower
spectral radii. However, for reasonable prior strengths (mild to moderate smoothing),
the convergence rates of OSL and DePierro's algorithm are still quite close to unity.
The EM update formula on the right-hand side of (3.1) follows Kaufman [27],
who was the rst to pose the EM algorithm as an optimization algorithm (namely, a
scaled steepest-ascent method). This representation allows for the inclusion of a line
search [27, 28] to accelerate the method's performance. Likewise, the MAP update
(3.2) can be enhanced by a line search.
Several other approaches for solving the maximum likelihood estimation problem
have been proposed. These include preconditioned conjugate gradient techniques
[27, 28, 34, 42] or truncated-Newton methods [27, 28]. The nonnegativity constraints
are maintained either be limiting the step length or by using a bending line search.
The paper [44] explores active set methods, while [43] enforces nonnegativity via
a quadratic penalty in the objective. In other work [29, 30, 31] a penalized least-squares
objective is used instead of the maximum likelihood. These problems are
solved by a preconditioned conjugate gradient and use specialized techniques to drive
the complementary slackness to zero.
There is considerable debate within the PET community regarding the appropriate
model for reconstruction. It has long been observed that the unregularized
maximum likelihood estimator gives grainy images. However if the EM algorithm is
stopped early, the resulting solution often produces images of acceptable quality. For
this reason some researchers argue that early termination is a form of smoothing, and
that no regularization is needed. Proponents of MAP argue that the approach allows
the user to control the amount of regularization through the parameter, and that the
regularized objective function is better conditioned. In either case, it has been observed
that EM-type algorithms may lead to nonuniform convergence. In particular,
the algorithms may converge slowly in \cold spots" (regions of low activity within
regions of activity) and in areas of isolated activity within cold spots. The use of an
interior-point algorithm oers the hope of more uniform convergence.
4. A primal-dual approach. The drawbacks of the EM algorithm motivate
our investigation into interior-point approaches for the ML and MAP reconstruction
problems. As is clear from (2.1), the objective function can be undened outside the
feasible region x 0. Thus the ML and MAP reconstruction problems would appear
to be \natural" candidates for interior-point algorithms. The reconstruction problem
is especially suited to an interior-point approach, because its output is a gray-scale
image. Whether a particular value is exactly \zero" or just very close to zero is
immaterial. Slight inaccuracies below the gray scale threshold are inconsequential;
obtaining an image rapidly is a neccessity.
Primal-dual methods have enjoyed considerable success in linear programming
[18, 32, 40], and have recently been proposed for nonlinear programming [5, 13, 41].
Although they are closely related to the logarithmic barrier method, primal-dual
methods may pose some advantages. In the logarithmic barrier method, the Lagrange
multiplier estimates may be inaccurate when the primal variables are not close to the
barrier trajectory [11]. Primal-dual methods oer the potential of improved \center-
ing" over barrier methods. Given the size of the current problem, the developments
presented here must be suitable for large-scale parallel computation.
In a manner similar to classical barrier methods, primal-dual methods attempt
to follow the \barrier trajectory," a smooth trajectory characterized by a barrier parameter
[12]. The points along the trajectory satisfy a perturbed
version of the KKT conditions:
Dening ng and our method
maintains (4.3) while attempting to solve (4.1), (4.2), that is
Xen en
0:
Given the point x
and the barrier parameter , the search direction
prescribed by Newton's method satises the \unsymmetric" primal-dual
equations [41]:
I
Elimination of the (1,2) block of the matrix in (4.5) yields the reduced system
where the \condensed" primal-dual matrix is given by
We have implemented an algorithm in which the primal and dual variables are
permitted to take separate steplengths:
The primal steplength x is chosen to ensure su-cient decrease in the merit function
log
Observe that F (x; ) is simply the logarithmic barrier function and that
8 C.A. JOHNSON AND A. SOFER
is identical to the right-hand side of (4.6) for . The unconstrained
minimizer x () of F (x; ) satises the perturbed KKT conditions (4.1)-(4.3) with
corresponding multiplier i Furthermore, the solution
of the condensed primal-dual Newton equation (4.6) is guaranteed to be a descent
direction of the merit function for > 0, since
and M is positive denite. We shall discuss in further detail the computation of the
primal search direction and step length.
The formula for the dual step length follows a suggestion by Conn, Gould, and
Toint (CGT) [5]. If lies component-wise in the interval
(where is a constant parameter that we have set to 100) then
otherwise nd 0 < < 1 such that
subject to k+1 being in the interval (4.9). These conditions on the dual step might
appear at rst glance to be overly restrictive but are actually designed to give maxi-
mum
exibility in the choice of k+1 . CGT use these bounds on and nonsingularity
of M to prove that, for any xed parameter value
, the minimization of F (x;
must be successful, that is, eventually a solution is found that satises the perturbed
KKT conditions (4.1)-(4.3).
In general it is neither necessary nor desirable to reach full subproblem conver-
gence. Rather, we have implemented a \short-step" algorithm in which only one
primal-dual step is usually needed before adjusting . Setting the barrier parameter
is an important consideration in primal-dual algorithms, and has a strong in
uence
on the convergence rate. A reduction in k is performed whenever the \-criticality"
conditions [5, 54] are satised:
k+1
are constant parameters. If the above conditions are satised, the
barrier parameter is reduced according to
where is a constant parameter such that
A consequence of (4.14) is that k cannot increase. Furthermore, since the minimization
of F (x; ) must be successful, a -critical solution (a weaker requirement) must
eventually be found. Thus it is impossible for k to be non-decreasing. Using this
argument, CGT prove that the algorithm must converge to a KKT solution [5].
In practice we nd that both the primal and dual direction vectors are well scaled,
and that x and are both typically close to 1. By far the most costly operations
are computing the primal direction p x and updating the gradient rF (x), as we shall
explore. In contrast, the costs of the line search for the primal steplength, the computation
of the dual search direction (4.7), and the dual line search (4.9) are relatively
insignicant. From empirical evidence in our computational studies, we have
found that a \short-step" algorithm with gradual reduction in achieves the fastest
convergence to the KKT conditions. Specically, we dene #
These parameter values enable the -critical conditions to be met after
only one primal-dual step for most subproblems.
4.1. Computing the primal direction. For large problems, factoring the condensed
primal-dual matrix M or even forming the Hessian r 2 f (x) would be prohibitive
due to the size of the matrix (376,000376,000 for even the smaller reconstructions
being considered in this paper) and the enormous amount of computation
that would be required. Thus we must consider methods for approximating the Newton
direction in (4.6). The approach we have successfully applied to this problem is
motivated by the truncated-Newton [6] method of unconstrained optimization. The
search direction is an approximate or truncated solution to the Newton equations
[47, 49]
An early-terminated conjugate gradient (CG) iteration [21] is used to obtain an approximate
solution to (4.15).
An equivalent statement of (4.15) is we seek to nd the direction p x that approximately
minimizes the quadratic Q (p x
x reasonable and
eective truncation point for (4.15), based on the monotonicity of Q (p x ), is proposed
in [48]; the CG is terminated at subiteration l if
x
x
The CG termination rule (4.16) has been an important component of the reconstruction
software in that it consistently yields a well-scaled primal direction vector as long
as s , where s is a threshold value below which stabilization is required (we
shall discuss the < s case in Section 4.2).
The CG method does not require storage of the Hessian or condensed primal-dual
matrix, but rather only application of matrix-vector products. From (2.3) we
can write the rst term of the matrix-vector product
for an arbitrary vector v 2 < n . Computationally, (4.17) consists of a forward transformation
(C T v) followed by a diagonal scaling (^y is already available from the computation
of rf (x)), followed by a backward transformation (premultiplication by C).
To be explicit, recalling (4.8), we have
C.A. JOHNSON AND A. SOFER
where r 2 R (x) v can be computed exactly without incurring signicant computational
expense. The forward-and-back-transformation operation in (4.17) dominates the
computational cost of a CG iteration. This operation is computationally similar to
computing the gradient, or one EM iteration.
Some authors advocate solving simultaneously for p x and p , using the full unsymmetric
primal-dual equations (4.5), or an equivalent symmetrized system [13, 14, 52,
61]. The unsymmetric primal-dual matrix in particular remains nonsingular, and its
condition number remains bounded as ! 0 [12, 41], when the standard conditions
of a constraint qualication, strict complementarity, and the second-order su-cient
conditions are satised at the solution. In our application, due to the size of our
problem, we must use an iterative method. We believe that solving a symmetric system
via a symmetric solver such as the CG would be more e-cient than solving the
full unsymmetric system via an unsymmetric iterative solver such as GMRES (even
though our symmetric system is ill-conditioned), since the the amount of work and
storage required per iteration in GMRES increases linearly with the iteration count.
An advantage of using the condensed system (4.6){(4.7) is that although the primal
search direction is computed inexactly, the equation for maintaining complementarity
(4.7) is maintained. In practice we nd that the resulting primal and dual direction
vectors are both well scaled, and that x and are typically close to 1.
4.1.1. Preconditioning. The use of a preconditioner with the CG is essential
for a competitive algorithm. Since every CG subiteration is as costly as a gradient
evaluation or EM iteration, it is highly desirable to obtain a quality direction vector
in as few CG iterations per subproblem as possible. We have investigated a number of
preconditioners, including FFT-based preconditioners that model the approximately
Toeplitz-block-Toeplitz nature of CC T with a circulant-block-circulant approximation
[2, 3], high-pass lter approximations to the FFT-based preconditioner [4], the EM
preconditioner XQ 1 [34], the exact diagonal of M , and diagonal Hessian approximations
[46].
Of the above preconditioners, by far the best-performing was the exact diagonal
of M , which can be computed at reasonable cost:
Note that the rst right-hand side term in (4.18) is similar in form to a backward
transformation, although a bit more expensive due to the squaring operations. We
have found that the preconditioned CG method using an exact diagonal preconditioner
in the form of (4.18) almost always requires using fewer than 10 iterations to
achieve (4.16), regardless of the size of the problem. In many cases, only 3 or 4 CG
iterations are required. Moreover, the directions produced using an exact diagonal
preconditioner are well scaled (usually resulting in primal step sizes of near 1), and
lead to rapid descent.
In contrast, the other preconditioners did not perform well. Already in the initial
subproblems they tended to yield a poorly-scaled search direction, which in turn,
resulted in small steplengths. Subsequent calls to the CG suered further from this
problem, and the algorithm made little progress. This behavior was particularly
surprising for the block-circulant FFT-based preconditioners. These preconditioners
perform very well in other reconstruction methods, especially in least-squares methods
where the block-circulant approximation is well matched to the Hessian structure. We
were motivated to try them for our problem because the ML Hessian is almost block
circulant. But because of the strong diagonal component in M and its spatially-
variant dependence on y, ^
y, x, and , shift-invariant Toeplitz models of M yield a
poor approximation in our method.
4.1.2. Line search. For ML and MAP reconstructions, knowledge of the structure
of the objective function can lead to a substantial reduction in the cost of implementing
a line search over a more naive approach. Specically, after the search
direction p x has been found, and once a forward transformation ^
been computed, it is possible to compute the objective function and rst and second
directional derivative values at the trial points x k
at nearly negligible cost.
To see this, note that ^
and therefore [27, 28]
Similar expressions exist for the directional rst and second derivatives [24].
After the initial forward transformation to compute ^
no further forward- or
back-transformation operations are required during the line search at any of the trial
points. The forward transformation ^
w can be re-used, so that only one backward
transformation is subsequently required to update the gradient. The above observations
and the well behaved convex nature of the objective function have permitted
us to implement a highly accurate but low-cost Newton line search. Due to the low
cost of each step we have chosen a relatively strict tolerance of 0:05 on the Wolfe
condition for termination of the line search: We nd this line search technique to be
highly eective and, in no small part, responsible for the positive results we report.
4.2. Stabilization. A well known property of the Hessian of the primal barrier
function is its increasingly ill-conditioned nature as ! 0 [45]. Analogous results
hold for the condensed primal-dual matrix: as the solution is approached the matrix
becomes increasingly ill-conditioned. (For a detailed analysis see the paper by Wright
[60]).
In [50], Nash and Sofer developed an approximation to the Newton direction
for the logarithmic barrier, that avoids the structural ill-conditioning of the barrier
Hessian and is suitable for large-scale problems. The direction is the sum of two
vectors, one in the null space of the Jacobian of the active constraints, and the other
orthogonal to it. The associated decoupling is based on a prediction of the binding
set at the solution.
We have recently adapted this approximation to the condensed Newton equations
arising in primal-dual methods. Although our derivation is valid for general nonlinear
constraints, we present it here for the special case of bound constraints in the context
of (4.6).
We will assume in the following that strict complementarity holds at the solution,
that is,
0g to be the index set of binding
constraints at the solution, and ^
n to be the number of binding constraints at the
solution. We will assume that 0 <
always the case in reconstructions of
practical interest. Dene I
0g the set of nonbinding constraints. Let x I
be the subvector of variables that are positive at the optimal solution, and x J the
subvector of variables that are zero at the optimal solution. Assume also that the
12 C.A. JOHNSON AND A. SOFER
variables are ordered so that the positive variables are rst, i.e.,
x I
x J
The Hessian of the objective function will then be similarly partitioned,
as will the condensed primal-dual matrix
I;J MJ ;J
I I H I;J
where X I , XJ , I and J are the diagonal matrices of the associated components
of x and .
We will assume that the sequence of iterates (x; ) generated by the primal-dual
satises the following properties, when is su-cently small:
Here we dene there exist constants 0 < l < u so that l kk u
for all su-ciently small > 0. We say that a vector or matrix is () if its norm
is (). We also dene = O() if there exists some positive constant u so that
kk u for all su-ciently small > 0.
We will also assume that near the solution the Hessian is reasonably well condi-
tioned, so that Now the diagonal terms of MJ ;J are O(1=), and become
unbounded as ! 0. In contrast, the diagonal terms of M I;I dier from those of the
reduced Hessian H I;I by O(), and the condition of M I;I thus re
ects that of the constrained
problem. The condensed primal-dual matrix M can then be shown to have
\large" eigenvalues of magnitude (1=), and
\small" eigenvalues that dier
from those of H I;I by O(), and have magnitude (1). The condensed primal-dual
matrix thus suers from the same structured ill-conditioning as the barrier Hessian.
For small values of we propose approximating the primal Newton direction p x ,
by a direction ~
whose null- and range-space components are computed as follows:
~
The system for computing the component ~
I
x involves the well conditioned matrix
I;I , and can be solved exactly or inexactly via the conjugate gradient method. The
computation of ~
x is straightforward. Thus, the ill-conditioning of the condensed
primal-dual is avoided. We will show now that under the assumptions above, ~
so that the accuracy of the approximation increases as the solution is
approached and the potential harm from ill-conditioning increases.
Using the well known formula for the inverse of a partioned matrix (see e.g.
[51, 61]) it follows that
~
I
I;I rF I +M I;J (XJ 1
I;I rF I (XJ 1
where
I;I
Now by denition
so that G
Note further, that
I
whereas
It follows that
~
I
and
~
so that ~
In [50], Nash and Sofer prove (for the case of the Newton direction arising from
the logarithmic barrier objective function) that, for su-ciently small ; the vector
computed using an approximation similar to (4.19) and (4.20) yields a descent direction
with respect to the logarithmic barrier objective function. The proof is readily
extended to the present primal-dual case; thus p x is a descent direction for the merit
function F (x; ). We have found that, for the present problem, the above approximation
to the Newton direction is useful for values of of order 10 4 or less.
Recently Wright [60] showed that the errors generated by backward-stable numerical
methods (various Cholesky factorizations and Gaussian elimination with partial
pivoting) for solving (4.6) are not magnied by the structured ill-conditioning. These
methods are inappropriate for our large problems which involve potentially millions of
variables. Instead we nd an approximate solution using a CG iteration. When working
in inexact arithmetic with large numbers of variables, the convergence rate of the
CG method depends on the condition of M [15]. Thus the structural ill-conditioning
in M can lead the CG iteration to spend an unnecessary amount of work in computing
Further, as we have observed, the criterion for terminating the CG may be overly
optimistic in an ill-conditioned system, so that the resulting direction is poorly scaled
as
The potential eect of ill-conditioning is illustrated through an example in Table
4.1. This example was encountered during development and motivated the incorporation
of stabilization into the algorithm. Starting at the subproblem
the primal steplength, dual steplength, and ncg (the number of CG it-
erations), are listed for both the non-stabilized and stabilized cases. This test was
terminated at T x=n 7:5 10 5 . Note that in the non-stabilized case, the number
of CG iterations from the rst subproblem in the test to termination is signicantly
lower in the stabilized test than the non-stabilized test. Note also that in many of the
non-stabilized subproblems, either the primal or dual steplength is small, indicating
a poorly scaled direction or loss of accuracy.
14 C.A. JOHNSON AND A. SOFER
Table
An example of the eect of stabilization. The number of CG iterations, ncg, is counted from
the beginning of the subproblem. The termination condition in this example is
non-stabilized stabilized
7.08E-5 0.392 1.000 46 3.25E-5
5.83E-5 1.000 0.166 51
4.77E-5 1.000 62There has been much recent interest in stabilization methods that do not require
a prediction of the active set [13, 14, 59]. These approaches are based on factorization
methods which are unsuitable for a problem as large as the present one. The
argument against stabilization methods that require a prediction set is that the active
set is unknown in interior-point methods. We argue that, close to the solution
in the emission tomography reconstruction problem, an accurate prediction of the
active set can be made. In our problem, the constraints have a simple interpretation.
The positive variables correspond to those voxels containing at least some radioactive
tracer, while the zero-valued variables correspond to those voxels that lack any tracer
activity. Close to the solution, when becomes su-ciently small that stabilization
is appropriate, the set of binding constraints is obvious and can be conservatively
identied with a -dependent threshold.
4.3. Extrapolation. Fiacco and McCormick showed that the solutions x () at
the perturbed KKT solutions form a unique dierentiable trajectory in [12]. The
perturbed KKT conditions (4.1){(4.3) dene a \central path" as ! 0. Thus, a
successful algorithm may be able to move both \along" and \toward" the path. As
discussed in [12], from the subproblem solutions fx ( l the trajectory
can be approximated as a polynomial
l=k r
c l l ;
where r is the degree of the approximating polynomial and c k r are r
vectors of coe-cients. Using the approximation in (4.21), we nd a direction x such
that
l=k r
c l l x
and set
to be a prediction to the next subproblem's primal solution. Here x k is the computed
(approximate) subproblem solution for Primal feasibility is maintained by
the steplength
is the maximum steplength that does not
violate non-negativity in x. Then, in the manner of (4.7), we compute a dual direction
vector according to
The dual vector is then moved according to
4x;
which requires another dual line search to minimize (4.10). The resulting point
serves as a starting point for the 1)st subproblem, a prediction
to the solution at k+1 . The extrapolated primal-dual method can be viewed as
a predictor-corrector algorithm, with the extrapolation (4.22 and 4.24) serving as the
\predictor" step, and the subproblem minimization serving as the centering or \cor-
rector" step [23]. The degree r of the approximating polynomial is 1 when predicting
the 3rd subproblem, 2 for the 4th, and 3 for the 5th and beyond.
We have experimented with line searches in conjunction with (4.22), but often
1, and hence the line search just yields
. For this reason, we have found that
(4.24) yields a more eective dual direction than does the equivalent of (4.7) in the
context of extrapolation. Although the extrapolated search direction x can often
be poorly scaled (i.e.,
1), we have observed that the directions produced are
always descent directions to the merit function and lead to a signicant decrease in
the objective function f . A number of reconstructions were performed in which
was computed by extrapolating the dual solution vector (rather than computing it via
(4.24)); the discouraging nature of the results led us to abandon direct extrapolation
of the dual vector in favor of (4.24) which is highly eective in comparison.
Following extrapolation, a gradient evaluation is required to update the vector
the primal-dual algorithm requires between 12 and 25
subproblems to perform a 3-D MAP reconstruction, extrapolation adds that many
gradient evaluation operations to the computational cost. So extrapolation is only
economical if it reduces the computational burden by at least as much as it adds.
Our experience has been that for some data sets, the cost of extrapolation is well
worthwhile but for other data sets the benets were only marginal. Extrapolation
thus appears to serve as somewhat of a safeguard against di-cult problems. In an
extrapolated primal-dual reconstruction, the convergence measure does not
decrease as monotonically as in a primal dual reconstruction without extrapolation.
Certain extrapolated steps seem to cause the algorithm to \get ahead of itself," but
this eect is transient. On the studies we've performed, the algorithm does ultimately
converge to an accurate solution with extrapolation.
4.4. Initialization. The choice of the initial barrier parameter may have a substantial
eect on the algorithm. If the parameter is too small, the rst subproblem
may have extreme di-culty due to ill conditioning; if the parameter is too large,
then many (unnecessary) subproblems will be required to solve the problem. Proper
initialization of the barrier parameter involves nding the most suitable point on
the barrier trajectory based on the initial solution x and the measurement data y.
Recalling the perturbed necessary conditions in (4.1), if the initial solution were to
be on the central path, it would satisfy
C.A. JOHNSON AND A. SOFER
Pre-multiplying by ^
T we arrive at
r
This suggests the following rule for initialization, which we nd quite eective:
r
Another, similar, initialization rule is motivated by the goal of nding an initial
value 0 so that
While (4.26) cannot be solved exactly, we can try to nd a 0 that results in a point ^
that is close to the barrier trajectory according to, say, the 2-norm. This motivation
leads to an alternative initialization rule [51]
During the course of development, both initialization rules were tried on certain data
sets. Although both initialization rules performed well, reconstructions initialized
with (4.25) usually reached the optimal solution in slightly less overall work than
those initialized with (4.27).
The initial estimate for ^
used most frequently was in each case a
positive uniform eld. A discussion on the rationale of using a uniform eld for
and on criteria for choosing the constant value of the primal initial solution may be
found in [24]. Alternative choices for the initial dual vector may be preferable, and
an investigation into this question may be worthwhile.
4.5. Termination. Given that subproblem termination is based on the -criti-
cality conditions (4.11) and (4.12), the closeness of each subproblem solution can be
measured by . If subproblems are solved exactly, jf [12]. The
-criticality conditions, however, are designed for a \short-step" algorithm in which
one truncated-Newton step should satisfy each subproblem for su-ciently small . To
ensure the accuracy of the nal solution, nal termination is based on the following
two requirements:
We have found that reasonably accurate solutions are ensured when "
The traditional view in tomographic reconstruction is that a highly accurate solution
is unnecessary. This view stems in part from the ill-posedness of the problem
and the computational cost of taking a reconstruction to full convergence. From empirical
evidence in our studies, the ability to perform certain imaging tasks such as
\cold spot detectability" improves with accuracy of the solution. Although the termination
criteria we propose above may not appear particularly strict, they are from
a tomographic reconstruction perspective.
Table
Properties aecting computation, memory, and storage costs for two dierent-sized reconstruction
problems. Gradient evaluation costs are based on a 2.5M-count study on 10 120-MHz IBM
RISC/6000 SP processors.
size class n N elements density storage cost of
in C in C cost of C gradient
thick-slice 376,882 5:36
thin-slice
5. Computational studies. To test our algorithm we have performed a number
of reconstructions on data acquired from a small animal scanner, and on data
generated by Monte Carlo simulations on the same animal scanner.
5.1. Size of the problem. Our studies involved two dierent-sized problems.
Raw coincidence data from the scanner can be binned into either \thick-slice" or
\thin-slice" measurement spaces, or both. \Thick-slice" reconstructions, in which
minutes for a gradient evaluation
using processors (120 MHz) on a 2.5M-count study. For
a \thin-slice" reconstruction with on the same
data and processors, a gradient evaluation requires 6.75 minutes. These properties
are summarized in Table 5.1. The cost of storing the full n N system matrix is
prohibitive, even for thick-slice reconstructions. Extensive exploitation of the sparsity
and symmetries inherent in the system matrix makes its storage and retrieval possible
[24, 25].
The dominant computational operations of the reconstruction problems are the
forward- and back-transformation operations that underlie EM iterations, gradient
evaluations, Hessian-vector products, and diagonal Hessian calculations. These operations
have been implemented in parallel via a data decomposition strategy that partitions
the \measurement-space" vectors y and ^
y across the processors. The \image-
space" vectors such as x and are replicated over all processors. Our data decomposition
is justiable under the observation that N >> n. On a data set with 2.5M
counts, at most 47% of the elements of y will be nonzero in the thick-slice case; at
most 4% in the thin-slice case. (The thin-slice conguration has over 10 times as
many lines of response as the thick-slice.) The dominant computational operations
have been implemented in such a way to exploit sparsity in y and further conserve
computation [24].
5.2. Cost metrics. We have devised metrics to measure the cost of an interior
point reconstruction. Dene the number of subproblems to be npr, the number of
truncated-Newton iterations nit, the number of conjugate gradient subiterations ncg.
The cost of one CG iteration (dominated by the Hessian-vector product) is equivalent
to the cost of one gradient calculation or EM iteration. One truncated-Newton iteration
requires, in addition to the ncg operations, one diagonal Hessian evaluation plus
one forward transformation and one backward transformation. The exact cost of these
operations varies depending on the size of the problem and number of counts, but we
shall approximate the cost of one truncated-Newton iteration to be the equivalent of
two gradient calculations beyond the cost of the conjugate gradients.
Using this approximation, the total cost of unextrapolated interior-point reconstructions
can be measured in units of equivalent number of gradient calculations (or
C.A. JOHNSON AND A. SOFER
Table
Summary of thick-slice primal-dual results and comparison with MAP-EM and LSEM. Extrapolation
was not used, and in all cases 2.
study f npr nit ncg ngr MAP-EM LSEM
A 2,465,770 19 19 110 148 1000 344
G 3,660,344 24 24 127 175 > 1000 724
average ngr 183
Table
Summary of thick-slice extrapolated primal-dual results and comparison with MAP-EM and
LSEM; in all cases 2.
study f npr nit ncg ngr MAP-EM LSEM
A 2,465,772 17 17 94 145 960 332
F 3,296,029
G 3,660,384 20 20 100 160 >1000 430
average ngr 156
EM iterations):
Extrapolation requires an additional gradient calculation following the extrapolation
in order to update the gradient vector. With extrapolation we modify the formula to
5.3. Computational results. We have performed a number of 3-D reconstructions
on data acquired from a small animal scanner and data generated by a Monte
Carlo simulation of the same small animal scanner. Reconstructions of seven datasets
were taken to full convergence, as dened by the termination criteria (4.28) and (4.29)
with . The various datasets used in our computational
studies represent a fairly diverse sample of the types of scans that might be
encountered in practice. The number of counts in the datasets used in these studies
ranged from 850K to 5.1M. The number of binding constraints at the optimal solution
ranged from approximately 20% to 80%.
Our main results are summarized in Tables 5.2 and 5.3 for the non-extrapolated
and extrapolated primal-dual cases, respectively. Studies A through D are reconstructions
of data acquired from a small animal PET scanner, while studies E through G
are reconstructions of Monte Carlo simulated data. These reconstructions were performed
in \thick-slice" mode (376,832 variables) with the regularization parameter
set at
In these tables, the column "MAP-EM" indicates the number
of DePierro MAP-EM iterations that were required to achieve the value of f in the
same row. The column "LSEM" indicates the number of iterations required for an
EM algorithm, where the search direction on the last term of (3.2) is enhanced by a
Table
Summary of thick-slice logarithmic barrier results and comparison with MAP-EM and LSEM.
Extrapolation was used on all data sets, and in all cases
study f npr nit ncg ngr MAP-EM LSEM
A 2,465,832 5 28 159 218 880 194
average ngr 265
line search. (To avoid excessive computation, the function values were only calculated
every iterations, and the nal count was rounded down, to favor this
method.) Since the cost of one gradient evaluation is equivalent to the cost of one EM
iteration, the numbers in the columns ngr and MAP-EM and LSEM can be compared
directly. We nd that the primal-dual method consistently reaches convergence much
more rapidly than either MAP-EM or LSEM.
Another interesting observation can be made in the comparison between Tables
5.2 and 5.3. Consider the number of EM iterations required to reach f for study C.
In
Table
5.2, the LSEM algorithm reached iterations. In Table
5.3 on the same data set, the LSEM algorithm reached
tions. Thus, the algorithm took 64 iterations to reduce the function value by only 10
units near the solution. MAP-EM did even worse, requiring 180 iterations to reduce
the function value by 10. This is in fact a typical example of the slow limit behavior
of the EM algorithm. In all studies, the EM method did not achieve the same convergence
results obtained by the primal-dual method at termination. The Lagrangian
gradient norm and complementary slackness values of the terminated MAP-EM and
LSEM iterates were consistently much higher than those of the terminated primal-dual
solution.
We have also performed these reconstructions using a stabilized logarithmic barrier
algorithm based on the method presented in [50] and specialized to the present
reconstruction problem. Many of the computational features of our logarithmic barrier
implementation are identical to our primal-dual implementation, e.g., truncated
Newton, line search, computation of the gradient, Hessian-vector product, etc. For
a more detailed discussion, see [24]. The logarithmic barrier results are summarized
and compared against MAP-EM in Table 5.4. Termination of the logarithmic barrier
was dened by (4.29) and
These termination criteria for the logarithmic barrier correspond to roughly the same
accuracy as (4.28) and (4.29) do for the primal-dual method. Being a \long-step"
method, the logarithmic barrier gives the user less control over the exact stopping
point than does the \short-step" primal-dual. All of the logarithmic barrier reconstructions
in Table 5.4 used extrapolation. In all logarithmic barrier reconstructions,
was reduced by a factor of 10 between subproblems.
The eect of extrapolation is illustrated in Figures 5.1 and 5.2. In Figure 5.1,
the equivalent number of gradient evaluations (ngr) to reach termination is plot-
C.A. JOHNSON AND A. SOFER
100 150 200 250 300 350
PD
barrier
f-f*
ngr
Fig. 5.1. \Distance" from optimal solution at termination, as measured by dierence in objective
function f f (where f is here dened to be the lowest objective function obtained per study),
versus work required to reach termination, as measured by ngr, the equivalent number of gradient
evaluations. The studies included are those listed in Table 5.2. PD stands for non-extrapolated
primal-dual, PDX for extrapolated primal-dual.
ngr
PD
barrier
Fig. 5.2. Average value of at subproblem termination versus average ngr (equivalent number
of gradient evaluations) for the seven studies listed in Table 5.2. PD stands for non-extrapolated
prmal-dual, PDX for extrapolated primal-dual.
ted against objective function \distance" f f , the dierence between the function
value of the terminated solution and the lowest function value obtained for that recon-
struction. In all seven test cases (those listed in Tables 5.2{5.4), the unextrapolated
primal-dual method achieved the lowest objective function value. Thus, f f is
zero for all unextrapolated primal-dual (PD) results but greater than zero for the
extrapolated primal-dual (PDX) and barrier results. The PDX results are clustered
in a region of lower ngr than the PD results. This indicates that extrapolation lowers
the computational expense to the solution at a slight deterioration in the nal
objective. Compared with the barrier method, either extrapolated or unextrapolated
primal-dual produces equivalent or better accuracy with less computation required.
In
Figure
5.2, the average number of equivalent gradient evaluations at subproblem
termination is plotted against the average value of for each subproblem. Both
averages (ngr and ) were taken from the same seven test cases of Tables 5.2{5.4.
Compared with either unextrapolated primal-dual (PD) or extrapolated primal-dual
Study C
MAP-EM
LSEM
PD
|f-f*|
ngr
Fig. 5.3. Improvement in objective function as a function of gradient evaluations, Study F.
PDX denotes extrapolated primal-dual, PD denotes unextrapolated primal-dual, both using
Study F
MAP-EM
LSEM
PD
|f-f*|
ngr
Fig. 5.4. Improvement in objective function as a function of gradient evaluations, Study C.
PDX denotes extrapolated primal-dual, PD denotes unextrapolated primal-dual, both using
(PDX), the logarithmic barrier is clearly on a slower trajectory. The PD and PDX
trajectories are quite similar until approximately 0:01, at which point the PD
curve \swings out", while the PDX curve continues to descent log-linearly. This result
conrms that the prediction (extrapolation) step becomes more accurate near the so-
lution, resulting in more rapid convergence. However, a comparison of the objective
functions indicates that the value of PDX is perhaps one step \ahead of itself,"
compared with the unextrapolated case.
The progress of the reconstruction on a study of a rat skull, Study C, is compared
for the various algorithms in Figure 5.3. The measure used is kf f k (plotted on
a logarithmic scale). In the initial iterations DePierro Map EM and LSEM progress
rapidly and are ahead of the primal dual method. However the interior-point methods
rapidly reach the DePierro and LSEM objective values, and hence on, surpass them.
In the primal-dual methods depicted the value of is 2. The methods achieve faster
initial progress using however the overall computational eort for full convergence
with this parameter setting is greater. The progress of the reconstruction in
another example, Study F, is compared in Figure 5.4.
We have also reconstructed a number of very large-scale \thin-slice" reconstructions
involving variables. Table 5.5 summarizes a number of properties of
22 C.A. JOHNSON AND A. SOFER
Table
Summary of thin-slice extrapolated results, including convergence measures and computational
costs to optimal solution.
study
f kr'k
F 1E-5 7,721,001 1.29E-9 2.87E-11 14 14 71 113
average ngr 115
these extrapolated primal-dual reconstructions at the converged solution. A smaller
group of datasets (the more visually \interesting" studies) were selected for the thin-
slice work, and certain reconstructions were repeated with dierent values of the
prior strength
. Thin-slice reconstructions seem to require a lower prior strength
than the corresponding thick-slice reconstructions. The most visually pleasing results
were from reconstructions using
which is 1/30 the prior strength that
was generally found to be most satisfactory in thick-slice reconstructions. The total
amount of work (as measured in ngr) required to reach termination in Table 5.5 is also
quite pleasing. The number of variables in a thin-slice reconstruction is approximately
3.7 times the number in thick-slice. The number of nonzero-valued measurements in
thin-slice mode is only marginally greater than in thick-slice mode, however, since
the number of counts is the same in both cases. These thin-slice reconstructions may
thus be better conditioned than their thick-slice counterparts.
In closing, we should comment that the tolerance we have used in our tests is
stricter than that usually necessary. Indeed, less accurate solutions may still give
acceptable images. When the EM method is applied to the (unregularized) ML ob-
jective, it is usually terminated after 50 or 100 iterations, and the images produced
are often good. Thus EM-ML remains a practical method that can sometimes reach
a solution of desirable image quality faster than an interior-point method. The difculty
with EM-ML is that its convergence is object-dependent [1]. Convergence in
areas of high activity amidst low activity or vice versa is notoriously slow, and a xed
termination rule based on (say) 50 or 100 iterations cannot guarantee acceptable image
quality. This has been observed in a number of reconstructions, including some
of high biomedical interest. In contrast to ML-EM, the primal-dual algorithm has
object-independent convergence characteristics. Furthermore it is
exible, and can be
adapted to solve a problem e-ciently both to the strict tolerance in the studies above
by setting a modest rate of decrease for the barrier parameter, say 2, and to a
looser tolerance by setting a more aggressive reduction rate such as
6. Conclusion. From the results of the previous Section, it is clear that the
primal-dual method can converge signicantly faster than the EM algorithm for regularized
ML reconstructions in emission tomography. The results also indicate that
the primal-dual method converges faster than the logarithmic barrier method. The
use of extrapolation in conjunction with the primal-dual method further reduces the
amount of computation required to achieve convergence.
Given that the negative regularized ML objective function that we minimize is
convex, approximately solving the reduced unsymmetric primal-dual Newton equa-
tions is appropriate. Symmetrizing the unsymmetric system, while potentially useful
for nonconvex problems, would in this case require solving for 2n variables without
avoiding the potential for ill-conditioning. Our stabilization technique avoids the
structural ill-conditioning of the condensed primal-dual matrix, and therefore solving
the reduced system poses no asymptotic di-culty as the barrier parameter approaches
zero. The computational e-ciency and relative simplicity of formation of the reduced
system of equations pose such a strong advantage that our choice of primal-dual
method almost seems obvious for this problem.
Since Newton's method converges quadratically near the solution, for a well-conditioned
system in the limit as ! 0, one truncated-Newton step per subproblem
should yield an increasingly accurate and well scaled direction to the subproblem
solution for k . As is decreased, the subproblem solutions should become \close" to
each other for a convex problem [14]. Yet, the example in Table 4.1 illustrates that the
direction produced by the early-terminated CG can in fact become less accurate for
smaller due to the structured ill-conditioning in M . In practice, we do not require
the accuracy of the test example in Table 4.1. Our termination conditions are dened
to be near the point on the trajectory where the stabilization approximation becomes
accurate enough to guarantee descent. These termination criteria are quite accurate
by the standards of the tomography community. Thus, although most reconstruction
problems are unlikely to be severely aected by ill-conditioning, the potential for slow
convergence near the solution due to ill-conditioning does exist. Our experience has
been that stabilization has been an eective safeguard against poor performance for
small values of the barrier parameter.
7.
Acknowledgments
. The study utilized the high-performance computational
capabilities of the IBM RISC/6000 SP system at the Division of Computer Research
and Technology, National Institutes of Health, Bethesda, MD. We are grateful to
Jurgen Seidel of the Department of Nuclear Medicine, National Institutes of Health,
for kindly providing us with the small animal data and Monte Carlo simulation data.
Our thanks go to two anonymous referees and the associate editor for their careful
reading and helpful comments.
--R
Noise properties of the EM algorithm: I.
Conjugate gradient methods for Toeplitz systems
A general class of preconditioners for statistical iterative reconstruction of emission computed tomography
Preconditioning methods for improved convergence rates in iterative reconstructions
A primal-dual algorithm for minimizing a nonconvex function subject to bound and linear equality constraints
Image reconstruction and restoration: overview of common estimation structures and problems
Maximum likelihood from incomplete data via the EM algorithm
On the convergence of an EM-type algorithm for penalized likelihood estimation in emission tomography
Numerical stability and e-ciency of penalty algorithms
Sequential Unconstrained Minimization Techniques
Stability of symmetric ill-conditioned systems arising in interior methods for constrained optimization
Matrix Computations
On use of the EM algorithm for penalized likelihood estimation
A generalized EM algorithm for 3-D Bayesian reconstruction from Poisson data using Gibbs priors
Image Reconstruction from Projections: the Fundamentals of Computerized Tomography
Methods of conjugate gradients for solving linear systems
Accelerated image reconstruction using ordered subsets of projection data
A practical interior-point method for convex programming
Nonlinear optimization for Volume
A system for the 3D reconstruction of retracted-septa PET data using the EM algorithm
Evaluation of 3D reconstruction algorithms for a small animal PET camera
Implementing and accelerating the EM algorithm for positron emission tomog- raphy
PET regularization for envelope guided conjugate gradients
Constrained reconstruction by the conjugate gradient method
A primal-dual interior point algorithm for linear programming
Probability measure estimation using
The importance of preconditioners in fast Poisson-based iterative reconstruction algorithms for SPECT
Practical tradeo
EM Reconstruction Algorithms for Emission and Transmission Tomography
A theoretical study of some maximum likelihood algorithms for emission and transmission tomography
Convergence of EM image reconstruction algorithms with Gibbs smoothing
A maximum a posteriori probability expectation maximization algorithm for image reconstruction in emission tomography
The superlinear convergence of a nonlinear primal-dual algorithm
Statistical Modeling and Fast Bayesian Reconstruction in Positron Tomog- raphy
Fast gradient-based methods for Bayesian reconstruction of transmission and emission PET images
Bayesian reconstruction of PET images: methodology and performance analysis
Analytic expressions for the eigenvalues and eigenvectors of the Hessian matrices of barrier and penalty functions
Preconditioning of truncated-Newton methods
Block truncated-Newton methods for parallel optimization
Barrier methods for large-scale quadratic programming
Maximum Likelihood Reconstruction for Emission Tomography
An infeasible interior-point method for linear complementarity problems
A statistical model for positron emission tomography
Noise properties of
Interior methods for constrained optimization
Stability of linear equation solvers in interior point methods.
--TR | applications of nonlinear programming;parallel computation;tomography;primal-dual methods;estimation;large-scale problems |
589052 | On the Convergence Theory of Trust-Region-Based Algorithms for Equality-Constrained Optimization. | In a recent paper, Dennis, El-Alem, and Maciel proved global convergence to a stationary point for a general trust-region-based algorithm for equality-constrained optimization. This general algorithm is based on appropriate choices of trust-region subproblems and seems particularly suitable for large problems.This paper shows global convergence to a point satisfying the second-order necessary optimality conditions for the same general trust-region-based algorithm. The results given here can be seen as a generalization of the convergence results for trust-regions methods for unconstrained optimization obtained by Mor and Sorensen. The behavior of the trust radius and the local rate of convergence are analyzed. Some interesting facts concerning the trust-region subproblem for the linearized constraints, the quasi-normal component of the step, and the hard case are presented.It is shown how these results can be applied to a class of discretized optimal control problems. | Introduction
. Trust-region algorithms have been proved to be efficient and
robust techniques to solve unconstrained optimization problems. An excellent survey
in this area is Mor'e [22]. Other classical references for convergence results are Carter
[3], Mor'e and Sorensen [23], Powell [26], and Shultz, Schnabel, and Byrd [29]. The
standard techniques to handle the trust-region subproblems are the dogleg algorithm
(Powell [25]), conjugate gradients (Steihaug [32] and Toint [33]), and Newton-like
methods for the computation of locally constrained optimal steps (Gay [15], Mor'e
and Sorensen [23], and Sorensen [30]). See also the book of Dennis and Schnabel [9].
Recent new algorithms to compute a locally constrained optimal step (in other words a
step that satisfies a fraction of optimal decrease on the trust-region subproblem) that
are very promising for large problems have been proposed by Rendl and Wolkowicz
[28] and Sorensen [31].
Since the mid eighties a significant effort has been made to address the equality-
constrained optimization problem. References are Celis, Dennis, and Tapia [4], Vardi
[34] (see also El-Hallabi [14]), Byrd, Schnabel, and Shultz [2], Powell and Yuan [27],
and El-Alem [13]. The fundamental questions associated with the application of trust-region
algorithms to equality-constrained optimization are the decomposition of the
step, the choice of the trust-region subproblems, and the choice of the merit function.
During the first stages of the research conducted in this area it was not clear how to
answer these questions properly. However, if we examine carefully the most recent
Department of Computational and Applied Mathematics, Rice University, Houston, Texas 77005-
USA. E-Mail: dennis@rice.edu. Support of this author has been provided by DOE contract
DOE-FG03-93ER25178, NSF cooperative agreement CCR-9120008, and AFOSR contract F49620-
9310212.
y Departamento de Matem'atica, Universidade de Coimbra, 3000 Coimbra, Portugal. This work
was developed while the author was a graduate student at the Department of Computational and
Applied Mathematics of Rice University. E-Mail: lvicente@mat.uc.pt. Support of this author has
been provided by INVOTAN (NATO scholarship), CCLA (Fulbright scholarship), FLAD, and NSF
cooperative agreement CCR-9120008.
references (Byrd and Omojokon [24], Dennis, El-Alem, and Maciel [7], El-Alem [12],
[13], and Lalee, Nocedal, and Plantenga [21]) we can observe the same decomposition
of the step (in its normal, or quasi-normal, and tangential components) and the same
trust-region subproblems (the trust-region subproblem for the linearized constraints
and the trust-region subproblem for the Lagrangian reduced to the tangent subspace).
This is explained in great detail in Section 2 of this paper. As in the unconstrained
case, the conditions that each component has to satisfy and the way they are computed
might of course differ from algorithm to algorithm. We can see also in these most
recent references that the merit function used is either the ' 2 penalty function without
constraint term squared (cases of [21], [24]) or the augmented Lagrangian function (in
Consider now the equality-constrained optimization (ECO) problem
minimize f(x)
subject to
n. The functions f and c i , are assumed to be at least twice
continuously differentiable in the domain of interest.
In [7], Dennis, El-Alem, and Maciel have considered a general trust-region-based
algorithm for the solution of the ECO problem (1.1). This general algorithm is very
much like the algorithm proposed by Byrd and Omojokon [24] 1 . As mentioned before,
each step s is decomposed as s n is the quasi-normal component of
the step, associated with trust-region subproblem for the linearized constraints and
s t is the tangential component, associated with the trust-region subproblem for the
Lagrangian reduced to the tangent subspace. Each component of the step is only
required to satisfy a fraction of Cauchy decrease on the corresponding trust-region
subproblem. Another key feature of this general algorithm is the choice of the augmented
Lagrangian as a merit function and the use of the El-Alem's scheme [11] to
update the penalty parameter. Under appropriate assumptions, it can be shown that
there exists a subsequence of iterates driving to zero the norm of the residual of the
constraints and the norm of the gradient of the Lagrangian reduced to the tangent sub-space
(see [7][Section 8]). It is important to remark that this global convergence result
is obtained under very mild conditions on the components of the step, on the multipliers
estimates, and on the Hessian approximations. Thus, the Dennis, El-Alem, and
Maciel [7] result is similar to the global result given by Powell [26] for unconstrained
optimization.
One of the purposes of this paper is to show global convergence to a point satisfying
the second-order necessary optimality conditions for this class of algorithms.
Our result is similar to the results established by Mor'e and Sorensen [23], [30] for
trust-region algorithms for unconstrained optimization. We accomplish this here by
imposing a fraction of optimal decrease on the tangential component s t of the step,
by using exact second-order information, and by imposing conditions on the quasi-
normal component s n and on the Lagrange multipliers.
1 The Thesis [24] was directed by Professor R. H. Byrd. The trust-region algorithm proposed here
is usually referred as the Byrd and Omojokon algorithm.
In [2], Byrd, Schnabel, and Shultz have proposed a trust-region algorithm for
equality-constrained optimization and established global convergence to a point satisfying
the second-order necessary optimality conditions. However this algorithm does
not belong to the class of trust-region algorithms considered here and their result is
obtained with the use of the (exact) normal component and the least-squares multipliers
update which we do not require in this paper. Other differences are that they use
the ' 1 penalty function as the merit function and the analysis is carried out by using
an orthogonal null-space basis. In recent papers, Coleman and Yuan [6] and El-Alem
[12] have proposed trust-region algorithms for which they prove global convergence to
points satisfying first-order and second-order necessary optimality conditions. Their
algorithms use the (exact) normal component, an orthogonal null-space basis, and the
least-squares multipliers update.
The conditions we need to impose to assure that a limit point of the sequence of
iterates satisfies the second-order necessary optimality conditions are
k is the quasi-normal component of the step s k , and
k is the trust-region radius. In the case where kC(x k )k is small compared with
the first condition implies that any increase of the quadratic model of the Lagrangian
from x k to x k +s n
k is O(ffi 2
To see why this is relevant recall that a fraction of optimal
decrease is being imposed on the tangential component s t
k yielding a decrease of O(ffi 2
on the quadratic model. The second condition is needed for the same reasons because
appears in the definition of predicted decrease. We show that both conditions
are satisfied when either (i) the (exact) normal component and the least-squares
multipliers are used; or (ii) the most reasonable choices of quasi-normal component
and multipliers are made for a class of discretized optimal control problems. The
former result is in agreement with the result obtained by El-Alem [12].
Gill, Murray, and Wright [17] and El-Alem [10] considered in their analyses that
k). In the latter work this assumption is used to prove local convergence
results, and in the former to establish properties of an augmented Lagrangian
merit function. We point out that this assumption implies that r x '(x
k is
since s k is O(ffi k ) and we assume that s n
k is O(kC(x k )k).
We also prove that if the algorithm converges to a point where the reduced Hessian
is positive definite, then the penalty parameter ae k is uniformly bounded and the
trust-region radius ffi k is uniformly bounded away from zero, a desired property of a
trust-region algorithm. In this case, particular choices of the multipliers and of the
components s n and s t lead us to a q-quadratic rate of convergence in x.
A detailed treatment of the global convergence theory is given in Vicente [35].
The structure of the trust-region subproblem for the linearized constraints can be
exploited to obtain some interesting results. We introduce a quasi-normal component
that satisfies a fraction of optimal decrease on the trust-region subproblem for the linearized
constraints. We show that the (exact) normal component shares this property.
We also prove that if the algorithm is well behaved (for instance if the trust radius is
uniformly bounded away from zero), then this subproblem has a natural tendency to
fall into the so-called hard case.
We review the notation used in this paper. The Lagrangian function associated
with the ECO problem (1.1) is defined by '(x;
is the Lagrange multiplier vector. The matrix rC(x) is given by
, where rc i (x) represents the gradient of the function c i (x). Let
be the Hessian matrices of f(x) and c i (x) respectively. We use
subscripted indices to represent the evaluation of a function at a particular point of
the sequences fx k g and f k g. For instance, f k represents f(x k ) and ' k is the same
as The vector and matrix norms used are the ' 2 norms, and I l represents
the identity matrix of order l. Finally, 1 (A) denotes the smallest eigenvalue of the
symmetric matrix A.
The structure of this paper is as follows. In Section 2, we introduce the trust-region
subproblems and outline the general trust-region algorithm and the general
assumptions. In Section 3, we present the global convergence theory. A class of
discretized optimal control problems is introduced in Section 4 as a justification for
the general form of our algorithms and theory. In Sections 5 and 6, we analyze
respectively the behavior of the trust radius and the local rates of convergence. The
trust-region subproblem for the linearized constraints is studied in Section 7. We end
this paper with some summary conclusions.
2. Algorithm and general assumptions. The trust-region algorithm analyzed
by Dennis, El-Alem, and Maciel [7] for the solution of the ECO problem (1.1),
consists of computing, at each iteration k, a step s k decomposed as s
the components s n
k and s t
are required to satisfy given conditions. If the step s k is
accepted, the algorithm continues by setting x k+1 to x . If the step is rejected
then x
2.1. Decomposition of the step. Suppose that kC k k 6= 0. The component
k is called the quasi-normal (or quasi-vertical) component of s k and is required to
satisfy a fraction of Cauchy decrease on the trust-region subproblem for the linearized
constraints defined by
subject to ks n k
where r 2 (0; 1) and ffi k is the trust radius. In other words, s n
k has to satisfy
k is the so-called Cauchy point for this trust-region subproblem,
i.e. c n
k is the optimal solution of
subject to c n 2 spanf\GammarC k C k
and therefore
The component s n
k also is required to satisfy the condition
ks n
where 1 is a positive constant independent of the iterate k of the algorithm. This
condition is saying that close to feasibility the quasi-normal component has to be
small.
In this paper, we require the quasi-normal component s n
k also to satisfy
where 2 is a positive constant independent of the iterates. The important consequence
of this condition is that if kC k k is small compared with ffi k , then any increase of the
quadratic model of the Lagrangian along the quasi-normal component s n
k is of O(ffi 2
The two choices of s n
k given in Sections 4.1 and 4.2 satisfy conditions (2.1), (2.2),
and (2.3). Other choices have been suggested in [7], [20].
The component s t
k is the tangential (or horizontal) component, and it must satisfy
i.e. it must lie in the null space N (rC T
k ) of rC T
k . Let W k be an n \Theta (n \Gamma m) matrix
whose columns form a basis for N (rC T
be a quadratic model of ' at is an approximation to r 2
Since for any s t 2 N (rC T
k ), there exists a s t 2 IR n\Gammam such that s
consider also
which is given by
with
If kg k k 6= 0, s t
k is required to satisfy a fraction of Cauchy decrease for the trust-region
subproblem
minimize
subject to ks n
Note that this is not a standard trust-region subproblem because s n
k might not be
orthogonal to N (rC T
might not be the center of the trust region.
The steepest-descent direction at
associated with
in the ' 2 norm is \Gamma g k .
If we take into account the scaling matrix W k , then the steepest-descent direction
in the kW k \Delta k norm is given by \Gamma(W T
k . We consider the steepest-descent
direction \Gamma g k for
ks n
and require
k to satisfy
where oe t ? 0, and
k is the Cauchy point for the ' 2 norm given by
with
ks n
This is equivalent to saying that max is the maximum step length along s n
allowed inside the trust region defined by ffi k . It is easy to verify that
The results given in this paper hold also if c t
k is defined along \Gamma(W T
provided the sequence fk(W T
are valid also if the coupled
trust-region constraint ks n
is decoupled as ks t k ffi k . In this latter case
the parameter r defining the quasi-normal component s n
k can have any positive value.
A step
k that satisfies this requirement can be computed by using Powell's dogleg
algorithm [25] or by the conjugate-gradient algorithm adapted for trust regions by
Steihaug [32] and Toint [33] (see also [7], [8], [21]).
In order to establish global convergence to a point satisfying the second-order
necessary optimality conditions, we need
k to satisfy a fraction of optimal decrease
on the following trust-region subproblem
minimize
subject to kW k
where
In other words, we require
k to satisfy the following conditions:
s
k is the optimal solution of the trust-region subproblem (2.5).
This can be accomplished by applying the GQTPAR routine of Mor'e and Sorensen
[23] or by using the algorithms recently proposed by Rendl and Wolkowicz [28] and
Sorensen [31].
If s t
k satisfies a fraction of optimal decrease on the trust-region subproblem (2.5),
then
ks k k ks n
If
k is only required to satisfy a fraction of Cauchy decrease, then ks k ks n
. We can combine both cases and write
ks ks n
It is also important to note that the definition of ~
assures that the fraction of
optimal decrease (2.6) implies the fraction of Cauchy decrease (2.4) provided
2.2. General trust-region algorithm. We introduce now the merit function
and the corresponding actual and predicted decreases. The merit function used is the
augmented Lagrangian
where ae is the penalty parameter. The actual decrease ared(s k ; ae k ) at the iteration k
is given by
The predicted decrease (see [7]) is the following:
pred(s
To update the penalty parameter ae k we use the scheme proposed by El-Alem [11].
The Lagrange multipliers k are required to satisfy
where 3 is a positive constant independent of k. This condition is not necessary for
global convergence to a stationary point.
The general trust-region algorithm is given below.
Algorithm 2.1 (General trust-region algorithm).
and r such that
ae ? 0, and r 2 (0; 1).
do
2.1 If kC
is given in (2.10), stop the
algorithm and use x k as a solution for the ECO problem (1.1).
2.2 Set s n
satisfying (2.1), (2.2), (2.3), and ks n
If kW T
satisfying (2.6).
k .
2.3 Compute k+1 satisfying (2.8).
2.4 Compute pred(s
If pred(s
then set ae
Otherwise set
ae:
2.5 If ared(s k ;ae k )
pred(s k ;ae k )
ks k k and reject s k .
Otherwise accept s k and choose ffi k+1 such that
2.6 If s k was rejected set x . Otherwise set x
It is important to understand that the role of ffi min is just to reset ffi k after a step
s k has been accepted. During the course of finding such a step the trust radius can be
decreased below ffi min . To our knowledge Zhang, Kim, and Lasdon [37] were the first
to suggest this modification. We remark that the rules to update the trust radius in
the previous algorithm can be much more complicated but those given suffice to prove
convergence results and to understand the trust-region mechanism.
As a direct consequence of the way the penalty parameter is updated, we have
the following result.
Lemma 2.1. The sequence fae k g satisfies
ae k ae
pred(s
In order to establish global convergence results, we use the general assumptions
given in [7]. These are Assumptions A.1-A.4. However for global convergence to a
point that satisfies the second-order necessary optimality conditions, we also need
Assumption A.5. We assume that for all iterations k, x k and x are in \Omega\Gamma where
\Omega is an open subset of IR n .
General assumptions
A.1 The functions f , c i , are twice continuously differentiable in \Omega\Gamma
A.2 The gradient matrix rC(x) has full column rank for all x 2 \Omega\Gamma
A.3 The functions f , rf , r are bounded in \Omega\Gamma
The matrix (rC(x) T rC(x)) \Gamma1 is uniformly bounded in \Omega\Gamma
A.4 The sequences fW k g, fH k g, and f k g are bounded.
A.5 The Hessian approximation H k is exact, i.e. H
are Lipschitz continuous in \Omega\Gamma
Assumptions A.3 and A.4 are equivalent to the existence of positive constants
9 such that jf(x)j 0 , krf(x)k 1 , kr 2 f(x)k 2 , kC(x)k 3 ,
2.3. Predicted decrease along the tangential component. Consider again
the trust-region subproblem (2.5). We can use the general assumptions and the
structure of this subproblem to obtain a lower bound on the predicted decrease
along the tangential component of the step.
It follows from the Karush-Kuhn-Tucker conditions that there exists a fl k 0
such that
positive semi-definite,
s
~
s
0:
(It turns out that these conditions are also sufficient for
s
k to solve the trust-region
subproblem (2.5), see Gay [15] and Sorensen [30].) As a consequence of this we can
s
where
Hence we have
3. Global convergence. Dennis, El-Alem, and Maciel [7] have proved under
Assumptions A.1-A.4 and conditions (2.1), (2.2), and (2.4) that
lim inf
0:
In this section we assume that s t
k satisfies the fraction of optimal decrease (2.6)
on the trust-region subproblem (2.5), as well as conditions (2.3), (2.8), and A.5 on
respectively, and show that (3.1) can be extended to
0:
The proof of (3.2) although simpler has the same structure as the proof given in [7].
We prove the result by contradiction, under the supposition that
for all k. We start by analyzing the fraction of Cauchy and optimal decrease conditions.
Lemma 3.1. Let the general assumptions hold. Then
and
and, moreover, since s t
k satisfies a fraction of optimal decrease for the trust-region
subproblem (2.5),
are positive constants independent of the iterate k.
Proof. The conditions (3.4) and (3.5) are an application of Powell's result (see
[26, Theorem 4], [22, Lemma 4.8]) followed by the general assumptions. The condition
(3.6) is a restatement of (2.11) with
The following inequality is needed in the forthcoming lemmas.
Lemma 3.2. If the general assumptions hold, then
positive constant independent of k.
Proof. The term q k
k ) can be bounded using (2.2), (2.3), and Assumption
A.4, in the following way:
ks n
On the other hand, it follows from (2.8) and krC T
that
If we combine these two bounds we get (3.7) with
The following technical lemma gives us upper bounds on the difference between
the actual decrease and the predicted decrease. The proof follows similar arguments
as the proof of Lemma 6.3 in [11].
Lemma 3.3. Let the general assumptions hold. There exist positive constants
independent of k, such that
ks k k 3
ks k
ae k
3 ks k k 3 ks k k 2
and
ks k
ae k
6 ks k k 3 ks k k 2
Proof. If we add and subtract '(x
for some 1
1). Again using the Taylor expansion we can write
1). Now we expand c i
This and the
general assumptions give us the estimate (3.8) for some positive constants
The inequality (3.9) follows from (3.8) and ae k 1.
The following three lemmas bound the predicted decrease. They correspond respectively
to Lemmas 7.6, 7.7, and 7.8 given in [7].
Lemma 3.4. Let the general assumptions hold. Then the predicted decrease in the
merit function satisfies
pred(s
(3.
and also
pred(s
for any ae ? 0.
Proof. The two conditions (3.10) and (3.11) follow from a direct application of
(3.7) and from the two different lower bounds (3.5) and (3.6) on q k (s n
Lemma 3.5. Let the general assumptions hold, and assume that kW T
ff min
ae ffl tol
min
ae 7 ffl tol
oe
9 ffl tol
oe
then the predicted decrease in the merit function satisfies either
pred(s
or
pred(s
for any ae ? 0.
Proof. From kW T
and the first bound on ff given by
(3.12), we get
Thus either kW T
us first assume that kW T
ffl tol . Then it follows from the second bound on ff given by (3.12) that
Using this, (3.10), and the third bound on ff given by (3.12), we obtain
pred(s
Now suppose that To establish (3.14), we combine (3.11) and the last
bound on ff given by (3.12) and get
pred(s
We can set ae to ae k\Gamma1 in Lemma 3.5 and conclude that, if kW T
ffl tol and kC k k ffffi k , then the penalty parameter at the current iterate does not need
to be increased. See Step 2.4 of Algorithm 2.1.
The proof of the next lemma follows the argument given in the proof of Lemma
3.5 to show that either kg k k ? 1ffl tol or fl k ? 1ffl tol holds.
Lemma 3.6. Let the general assumptions hold, and assume that kW T
(3.12), then there exists a constant
pred(s
Proof. By Lemma 3.5 we know that either (3.13) or (3.14) holds. Now we set
. In the first case we use kg
pred(s
In the second case we use
pred(s
Hence (3.15) holds with
ae 6 ffl tol
min
ae 7 ffl tol
oe
9 ffl toloe
Next we prove under the supposition (3.3), that the penalty parameter ae k is
bounded.
Lemma 3.7. Let the general assumptions hold. If kW T
for all k, then
ae k ae ;
where ae does not depend on k, and thus fae k g and fL k g are bounded sequences.
Proof. If ae k is increased at iteration k, then it is updated according to the rule
ae:
We can write
ae ki
By applying (3.4) to the left hand side and (3.5) and (3.7) to the right hand side, we
obtain
aei
If ae k is increased at iteration k, then from Lemma 3.5 we certainly know that kC k k ?
Now we use this fact to establish that
We have proved that fae k g is bounded. From this and the general assumptions we
conclude that fL k g is also bounded.
We can prove also under the supposition (3.3), that the trust radius is bounded
away from zero.
Lemma 3.8. Let the general assumptions hold. If kW T
for all k, then
where does not depend on k.
Proof. If s k\Gamma1 was an acceptable step, then ffi k ffi min . If not then ks
and we consider the cases kC
(3.12). In both cases we use the fact
ared(s
pred(s
Case I. kC From Lemma 3.6, inequality (3.15) holds for
Thus we can use ks
ared(s
pred(s
)ks
Thus ks
Case II. kC . In this case from (2.9) and (3.4) with
pred(s
where rg. Again we use ae
and this time the last two lower bounds on pred(s
pred(s
ks
ae
ks
ae
ks
Hence ks
The result follows by setting ffi g.
The next result is needed also for the forthcoming Theorem 3.1.
Lemma 3.9. Let the general assumptions hold. If kW T
for all k, then an acceptable step is always found in finitely many trial steps.
Proof. Let us prove the assertion by contradiction. Assume that for a given k,
k. This means that lim k!+1 all steps are rejected
after iteration k. See Steps 2.5 and 2.6 of Algorithm 2.1. We can consider the cases
and appeal to arguments similar
to those used in Lemma 3.8 to conclude that in any case
pred(s
where 15 is a positive constant independent of the iterates. Since we are assuming
that lim k!+1
ared(s k ;ae k )
pred(s k ;ae k )
1 and this contradicts the rules
that update the trust radius. See Step 2.5 of Algorithm 2.1.
Now we finally can state our first asymptotic result.
Theorem 3.1. Under the general assumptions, the sequence of iterates fx k g
generated by the Algorithm 2.1 satisfies
lim inf
0:
Proof. Suppose that there exists an ffl tol ? 0 such that kW T
for all k.
At each iteration k either kC k k ffffi k or kC k k ? ffffi k , where ff satisfies (3.12). In
the first case we appeal to Lemmas 3.6 and 3.8 and obtain
pred(s
we have from ae k 1, (2.9), (3.4), and Lemma 3.8, that
pred(s
Hence there exists a positive constant 16 not depending on k such that pred(s k
From Lemma 3.9, we can ignore the rejected steps and work only with successful
iterates. So, without loss of generality, we have
Now, if we let k go to infinity, this contradicts the boundedness of fL k g.
From this result we can state our global convergence result: existence of a limit
point of the sequence of iterates generated by the algorithm satisfying the second-order
necessary optimality conditions. This result generalizes those obtained for unconstrained
optimization by Sorensen [30] and Mor'e and Sorensen [23].
Theorem 3.2. Let the general assumptions hold. Assume that W (x) and (x)
are continuous functions and
If fx k g is a bounded sequence generated by Algorithm 2.1, then there exists a limit
point x such that
positive semi-definite on N (rC(x ) T ).
Moreover, if (x ) is such that r x '(x ; (x satisfies the second-order
necessary optimality conditions.
Proof. Let fk i g be the index subsequence considered in (3.16). Since fx k i g is
bounded, it has a subsequence fx k j g that converges to a point x and for which
lim
0:
Now from this and the continuity of C(x), we get C(x we use the
continuity of W (x) and rf(x) to obtain
Since 1 (\Delta) is a continuous function, we can use (2.10), lim j!+1 the
continuity of W (x), (x), and of the second derivatives of f(x) and c i (x),
to obtain
0:
This shows that r 2
positive semi-definite on N (rC(x ) T ).
The continuity of an orthogonal null space basis has been discussed in [1], [5], [16].
A class of nonorthogonal null space basis is described in Section 4.1.
The equation r x '(x ; (x consistent updates of the Lagrange
multipliers like the least-squares update (4.7) or the adjoint update (4.3).
4. Examples.
4.1. A class of discretized optimal control problems. We now introduce an
important class of ECO problems where we can find convenient matrices W k , quasi-
normal components s n
k , and multipliers k satisfying all the requirements needed for
our analysis. The numerical solution of many discretized optimal control problems
involves solving the ECO problem
subject to C(y;
y
(see [8], [19], [20]). The variables in y are
the state variables and the variables in u are the control variables. Other applications
include parameter identification, inverse, and flow problems and design optimization.
In many situations there are bounds on the control variables, but this is not considered
here. Another interesting aspect of these problems is that rC(x) T can be partitioned
as
where C y (x) is a square matrix of order m.
In this class of problems the following assumption traditionally is made:
The partial Jacobian C y (x) is nonsingular and its inverse is uniformly
bounded in \Omega\Gamma
As a consequence of this, the columns of
\GammaC
I n\Gammam
form a basis for the null space of rC(x) T .
The usual choice for k in these problems is the so-called adjoint multipliers
It follows directly from the continuity of rC(x) and the uniformly boundedness of
continuously with x. Furthermore
is a continuous function of x with bounded derivatives.
Using the structure of the problem we can define the quasi-normal component s n
(see references [8], [19], [20]) as
where
kCy
As we will see in Section 7, the quasi-normal component (4.4) satisfies a fraction of optimal
decrease and hence a fraction of Cauchy decrease on the trust-region subproblem
for the linearized constraints.
Other choices for quasi-normal components are given in [20]. All these quasi-
normal components are of the form
Lemma 4.1. If s n
verifies (4.5) and k is given by (4.3), then conditions (2.3)
and (2.8) are satisfied.
Proof. From (4.3) and (4.5) we can see that
/r
and condition (2.3) is trivially satisfied. Condition (2.8) follows from the existence of
bounded derivatives for
4.2. The normal component and the least-squares multipliers. Consider
again the general ECO problem (1.1). If the component s n
k of the step s k is orthogonal
to the null space of rC T
k , then it is a multiple of rC k (rC T
. If we also
require that s n
lies inside the trust region of radius rffi k , then it is given by
. If the quasi-normal component s n
k of the step is
given by (4.6), then it is called normal. As we will see in the Section 7, the normal
component (4.6) satisfies a fraction of optimal decrease and hence a fraction of Cauchy
decrease on the trust-region subproblem for the linearized constraints.
Lemma 4.2. The quasi-normal component (4.6) and the least-squares update
satisfy conditions (2.3) and (2.8).
Proof. It can be easily confirmed that r x ' T
The condition (2.8) holds
since bounded derivatives in \Omega\Gamma
5. The behavior of the trust radius. In Sections 5 and 6 we no longer need
to consider that the tangential component s t
k satisfies a fraction of optimal decrease
on the trust-region subproblem (2.5). It suffices to assume the fraction of Cauchy
decrease condition (2.4). We assume that the component s n
k satisfies conditions (2.1)
and (2.2).
We need to strengthen conditions (2.3) and (2.8) in the following way:
ks k k;
ks k k;
ks n
ks k k;
3 , and 0
4 are positive constants independent of the iterates. The choices
of s n
k and k suggested in Section 4 satisfy these requirements as well. See Lemmas
4.1 and 4.2 for the first two conditions. It is obvious that the normal component (4.6)
satisfy (5.3). The quasi-normal component (4.4) also satisfies (5.3) (see [35][Lemma
5.6.1]).
The next theorems show that if lim k!+1 x
positive
definite on N (rC(x ) T ), then the penalty parameter ae k is uniformly bounded and
the trust radius ffi k is uniformly bounded away from zero.
Theorem 5.1. Let the general assumptions hold and W (x) and (x) be continu-
ous. If fx k g converges to x and r 2
positive definite on N (rC(x
then fae k g is a bounded sequence.
Proof. First since r 2
positive definite on N (rC(x
are continuous functions of x, there exists a
neighborhood N (x ) of x and a fl ? 0 such that for any x in N (x ),
xx '(x; (x))W (x)
fl:
Since
ks t
Thus for all k such that x k 2 N (x ) we
flks t
ks t
and this implies
ks t
Now by using (3.5) and (5.4), we have for all k such that x k 2 N (x ), that
17 ks t
where
1+r
g.
Now let kC k k ff 0 ks k k where the positive constant ff 0 is defined later. Using
similar arguments as in Lemma 3.2, it follows from (2.2), (5.1), (5.2), kC k k ff 0 ks k k,
and Assumption A.4 that
ks k k;
3 .
From (2.2) and kC k k ff 0 ks k k we also get
ks
ks n
ks t
2ks n
ks
which together with (5.5) and (5.6) implies
pred(s ks
ks
ks k k
for all ae ? 0. We now need to impose the following condition on ff
Now we set ae = ae k\Gamma1 in (5.7) and conclude that the penalty parameter does not
need to be increased if kC k k ff 0 ks k k (see Step 2.4 of Algorithm 2.1). Hence, if ae k is
increased then kC ks k k holds, and by using (5.1)-(5.3) we obtain:
ks k k;
with 00
3 . Recall from the proof of Lemma 3.7 that if ae k is
increased then
ae
r
ks k k
oe
ae 4 )ks k k kC k k;
which in turn implies
ae
r
ae k 00
ae 4
This completes the proof of the Theorem.
Theorem 5.2. Let the general assumptions hold and W (x) and (x) be continu-
ous. If fx k g converges to x and r 2
positive definite on N (rC(x
then ffi k is uniformly bounded away from zero and eventually all iterations are successful
Proof. The proof of the theorem is based on the boundedness of fae k g. We consider
the cases kC ks k k and kC k k ff 0 ks k k, where ff 0 satisfies (5.8).
ks k k, then from (2.7), (2.9), and (3.4), we find that
pred(s
ks
where
g. In this case it follows from (3.9), (5.10), and ae k 1
that
pred(s
ks
Now, suppose that kC k k ff 0 ks k k. From (5.7) with
pred(s
ks k
Now we use (3.9) and ae k ae to get
pred(s
ks
It follows from Theorem 8.4 in [7] that
lim inf
0:
From this result, the continuity of C(x), and the convergence of fx k g we obtain
Finally from (5.11), (5.12), and the limits lim k!+1 x
finally get
lim
pred(s
which by the rules for updating the trust radius in Step 2.5 of Algorithm 2.1 shows
that ffi k is uniformly bounded away from zero.
6. Local rate of convergence. In order to obtain q-quadratic local rates of
convergence, we require the reduced tangential component s t
k to satisfy (2.4) and the
following
if
H k is positive definite and k
6.1. Discretized optimal control formulation. Consider again problem (4.1)
and assume that this problem has the structure described in Section 4.1. We can now
use Theorem 5.2 to obtain a local rate of convergence.
Theorem 6.1. Suppose that the ECO problem is of the form (4.1). Let the
general assumptions and assumption (4.2) hold and assume that fx k g converges to x .
In addition to this, let s t
k , and k be given by (6.1), (4.4) and (4.3).
If r 2
xx '(x ; ) is positive definite on N (rC(x
then x k converges q-quadratically to x .
Proof. It can be shown by appealing to Theorem 8.4 in [7] that r x '(x ;
It follows from Theorem 5.2 that ffi k is uniformly bounded away from zero. Thus
there exists a positive integer k such that for all k k,
\GammaC y
. Now the rate of convergence follows from [19].
6.2. Normal component and least-squares multipliers. Consider the general
ECO problem (1.1) again and suppose that the quasi-normal component is the
normal component (4.6) and k is given by (4.7).
We can now use Theorem 5.2 to obtain the desired local rate of convergence. It
is assumed that the orthogonal null-space basis is a continuous function of x.
Theorem 6.2. Let the general assumptions hold and assume that fx k g converges
to x . In addition to this, let
k , and k be given by (6.1), (4.6), and (4.7).
If r 2
xx '(x ; ) is positive definite on N (rC(x
then x k converges q-quadratically to x .
Proof. It can be shown by appealing to Theorem 8.4 in [7] that r x '(x ;
It follows from Theorem 5.2 that ffi k is uniformly bounded away from zero. Thus
there exists a positive integer k such that for all k k,
. The q-quadratic rate of convergence follows from [18], [36].
7. The trust-region subproblem for the linearized constraints. In this
section we investigate a few aspects of the trust-region subproblem for the linearized
constraints
subject to ks n k
First we prove that the normal component (4.6) and the quasi-normal component
always give a fraction of optimal decrease on this trust-region subproblem.
Theorem 7.1. Let the general assumptions hold. Then:
(i) The normal component (4.6) satisfies a fraction of optimal decrease on the
trust-region subproblem for the linearized constraints, i.e. there exists a positive
constant fi n
1 such that
where s
k is the optimal solution of (7.1).
(ii) In addition, assume assumption (4.2). The quasi-normal component (4.4)
satisfies the fraction of optimal decrease (7.2).
Proof. (i) If krC k (rC T
solves (7.1), and the result
holds for any positive value of fi n
1 in (0; 1]. If this is not the case, then
since krC k (rC T
We also have
ks
ks
ks
since krC k (rC T
ks
k. Combining this last inequality with
(7.3) we get
and this completes the proof of (i).
(ii) If kC y
solves (7.1), and (7.2) holds for any positive
value of fi n
1 . If this is not the case, we have
is the uniform bound on kC y Now the rest of the proof follows as
in (i).
As a consequence of this theorem, we have immediately that the normal component
(4.6) and the quasi-normal component (4.4) give a fraction of Cauchy decrease
on the trust-region subproblem for the linearized constraints.
To compute a step s n
k that gives a fraction of optimal decrease on the trust-region
subproblem for the linearized constraints we can also use the techniques proposed in
[23], [28], [31].
In the next theorem we show that the trust-region subproblem (7.1), due to its
particular structure, tends to fall in the hard case in the latest stages of the algorithm.
This result is relevant in our opinion since the algorithms proposed in [23], [28], [31]
deal with the hard case.
The trust-region subproblem (7.1) can be rewritten as
subject to ks n k
The matrix rC k rC T
k is always positive semi-definite and, under the general assump-
tions, has rank m. Let E k (0) denote the eigenspace associated with the eigenvalue
0g. The hard case is defined by the two
following conditions:
(a) (v k
(b) k(rC k rC T
Theorem 7.2. Under the general assumptions, if lim k!+1
exists a k h such that, for all k k h , the trust-region subproblem (7.5) falls in the hard
case as defined above by (a) and (b).
Proof. First we show that (a) holds at every iteration of the algorithm. If v k 2
Multiplying both sides by (rC T
k gives us
Thus (v k
Now we prove that there exists a k h such that (b) holds for every k k h . Since
is a monotone strictly decreasing function of
lim
is equivalent to g k () ! rffi k , for all ? 0. Also, from the singular value decomposition
of rC k , we obtain
lim
Hence holds for all ? 0 if and only if krC k (rC T
Now since lim k!+1
there exists a k h such that kC k
, and this
completes the proof of the theorem.
We complete this section with the following corollary.
Corollary 7.1. Under the general assumptions, if lim k!+1 kC k and the
trust radius is uniformly bounded away from zero, then there exists a k h such that, for
all k k h , the trust-region subproblem (7.5) falls in the hard case as defined above by
(a) and (b).
Proof. If lim k!+1 kC k and the trust radius is uniformly bounded away from
zero then lim k!+1
Theorem 7.2 can be applied.
8. Concluding remarks. In Theorems 3.1 and 3.2 we have established global
convergence to a point satisfying the second-order necessary optimality conditions
for the general trust-region-based algorithm considered in this paper. In Theorem
5.2 we have proved that the trust radius is, under sufficient second-order optimality
conditions, bounded away from zero. With the help of this result we analyzed local
rates of convergence for different choices of steps and multipliers. We believe that
these results complement the theory developed by Dennis, El-Alem, and Maciel in [7]
that proves global convergence to a stationary point. We have also given a detailed
analysis of the trust-region subproblem for the linearized constraints.
Acknowledgments
. We thank Mahmoud El-Alem with whom we had many
discussions about the contents of this paper. We are also grateful to our referees for
their careful and insightful reading of this paper.
--R
Continuity of the null space basis and constrained optimiza- tion
A trust region algorithm for nonlinearly constrained optimization
On the global convergence of trust region algorithms using inexact gradient information
A trust region strategy for nonlinear equality constrained optimization
A note on the computation of an orthonormal basis for the null space of a matrix
A new trust region algorithm for equality constrained optimiza- tion
A global convergence theory for general trust-region-based algorithms for equality constrained optimization
Numerical Methods for Unconstrained Optimization and Nonlinear Equations
A Global Gonvergence Theory for a Class of Trust Region Algorithms for Constrained Optimization
A global convergence theory for arbitrary norm trust-region algorithms for equality constrained optimization
Computing optimal locally constrained steps
Properties of a representation of a basis for the null space
Some theoretical properties of an augmented Lagrangian merit function
Newton's method for constrained optimization
Projected sequential quadratic programming methods
Analysis of inexact trust-region interior-point SQP algorithms
On the implementation of an algorithm for large-scale equality constrained optimization
Trust Region Algorithms for Optimization with Nonlinear Equality and Inequality Constraints
A new algorithm for unconstrained optimization
A trust region algorithm for equality constrained optimization
A semidefinite framework for trust region subproblems with applications to large scale minimization
A family of trust-region-based algorithms for unconstrained minimization with strong global convergence properties
Newton's method with a model trust region modification
The conjugate gradient method and trust regions in large scale optimization
Towards an efficient sparsity exploiting Newton method for minimization
A trust region algorithm for equality constrained minimization: convergence properties and implementation
Numerical Methods for Nonlinearly Constrained Optimization
An improved successive linear programming algorithm
--TR
--CTR
Zhensheng Yu , Changyu Wang , Jiguo Yu, Combining trust region and linesearch algorithm for equality constrained optimization, Journal of Computational and Applied Mathematics, v.14 n.1-2, p.123-136, 1 January 1986
Detong Zhu, Nonmonotonic back-tracking trust region interior point algorithm for linear constrained optimization, Journal of Computational and Applied Mathematics, v.155 n.2, p.285-305, 15 June | equality-constrained optimization;second-order necessary optimality conditions;trust regions;local rate of convergence;hard case;SQP methods |
589060 | Algorithms for Constrained and Weighted Nonlinear Least Squares. | A hybrid algorithm consisting of a Gauss--Newton method and a second-order method for solving constrained and weighted nonlinear least squares problems is developed, analyzed, and tested. One of the advantages of the algorithm is that arbitrarily large weights can be handled and that the weights in the merit function do not get unnecessarily large when the iterates diverge from a saddle point. The local convergence properties for the Gauss--Newton method are thoroughly analyzed and simple ways of estimating and calculating the local convergence rate for the Gauss--Newton method are given. Under the assumption that the constrained and weighted linear least squares subproblems attained in the Gauss--Newton method are not too ill conditioned, global convergence towards a first-order KKT point is proved. | Introduction
. Assume that f : R n continuously differentiable
function and that diagonal matrix with weights
We will discuss the Gauss-Newton method and a second order method for
solving the problem
min
x2R
denotes the 2-norm. For simplicity, and without loss of generality, we
assume that the weights are normalized and sorted such that
The normalization is easily done by first sorting out the zero weights, reducing the
problem and then dividing the remaining nonzero weights with the smallest positive
weight.
To our knowledge all existing algorithms for solving (1.1) are based on the un-weighted
problem
min
Assume that the ordinary Gauss-Newton method
is used to solve (1.3). The search direction, p, is then got by solving
min p2
(1.
rg. Note that (1.4) is solved as an unweighted problem and thus the
condition of this problem is determined by kKk kK y k where K y is the pseudo inverse
of K.
If we on the other hand linearize (1.1), without explicitly multiplying
with the weights, we solve the weighted linear least squares problem
to obtain the search direction p. The condition for the problem (1.5) is mainly determined
by kBk kJk where . For a more detailed discussion on condition
numbers for (1.5) see [11]. The problem (1.4) may be very ill conditioned (regarded
as an unweighted linear least squares problem) despite the fact that (1.5) is well conditioned
(regarded as a weighted linear least squares problem). Obviously it is very
important to look at (1.1) as the class of weighted nonlinear least squares problem.
Another important advantage of using (1.1) instead of (1.3) is that the former
defines a more general problem class than the latter. This is evident if we allow the
weights to be infinitely large. To be more precise, we define the vector -
the equations
weights correspond to zero elements in M . Note that
is the Lagrange multiplier corresponding to the ith constraint and
consequently - i is not defined by (1.6). We will return to the proper way of calculating
these Lagrange multipliers. Problem (1.1) is rewritten, using (1.6), as
Hence, by allowing infinite weights, our original problem formulation (1.1) defines the
class of weighted nonlinear least squares problems with nonlinear equality constraints.
To be even more specific we assume that we have p infinite weights such that
Problem (1.7) can now be stated as
\Theta f T
An equivalent formulation of problem (1.8)
using - is
min
2 . Of course, we could have started by defining our problem as
the one in (1.9) instead of (1.5) (without the need of (1.7) and (1.8)) but then the
notations would get unnecessarily complicated.
In the next section we describe the Gauss-Newton method for solving (1.1). The
local convergence properties of the Gauss-Newton method is analyzed in Section 3
and in Section 4 we show that, under certain assumptions on nondegeneracy, global
convergence is achieved. If the Gauss-Newton method is too slow or does not converge
and second derivatives are available at a reasonable cost then the Newton method
may be used to solve (1.1). However, when there are large and possibly infinite
weights a pure Newton method based on forming the Hessian of g(x) may not work
or, with infinite weights, is not even defined. The natural approach is then to use the
Perturbation method [9] that we will call the generalized Newton-Raphson method
(the gNR method). In Section 5 we construct and analyze an algorithm for solving
(1.1) based on the gNR method. Computational experiments are presented in Section
6 and finally we discuss our results and give hints of possible future work.
2. The Gauss-Newton Method Using the System Equations. In the Gauss-Newton
method, the nonlinear least squares problem (1.1) is linearized around the
current iteration point, x k , and the search direction, p k , is computed as the solution
to
The next iterate is x
the steplength. In the presence of large weights, possibly infinite, it is adequate to
reformulate (2.1) as
\Gammaf#
where we for simplicity have dropped the iteration index k. There are several names
to the linear system of equations in (2.2) such as the equilibrium equations, the system
equations or the augmented system equations. We call (2.2) the system equations and
the matrix in (2.2) is called the system matrix. A less obvious reason for using (2.2)
is that the elements in - corresponding to infinite weights are approximations to the
Lagrange multipliers and that - can be used in a second order method as described
in Section 5.
The following lemma gives the relevant conditions for the system matrix to be
nonsingular.
Lemma 2.1. The system matrix in (2.2) is nonsingular if and only if the rows in
J that correspond to infinite weights are linearly independent and J has full column
rank.
There exist several stable algorithms that solve (2.2), see e.g. [5] for further
references. We have chosen to use the modified QR decomposition, see [5], and the
reasons are the following. The modified QR decomposition is simple and easy to
compute and it is identical to the ordinary QR decomposition when the weights are
equal. The modified QR decomposition is also easily reused in the second order gNR
method, see Section 5.
The modified QR decomposition of J 2 R m\Thetan is defined as
R#
n\Thetan is an upper triangular matrix and \Pi is a permutation
matrix. The decomposition in (2.3), with Q and R nonsingular, exists if and only if
the system matrix in (2.2) is nonsingular (see Lemma 2.1).
The system equations are solved with the modified QR decomposition in the
following way. Using the decomposition (2.3) in (2.2) we get6 6 4
R#
If we make the partition Mm\Gamman ) the solution to (2.4) is
m\Gamman
3. The local rate of convergence for the Gauss-Newton method. In this
section we will describe the local convergence properties of the Gauss-Newton method
described in the previous section. Our analysis depends much upon the perturbation
analysis of the constrained and weighted linear least squares problem done in [11, 12].
After having defined the inverse of the system matrix, using the same notation as in
[11], we state and prove two important theorems on the local convergence rate for
projected residual). In fact, the local convergence properties of
these two quantities are, as we shall see, very similar. Finally we show that J k p k and
the local convergence rate for x
x and J k p k are independent of the parametrization
in R n .
Assuming that b x is a solution of (1.1), we define b
and the corresponding
notation for other quantities evaluated at b
x.
A necessary condition for our algorithm to converge without regularization is that
the system matrix in (2.2) has full rank and it is convenient to make the following
definition.
Definition 3.1. If the system matrix in (2.2) is nonsingular at x we say that x
is a nondegenerate point.
At a nondegenerate point the inverse of the system matrix in (2.2) is given by
is a generalized inverse of J , see [11]. From (2.5) we immediately
get
The following theorem describes the local behaviour of x
Theorem 3.1. Assume that fp k g are generated by solving (2.2) and that all
points x are nondegenerate. If b x is the solution of (1.1) and b
- is the
vector - from (2.2) at b x then
R 1f 00
Proof. From x
f)
Using the Taylor expansion
the first term in (3.4) can be expressed as
To express the second term, \GammaB k
f , in (3.4) we use the perturbation identity (2.2) p.
in [11] which says that
Using the identity
Z 1[f 00
equation (3.6) becomes
The equations (3.5) and (3.7) inserted into (3.4) gives the theorem.
The Gauss-Newton method can be written as
and with b
From Theorem 3.1 we conclude that
and from [8] we get the following theorem.
Theorem 3.2. Define
and - i as the eigenvalues of H x . Then
lim sup
xk
xk
It is easy to get an estimation of the local convergence rate if we use the matrix
defined by (3.2), because then b
R
R \GammaT \Pi T .
A useful quantity for estimating how close x k is to the solution, b x, is the projected
residual is the oblique projection of f k onto R(J k ).
The following theorem shows that J k p k locally has the same convergence behaviour
as
Theorem 3.3. Assume that fp k g are generated by solving (2.2) and that all
points x are nondegenerate. If -(x k ) is the vector - from (2.2) at x k
then
R 1f 00
Proof. Denote the projection J k B k by P k . Then we have
Using the Taylor expansion
multiplying with P k+1
we obtain
I , the equality B k s holds, and we can identify the last term in
equation (3.14) as
From (3.1) we get
and hence
From the perturbation identity (2.1) p. 16 in [11] we get
Using (3.18) and the fact that Y k+1 ffiJ k B k
the equation (3.17) becomes
where the last equality follows from (3.16). The identities -
\Gammap k together with a Taylor expansion of (ffiJ k ) T give
The theorem follows by inserting (3.15) and (3.19) into (3.14).
The matrix corresponding to H x for the projected residual, s k , is
B:
and it is easy to show that H x and H s has the same nonzero eigenvalues. Hence, we
have from Theorem 3.3 the following corollary.
Corollary 3.1. Define B k from the inverse of the system matrix in (3.1). If
lim sup
ks
are the eigenvalues of the matrix H x defined in (3.11).
The relation (3.12) can also be used to determine when ks k+1 k=ks k k reflects
the linear convergence rate and if a second order method should be used. If the
convergence of the Gauss-Newton method is slow we use a higher order method if2
see also Algorithm 6.1.
Several of the above quantities are invariant under a change of parametrization
and as an example we have the following theorem.
Theorem 3.4. The matrix
B:
is independent of the parametrization in R n .
Proof. Assume that
f(x(')). We want to show that
y
B y is the generalized inverse of ry( b
'). Now, consider the Taylor expansion
where
\Delta' T x 00
\Delta' T x 00
By comparing the Taylor expansion (3.21) above with the Taylor expansion
using that J T
that
From (3.23) we finally get
y
which proves the theorem.
A consequence of Theorem 3.4 is that the local convergence for is the same
as for x
The main argument for choosing Jp as a measure of the closeness to the solution
is the following theorem which is a direct consequence of (3.23).
Theorem 3.5. The projection of f on R(J), is independent of the
parametrization in R n .
4. Global convergence. In this section we assume that x k , where k is the
iteration index, is nondegenerate and that p k is the solution of (2.2) at x k . If nothing
else is stated we assume that all limits denoted by ! are when k ! 1, and that all
sums with no explicitly stated upper or lower limit are from one to infinity.
4.1. The merit function. As a merit function we have chosen
The goal is to find a matrix D k of merit weights and a step length ff k , at each
iteration, such that global convergence towards a first order Kuhn-Tucker point can
be proved. To compute D k we will use the approximation
of D). For a fixed matrix D, we define Obviously
a sufficient condition on p k to be a descent direction to \Phi(x; D) at x k is that OE 0
We realize that we can determine a good matrix, \Upsilon(x k ), of merit weights by
solving
min
k\Upsilonk
s:t:
where ffi is a small positive constant and - i is a lower limit for the weights determined
by some previously computed weights, see below. There is always a solution to (4.2)
because
lim
argf min ff
Note that keeping the weights not too large is important in practice but for the global
convergence it is only the constraints in (4.2) that must be satsified. We will now
describe the algorithm for computing the merit weights D k , using \Upsilon(x k ), such that
does not become unnecessarily large. We first describe a method for solving (4.2)
and then an algorithm for computing the actual merit weights D k .
When solving (4.2) we have chosen to use the max-norm since this gives a simple
algorithm. The problem (4.2) can be rewritten as
min kuk1
where u is the diagonal in \Upsilon(x k ), ! is the diagonal in W , and y
with that when Jp is given the problem (4.3) consists of only vectors
and no matrices. The first step in our algorithm is to reduce (4.3) such that u
We then get a new problem
y are the corresponding parts of u; -; ! and y left after the reduction
ae we are ready with the solution -
-. Otherwise
we choose -
y, where e is a vector of ones and thus attain equality in the
constraints. If -
respectively. Again we
can reduce the problem to a copy of (4.4) but where the vectors are shorter and ae
is smaller. The procedure is then repeated until the whole of u is found. It is easily
realized that the infinite weights in ! do not change the algorithm and the algorithm
will terminate with a solution of (4.4).
We determine the actual merit weights D k from the solution \Upsilon(x k ) of (4.2). The
weights may get large close to a saddle point and when the iterates diverge from this
saddle point (that is always the case with the Gauss-Newton method) we would like the
weights to decrease. This is accomplished by saving, say t, older versions,
the merit weight matrices. Initially, at iteration
and at the kth iteration we update
m ), as in Algorithm 4.1.
Algorithm 4.1.
Solve (4.2) for the vector u(x k ).
If d (k)
i be the new
sequence - (1)
In Algorithm 4.2 our Gauss-Newton algorithm is described with line search and
quadratic merit function.
Algorithm 4.2.
Initiate the start vector x k .
while not convergence
Compute
Compute using the modified QR decomposition of J k .
Determine D k from Algorithm 4.1.
Determine the step length ff k such that
4.2. Proving global convergence. We will need the following two technical
lemmas to prove that our algorithm is globally convergent. In the lemmas we use d k
as an arbitrary diagonal element in D k .
Lemma 4.1. Assume that d k - 0; and that fd k g is bounded. Let
be the subsequence of fd k g such that d k j+1 ? d k j . Then the positive series
converges if and only if
converges.
Proof. Take
a
a
where a
converges too. Now assume that b +
N converges to b
N . Hence,
k g is a bounded
sequence that increases to a limit b \Gamma and
converges to b
Lemma 4.2. Assume that an arbitrary component, d k , in the diagonal of D k
stays bounded as k ! 1 and let v k be the corresponding diagonal element in V t .
Then lim and the series
converges.
Proof. Let us first exclude the trivial case that v k becomes equal to the upper
bound ! for a finite k.
The sequence fv k g is an increasing infinite sequence. Hence, lim v k exists and is
denoted v. Take
ffl be an arbitrary small but
fixed positive number. Then d k ? -
k-values. Hence, v ? -
and since ffl ? 0 was arbitrary this implies that v -
d. From d k - v k it follows that
thus we have v -
d - d - v and consequently d k ! v.
Let fd i k g be the subsequence of fd k g with d i k+1 ? d i k . From lemma 4.1 we know
that the series
converges if and only if
converges. Let
us now prove that the latter series converges. From v i k - d i k it follows that
and hence
Since
a subserie of
increases to v, the series
converges. Since
positive series it is sufficient to prove
that it is bounded. Hence, it only remains to prove that the series
converges. Since d the saved older weights are updated in step i 1 . When
we reach d i 1+t
there have been t updates and v i 1+t
equals one of the earlier d
t. In this way we can eliminate both this v i 1+t and the corresponding d i j . In
the same way it is seen that v i 1+t+1
equals one of the d i j . That pair can also be
eliminated from the series. We go on and eliminate elements in this way to get
t. Thus the positive series
bounded and so converges.
That completes the proof.
Our main global convergence theorem covers both bounded and unbounded sequences
of merit weights.
Theorem 4.3. Let fx k g and fD k g be generated by Algorithm 4.2. Assume that
is bounded and that the system matrix in (2.2) is nonsingular in the closure of
g. Then the sequence fx k g has either finite termination at a KKT point or an
accumulation point that is a KKT point of (1.1).
Proof. It is trivial that there is finite termination just at KKT points. Let us now
assume that we have an infinite sequence. Algorithm 4.2 implies that it is sufficient
to consider the following two cases :
These cases will now be treated separately.
There exist a subsequence fx i k g of fx k g such that kD i
is bounded it its possible to choose a subsequence fx j k g of fx i k g such
that x
x for some e
x. From Algorithm 4.2 it follows that kD k k ! 1 only when
is continuous for all points in the closure of fx k g except
KKT points, e
x is both an accumulation point of fx k g and a KKT point.
ii) From the inequality
one can prove that a point e x cannot be an accumulation point of fx k g if there exist
(The proof of (4.5) is a trivial extension of a similar proof in [7] pp. 21-22.)
From Lemma 4.2 we know that
converges and from the Goldstein-
Armijo condition in Algorithm 4.2 then, for a given D k , it follows that for every point
e
x in the closure of fx k g, that is not a KKT point, there exists constants ffl ? 0
that (4.5) is satisfied. Hence, only KKT points remain as possible
accumulation points. That proves the theorem in case ii).
4.3. Line search. We have chosen to keep things simple and therefore we use
a standard cubic interpolation from [3] to approximate the minimum of our merit
function OE(ff). Another, more efficient, line search algorithm can be found in [6].
4.4. Regularization. We use a simple form of subspace minimization described
for the unweighted and constrained case in [7]. We have not been able to prove a
general global convergence result as the one in Theorem 4.3 but as we shall see in the
computational experiments our regularization seems to work appropriately.
5. The generalized Newton-Raphson method. A constrained Newton method
for solving (1.9) can be based on the quadratic subproblem
G)p
are first order approximations
of the Lagrange multipliers. The solution, -
p, to (5.1) is given by the linear
system of equations
G
\Gammaf#
The main disadvantage with using (5.2) is that for very large weights in W 2 the
quadratic subproblem (5.1) and the matrix in (5.2) may be very ill conditioned.
To avoid the ill conditioning due to large weights in W 2 we solve
\Gammaf#
and - is from (2.2). This method is the generalized Newton-Raphson
method [9], or just the gNR method.
The gNR method has an interesting theoretical motivation. Assume that we have
reached a point x k . From the first order approximation (1.5) it is known that x k
solves the perturbed problem
min
is a projection onto R(J k ). Hence, we know
the solution x k of (5.4) and want to compute the solution of the perturbed problem
min
Then we can use the quadratic approximation of z(x) at x k to compute a solution of
problem (5.5) whose error is O(kP k f k k 2 ). If we change back to the original notations
in f(x), this perturbed solution is found by solving problem (5.3) for
From (5.3) it is seen that there exists a matrix N k such that
x as the solution to (1.1) and
Take x . Then from the quadratic approximation in (5.6) we get
From (5.6) it is also seen that J k N k only depends on the surface and not on the
parameterization in x and consequently J k p k is independent of the parameterization
in R n . The generalized Newton-Raphson method is in fact the only quadratically
convergent method with J k p k independent of the parametrization. To see this we
assume that there exists another method which computes e
e
. The series expansion (5.6) is unique and we have J k
e
which implies that e
If we define Z 1 as a matrix whose columns span the null space of J 1 we call p a
descent direction if p T Z T
drawback with both the constrained Newton
method based on (5.2) and the gNR method is that a nonsingular matrix in (5.2)
or (5.3) is not sufficient for p to be a descent direction. However, we use the gNR
method only when we are close to the solution, see (3.20) and Algorithm 6.1, and
therefore we use the gNR method undamped. From (2.2) we get -, needed for G, and
the Gauss-Newton search direction and if the matrix in (5.3) is singular we use the
already available Gauss-Newton direction.
If we use the modified QR decomposition to solve (2.2) it is possible to reduce the
size of the system in (5.3). Ignoring the permutation matrix it is possible to rewrite
R#
Now implies that Q and we can reduce (5.7) to
R T \GammaG
\Gamma-
are the first n elements in Q T - and Q
respectively.
The matrix in (5.8) may be indefinite and we must either use a stable method
for indefinite systems, see e.g. [4], or add some condition on the submatrices in (5.8).
One possibility of the latter kind is to assume that R is well conditioned and use R T
to reduce (5.8) to
R T \GammaG
"\Gamma-
The solution is the matrix R+MnR \GammaT G is nonsingular,
if not we take a Gauss-Newton step.
6. Computational experiments. The algorithm we use in our tests is shown
below.
Algorithm 6.1.
Initialize
while
Determine the Jacobian J and the vector f .
Compute the GN direction p and - by solving (2.2).
If regularization was needed then Second := false.
If Close and Second and Rate ? 0:5
Compute the gNR direction, p gNR , by solving (5.3).
If the matrix in (5.3) is nonsingular then
If GN
Compute the merit weights by Algorithm 4.1.
Determine the step length ff using the line search described
in Section (4.3) with the merit function OE(ff).
x
To use a pure GN method then the variable Second has a fixed value of false.
We have tested our algorithm on three different problems described in the Ap-
pendix; Schittkowski 308 [10], Boggs 2 and Boggs 8 [2]. The intention with the tests is
not to show that the algorithms are faster than other existing algorithms but to show
how our algorithms handles large weights and inadequate models (ill conditioning in
the linear problems). Another important aim with the tests is to verify our theoretical
results on the local convergence rate. Therefore it has been natural to use small and
simple test problems.
We define as two different
measures of the convergence rate for the Gauss-Newton method. We emphasize that
k is an excellent way of estimating the convergence rate when regularization is not
needed and when b x is not known.
The first problem, Schittkowski 308, is first solved with the Gauss-Newton method
and the result is in Tab. 1. The largest weight is 10 20 and if the weights are multiplied
explicitly with f , forming then the algorithm breaks down because of
numerical instability. Note the slow growth of the merit weights. The first problem
Schittkowski 308 with the Gauss-Newton method.
5 7.3e-3 6.6e-5 2.7e-3 4.6e-2 4.4e-2 3.0 99 1.0
6 3.4e-4 3.0e-6 1.2e-4 4.6e-2 4.6e-2 2.9 3.9 1.0
8 7.2e-7 6.5e-9 2.7e-7 4.6e-2 4.7e-2 3.0 1.0e+2 1.0
9 3.3e-8 3.0e-10 1.2e-8 4.6e-2 4.6e-2 3.0 1.0e+2 1.0
Table
Schittkowski 308 with the gNR method.
5 5.9e-5 5.6e-12 2.2e-5 1.0
6 2.9e-11 2.3e-16 1.1e-11 1.0
solved with the gNR method is showed in Tab. 2. The asterisk indicates that the
gNR method was used in that step. The second problem, Boggs 2, is a constrained
problem and it has been solved with the Gauss-Newton method, Tab. 3, and the gNR
method, Tab. 4. All the merit weights for the Gauss-Newton method were equal to
one and are not shown in the Tab. 3. The remaining two test problems illustrate the
regularization. The rank of the problem is shown under the headline Rank. In Tab.
5 the second test problem, Boggs 2, is solved with the Gauss-Newton method when
the Jacobian is rank deficient at the starting point. In the third problem, Boggs 8,
the Jacobian at the solution is rank deficient and the result is shown in Tab. 6.
7. Discussion. We claim that we have developed an efficient and fairly robust
algorithm for solving (1.1) (with possibly infinite weights as discussed in the intro-
duction). However, it is difficult for us to measure the effectiveness of the algorithm
Boggs 2 with the Gauss-Newton method.
3 1.2e-2 2.8e-2 2.3 0.13 0.50 2.6 1.0
5 2.4e-4 2.5e-4 7.3e-3 0.10 1.9e-2 1.6e-2 1.0
6 6.4e-5 6.6e-5 3.0e-4 0.27 4.2e-2 1.6e-2 1.0
Table
Boggs 2 with the gNR method.
3 1.2e-2 2.8e-2 2.3 1.0
5 2.4e-4 2.5e-4 7.3e-3 1.0
6 6.4e-5 6.6e-5 3.0e-4 1.0
9 8.2e-16 1.0e-15 1.9e-15 1.0
because there are, to our knowledge, no other algorithm that can solve such a general
problem as (1.1).
The local convergence properties are well understood for the Gauss-Newton algo-
rithm. It is especially interesting that the local convergence results are valid for the
whole problem class defined by (1.1) and that they are independent of the parametrization
in R n .
The merit function is especially suited for our weighted and constrained problem
and our technique for choosing the merit weights is effective and do not lead to
unnecessary large weights.
Boggs 2, Gauss-Newton and rank deficient at the starting point.
9 0.47 0.71 0.14 1.0 0.12 3
As for robustness, we have shown that our algorithm is globally convergent when
the iteration points are nondegenerate. It remains to find a way to regularize when
the rows in J corresponding to very large weights become (almost) linearly dependent.
We believe that this is a difficult and challenging problem to solve.
Appendix
. Test problems. In this appendix we define our three test problems
and the weight sequences. We also give the starting points, x start ; solutions, b
the residuals f(bx). The examples are from [10] and [2] and includes unconstrained as
well as constrained problems.
Schittkowski 308 [10]
An unconstrained problem which we have modified by incorporation of weights.
A constrained problem where the Jacobian is rank deficient at the second
Boggs 8, Gauss-Newton and rank deficient at the solution.
28 1.1 1.0 0.65 1.0 3.4e-10 5
29 1.1 0.99 1.6 3.5 9.5e-7 4
53 4.2e-9 0.25 0.50 3.5 1.0 4
54 1.0e-9 0.25 0.50 3.5 1.0 3
starting point, x start2 .
Boggs 8 [2]
A constrained problem where the Jacobian is rank deficient at the solution.
--R
A Strategy for Global Convergence in a Sequential Quadratic Programming Algorithm
Numerical methods for unconstrained optimization and nonlinear equations
Matrix Computations
Modifying the QR decomposition to weighted and constrained linear least squares
Iterative solution of nonlinear equations in several variables
A comparision of some algorithms for the nonlinear least squares problem
More Test Examples for Nonlinear Programming Codes
Perturbation theory and condition numbers for generalized and constrained linear least squares problems
--TR
--CTR
Hiroshi Hosobe, Hierarchical nonlinear constraint satisfaction, Proceedings of the 2004 ACM symposium on Applied computing, March 14-17, 2004, Nicosia, Cyprus | nonlinear least squares;parameter estimation;optimization;weights |
589066 | On the Convergence of Pattern Search Algorithms. | We introduce an abstract definition of pattern search methods for solving nonlinear unconstrained optimization problems. Our definition unifies an important collection of optimization methods that neither compute nor explicitly approximate derivatives. We exploit our characterization of pattern search methods to establish a global convergence theory that does not enforce a notion of sufficient decrease. Our analysis is possible because the iterates of a pattern search method lie on a scaled, translated integer lattice. This allows us to relax the classical requirements on the acceptance of the step, at the expense of stronger conditions on the form of the step, and still guarantee global convergence. | Introduction
. We consider the familiar problem of minimizing a continuously
di#erentiable function f : R n
# R. Direct search methods for this problem
are methods that neither compute nor explicitly approximate derivatives of f . Our
interest is in a particular subset of direct search methods that we will call pattern
search methods. Our purpose is to generalize these methods and to present a global
convergence theory for them. To our knowledge, this is the first convergence result
for some of these methods and the first general convergence theory for all of them.
Examples of pattern search methods include such classical direct search algorithms
as coordinate search with fixed step sizes, evolutionary operation using factorial
designs (first proposed by G. E. P. Box [2, 3, 13]), and the original pattern search
algorithm of Hooke and Jeeves [7]. A more recent example is the multidirectional
search algorithm of Dennis and Torczon [6, 15]. For some time, it has been apparent
to us that the unifying theme that distinguishes these algorithms from other direct
search methods is that each of them performs a search using a "pattern" of points
that is independent of the objective function f . This informal insight is the basis for
our general definition of pattern search methods-it turns out that each of the above
pattern search methods is an instance of our general model.
Formally, our definition of pattern search methods requires the existence of a
lattice T such that if {x 1 , . , xN } are the first N iterates generated by a pattern
search method, then there exists a scale factor #N such that the steps {x 1
lie in the scaled lattice #N T . The lattice depends on the
pattern that defines the individual method and on the initial choice of the step length
control parameter, but it is independent of the objective function f . The scaling
# Received by the editors June 23, 1993; accepted for publication (in revised form) September 20,
1995. This research was sponsored by Air Force O#ce of Scientific Research grants 89-0363, F49620-
93-1-0212, and F49620-95-1-0210; United States Air Force grant F49629-92-J-0203; and Department
of Energy grant DE-FG005-86ER25017. This research was also supported in part by the Geophysical
Parallel Computation Project under State of Texas contract 1059.
http://www.siam.org/journals/siopt/7-1/25078.html
Department of Computer Science, The College of William &Mary, Williamsburg, VA 23187-8795
(va@cs.wm.edu). This work was completed while the author was in the Department of Computational
and Applied Mathematics and the Center for Research on Parallel Computation, Rice University,
Houston,
depends solely on the sequence of updates that have been applied to the step length
control parameter.
Despite isolated convergence results [4, 11, 16] for certain individual pattern search
methods, a general theory of convergence for the class of such methods remained
elusive for some time. The standard convergence theory for line search and trust
region methods depends crucially on some notion of su#cient decrease, but pattern
search methods do not enforce any such notion. Therefore, attempts such as [18]
to apply the standard theory to pattern search methods arbitrarily introduce some
notion of su#cient decrease, thereby modifying the original algorithms. Thus, the
challenge was to develop a general convergence theory for pattern search methods
without redefining what they are.
Our convergence analysis is guided by that found in Torczon [16] for the multidirectional
search algorithm; however, the present level of abstraction makes the
important elements of that analysis easier to appreciate. The present paper also
includes a correction to the specification of the scaling factors found in [16].
There are three key points to our analysis. First, we show that pattern search
methods are descent methods. Second, we prove that pattern search methods are
gradient-related methods in the sense of [10]. Finally, we demonstrate that pattern
search methods cannot terminate prematurely due to inadequate step length control
mechanisms. The crucial element of this analysis is the fact that pattern search
methods are able to relax the conditions on accepting a step by enforcing stronger
conditions on the step itself. The lattice T , together with the way in which the step
length control parameter is updated, prevent a pathological choice of steps: steps of
arbitrary lengths along arbitrary search directions are not permitted.
We are able to guarantee that, if the function f is continuously di#erentiable, then
an explicit representation of the gradient or the
directional derivative. In particular, we prove global convergence for pattern search
methods despite the fact that they do not explicitly enforce a notion of su#cient
decrease on their iterates, such as fraction of Cauchy decrease, fraction of optimal de-
crease, or the Armijo-Goldstein-Wolfe conditions. However, our convergence analysis
does share certain characteristics with the classical convergence analysis of both line
search and trust region methods. This connection is both subtle and unexpected.
Our convergence analysis for pattern search methods makes it clear why these
methods are as robust as their proponents have long claimed, while clarifying some of
the limitations that have long been ascribed to them. In addition, having identified
the common structure of these methods, it is now possible to develop new pattern
search methods with guaranteed global convergence.
In section 2 we establish the notation and general specification of pattern search
methods. In section 3 we prove that if the function to be minimized is continuously dif-
ferentiable, then pattern search methods guarantee that lim inf k#f(x k
In addition, we identify the modifications that must be made to pattern search methods
to obtain the stronger result lim k#f(x k In section 4 we show that
the classical pattern search methods mentioned above, as well as the newer multidirectional
search algorithm of Dennis and Torczon, conform to the general specification
for pattern search methods. In section 5, we give some concluding remarks; section 6
contains technical results needed for the proofs of section 3.
Notation. We denote by R, Q, Z, and N the sets of real, rational, integer, and
natural numbers, respectively.
All norms are Euclidean vector norms or the associated operator norm. We define
ON THE CONVERGENCE OF PATTERN SEARCH ALGORITHMS 3
2. Pattern search methods. We begin by introducing the following abstraction
of pattern search methods. We defer to section 4 demonstrations that the pattern
search methods mentioned above fall comfortably within this abstraction.
2.1. The pattern. To define a pattern we need two components, a basis matrix
and a generating matrix.
The basis matrix can be any nonsingular matrix B # R n-n .
The generating matrix is a matrix C k # Z n-p , where p > 2n. We partition the
generating matrix into components
(1)
We require that M k # M # Z n-n , where M is a finite set of nonsingular matrices,
and that L k # Z n-(p-2n) and contains at least one column, the column of zeros.
A pattern P k is then defined by the columns of the matrix P Because
both B and C k have rank n, the columns of P k span R n . For convenience, we use the
partition of the generating matrix C k given in (1) to partition P k as follows:
(2)
Given # k # R, # k > 0, we define a trial step s i
k to be any vector of the form
denotes a column of C
]. Note that Bc i
determines the direction
of the step, while # k serves as a step length parameter.
At iteration k, we define a trial point as any point of the form x i
x k is the current iterate.
2.2. The exploratory moves. Pattern search methods proceed by conducting
a series of exploratory moves about the current iterate before declaring a new iterate
and updating the associated information. These moves can be viewed as sampling the
function about the current iterate x k in a well-defined deterministic fashion in search of
a new iterate x with a lower function value. The individual pattern search
methods are distinguished, in part, by the manner in which these exploratory moves
are conducted. To allow the broadest possible choice of exploratory moves and yet still
maintain the properties required to prove convergence for the pattern search methods,
we place two requirements on the exploratory moves associated with any particular
pattern search method. These requirements are given in the following Hypotheses on
exploratory moves. (Please note an abuse of notation that is nonetheless convenient:
means that the vector y is contained in the set of columns of the matrix A.)
Hypotheses on exploratory moves.
1.
2. If min{f(x k
The choice of exploratory moves must ensure two things:
1. The direction of any step s k accepted at iteration k is defined by the pattern
its length is determined by # k .
2. If simple decrease on the function value at the current iterate can be found
among any of the 2n trial steps defined by # k B# k , then the exploratory
moves must produce a step s k that also gives simple decrease on the function
value at the current iterate. In particular, f(x k need not be less than
or equal to min{f(x k
Thus, a legitimate exploratory moves algorithm would be one that somehow
guesses which of the steps defined by # k P k will produce simple decrease
and then evaluates the function at only one such step. (And that step may
be contained in # k BL k rather than in # k B# k .) At the other extreme, a
legitimate exploratory moves algorithm would be one that evaluates all p
steps defined by # k P k and returns the step that produced the least function
value.
These are the properties of the exploratory moves that enable us to prove
lim inf
even though we only require simple decrease on f . Thus we avoid the necessity
of enforcing either fraction of Cauchy decrease, fraction of optimal decrease, or the
Armijo-Goldstein-Wolfe conditions on the iterates. To obtain
lim
we need to place stronger hypotheses on the exploratory moves as well as place a
boundedness condition on the columns of the generating matrices. These extensions
will be discussed further in section 3.3.2.
2.3. The generalized pattern search method. Algorithm 1 states the generalized
pattern search method for unconstrained minimization.
Algorithm 1. The Generalized Pattern Search Method.
For
(a) Compute
(b) Determine a step s k using an exploratory moves algorithm.
(c) Compute
(d) If # k > 0 then x
Update C k and # k .
To define a particular pattern search method, it is necessary to specify the basis
matrix B, the generating matrix C k , the exploratory moves to be used to produce a
step s k , and the algorithms for updating C k and # k .
2.4. The updates. Algorithm 2 specifies the requirements for updating # k .
The aim of the updating algorithm for # k is to force # k > 0. An iteration with
otherwise, the iteration is unsuccessful. Again we note that to
accept a step we only require simple, as opposed to su#cient, decrease.
Algorithm 2. Updating # k .
Given # Q, let # w0 and # k # w1 , . , # wL
(a) If # k # 0 then #
(b) If # k > 0 then #
The conditions on # and # ensure that 0 < # < 1 and # i # 1 for all # i #. Thus,
if an iteration is successful it may be possible to increase the step length parameter
k is not allowed to decrease. Not surprisingly, this is crucial to the success
of the analysis. Also crucial to the analysis is the relationship (overlooked in [16])
between # and the elements of #.
ON THE CONVERGENCE OF PATTERN SEARCH ALGORITHMS 5
The algorithm for updating C k depends on the pattern search method. For theoretical
purposes, it is su#cient to choose the columns of C k so that they satisfy (1) and
the conditions we have placed on the matrices M k # M # Z n-n and L k # Z n-(p-2n) .
3. The convergence theory. Having set up the machinery to define pattern
search methods, we are now ready to analyze these methods. This analysis produces
theorems of several types. The first, developed in section 3.1, demonstrates an algebraic
fact about the nature of pattern search methods that requires no assumption
on the function f . This theorem is critical to the proof of the convergence results
for it shows that we only need require simple decrease in f to ensure global conver-
gence. The second theorem, developed in section 3.2, describes the limiting behavior
of the step length control parameter # k if we place only a very mild condition on the
function f and exploit the interaction of the simple decrease condition for the generalized
pattern search method with the algorithm for updating # k . Finally, the third
and fourth theorems, developed in section 3.3, give the global convergence results.
The first theorem guarantees lim inf k#f(x k generalized pattern
search method that satisfies the specifications given in section 2. This is significant
since the theorem applies to all the pattern search methods we discuss in section 4
without the need to impose any modifications on the methods as originally stated.
The second theorem is equivalent to convergence results for line search and trust-region
globalization strategies. We can guarantee lim k#f(x k but to do
so requires placing stronger conditions on the specifications for generalized pattern
search methods. We could certainly impose these stronger conditions on the pattern
search methods presented in section 4-none of them are unreasonable to suggest or
to enforce-but we would do so at the expense of attractive algorithmic features found
in the original methods.
3.1. The algebraic structure of the iterates. The results found in this section
are purely algebraic facts about the nature of pattern search methods; they are
also independent of the function to be optimized. It is the algebraic structure of the
iterates that allows us to prove global convergence for pattern search methods without
imposing a notion of su#cient decrease on the iterates.
We begin by showing in what sense # k is a step length parameter.
Lemma 3.1. There exists a constant # > 0, independent of k, such that for any
trial step s i
produced by a generalized pattern search method (Algorithm 1) we
have
Proof. From (3) we have s i
k . The conditions we have placed on the
generating matrix C k ensure that c i
the smallest singular value of B. Then
The last inequality holds because at least one of the components of c i
k is a nonzero
integer, and hence #c i
k # 1.
From Lemma 3.1 we can see that the role of # k as a step length parameter is to
regulate backtracking and thus prevent excessively short steps.
Theorem 3.2. Any iterate xN produced by a generalized pattern search method
6 VIRGINIA TORCZON
(Algorithm 1) can be expressed in the following form:
z k ,
where
. x 0 is the initial guess,
. # , with # N and relatively prime, and # is as defined in the
algorithm for updating # k (Algorithm 2),
. r LB and r UB depend on N ,
. # 0 is the initial choice for the step length control parameter,
. B is the basis matrix, and
. z k # Z n ,
Proof. The generalized pattern search algorithm, as stated in Algorithm 1, guarantees
that any iterate xN is of the form
s k .
(We adopt the convention that s iteration k is unsuccessful.) We also know
that the step s k must come from the set of trial steps s i
p. The trial steps
are of the form s i
k .
Consider the step length parameter # k . For any k # 0, the update for # k given
in Algorithm 2 guarantees that # k is of the form
(Recall that We have also placed the following
restrictions on the form of # and
L. We can thus rewrite (5) as:
where r k # Z. Let
Then from (4) and (6) we have
Since # is rational, we can express # as #
, where # N are relatively prime.
Then, using (7),
z k ,
where z k # Z n .
Theorem 3.2 synthesizes the requirements we have placed on the pattern, the
definition of the trial steps, and the algorithm for updating # k . Note that this means
that for a fixed N , all the iterates lie on a translated integer lattice generated by x 0
and the columns of # rLB # -rUB # 0 B.
ON THE CONVERGENCE OF PATTERN SEARCH ALGORITHMS 7
3.2. The limiting behavior of the step length control parameter. The
next theorem combines the strict algebraic structure of the iterates with the simple decrease
condition of the generalized pattern search algorithm, along with the algorithm
for updating # k , to give us a useful fact about the limiting behavior of # k .
Theorem 3.3. Assume that L(x 0 ) is compact. Then lim inf k#
Proof. The proof is by contradiction. Suppose 0 < #LB # k for all k. From (6)
we know that # k can be written as #
The hypothesis that #LB # k for all k means that the sequence {# rk
} is bounded
away from zero. Meanwhile, we also know that the sequence {# k } is bounded above
because all the iterates x k must lie inside the set L(x 0 and the
latter set is compact; Lemma 3.1 then guarantees an upper bound #UB for {# k }.
This, in turn, means that the sequence {# rk
} is bounded above. Consequently, the
sequence {# rk
} is a finite set. Equivalently, the sequence {r k } is bounded above and
below.
Let
Then (8) now holds for the bounds given in (9), rather than (7), and we see that for
all k, x k lies in the translated integer lattice G generated by x 0 and the columns of
The intersection of the compact set L(x 0 ) with the translated integer lattice G is
finite. Thus, there must exist at least one point x # in the lattice for which x
for infinitely many k.
We appeal to the simple decrease condition in the generalized pattern search
method (Algorithm 1 (d)), which guarantees that a lattice point cannot be revisited
infinitely many times since we accept a new step s k if and only if f(x k ) >
Thus there exists an N such that for all k # N , x which implies that #
We now appeal to the algorithm for updating # k (Algorithm 2 (a)) to see that
thus leading to a contradiction.
3.3. Global convergence. Throughout the discussion in this section, we assume
that f is continuously di#erentiable on a neighborhood of L(x 0 ); however, this
assumption can be weakened, using the same style of argument found in [16].
3.3.1. The general result. To prove Theorem 3.5 we need Proposition 3.4. We
defer the proof of Proposition 3.4 to section 6 in part because we wish to discuss there
several other issues that are tangential to the proof of Theorem 3.5. It is also the case
that the proofs for the results in section 6 are similar to those given for the equivalent
results found in [16], though now restated more succinctly in terms of the machinery
developed in section 2.
Proposition 3.4. Assume that L(x 0 ) is compact, that f is continuously di#er-
entiable on a neighborhood of L(x 0 ), and that lim inf k#f(x k )#= 0. Then there
exists a constant #LB > 0 such that for all k, # k > #LB .
We emphasize that the existence of a positive lower bound #LB for # k is guaranteed
only under the null hypothesis that lim inf k#f(x k )#= 0.
Theorem 3.5. Assume that L(x 0 ) is compact and that f is continuously di#er-
entiable on a neighborhood of L(x 0 ). Then for the sequence of iterates {x k } produced
by the generalized pattern search method (Algorithm 1),
lim inf
Proof. The proof is by contradiction. Suppose that lim inf k#f(x k )#= 0.
Then Proposition 3.4 tells us that there exists #LB > 0 such that for all k, # k #LB .
But this contradicts Theorem 3.3.
3.3.2. The stronger result. We can strengthen the result given in Theorem 3.5
at the expense of wider applicability. To begin with, we must add three further
restrictions: one on the pattern matrix, one on the Hypotheses on exploratory moves,
and one on the limiting behavior of the step length control parameter # k .
First, we must ensure that the columns of the generating matrix C k are bounded
in norm, i.e., that there exists a constant C > 0 such that for all k, C > #c i
k # for all
p. Given this bound, we can place an upper bound, in terms of # k , on the
norm of any trial step s i
k .
Lemma 3.6. Given a constant C > 0 such that for all k, C > #c i
k # for all
there exists a constant # > 0, independent of k, such that for any trial
step s i
k produced by a generalized pattern search method (Algorithm 1) we have
Proof. From (3) we have s i
k . Then #s i
C||B|| .
Note that the columns of M k # M are bounded by the assumption that |M| <
+#; we use this fact in the proof of Proposition 6.4. The stronger boundedness
condition on the columns of C is needed to monitor the behavior
of L k .
Second, we must replace the original Hypotheses on exploratory moves with a
stronger version, as given below. Together, Lemma 3.6 and the Strong hypotheses
on exploratory moves allow us to tie decrease in f to the norm of the gradient when
the step sizes get small enough. This is the import of Corollary 6.5, which is given in
section 6.
Strong hypotheses on exploratory moves.
1.
2. If min{f(x k
Third, we require that lim k# We can use the algorithm for updating
to ensure that this condition holds. For instance, we can force # k
to be nonincreasing by requiring w which when taken together
with Theorem 3.3 guarantees that lim k# All the algorithms we consider
in section 4, except the multidirectional search algorithm, enforce this condition by
limiting
However, it is not necessary to force the steps to be nonin-
creasing; we need only require that in the limit the step length control parameter goes
to zero, which, in conjunction with Lemmas 3.1 and 3.6, has the e#ect of ultimately
forcing the steps to zero.
Theorem 3.7. Assume that L(x 0 ) is compact and that f is continuously differentiable
on a neighborhood of L(x 0 ). In addition, assume that the columns of
the generating matrices are bounded in norm, that lim k# and that the
generalized pattern search method (Algorithm 1) enforces the Strong hypotheses on
exploratory moves. Then for the sequence of iterates {x k } produced by the generalized
pattern search method,
lim
ON THE CONVERGENCE OF PATTERN SEARCH ALGORITHMS 9
Proof. The proof is by contradiction. Suppose lim sup k#f(x k )#= 0. Let
be such that there exists a subsequence #f(xm i
)#. Since
lim inf
given any 0 < #, there exists an associated subsequence l i such that
)#.
Then, since # k # 0, we can appeal to Corollary 6.5 to obtain for
su#ciently large,
Then the telescoping sum
l i
gives us
Since f is bounded below, f(xm i
because #f is uniformly continuous,
for i su#ciently large. However,
)# 2#.
Since equation (10) must hold for any #, 0 < #, we have a contradiction (e.g., try
The proof of Theorem 3.7 is almost identical to that of an equivalent result for
trust-region methods that was first given by Thomas [14] and which is included, in a
more general form, in the survey by Mor-e [8].
One final note: the hypotheses of Theorem 3.7 suggest that in the absence of
any explicit higher-order information about the function to be minimized, it makes
sense to terminate a generalized pattern search algorithm when # k is less than some
reasonably small tolerance. In fact, this is a common stopping condition for algorithms
of this sort and the one implemented for the multidirectional search algorithm [17].
4. The particular pattern search methods. In section 2 we stated the conditions
an algorithm must satisfy to be a pattern search method. We now illustrate
these conditions by considering the following specific algorithms:
. coordinate search with fixed step lengths,
. evolutionary operation using factorial designs [2, 3, 13],
. the original pattern search method of Hooke and Jeeves [7], and
. the multidirectional search algorithm of Dennis and Torczon [6, 15].
We will show that these algorithms satisfy the conditions that define pattern search
methods and thus are special cases of the generalized pattern search method presented
as Algorithm 1. Then we can appeal to Theorem 3.5 to claim global convergence for
these methods.
There are other algorithms for which the abstraction and accompanying analysis
holds-including various modifications to the algorithms presented-but we shall
confine our investigation to these, the best known of the pattern search methods, to
illustrate the power of our abstract approach to pattern search methods.
4.1. Coordinate search with fixed step lengths. The method of coordinate
search is perhaps the simplest and most obvious of all the pattern search methods.
Davidon describes it concisely in the opening of his belated preface to Argonne National
Laboratory Research and Development Report 5990 [5]:
Enrico Fermi and Nicholas Metropolis used one of the first digital
computers, the Los Alamos Maniac, to determine which values of
certain theoretical parameters (phase shifts) best fit experimental
data (scattering cross sections). They varied one theoretical parameter
at a time by steps of the same magnitude, and when no such
increase or decrease in any one parameter further improved the fit
to the experimental data, they halved the step size and repeated the
process until the steps were deemed su#ciently small. Their simple
procedure was slow but sure.
This simple search method enjoys many names, among them alternating direc-
tions, alternating variable search, axial relaxation, and local variation. We shall refer
to it as coordinate search.
Perhaps less obvious is that coordinate search is a pattern search method. To see
this, we begin by considering all possible outcomes for a single iteration of coordinate
search when shown in Fig. 1. We mark the current iterate x k . The x i
's
denote trial points considered during the course of the iteration. The next iterate x k+1
is marked. Solid circles indicate successful intermediate steps taken during the course
of the exploratory moves while open circles indicate points at which the function was
evaluated but that did not produce further decrease in the value of the objective
function. Thus, in the first scenario shown a step from x k to x 1
k resulted in a decrease
in the objective function, so the step from x 1
k to x k+1 was tried and led to a further
decrease in the objective function value. The iteration was then terminated with a
new point x k+1 that satisfies the simple decrease condition f(x k+1 ) < f(x k ). In the
worst case, the last scenario shown, 2n trial points were evaluated
k , and
producing decrease in the function value at the current iterate x k . In
this case, x and the step size must be reduced for the next iteration.
We now show this algorithm is an instance of a generalized pattern search method.
4.1.1. The matrices. Coordinate search is usually defined so that the basis
matrix is the identity matrix; i.e., However, knowledge of the problem may
lead to a di#erent choice for the basis matrix. It may make sense to search using
a di#erent coordinate system. For instance, if the variables are known to di#er by
several orders of magnitude, this can be taken into account in the choice of the basis
matrix (though, as we will see in section 6.2, this may have a significant e#ect on the
behavior of the method).
The generating matrix for coordinate search is fixed across all iterations of the
method. The generating matrix C contains in its columns all possible combi-
ON THE CONVERGENCE OF PATTERN SEARCH ALGORITHMS 11
Fig. 1. All possible subsets of the steps for coordinate search in R 2 .
#k
z }| {
Fig. 2. The pattern for coordinate search in R 2 with a given step length control parameter # k .
nations of {-1, 0, 1}. Thus, C has columns. In particular, the columns of C
contain both I and -I, as well as a column of zeros. We define
of the remaining 3 n
columns of C. Since C is fixed across all iterations of the
method, there is no need for an update algorithm.
For
.
Thus, when possible trial points defined by the pattern
given step length # k , can be seen in Fig. 2. Note that the pattern includes all the
possible trial points enumerated in Fig. 1.
4.1.2. The exploratory moves. The exploratory moves for coordinate search
are given in Algorithm 3, where the e i 's denote the unit coordinate vectors.
Algorithm 3. Exploratory Moves Algorithm for Coordinate Search.
Given x k , # k , f(x k ), and B, set s
For do
(a) s i
. Compute
(b) If f(x i
Otherwise,
k . Compute
(ii) If f(x i
k .
Return.
The exploratory moves are executed sequentially in the sense that the selection of
the next trial step is based on the success or failure of the previous trial step. Thus,
while there are 3 n possible trial steps, we may compute as few as n trial steps, but
we compute no more than 2n at any given iteration, as we saw in Fig. 1.
From the perspective of the theory, there are two conditions that need to be met
by the exploratory moves algorithm. First, as Figs. 1 and 2 illustrate, all possible
trial steps are contained in # k P .
The second condition on the exploratory moves is the more interesting; coordinate
search demonstrates the laxity of this second hypothesis. For instance, in the first
scenario shown in Fig. 1, decrease in the objective function was realized for the first
trial step
so the second trial step
was tried and accepted. It is certainly possible that greater decrease in the value of
the objective function might have been realized for the trial step
which is defined by a column in the matrix M (the step s 2
k is defined by a column in the
matrix L), but s #
k is not tried when simple decrease is realized by the step s 1
k . However,
in the worst case, as seen in Fig. 1, the algorithm for coordinate search ensures that
all 2n steps defined by # k B# k B[M -M are tried before returning
the step s In other words, the exploratory moves given in Algorithm 3 examine
all 2n steps defined by # k B# unless a step satisfying
4.1.3. Updating the step length. The update for # k is exactly as given in
Algorithm 2. As noted by Davidon, the usual practice is to continue with steps of
the same magnitude until no further decrease in the objective function is realized, at
which point the step size is halved. This corresponds to setting
Thus,
This su#ces to verify that coordinate search with fixed step length is a pattern
search method. Theorem 3.5 thus holds. The exploratory moves algorithm for coordinate
search would need to be modified to satisfy the Strong hypotheses on exploratory
moves for the conditions of Theorem 3.7 to be met.
ON THE CONVERGENCE OF PATTERN SEARCH ALGORITHMS 13
4.2. Evolutionary operation using factorial designs. In 1957 G. E. P. Box
[2] introduced the notion of evolutionary operation as a method for increasing industrial
productivity. The ideas were developed within the context of the on-line
management of industrial processes, but Box recognized that the technique had more
general applicability. Subsequent authors [3, 13] argued that the basic technique was
readily applicable to general unconstrained optimization and it is within this context
that we examine the ideas here.
In its simplest form, evolutionary operation is based on using two-level factorial
designs: evaluate the function at the vertices of a hypercube centered about the
current iterate. (G. E. P. Box refers to this as one of a variety of "pattern of variants"
[2].) If simple decrease in the value of the objective function is observed at one of
the vertices, it becomes the new iterate. Otherwise, the lengths of the edges in the
hypercube are halved and the process is repeated.
4.2.1. The matrices. As with coordinate search, the usual choice for the basis
matrix is though, as with coordinate search, other choices may be made to
reflect information known about the problem to be solved.
The generating matrix for evolutionary operation is fixed across all iterations of
the method. The generating matrix C contains in its columns all possible
combinations of {-1, 1}; to this we append a column of zeros. Thus C has
columns.
We take M to be any linearly independent subset of n columns of C; -M necessarily
will be contained in C. Once again, L is fixed and consists of the remaining
columns of C.
There is no need for an algorithm to update C since the generating matrix is
fixed.
4.2.2. The exploratory moves. The exploratory moves given in Algorithm 4
are simultaneous in the sense that every possible trial step s i
computed at each iteration. It is then the case that every trial step s i
k is contained in
. The second observation of note is that since
of our choice of M (and thus, by extension, our choice of #). Furthermore, we are
guaranteed that the Strong hypotheses on exploratory moves are satisfied.
Algorithm 4. Exploratory Moves Algorithm for Evolutionary Operation
Given x k , # k , f(x k ), B, and
, set s
For do
(a) s i
k . Compute
(b) If f(x i
k .
Return.
4.2.3. Updating the step length. The algorithm for updating # k is exactly
as given in Algorithm 2, with # usually set to 1/2 and
Since we have shown that evolutionary operation satisfies all the necessary re-
quirements, we can therefore conclude that it, too, is a pattern search method, so
Theorem 3.5 holds. The algorithm, as stated above, also satisfies the conditions of
Theorem 3.7.
14 VIRGINIA TORCZON
Fig. 3. The pattern step in R 2 , given x k #= x k-1 , k > 0.
4.3. Hooke and Jeeves' pattern search algorithm. In addition to introducing
the general notion of a "direct search" method, Hooke and Jeeves introduced the
pattern search method, a specific kind of search strategy [7]. The pattern search of
Hooke and Jeeves is a variant of coordinate search that incorporates a pattern step
in an attempt to accelerate the progress of the algorithm by exploiting information
gained from the search during previous successful iterations.
The Hooke and Jeeves pattern search algorithm is opportunistic. If the previous
iteration was successful (i.e., # k-1 > 0), then the current iteration begins by conducting
coordinate search about a speculative iterate x k
the current iterate x k . This is the pattern step. The idea is to investigate whether
further progress is possible in the general direction x k - x k-1 (since, if x k #= x k-1 ,
then x k - x k-1 is clearly a promising direction).
To make this a little clearer, we consider the example shown in Fig. 3. Given
x k-1 and x k (we assume, for now, that k > 0 and that x k #= x k-1 ), the pattern search
algorithm takes the step x k - x k-1 from x k . The function is evaluated at this trial
step and the trial step is accepted, temporarily, even if f(x k
The Hooke and Jeeves pattern search algorithm then proceeds to conduct coordinate
search about the temporary iterate x k Thus, in R 2 , the exploratory
moves are exactly as shown in Fig. 1, but with x k substituted for x k .
If coordinate search about the temporary iterate x k
then the point returned by coordinate search about the temporary iterate is accepted
as the new iterate x k+1 . If not, i.e., then the
pattern step is deemed unsuccessful, and the method reduces to coordinate search
about x k . For the two dimensional case, then, the exploratory moves would simply
resort to the possibilities shown in Fig. 1.
If the previous iteration was not successful, so x
the iteration is limited to coordinate search about x k . In this instance, though, the
updating algorithm for # k will have reduced the size of the step (i.e., #
The algorithm does not execute the pattern step when
To express the pattern search algorithm within the framework we have developed,
we use all the machinery required for coordinate search. Once again, the basis matrix
is usually defined to be We append to the generating matrix another set of
columns to capture the e#ect of the pattern step and we change the exploratory
moves algorithm, as detailed below.
4.3.1. The generating matrix. Recall that the generating matrix for coordinate
search consists of all possible combinations of {-1, 0, 1} and is never changed.
For the Hooke and Jeeves pattern search method, we allow the generating matrix to
change from iteration to iteration to capture the e#ect of the pattern step. We append
ON THE CONVERGENCE OF PATTERN SEARCH ALGORITHMS 15
another set of 3 n columns, consisting of all possible combinations of {-1, 0, 1}, to the
initial generating matrix for coordinate search. Thus C k has columns. The
additional 3 n columns allow us to express the e#ect of the pattern step with respect
to x k , rather than with respect to the temporary iterate x k which is
how the Hooke and Jeeves pattern search method usually is described. The matrix
M is unchanged; is allowed to vary, though
only in the 3 n columns associated with the pattern step. For
.
For notational convenience, we require that the last column of C 0 , which we denote
as c p
0 , be the column of zeros. In both the algorithm for updating C k (Algorithm 5)
and the algorithm for the exploratory moves (Algorithm 6), we use the column c p
k to
measure the accumulation of a sequence of successful pattern steps. This can be seen,
in (12), for our example from Fig. 3. In this example, we have the generating matrix
.
The pattern step represented by the vector (1 1) T , seen in the last
column of C k . Note that the only di#erence between the columns of C 0 given in (11)
and the columns of C k given in (12) is that (1 1) T has been added to the last 3 2
columns of C k .
The algorithm for updating the generating matrix updates the last 3 n columns
of C k ; the first 3 n columns remain unchanged, as in coordinate search. The purpose
of the updating algorithm is to incorporate the result of the search at the current
iteration into the pattern for the next iteration. This is done using Algorithm 5. Note
the distinguished role of c p
k , the last column of C k , which represents the pattern step
Algorithm 5. Updating C k .
For do
k .
Return.
Since (1/# k )s k is necessarily a column of C k and C 0 # Z n-p , an argument by
induction shows that the update algorithm for C k ensures that the columns of C k
always consist of integers.
4.3.2. The exploratory moves. In Algorithm 6, the e i 's denote the unit co-ordinate
vectors and c p
k denotes the last column of C k . We set
is defined when
A useful example for working through the logic of the algorithm can be found in
[1], though the presentation and notation di#er somewhat from that given here.
Algorithm 6. Exploratory Moves Algorithm for Hooke and Jeeves.
Given x k , # k , f(x k ), B, and # k-1 , set #
For do
(a)s i
k . Compute
(b)If f(x i
k .
Otherwise,
k . Compute
k .
For do
(a)s i
k . Compute
(b)If f(x i
k .
Otherwise,
k . Compute
k .
Return.
All possible steps are contained in # k P k since C k contains columns that represent
the "pattern steps" tried at the beginning of the iteration. And, once again, the
exploratory moves given in Algorithm 6 examine all 2n steps defined by # k B# unless
a step satisfying
Since we have shown that the pattern search algorithm of Hooke and Jeeves
satisfies all the necessary requirements, we can therefore conclude that it, too, is a
special case of the generalized pattern search method and Theorem 3.5 holds.
4.4. Multidirectional search. The multidirectional search algorithm was introduced
by Dennis and Torczon in 1989 [15] as a first step towards a general purpose
optimization algorithm with promising properties for parallel computation. While
subsequent work led to a class of algorithms (based on the multidirectional search
algorithm) that allows for more flexible computation [6, 17], one of the unanticipated
results of the original research was a global convergence theorem for the multidirectional
search algorithm [16].
The multidirectional search algorithm is a simplex-based algorithm. The pattern
of points can be expressed as a simplex (i.e., points or vertices) based at the
current iterate; as such, multidirectional search owes much in its conception to its
predecessors, the simplex design algorithm of Spendley, Hext, and Himsworth [12] and
the simplex algorithm of Nelder and Mead [9]. However, multidirectional search is a
di#erent algorithm-particularly from a theoretical standpoint. Convergence for the
Spendley, Hext, and Himsworth algorithm can be shown only with some modification
of the original algorithm, and then only under the additional assumption that the
function f is convex. There are numerical examples to demonstrate that the Nelder-Mead
simplex algorithm may fail to converge to a stationary point of the function
because the uniform linear independence property (discussed in section 6.2), which
plays a key role in the convergence analysis, cannot be guaranteed to hold [15].
The multidirectional search algorithm is described in detail in both [6] and [16].
The formulation given here is di#erent and, in fact, introduces some redundancy that
can be eliminated when actually implementing the algorithm. However, the way of
expressing the algorithm that we use here allows us to make clear the similarities
between this and other pattern search methods.
4.4.1. The matrices. It is most natural to express multidirectional search in
terms of multiple basis matrices B k and a fixed generating matrix C, which is at odds
with our definition for generalized pattern search methods. As we shall see, however,
ON THE CONVERGENCE OF PATTERN SEARCH ALGORITHMS 17
it is possible to convert the more natural specification to one that conforms to our
requirements for a pattern search method.
The multidirectional search algorithm centers around a family of basis matrices
B that consists of all matrices representing the edges adjacent to each vertex in a
nondegenerate n-dimensional simplex that the user is allowed to specify. Since the
ordering of the columns is not unique and typically not preserved in the implementation
of the method, we consider all possible representations of the columns of the
matrices associated with the edges adjacent to the (n+1) vertices of the simplex. We
then add the negatives of these (n basis matrices to account for the e#ect of the
reflection step allowed by the multidirectional search algorithm. Thus the cardinality
of the set B is
Fortunately, there is no need to construct this unwieldy number of basis matrices
to initialize the method. We can update the basis matrix after each iteration k
by reconstructing the new basis matrix B k+1 , given the outcome of the exploratory
moves, from the trial points x i
during the course of the
exploratory moves. This procedure is given in Algorithm 7. The scalar scale is
chosen during the course of the exploratory moves (see Algorithm 8) to ensure that
factoring out any change in the size of the simplex introduced by a
change in # k . This has the further e#ect of preserving the role of # k as a step length
parameter.
Algorithm 7. Updating B k .
Given
scale, best, and x i
For (best - 1) do
best
For (best do
best
Otherwise
For do
Return.
Given this use of a family of basis matrices to help define the multidirectional
search algorithm, the generating matrix is then the fixed matrix
Thus, C contains To ensure that C # Z n-p , we
Z. Furthermore, to ensure that the role of # k as a step length parameter
is not lost with the introduction of the expansion step represented by -I, we require
- #. The algorithm is defined so that # w1 , # w2
This
requires the further restriction that # N. Again, this is not an onerous restriction.
Multidirectional search usually is specified so that 2.
Now, to bring this notation into conformity with our definition for a generalized
pattern search method, observe that we can represent all possible basis matrices B #
in terms of a single reference matrix B # B so that
A convenient feature of using the edges of a simplex to form the set of basis matrices
is that the matrices -
consist only of elements from the set {-1, 0, 1}. The matrices
are necessarily nonsingular because of the nondegeneracy of the simplex. We use
to represent the set of matrices -
and observe that since B is a finite set, the set
B is also finite.
We then observe that
Thus we can define the pattern in terms of the single reference matrix B and the
redefined generating matrix
with
B. We also have L k # [-
0] and since - # Z,
4.4.2. The exploratory moves. The exploratory moves for the multidirectional
search method are given in Algorithm 8; the e i 's denote the unit coordinate
vectors. We use the notion of B k # B for consistency with the update algorithm given
in Algorithm 6, but we could just as easily substitute B -
in the algorithm
given below.
Algorithm 8. Exploratory Moves Algorithm for Multidirectional
Search.
Given
# N, set s
For do
(a) s i
k . Compute
(b) If f(x i
k , and best = i.
For do
(a) s i
k . Compute
(b) If f(x i
k , and best = i.
For do
(a) s i
(b) If f(x i
Return.
Clearly, s k # k P k . Since the exploratory moves algorithm considers all steps
of the form # k B# k , unless simple decrease is found after examining only the steps
defined by # k BM k , this guarantees we satisfy the condition that if min{f(x k +y), y #
4.4.3. Updating the step length. The algorithm for updating # k is that given
in Algorithm 2. In this case, while # usually is set to 1/2 so that
and include an expansion factor usually equals
one. Thus usually 2. The choice of # k # is made during the
execution of the exploratory moves.
Since we have shown that the multidirectional search algorithm satisfies all the
necessary requirements, we conclude that it is also a pattern search method and thus
Theorem 3.5 applies. Note that since we allow - > 1, which is a useful algorithmic
feature, we cannot guarantee that lim k# and so Theorem 3.7 does not
automatically apply.
ON THE CONVERGENCE OF PATTERN SEARCH ALGORITHMS 19
5. Conclusions. We have presented a framework in which one can analyze pattern
search methods. This framework abstracts and quantifies the similarities of the
classical pattern search methods and enables us to prove lim inf k#f(x k
for this class of algorithms. We also specify the conditions under which the limit
can be shown to hold.
These convergence results are perhaps surprising, given the simplicity of pattern
search methods, but derive from the algebraic rigidity imposed on the iterates produced
by pattern search methods. This is gratifying, since while this rigidity originally
was introduced as a heuristic for directing the exploratory moves, it turns out to be
the key to proving convergence as well. This analysis also highlights just how weak
the conditions on the acceptance of the step can be and yet still allow a global convergence
analysis, an observation that may prove useful in the analysis of other classes
of optimization methods.
6. Technical results. We deferred the proof of Proposition 3.4 for several rea-
sons. First, many of the results in this section are generalizations of similar results to
be found in [16]. The abstraction in section 2 leads to more succinct proofs. Second,
the proof of Proposition 3.4 is closely related to that of several other results presented
in this section and requires us to introduce several additional notions.
We return to our definition of the pattern as to show that the pattern
contains at least one direction of descent whenever #f(x k ) #= 0.
Recall that we require the columns of C k to contain both M k and -M k . Thus,
can be partitioned as follows:
We now elaborate on these requirements. Since M k is an n-n nonsingular matrix
and B is nonsingular, we are guaranteed that BM k forms a basis for R n . Further,
we are guaranteed that at any iteration k, if #f(x k ) #= 0, x k - Bc i
is a direction of
descent for at least one column c i
k contained in the block # k .
6.1. Descent methods. Of course, the existence of a trial step in a descent
direction is not su#cient to guarantee that decrease in the value of the objective
function will be realized. To guarantee that a pattern search method is a descent
method, we need to guarantee that in a finite number of iterations the method produces
a positive step size # k that achieves decrease on the objective function at the
current iterate. We now show that this is the case.
Lemma 6.1. Suppose that f is continuously di#erentiable on a neighborhood of
there exists q # Z, q # 0 such that # k+q > 0 (i.e., the
q)th iteration is successful).
Proof. A key hypothesis placed on the exploratory moves is that if descent can
be found for one of the trial steps defined by # k B# k , then the exploratory moves
returns a step that produces descent.
Because BC k has rank n, if there exists at least one trial
direction d i
loss of generality. Thus, there exists an h k > 0
such that for 0 < h # h k ,
If at iteration k, # k > h k , then the iteration may be unsuccessful; that is, #
When the iteration is unsuccessful, the generalized pattern
search method sets x and the updating algorithm sets #
is strictly less than one, there exists q # Z, q # 0, such that # q # k # h k . Thus we are
guaranteed descent, i.e., a successful iteration, in at most q iterations.
6.2. Uniform linear independence. The pattern P k guarantees the existence
of at least one direction of descent whenever #f(x k ) #= 0. We now want to guarantee
the existence of a bound on the angle between the direction of descent contained in
B# k and the negative gradient at x k (whenever #f(x k ) #= 0). We will show, in fact,
that this bound is uniform across all iterations of the pattern search algorithm. To
do so, we use the notion of uniform linear independence [10].
Lemma 6.2. For a pattern search algorithm, there exists a constant # > 0 such
that for all k # 0 and x #= 0,
Proof. To demonstrate the existence of #, we first consider the simplest possible
I and and use this to derive a bound for any
choice of B and C k that satisfies the conditions we have imposed.
Lemma 6.3. Suppose
where the e j 's are the unit coordinate vectors.
I and
min
cos #(y) =# n
Proof. We have |y T e j
are guaranteed that |y j | # 1/ # n for some j, so |y T e j | # 1/ # n for some j. Thus
cos #(y) # 1/ # n.
Now note that cos #(y) attains this lower bound for any y
Thus, if the pattern search is restricted to the coordinate directions defined by
gives the lower bound on the absolute value of the cosine of
the angle between the gradient and a guaranteed direction of descent. We now use
the bound for this particular case to derive a bound for the general case.
Assume a general basis matrix B and a general matrix M k # M, where |M| <
+#. We adopt the notation BM
k ]. Then for any x #= 0 we have the
If we set -T w, we have
ON THE CONVERGENCE OF PATTERN SEARCH ALGORITHMS 21
where #(BM k ) is the condition number of the matrix BM k . Thus, we have
To ensure a bound # that is independent of the choice of any particular matrix
simply observe that the set M is required to be finite. Thus, # is taken
to be
M#M
.
The bound given in (14) points to two features that explain much about the behavior
of pattern search methods. Since we never explicitly calculate-or approximate-
the gradient, we are dependent on the fact that in the worst case at least one of our
search directions is not orthogonal to the gradient; # gives us a bound on how far
away we can be. Thus, as either the condition number of the product BM k increases,
or the dimension of the problem increases, our bound on the angle between the search
direction and the gradient deteriorates. This suggests two things. First, we should
be very careful in our choice of B and M for any particular pattern search method.
Second, we should not be surprised that these methods become less e#ective as the
dimension of the problem increases.
Nevertheless, even though pattern search methods neither require nor explicitly
approximate the gradient of the function, the uniform linear independence condition
demonstrates that the pattern search methods are, in fact, gradient-related methods,
as defined by Ortega and Rheinboldt [10], which is one reason why we can establish
global convergence.
6.3. The descent condition. Having introduced the notion of uniform linear
independence with the bound #, we are now ready to show that pattern search methods
reduce # k only when necessary to find descent. To do this we will show that once
the steps s i
are small enough, then a successful step must be returned
by the exploratory moves algorithm. Lemma 3.1 allows us to restate this condition
in terms of # k . We use the result to prove Proposition 3.4.
Proposition 6.4. Suppose that L(x 0 ) is compact and f is continuously di#er-
entiable on a neighborhood of L(x 0 ). Given # > 0,
Suppose also that x 0
# . Then there exists # > 0, independent of k, such that if
# and # k < #, then the kth iteration of a generalized pattern search method
(see Algorithm 1) will be successful (i.e., # thus
Proof. We restrict our attention to the steps defined by the columns of # k B# k .
This is su#cient since the Hypotheses on exploratory moves ensure that a step s k
satisfying the simple decrease condition # k > 0 must be returned if a trial step defined
by a column of # k B# k satisfies the simple decrease condition.
If s i
is a step defined by # k B# k (we assume that P k is partitioned
as in (2) so that the first 2n columns of P k contain the columns of B# k #
independent of k,
22 VIRGINIA TORCZON
since M k # M # Z n-n and M is a finite set of matrices. Together, (15) and
Lemma 3.1 yield
allows us to define
dist (L(xN are compact and disjoint, we
know that d > 0. If # k < d/2# , then #s i
Thus x i
k lies in the interior of L(x 0 ) for all precisely, for all
lies in the ball B(x k , d/2) # L(x 0 ).
x#f(x)#. By design, # > 0. Since #f is continuous on a
neighborhood of L(x 0 ), #f is uniformly continuous on a neighborhood of L(x 0 ).
Thus, there exists a constant r > 0, depending only on # and the # from (13), such
that
We define
min d, r .
We are now assured that if
then
and
We are ready to argue that if at any iteration k # N , x k
# and (17) is satisfied,
then an acceptable step will be found.
Choose a trial point x i
and
The definitions
of# and the pattern P k , together with Lemma 6.2, guarantee the
existence of at least one such x i
k .
Since (17) holds by assumption, (18) also holds. We can apply the mean value
theorem to obtain f(x i
Consider the first term on the right-hand side of (20). Our choice of x i
k gives us
ON THE CONVERGENCE OF PATTERN SEARCH ALGORITHMS 23
Furthermore, since #f(x k ) T
Now consider the second term on the right-hand side of (20). The Cauchy-Schwarz
inequality gives us
Combine (21) and (22) to rewrite (20) as
holds by assumption, (19) also holds. We then have
Thus, when # k < #, f(x i
for at least one s i
k defined by # k Bc i
2n. The Hypotheses on exploratory moves guarantee that if min{f(x k
and the algorithm for updating # k (Algorithm 2) ensures that # k+1 # k .
Proposition 6.4 guarantees that if # k is small enough, a generalized pattern search
method realizes simple decrease because there exists at least one step among the 2n
steps defined by # k B# k that gives decrease as a function of the norm of the gradient
at the current iterate, as shown in (23); the Hypotheses on exploratory moves then
ensure that the exploratory moves algorithm must return a step that satisfies at least
simple decrease. However, there are no guarantees that the step returned by an
exploratory moves algorithm satisfies more than the simple decrease condition.
To tie the amount of actual decrease to the norm of the gradient, we must place
much stronger conditions on the generalized pattern search method, as discussed in
section 3.3.2. Once we have done so, Corollary 6.5 follows more or less immediately
from Proposition 6.4.
Corollary 6.5. Suppose that L(x 0 ) is compact and f is continuously di#eren-
tiable on a neighborhood of L(x 0 ). Suppose that the columns of the generating matrix
are bounded in norm and that the generalized pattern search method (Algorithm 1)
enforces the Strong hypotheses on exploratory moves. Given # > 0, let
Suppose also that x 0
# . Then there exist # > 0 and # > 0, independent of k, such
that for all but finitely many k, if x k
# and # k < #, then
Proof. From Proposition 6.4, (23) says that for k #
(Lemma 6.1 guarantees the existence of N), there exists at least one trial step s i
such that once # k < #, where # is as defined in (16), we have
The Strong hypotheses on exploratory moves give us
Lemma 3.1 ensures that
Lemma 3.6, which holds only when the columns of the generating matrix are bounded
in norm, gives us
We define #
2 # to complete the proof.
We now prove Proposition 3.4.
Proof. By assumption, lim inf k#f(x k )#= 0. Then we can find N 1 and
such that for all k # N 1 , x k
guarantees the existence of N
From Proposition 6.4 we are assured of # > 0 such that if # k #, then the
iteration will be successful. Given # 0 , there exists a constant q # Z, q # 0, such
that # q # 0 #, where # (0, 1) and is as defined in the algorithm for updating # k
(Algorithm 2). Thus, for k # N , # q+1 # 0 < # k .
Acknowledgments
. This paper benefited from conversations with J. E. Dennis,
Stephen Nash, Michael Trosset, Lu-s Vicente, and especially Michael Lewis. In partic-
ular, discussions with Michael Lewis were critical in the distillation of the abstraction
for pattern search methods found in section 2 and in the development of the analytic
arguments for the algebraic structure of the iterates found in section 3.1.
The review and comments made by an anonymous referee, Danny Ralph (the
second referee, who agreed to reveal his identity for this acknowledgment), and Jorge
Mor-e, the editor, are gratefully acknowledged. In particular, the observation by Danny
Ralph that the pattern contains only vectors in a fixed lattice led to a more general
result and a much more elegant presentation.
--R
Analysis and Methods
Variable metric method for minimization
"Direct search"
A simplex method for function minimization
Iterative Solution of Nonlinear Equations in Several Variables
Computational Methods in Optimization: A Unified Approach
Sequential application of simplex designs in optimisation and evolutionary operation
in Numerical Methods for Unconstrained Optimization
Sequential Estimation Techniques for Quasi-Newton Algorithms
A Direct Search Algorithm for Parallel Machines
On the convergence of the multidirectional search algorithm
PDS: Direct Search Methods for Unconstrained Optimization on Either Sequential or Parallel Machines
Positive basis and a class of direct search techniques
--TR
--CTR
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589078 | Degenerate Nonlinear Programming with a Quadratic Growth Condition. | We show that the quadratic growth condition and the Mangasarian--Fromovitz constraint qualification (MFCQ) imply that local minima of nonlinear programs are isolated stationary points. As a result, when started sufficiently close to such points, an $L_\infty$ exact penalty sequential quadratic programming algorithm will induce at least R-linear convergence of the iterates to such a local minimum. We construct an example of a degenerate nonlinear program with a unique local minimum satisfying the quadratic growth and the MFCQ but for which no positive semidefinite augmented Lagrangian exists. We present numerical results obtained using several nonlinear programming packages on this example and discuss its implications for some algorithms. | Introduction
Recently, there has been renewed interest in analyzing and modifying sequential
quadratic programming (SQP) algorithms for constrained nonlinear optimization
for cases where the traditional regularity conditions do not hold [14,13,
20,25]. This research has been motivated by the fact that large-scale nonlinear
programming problems tend to be almost degenerate (have large condition
numbers for the Jacobian of the active constraints). It is therefore important
to establish to what extent the convergence properties of the SQP methods are
dependent on the ill-conditioning of the constraints. In this work, we term as
degenerate those nonlinear programs (NLPs) for which the gradients of the active
constraints are linearly dependent. In this case there may be several feasible
Lagrange multipliers.
Many of the previous analysis and rate of convergence results for degenerate
NLP are based on the validity of second-order conditions. These are essentially
equivalent to the condition in unconstrained optimization that, for a critical
point of a function f(x) to be a local minimum, f xx 0 is a necessary condition
and f xx 0 is a sufficient condition. Here is the positive semidefinite ordering.
The place of f xx in constrained optimization is taken for these conditions by L xx ,
the Hessian of the Lagrangian, which is now required to be positive definite on
the critical cone for one or all of the Lagrange multipliers [6,21].
Department of Mathematics, University of Pittsburgh
This work was completed while the author was the Wilkinson Fellow at the Mathematics
and Computer Science Division, Argonne National Laboratory, 9700 South Cass Avenue,
Argonne, IL 60439. This work was supported by the Mathematical, Information, and Computational
Sciences Division subprogram of the Office of Advanced Scientific Computing, U.S.
Department of Energy, under Contract W-31-109-Eng-38.
This work differs from previous approaches in that we assume only that
1. At a local solution x of the constrained nonlinear program, the first-order
Mangasarian-Fromowitz constraint qualification holds.
2. The quadratic growth condition (QG) [4,16] is satisfied:
for some oe ? 0 and all x feasible in a neighborhood of x .
3. The data of the problem are twice continuously differentiable.
These assumptions are equivalent to a weaker form of the second-order sufficient
conditions [15,4] which do not require the positive semidefinitenes of the Hessian
of the Lagrangian on the entire critical cone.
We prove that these conditions guarantee that x is the only local stationary
point (3) of the nonlinear program. This is an important issue because it guarantees
that descent-like algorithms will not stop arbitrarily close to x , except
at x . This extends a result from [21] that required some second-order sufficient
conditions to be satisfied for all multipliers. In particular, our work implies that
if MFCQ holds and the second-order sufficient conditions hold for one multiplier,
then x is a strict local minimum and an isolated stationary point.
We also show that, under the same assumptions, the L1 exact penalty sequential
quadratic program (SQP) induces at least Q linear convergence [19] of
the penalized objective to f(x ) and R-linear convergence of the iterates. Finally,
we provide an example of a nonlinear program that satisfies our assumptions for
which it is not possible to construct an augmented Lagrangian such that x will
be an unconstrained local minimum. This may present an adverse case to algorithms
based on this assumption, such as Lagrange multiplier methods. However,
we show that it is possible to construct a nondifferentiable function that has x
as its minimum, namely the L1 penalty function (which can also be inferred
from the results in [4]). We describe our computational experience with several
nonlinear programming packages applied to this example and discuss the
expected and observed behavior of Lagrangian multiplier methods.
Our convergence analysis for the L1 exact penalty function suggests that it is
possible to construct a convergence theory with much more general second-order
conditions. This may result in algorithms with superior robustness, because their
properties depend on significantly fewer assumptions.
1.1. Previous Work, Framework, and Notations
We deal with the NLP problem
f(x) subject to g(x) 0; (2)
are twice continuously differentiable.
We call x a stationary point if the following conditions hold for some 2 IR
Degenerate Nonlinear Programming with a Quadratic Growth Condition 3
Here L is the Lagrangian function
If certain regularity conditions hold (discussed below), then a local solution
x of (2) is a stationary point. In that case (3) are referred to as the KKT
(Karush-Kuhn-Tucker) conditions.
Since our analysis will be limited to a neighborhood of a point x that is a
strict minimum, we will assume that all constraints are active at x , or g(x
Such a situation can be obtained by simply dropping the constraints i for
which 0, since this relationship holds in an entire neighborhood of x .
This does not reduce the generality of our results, but it simplifies the notation
because now we do not have to refer separately to the active set.
The regularity condition, or constraint qualification, ensures that a linear approximation
of the feasible set in the neighborhood of x captures the geometry
of the feasible set. Often in local convergence analysis of constrained optimization
algorithms, it is assumed that the constraint gradients rg i
are linearly independent, so that the Lagrange multiplier in (3) is unique. We
assume instead the Mangasarian-Fromowitz constraint qualification (MFCQ):
n . (5)
It is well known [9] that MFCQ is equivalent to boundedness of the set M(x )
of Lagrange multipliers that satisfy (3), that is,
Note that M(x ) is certainly polyhedral in any case.
The critical cone at x is [6,22]
We briefly review the some of the second-order conditions in the literature,
although they are not an assumption for our analysis but only a basis for com-
parison. In the framework of [6], the second-order sufficient conditions for x to
be an isolated local solution of (2) are:
9 2 M(x
If these conditions hold at x for some , then the quadratic growth condition
is satisfied, irrespective of the validity of the first-order constraint qualification
[6,7]. However, this does not imply that x is an isolated stationary point, as
shown by a simple example [21], which may prevent an optimization algorithm
that uses only first derivative information from reaching x even when started
arbitrarily close to x .
In [21] it is shown that if MFCQ holds, and the relation (8) is satisfied for
all 2 M(x ) then x is an isolated stationary point and a minimum of (2).
Also, with these conditions, the exact solution is Lipschitz stable with respect
to perturbations. By compactness of M(x ), we can choose oe independently of
in this case. In [1] it is proven that, under these assumptions, the L1 exact
penalty SQP will converge Q-linearly to f(x ), when the descent direction is
computed by a QP using only first-order information.
A refinement of the second-order conditions was introduced in [15]. In the
presence of MFCQ, those conditions require that
Further analysis shows that, in presence of MFCQ, these conditions are necessary
and sufficient for the quadratic growth condition to hold [4,15,16,22]. Also, the
exact solution is Lipschitz stable with respect to certain classes of perturbations
[22], though not to any perturbation (see an example in [10, p.308]). In this
paper we assume only the quadratic growth condition and MFCQ, and thus we
do not use the perturbation results.
If the condition (9) holds, but (8) does not, then there is no positive semidefinite
augmented Lagrangian, as we will show with an example. This is an interesting
aspect since it invalidates the usual working assumption of Lagrange
multiplier methods [3].
Finally, we review some of the facts concerning the L1 nondifferentiable
exact penalty function:
We are looking for an unconstrained minimum of the function
where c OE is a sufficiently large constant. Descent directions d of OE(x) at the point
x can be obtained by solving the following quadratic program (QP) [3]:
subject to g j (x)
where H is some positive definite matrix and g 0 In this paper the analysis
will be restricted to the case although the same results apply for any
other positive definite matrix.
At the current point x k of an iterative procedure that attempts to determine
x , the QP (11) generates the descent direction d k . The next iterate is x
obtained by a line search procedure. Usual stepsize rules
are the minimization rule, the limited minimization rule, and the Armijo rule
[3]. For these rules, any limit point of fx k g is a stationary point of OE(x), and the
descent procedure is therefore globally convergent in this sense [3].
If, in addition,
Degenerate Nonlinear Programming with a Quadratic Growth Condition 5
for some 2 M(x ), then x is a stationary point of OE(x) [2]. A suitable value
for c OE is not available in the early stages of the algorithm, but a good estimate
can be found via an update scheme [2]. Here we assume that c is constant and
for all 2 M(x ), where fl is some prescribed safety factor.
Consider the quadratic program
subject to g j (x)
We denote the unique solution of this program by d(x) and the set of its multipliers
by M(x). At x (14) has the same multiplier set as (2), which are both
denoted by M(x ). Since MFCQ is satisfied at x , this QP is feasible in a neighborhood
of x . The KKT conditions for this QP require
With these notations, d(x If the QP (14) were unconstrained, then its
solution would be We name a descent-like algorithm a sequential
quadratic program that solves instances of the above QP.
At x , the QP (14) satisfies MFCQ and some second-order sufficient con-
ditions. From [21] there exists c d such that, in a neighborhood of x we have
there exists 2 M(x ) such that
Therefore, from the definition of c OE , there exists a neighborhood of x such that
for all multipliers 2 M(x). For such x, it can be verified by inspection that
is a solution of (11) [2, p. 195]. We therefore concentrate on the
QP (14), because, if c OE is large enough and we are sufficiently close to x , it
generates the same descent direction as (11), thus sharing its global convergence
property.
For some function h : IR
k we denote by c 1h , c 2h bounds depending
on the first and second derivatives of h. The positive and negative parts of h(x)
are h
componentwise. With this notation
6 Mihai Anitescu
2. Stationary Points of NLPs Satisfying MFCQ
In this section, we assume that x is in a sufficiently small neighborhood of x ,
whose size or properties are specified in each of the following results. In particular
the standing assumptions hold on all neighborhoods considered here and
Here p with one of the vectors satisfying (5),
suitable neighborhood of x .
Lemma 1. There exist
a neighborhood W (x ) such that
Here P (x) is the usual L1 penalty function (10).
Proof. We have by Taylor's theorem
We choose
For 0 ff ff P we have
The claim follows after choosing c proof of the following lemma can be inferred from [4]. We include it here
for completeness.
Lemma 2. There exists a c OE such that
for all x in a neighborhood of x .
Proof. Let r ? 0 be such that B(x ; r) ae W (x ). We choose r
2 such that
This is always possible because
We then have that, for any x 2 B(x
and thus x r). By the intermediate value theorem, we have that
implying in turn that rg i
Degenerate Nonlinear Programming with a Quadratic Growth Condition 7
Take now
If x is infeasible, then ff 1 ? 0 and there exists i such that g i
applies to give
If x is feasible, then ff and the previous bound still applies.
From the quadratic growth assumption (1) and the feasibility of x+ ff 1 p, we
must have that
or
By (23) and Taylor's theorem we have
c P
Choose
c P
c P
Then by (23)
c P
c P
c P
c P
Using (25), (24) and (26) we get
The conclusion follows, because
from the Cauchy-Schwartz inequality.
We can assume that c OE from the previous lemma satisfies (17), or otherwise
we replace it with the right-hand side of (17) and the conclusion of the lemma
still holds for the new c OE .
To prove the following results, we will use the results from [12] concerning
sets defined by linear inequalities:
For such a set, denote by d(x; P) the distance from a point x 2 IR
n to the set P.
Also, denote by dP (x) the maximum value of the infeasibility:
Then there exists a number (P) ? 0 such that
If we have equality constraints, we recast them as two inequality constraints.
The following lemma uses the fact that M(x ) is polyhedral and can thus be
expressed in the form (27).
Lemma 3. Let I be an index set such that there exists a multiplier
with
lambda there exists a constant c I such that 8 2 M(x ) there
exists a 2 M(x ) with
and such that jj \Gamma jj c I jj I jj 1 .
For a vector we have denoted by I the restriction of the vector to the index
set I.
Proof. Let M I (x ) be the set of all 2 M(x ) such that I = 0. Then
0: (32)
From our assumptions, M I (x ) is not empty. By eventually rescaling the x space,
we can assume, without loss of generality, that the vectors defining the equality
constraints in (30) are of norm 1; otherwise, if all entries are 0, we remove that
row, and the feasible set remains unchanged. M I can be described in the
alternative, way:
I 0; (35)
I 0; (36)
0; (37)
where each row is described by a unit vector, which puts the set in the form
(27). Thus from [12] there exists a (M I ) ? 0 such that
(M I )d(; M I ) dMI (): (38)
However, since 2 M(x ) is a valid multiplier set, we have that only the constraints
I 0, (35), are violated. Thus . The conclusion follows
from (38) by taking c I
. The proof is complete.
We define
Iaef1;::;mg
c I ; for feasible M I (x
Lemma 4. There exists a neighborhood W of x such that, 8x 2 W; 2 M(x),
implies that there exists a 2 M(x ) with I = 0.
Degenerate Nonlinear Programming with a Quadratic Growth Condition 9
Proof. Assume the contrary. Then there exists a sequence x k ! x such that
there exists k 2 M(x) and an index set I for which I = 0, but I 6= 0,
there is only a finite set of index sets, we can extract an
infinite subsequence for which the above happens for a fixed set I. By extracting
another subsequence, we can assume that k is convergent, from (16) and the
fact that M(x ) is compact.
But then k ! 2 M(x ) and I = 0, a contradiction.
From here on we will use extensively that, for h twice continuously differen-
tiable, we have
is a continuous function with / 3h Indeed by
Taylor's theorem we have that there exist continuous functions / 1
3h
and
3h
3h
3h
and
The relation (40) now follows by comparing the last two equations.
Theorem 1. There exists a constant c oe ? 0 such that in a neighborhood of x
we have that
where (d; ) is the solution of the QP (14).
Proof. From (16), there exists a 2 M(x ) such that jj \Gamma jj c d jjx \Gamma x jj.
Let I be the set of indices i for which We have jj I jj
c d jjx \Gamma x jj. From (39) and Lemmas 3 and 4 there exists a ~ 2 M(x ) with
~
As a result
and The important consequence of this fact, using the complementarity
relations from (15), is that
\Gamma( ~
Indeed,
all the above equalities are 0.
From Lemma 2 we have that
Here (d; ) is a solution of (15), and ~
We also used
(40). We now employ the identity ab
(42), and Taylor's theorem for rg(x) to get, by continuing from the previous
equation,
We denote
From
sufficiently close to x . By using 1 , (40),
(43) and (42), we get
(x). Using the above bound in (52), together
with \Gammad T
Degenerate Nonlinear Programming with a Quadratic Growth Condition 11
We can now choose a sufficiently small neighborhood of x such that /(jjx \Gamma
x jj) oeand subtract the last term of the last relation from the lower bound
We take and with this new notation, we
get that
We treat jjx \Gamma x jj as a variable and, by using the formulas for the quadratic
equation, we get that
oe
By using the arithmetic-quadratic mean inequality, we get that
Choosing
oe c OE g (67)
we prove the claim.
Corollary 1. x is an isolated stationary point.
Proof. Let x be another stationary point of the NLP in the neighborhood of x
where the above theorem holds. Therefore there exists a 2 M(x) satisfying
(3). Hence solution of (15) and is the unique solution of the
strictly convex QP (14). Since is feasible from (14) and P
(x). Now from the complementarity conditions in (15) we get T
From the previous theorem we get which proves the claim.
Corollary 2. If the second-order sufficient condition (8) is satisfied for one
multiplier, and if MFCQ holds at x , then x is an isolated stationary point.
Proof. Since x is satisfies the quadratic growth condition (1) under these assumptions
[6,7] and MFCQ holds, Corollary 1 applies.
3. An Example Without a Locally Convex Augmented Lagrangian
Consider the matrix
Take
We then have that u T 1. Since the vector
corresponds to the positive eigenvalue, we have that for any u at an
angle of at most
6 from u 0 , u T Qu 1jjujj 2 . Consider now the rotation matrix
since Q 0 and Q 2 have the same axes of symmetry, but with the eigenvalues
switched. Also, for any u 2 IR 2 , there exists a k such that u T Q k u 1jjujj 2 , since
the wide cones centered at the axis of the positive eigenvalues of Q k now sweep
the entire IR 2 .
Consider now the optimization problem
minz subject to z
By the previous observation, we have that z 1(x 2 +y 2 ) on the feasible set;
thus z 0. Clearly, the only solution of the problem is (0; 0; 0). Since z z 2, if
4, we have that z 1(x 2 +y 2 +z 2 ), for all x;
Therefore at x the quadratic growth condition is satisfied for the
above NLP, with constant 1. Obviously, MFCQ holds at (0; 0; 0), and a simple
calculation shows that
a multiplier of (70). In particular,
at least one multiplier has to be positive. Also, at (0; 0; 0), all constraints are
active and their gradients are (0; 0; \Gamma1) for any of them. As a result, the linear
constraints in (8) now become either z 0 or z = 0, with at least one being
Therefore the critical cone at x is 0g. Also, from
(3), if 2 M(x ), then
Assume that there is a choice 2 M(x ) such that L xx , the Hessian of the
Lagrangian, is positive semidefinite on the critical cone:
y
zA 0; 8(x; y; z); such that z = 0: (71)
This is equivalent to X
Since our construction is invariant to rotations with (U T
that the positive semi-definiteness holds for any circular permutation oe of this
multiplier set: X
We denote by A c (4) the set of circular permutations of four elements. Since the
set of positive definite matrices is a convex cone, and
we must have
which is impossible. Therefore L xx cannot be positive semidefinite on the critical
cone for any choice 2 M(x ). Hence the second-order conditions from [6,21]
will not hold for any choice of the multipliers.
Degenerate Nonlinear Programming with a Quadratic Growth Condition 13
3.1. Augmented Lagrangian Approaches
Here we discuss the expected behavior of augmented Lagrangian techniques when
applied to this example. For these methods, the inequalities of the NLP (2) are
converted into equalities [3,5]. The feasible set can be represented as [5]
The NLP is replaced by a bound-constrained optimization problem. The equality
constraints are incorporated in the objective function based on an estimate of
the multipliers and a penalty term,
subject to t i 0;
Here is the barrier parameter. The objective function in (76) is the augmented
Lagrangian. The problem is subjected to an additional trust-region constraint
[5] to enforce global convergence.
The desired outcome is to have bounded away from zero and the trust-region
inactive as approaches M(x ) and the solution of the above problem
approaches x .
If that happens for our example, then, by a continuity argument following
the lower boundedness of , should be a solution of (76) for an
appropriate choice of ; . Since (76) has linearly independent gradients of the
constraints, both the first and second order necessary conditions must hold [7].
The first order necessary condition results in
where , with components i 0, are the multipliers associated with the variables
As a result 2 M(x ). The second order necessary conditions require
that
I 4
be positive semidefinite, at least on the subspace of (ffix; ffi t) with
results in
or
We proved that the last matrix cannot be positive semidefinite for our example
and we thus get a contradiction. This shows that, either the trust region
will be active arbitrarily close to x , or ! 0.
14 Mihai Anitescu
This also shows that the Hessian of the augmented Lagrangian of the equality
constrained problem
F xx +X
is not positive semidefinite and thus the augmented Lagrangian of the equality
constrained problem cannot be locally convex.
4. Linear Convergence of the SQP with Nondifferentiable Exact
Penalty P (x)
The points x considered in thus subsection are assumed to be sufficiently close
to x . The notation d and 2 M(x) will refer to the solutions of (14) and (15).
Also, P (x) is the L1 penalty function (10) and
4.1. Proof of the Technical Results
Lemma 5.
Proof. Since d is a feasible point of (14), we have that rg i (x) T d \Gammag i (x); 8i 2
mg. By Taylor's remainder theorem
Hence
This completes the proof.
Lemma 6. There exist
ff]:
Proof. Writing the KKT conditions for (14), we obtain
and, hence,
(d) T d +rf(x) T d
(d)
Degenerate Nonlinear Programming with a Quadratic Growth Condition 15
since, by the complementarity conditions satisfied by the solution of (14), T rg(x) T
m. Therefore, since g i
\Gamma(d)
\Gamma(d)
by (10), (17). By Taylor's remainder theorem,
Hence, for ff 2 [0; 1],
ff(\Gamma(d)
from (79) and Lemma 5. Therefore, for ff 2 [0; 1],
The result of the statement follows by choosing
and
Lemma 7. There exists a constant c 5 such that, 8() 2 M(x),
Proof. From (15) and the definition of the Lagrangian (4) it follows, using Tay-
lor's theorem, that, for a sufficiently small neighborhood of x,
g. Also, by (16), we can choose 2 M(x ) such that
we have that
and, thus
Therefore
The conclusion of the lemma follows by choosing c 1g.
4.2. Nondifferentiable Exact Penalty Algorithms and the Linear Convergence
Theorem
The linearization algorithm [3, p.372] has the following form:
1.
2. Compute d k from (11).
3. Choose ff k from a line search procedure, and set x
4. return to Step 2.
The stepsize ff k is chosen by one of the following procedures [3, p.372].
(a) Minimization rule Here ff k is chosen such that
(b) Limited minimization rule Here a fixed scalar s ? 0 is selected, and ff k is
chosen such that
(c) Armijo rule Here fixed scalars s, , and oe with s ? 0, 2 (0; 1), and oe 2 (0; 1)
are chosen and we set ff is the first nonnegative integer
m for which
It can be shown that the Armijo rule yields a stepsize after a finite number of
iterations.
The following theorem establishes the convergence properties of the linearization
algorithm. The global convergence properties, established in [2, Prop. 4.3.3],
are also stated here for completeness.
Theorem 2. Let x k be a sequence generated by the linearization algorithm,
where the stepsize ff k is chosen by the minimization rule, limited minimization
rule or the Armijo rule. Then any accumulation point of the sequence x k
is a stationary point of is a strict
local minimum of the problem (2) satisfying the local quadratic growth (1) and
the Mangasarian-Fromowitz constraint qualification (5), then OE(x k
Q-linearly and x k ! x R-linearly.
Proof. The first part is an immediate consequence of [2, Prop. 4.3.3]. We prove
the linear convergence statement only for the Armijo rule, the proof being similar
for the other stepsize selection mechanisms. By Lemma 6
for all ff 2 [0; ff]. Since m k is the smallest integer m for which
Degenerate Nonlinear Programming with a Quadratic Growth Condition 17
it follows that m s
ff. This therefore ensures that the stepsize is at least
ff
for k sufficiently large. As a result of Lemma 6, we have that
On the other hand, by Lemma 7 we have that
By Theorem (1) and the previous relation it follows that there exists c
c6
by using Lemma 6 and where
. After some obvious manipulation, it
follows that
which proves the Q-linear convergence [19] of the sequence OE(x k ) to OE(x ) with
a linear rate of at most ffi\Gamma1
. Therefore
lim sup
From Lemma 2
Therefore
lim sup
which proves the R-linear convergence [19] to 0 of the sequence x . The
proof is complete.
Following the techniques from [1], we can extend the result for the case where
the matrix H of the QP is not I but changes from iteration to iteration. The
only condition is that the sequence of strictly convex H k be uniformly upper
and lower bounded.
Iteration OE(x k )\GammaOE(x )
9 4.00
Table
1. Rates of convergence for the L1 penalty algorithm
Iteration (New) Penalty Parameter Trust Region Radius jjjj 1
43 1e-4 1.1 e-02
268 1e-14 1.93
283 1e-16 4.41 e02
336 STOP
Table
2. Reduction of the penalty parameter for LANCELOT
5. Numerical Experiments with Degenerate NLP
We experimented with several nonlinear programming packages on the example
from Section 3. Certainly, comparing the behavior of NLP algorithms on a
unique degenerate example cannot result in a complete characterization. Nev-
ertheless, it may be of interest to determine whether methods using augmented
Lagrangians will really encounter problems when solving an example without
a positive semidefinite augmented Lagrangian. We also desire to validate the
theoretical conclusions of the preceding sections.
We have shifted the origin for our example, to avoid one step convergence of
algorithms that start at 0; 0; 0 by default. The algebraic form of the example is
minz
From our analysis, we have that w is a minimum satisfying the
quadratic growth condition (1) with z \Gamma 0
feasible (x; . The feasible set is described in Figure 5. In the lateral
view, the quadratic growth at (1; 1; 0) is fairly obvious from the curvature of the
ridges that appear at the intersection of two constraints. From the shape of the
feasible set it is also clear that (1; 1; 0) is the unique stationary point of the NLP.
Among the solvers we used, MINOS [17] and SNOPT [11] use quasi-Newton
methods that do not require second-order derivatives of the constraints. They
Degenerate Nonlinear Programming with a Quadratic Growth Condition 19
Feasible set: view from above. Center: (1,1,0)
lateral view.
Fig. 1. Feasible set of the nonlinear program (89) (1,1,0) is the local minimum satisfying the
quadratic growth condition (1). The jagged edges in the lateral view are a meshing effect.
also use an augmented Lagrangian as a merit function. DONLP2 [23] solves a
linear system instead of a Quadratic Program at each iteration and uses an L 1
penalty function. LANCELOT [5] uses an augmented Lagrangian technique in
conjunction with a trust-region. FilterSQP [8] also uses a trust region approach
but with a special classification of the relative merits of the iterates instead of
Nonlinear solver jjx final \Gamma x jj 2
Iterations Message at termination
FilterSQP 5.26e-09 28 Convergence
LANCELOT 8.65e-07 336 Step size too small
LINF 1.05e-08 28 Step size too small
LOQO 1.60e-07 200 Iteration limit
LOQO 5.50e-07 1000 Iteration limit
MINOS 4.76e-06 27 Current point cannot be improved
SNOPT 3.37e-07 3 Optimal Solution Found
Table
3. Runs with various nonlinear solvers on the problem (89)
a penalty or merit function. LOQO [24] is an interior-point approach. Finally,
LINF is an ad hoc Matlab implementation of the L1 exact penalty function
described in the preceding section, with an Armijo rule. The latter algorithm is
started at (0; 0; 0). All runs, except for the L1 penalty and FilterSQP algorithms,
were done on the NEOS server [18], where additional documentation can be
found for all of the above solvers.
For such a small example the time of execution is not relevant in comparing
the behavior of the solvers. Since the solution of the problem is known, we chose
as a criteria for comparison the best achievable solution. We set all relevant
tolerances to 1e \Gamma 16, via the AMPL interface of NEOS. Smaller tolerances may
interfere with the machine precision, though most of the solvers gave comparable
answers even when the tolerances are set to 1e \Gamma 20. Larger tolerances (1e \Gamma 12-
resulted in very similar results. Whenever allowed, we also changed
other limiting parameters until an intrinsic stopping decision was issued. The
only exception was DONLP2 which converged to all digits in the mantissa with
the default settings.
Table
1 shows the ratios OE(x k )\GammaOE(x )
at various iterations for our implementation
LINF. All are close to 4:00, which is consistent with the Q-linear
convergence claim for OE(x).
Table
2 shows that LANCELOT decreases succesively the value of the penalty
parameter (by 16 orders of magnitude), until it stops with the message 'Step size
too small'. This was indeed one of the alternatives allowed by our analysis in
Subsection 3.1 0). This is an undesirable outcome since the subproblems
may become harder to solve.
The results for all runs are illustrated in Table 3. It can be seen that the
solvers that use augmented Lagrangians MINOS, SNOPT, LANCELOT exhibit
an error of at least one order of magnitude larger compared to all other algo-
rithms. However, one would expect that SNOPT and MINOS would have had
at least as good a behavior as LINF if they would use a different merit function,
since the nature of the QP solved is very similar to (14). Increasing the iteration
limit in LOQO did not result in a better outcome. It is interesting to note that
the outcome in FilterSQP and LINF differ by only a factor of 2 in the same
number of iterations, though FilterSQP uses second-order information whereas
LINF does not. Both LINF and FilterSQP solve quadratic programs at each
Degenerate Nonlinear Programming with a Quadratic Growth Condition 21
iteration. DONLP2 has a remarkable behavior, though further investigation is
necessary to determine whether this has some general implications.
It is impossible to draw a general conclusion from one example. However,
there seems to be an adverse bias for methods using augmented Lagrangians on
degenerate NLPs as the one above. We are not advocating the use of LINF on
general NLP, since its similarity to steepest descent makes it very sensitive to
ill-conditioning. But the fact that it gives an outcome comparable to the one of
solvers using second-order information shows that, for better results, a different
way of incorporating second-order derivatives may be necessary.
6. Conclusions
In this work we analyze the behavior of nonlinear programs in presence of constraint
degeneracy: linear dependence of the gradients of the active constrains.
The problems of interest exhibit minima with a quadratic growth property that
satisfy the Mangasarian-Fromowitz constraint qualification. The novelty of our
approach is that, while studying the SQP convergence properties, we do not
assume the positive semidefiniteness of the Hessian of the Lagrangian on the
critical cone for any of the feasible Lagrange multipliers. Our conditions are
equivalent to a weak second-order sufficient condition [15,22].
We prove that, under these assumptions, if the data of the problem are twice
continuously differentiable, the target minimum will be an isolated stationary
point of the NLP. We also show that, when started sufficiently close to the
minimum, the L1 exact penalty SQPs induce Q-linear convergence of the values
of the penalized objective R-linear convergence of the
iterates. This shows that such methods are robust with respect to constraint
degeneracy.
We give an example of a nonlinear program with a unique minimum that
satisfies our conditions for which the Hessian of the Lagrangian is not positive
semidefinite on the critical cone for any feasible choice of the multipliers. The
direct consequence of this fact is that there is no augmented Lagrangian that
will be positive semidefinite at the solution. Therefore, Lagrange multipliers
algorithms will have to drive the penalty parameter to zero for such examples
unless the trust region is active even at convergence.
We provide our computational experience with this small nonlinear program.
As a criteria for comparison we used the best achievable solution, which was obtained
after tuning the parameters of the algorithms. We observed that, for
this example, algorithms that use augmented Lagrangians resulted in errors of
one order of magnitude or larger when compared to the other approaches. The
Lagrange multiplier package that we used (LANCELOT [5]), was confined to decrease
substantially the value of the penalty parameter (16 orders of magnitude),
which is one of the outcomes allowed by our analysis. The linear convergence
results concerning the L1 penalty function were also validated by our experiments
22 Mihai Anitescu
Undoubtedly, such a small experiment is insufficient to draw any conclusions,
especially about the approaches for which we have no theory under these assump-
tions, such as interior-point algorithms. However, both from our theory and our
experiments, it does appear that methods that use augmented Lagrangians are
less robust with respect to constraint degeneracy when compared to SQP.
We believe that attempting to develop a convergence theory in absence of the
usual second-order conditions is interesting because it may result in algorithms
that are more robust by virtue of the fact that their properties depend on fewer
assumptions. However, how to improve on the current results, and especially
how to define reliable variants of the Newton method (if possible) for this case,
is a subject of future research.
Acknowledgments
Thanks to Stephen Wright, Jorge Mor'e and Danny Ralph for the many discussions
on the subject. David Gay, Sven Leyffer and Chi-Jen Lin have kindly
provided me information and support for the numerical examples.
--R
"On the rate of convergence of sequential quadratic programming with non-differentiable exact penalty function in the presence of constraint degeneracy"
New York
"Second-order sufficiency and quadratic growth for nonisolated minima"
LANCELOT: A Fortran Package for Large-Scale Nonlinear Optimization
Introduction to Sensitivity and Stability Analysis in
Practical Methods of Optimization
"Nonlinear programming without a penalty function"
"A necessary and sufficient regularity condition to have bounded multipliers in nonconvex programming"
"Differential stability in nonlinear programming"
"User's guide for SNOPT 5.3: A Fortran package for large-scale nonlinear programming"
"The relaxation method for solving systems of linear inequalities"
"Stabilized sequential quadratic programming"
"Stability in the presence of degeneracy and error estima- tion"
"Necessary and sufficient conditions for a local minimum.3: Second order conditions and augmented duality"
"On Sensitivity analysis of nonlinear programs in Banach spaces: the approach via composite unconstrained optimization"
The NEOS Guide.
Iterative Solutions of Nonlinear Equations in Several Vari- ables
"Superlinear convergence of an interior-point method despite dependent constraints"
"Generalized equations and their solutions, Part II: Applications to non-linear programming"
"Sensitivity analysis of nonlinear programs and differentiability properties of metric projections"
"An SQP method for general nonlinear programs using only equality constrained subproblems"
"An interior-point code for quadratic programming"
"Superlinear convergence of a stabilized SQP method to a degenerate so- lution"
--TR
--CTR
Jin-Bao Jian, A Superlinearly Convergent Implicit Smooth SQP Algorithm for Mathematical Programs with Nonlinear Complementarity Constraints, Computational Optimization and Applications, v.31 n.3, p.335-361, July 2005 | degeneracy;sequential quadratic programming;nonlinear programming;quadratic growth |
589079 | Bounds for Linear Matrix Inequalities. | For iterative sequences that converge to the solution set of a linear matrix inequality, we show that the distance of the iterates to the solution set is at most \( O(\epsilon ^{2^{-d}}) \). The nonnegative integer d is the so-called degree of singularity of the linear matrix inequality, and $\epsilon $ denotes the amount of constraint violation in the iterate. For infeasible linear matrix inequalities, we show that the minimal norm of $\epsilon $-approximate primal solutions is at least \( 1/O(\epsilon ^{1/(2^{d}-1)}) \), and the minimal norm of $\epsilon $-approximate Farkas-type dual solutions is at most \( O(1/ \epsilon ^{2^{d}-1}) \). As an application of these error bounds, we show that for any bounded sequence of $\epsilon $-approximate solutions to a semidefinite programming problem, the distance to the optimal solution set is at most \( O(\epsilon ^{2^{-k}}) \), where k is the degree of singularity of the optimal solution set. | Introduction
Linear matrix inequalities play an important role in system and control theory,
see the book by Boyd et al. [3]. Recently, considerable progress has been made
in optimization over linear matrix inequalities, i.e. semi-definite programming,
see [1, 6, 8, 9, 16, 19, 18, 23, 25] and the references cited therein.
We study the linear matrix inequality (LMI)
ae
means positive semi-definiteness, B is a given (real) symmetric
matrix and A is a linear subspace of symmetric matrices.
The LMI (1) is in conic form, see e.g. [17, 23]. Since we leave complete
freedom as to the formulation of A, it is in general not difficult to fit a given
LMI into conic form. Consider for instance a linear matrix inequality
are given symmetric matrices. This is a conic form LMI
and A is the span of fF g.
Recently developed interior point codes for semi-definite programming make
it possible to solve LMIs numerically. Such algorithms generate sequences of
increasingly good approximate solutions, provided that the LMI is solvable.
For a discussion of interior point methods for semi-definite programming, see
e.g. [8, 23]. A typical way to measure the quality of an approximate solution, is
by evaluating its constraint violation.
For instance, if we denote the smallest eigenvalue of an approximate solution
~
X), then we may say that ~
X violates the constraint 'X - 0' by an
amount of [\Gamma- min ( ~
X)]+ , where the operator [\Delta] + yields the positive part. In fact,
X)]+ is the distance, measured in the matrix 2-norm, of the approximate
solution ~
X to the cone of positive semi-definite matrices. The matrix 2-norm
is a convenient measure for the amount by which the positive semi-definiteness
constraint is violated, but other matrix norms can in principle be used as well.
Similarly, we say that ~
X violates the constraint 'X 2 B +A' by an amount
of dist( ~
denotes the distance function (for a given
norm). The total amount of constraint violation in ~
X, i.e.
is called the backward error of ~
X with respect to the LMI (1). The backward
error indicates how much we should perturb the data of the problem, such that
~
X is an exact solution to the perturbed problem.
However, the backward error does not (immediately) tell us the distance from
~
X to the solution set of the original LMI; this distance is called the forward error
of ~
X .
knowing any exact solution, there is no straightforward way to
estimate the forward error. For linear inequality and equation systems however,
the forward error and backward error are of the same order of magnitude, see
Hoffman [7]. The equivalence between forward and backward errors holds also
true for systems that are described by convex quadratic inequalities, if a Slater
condition holds, see Luo and Luo [12]. In these cases, we have a relation of the
which is called a Lipschitzian error bound. For systems of convex quadratic
inequalities without Slater's condition, an error bound of the form
was obtained by Wang and Pang [26]. They also showed that d -
where n is the dimension of the problem. Error bounds for systems with a
nonconvex quadratic inequality are given in Luo and Sturm [14], and references
cited therein.
An error bound of the form (3) is called a H-olderian error bound. A
H-olderian error bound has been demonstrated for analytic inequality and equation
systems, if the size of the approximate solutions is bounded by a fixed
constant, see Luo and Pang [13]. However, there are no known positive lower
bounds on the exponent fl, except in the linear and quadratic cases that are
mentioned above, or when a Slater condition holds [4], For a comprehensive
survey of error bounds, we refer to Pang [20].
Some issues on error bounds for LMIs and semi-definite programming were
recently addressed by Deng and Hu [4], Goldfarb and Scheinberg [5], Luo, Sturm
and Zhang [16] and Sturm and Zhang [24]. Deng and Hu [4] derived upper
bounds on the Lipschitz constant (or condition number) for LMIs, if Slater's
condition holds. Luo Sturm and Zhang [16] and Sturm and Zhang [24] prove
some Lipschitzian type error bounds for central solutions for semi-definite programs
under strict complementarity. Goldfarb and Scheinberg [5] prove Lipschitz
continuity of the optimal value function for semi-definite programs.
In this paper, we show for LMIs in n \Theta n matrices, that (3) holds for a certain
the so-called degree of singularity, provided that the size
of the approximate solutions is bounded. We interpret the degree of singularity
in the context of Ramana-type regularized duality. It is basically the number
of elementary regularizations that are needed to obtain a fully regularized dual.
Under Slater's constraint qualification, the irregularity level d is zero. (Notice
that this is also true for convex quadratic systems, see Wang and Pang [26].) The
degree of singularity of the optimal solution set of a semi-definite programming
problem is at most one, if strict complementarity holds. The concept of singularity
degrees thus embeds the Slater and strict complementarity conditions in
a hierarchy of singularity for LMIs.
This paper is organized as follows. In Section 2, we introduce the concept of
regularized backward errors, which is closely related to the concept of minimal
cones [2]. In this section, we also show that there is a close connection between
the regularized backward error and the forward error. We will then estimate
in Section 3 how the regularized backward error depends on the usual backward
error. In Section 4, we apply the error bound for LMIs to semi-definite
programming problems. The paper is concluded in Section 5.
Notation. Let S n\Thetan denote the space of n \Theta n real symmetric matrices.
The cone of all positive semi-definite matrices in S n\Thetan is denoted by S n\Thetan
we
. The interior of S n\Thetan
is the set of
positive definite matrices S n\Thetan
++ , and we write X - 0 if and only if X 2 S n\Thetan
++ .
We let N := n(n + 1)=2 denote the dimension of the real linear space S n\Thetan .
The standard inner product for two symmetric matrices X and Y is
tr XY . The matrix norm kXk 2 is the operator 2-norm that is associated with
the Euclidean norm for vectors, namely
For symmetric matrices, kXk 2 is the eigenvalue of X that has the largest absolute
value.
2 The regularized backward error
A denote the smallest linear subspace containing B +A, i.e.
We are naturally interested in the intersection of this linear subspace with the
cone of positive semi-definite matrices. It holds that
A " S n\Thetan
the above characterization is a special case of a duality theorem for convex
cones.
The general theorem states that given a linear subspace L and a convex cone
!+ g, it holds that
see Corollary 2 in Luo, Sturm and Zhang [15] and Corollary 2.2 in [23]. This
result generalizes a classical duality theorem of Gordon and Stiemke for linear
inequalities.
To see why (5) is a special case of (6), we must interpret S n\Thetan
as a convex
cone in ! N . This can be established by choosing an orthonormal basis of S n\Thetan ,
say an orthonormal set of symmetric matrices fS[1];
1)=2 is the dimension of S n\Thetan . We can then associate with any matrix
n\Thetan a coordinate vector x 2 ! N into this basis, and vice versa. Namely,
we let x
Due to the
orthonormality of the basis, we have y, for all matrices X;Y 2 S n\Thetan
with coordinate vectors x; y 2 ! N .
As a convention, we use upper-case symbols, like X and B, for symmetric
matrices, and we implicitly define the corresponding lower-case symbols, like x
and b, to be the associated coordinate vectors, as described above. Furthermore,
we use calligraphic letters, such as S n\Thetan
, to denote sets. With the established
one-to-one correspondence between S n\Thetan and ! N in mind, we do not only use
S n\Thetan
for the set of positive semi-definite matrices in S n\Thetan , but also for the set
of coordinate vectors of positive semi-definite matrices, which is a convex cone
in the Euclidean space ! N . We will also use such a convention for other sets of
symmetric matrices. In particular, we reformulate (4) as
where Img b ae ! N is the line of all multiples of b. The orthogonal complement
of -
A is
The all-zero matrix is obviously the only matrix that is both positive and
negative semi-definite, i.e. S n\Thetan
" \GammaS n\Thetan
f0g. Also, the cone of positive semi-definite
matrices is self-dual, i.e. (S n\Thetan
. Thus, taking
and
A ? in (6) yields (5).
Relation (5) states that if -
A and S n\Thetan
intersect only at the origin, then
there exists a positive definite matrix Z 2 -
A ? . Consider now a sequence of
increasingly accurate solutions fX(ffl)
notice that the parameter ffl measures the backward error in X(ffl). It follows
that since Z?(B + A), we must have jZ ffl O(ffl). Using the fact that
positive definite, this implies that O(ffl). The
above reasoning establishes the relation
A " S n\Thetan
which is an error bound for the case that -
A intersects the semi-definite cone
only at the origin.
Assume now that -
A " S n\Thetan
A " S n\Thetan
applying a basis transformation if necessary, we may assume without loss of
generality that we can partition X as
Using this notation, we can partition an arbitrary matrix X 2 S n\Thetan as
U XN
A " S n\Thetan
suppose without loss of generality
that X is of the form (9). Then it holds for all
A " S n\Thetan
and
Proof. Suppose to the contrary that XN is not the all-zero matrix, and let
\Theta 0 y T
be such that XN yN 6= 0. Then for any ff 2 !,
where we used the fact that X is positive semi-definite. Consequently, we have
for all ff ? 0 that
A " S n\Thetan
which contradicts the fact that by definition, X is in the relative interior of
A " S n\Thetan
. We have now shown by contradiction that
positive semi-definite, it follows that also
A face of S n\Thetan
is by definition a cone of the form
n\Thetan
where Z is a given positive semi-definite matrix. In particular, if
then
n\Thetan
ae
oe
and X is in the relative interior of face(S n\Thetan
0). The facial structure of S n\Thetan
has been studied in detail by Lewis [11]
and Pataki [21].
A " S n\Thetan
suppose without loss of generality
that X is of the form (9). Then
relint
S n\Thetan
I
Proof. The lemma holds trivially true if (B +A) " S n\Thetan
now
that there exists
A, there exists t 2 ! such
that X \Gamma tB 2 A. However, for all ff ? 0 satisfying fft ? \Gamma1, we
S n\Thetan
I
where we used Lemma 1. This shows that
S n\Thetan
I
Using Lemma 1 once again, the lemma follows from the above relation. Q.E.D.
Due to the above result, the face
face
S n\Thetan
I
is sometimes called the minimal cone [2] or the regularized semi-definite cone [15]
for the affine space B +A.
The backward error of X(ffl) with respect to the regularized system
is naturally defined as
The following lemma states, among others, that if fX(ffl)
then the regularized backward error is of the same order as the forward error
A " S n\Thetan
suppose without loss of generality
that X is of the form (9). If fX(ffl) is such that
for all ffl ? 0, then (B+A)"S n\Thetan
;. Moreover, there exists
such that
Proof. As is well known, the backward and forward error for a system of
linear equations are of the same order [7]. Therefore, the relations
imply that
This bound implies the existence of fY (ffl) 0g, such that
Using also the fact that X
B is positive definite, it follows that
with
Notice that
A, there must exist t 2 ! such that
be such that
and hence
dist
Under Slater's condition, i.e. if (B
Hoffman's error bound [7] for systems of linear inequalities and equations to
LMIs. Notice in particular that no boundedness assumptions are made, i.e. the
error bound holds globally over S n\Thetan . However, the lemma requires a scaling
which is not needed in case of linear inequalities and equations.
The following example shows that this scaling factor is essential in the case of
LMIs.
Example 1 Consider the LMI in S 2\Theta2 with
ae
oe
i.e. we want to find find x 11 and x 12 such that x 11 - jx 12 j 2 . This LMI obviously
has positive definite solutions (the identity matrix for instance). Therefore,
the regularized backward error is identical to the usual backward error. The
approximate solution
has backward error ffl ? 0. However, X(ffl)
if and only
if
y 22
which shows that the distance of X(ffl) to (B
is bounded from below
by a positive constant as ffl # 0. However, we have X(ffl)=(1+ ffl) 2 (B+A)"S 2\Theta2
which agrees with the statement of Lemma 3.
Below are more remarks on the regularized error bound of Lemma 3.
states that the mere existence of fX(ffl)
(12) for all ffl ? 0 implies that (B
even though X(ffl) is not
necessarily bounded for ffl # 0. In the case of weak infeasibility, i.e. if
dist(B +A;S n\Thetan
we can therefore conclude that if X(ffl) satisfies (7) then
lim inf
ffl#0
is a bounded sequence with
then also kX (k)
as follows from Lemma 1. Letting
it follows from Lemma 3 and the boundedness of the sequence fX
3 Regularization steps
In order to bound the regularized backward error (11) in terms of the original
backward error (2), we use a sequence of regularization steps.
In the preceding, we have partitioned n \Theta n matrices according to the structure
of X , given by (9). In this section, we will also partition n \Theta n matrices
into blocks, but with respect to a possibly different eigenvector basis; the sizes of
the blocks can be different as well. We will denote the blocks by the subscripts
We will also encounter the dual cone of a face of S n\Thetan
face
S n\Thetan
I
ae -
oe
Obviously, we have
relint face
S n\Thetan
I
ae -
oe
In the following, we will allow the possibility that are X 22
are non-existent. For this case, we use the convention that kX 12
A be a linear subspace of S n\Thetan , and suppose that fX(ffl)
is such that
for all
S n\Thetan
I
It holds that
ffl Z 11 - 0 if and only if
S n\Thetan
I
only if
S n\Thetan
I
ffl For the remaining case that 0 6= Z 11 6- 0, let
\Theta
be an orthogonal
matrix such that Z 11
Proof. The first two cases, i.e. Z are immediate applications
of (6). It remains to consider the case that Z 11 is a nonzero but singular, positive
semi-definite matrix.
- ffl, there must exist Y (ffl), such that
for all ffl ? 0. This implies that Z?(X(ffl)+Y (ffl)) because Z 2 -
A ? , and therefore
Z
'-
where we used the Cauchy-Schwartz inequality. Recall now that
so that we further obtain
Z
Since Z 11 is positive semi-definite and - min (X(ffl)) - \Gammaffl, we have
where we used Z 11 in the first identity, and (15) in the last identity.
Recalling that Q T
easily follows from the above relation that
Finally, since - min (X(ffl)) - \Gammaffl, we know that X 11 (ffl) + fflI is positive semi-
definite, and hence
where we used (16). This completes the proof. Q.E.D.
For a given linear subspace -
A ' S n\Thetan , we define the level of singularity
by recursively applying the construction of Lemma 4. This procedure is
outlined below:
Procedure 1 Definition of the level of singularity of a linear subspace -
S n\Thetan .
Otherwise, proceed with Step 2.
be such that Z (0)
ae
\Theta
A
oe
A ?
S n\Thetan
I
If Z (d)
proceed with Step 4.
be such that Z (d)
I
Let
ae
\Theta -
A d
oe
return to Step 3.
In the above procedure, we start with the full dimensional cone S n\Thetan
, and in
the first iteration we determine a face of this cone. Next, we arrive at a face of
this face, and so on. We claim that this procedure finally arrives at the minimal
cone. To see this, notice that at any given step
above, we
perform a regularization step as described in Lemma 4. Recall from (5) that
A " S n\Thetan
f0g, and this case has already
been treated in Section 2. In any other case, we have Z (d( -
It is easily
seen from Lemma 4 that if X 2 -
A " S n\Thetan
This means that
all nonzeros of X must be contained in the (final) 11 block for -
A
. On the
other hand, since Z (d( -
in the above procedure, it follows from (6) that
there exists ~
A " S n\Thetan
such that ~
we just showed that X
A " S n\Thetan
, we must have
~
A " S n\Thetan
Hence, the face in the final iteration is the minimal
cone. For -
(9).
By applying a basis transformation if necessary, we may assume without loss
of generality that there is a (d( -
partition, such that
0:
Above, we used a Matlab-type 1 notation, thus means
denotes the block on the ith row and jth column in the (d( -
is symmetric, we only specified the upper
block triangular part of Z. The relation between the (d( -
partition in (18) and the 2 \Theta 2 partition in iteration
is that
The minimal cone is the set of matrices X for which
In iteration
of Procedure 1, we arrive at the face where
which indeed includes the minimal cone.
Remark that the 3rd row and column in the 3 \Theta 3 block form of (18) are
non-existent for
A), i.e. for
Based on Lemma 4, we can now estimate the regularized backward error.
A " S n\Thetan
loss of generality that X is of the form (9). If
is such that for all
then
with
is the degree of singularity of -
A.
Proof. Applying Lemma 4 in iteration of Procedure 1, we have that
1 MATLAB is a registered trademark of The MathWorks, Inc.
where we used X ffl as a synonym for X(ffl). Suppose now that in iteration d 2
where
It then follows from Lemma 4 that (19) also holds for
By induction. we obtain that (19) holds for
(ffl), the lemma follows. Q.E.D.
We arrive now at the main result of this paper, namely an error bound for
LMIs.
Theorem 1 Let -
is such that kX(ffl)k is
bounded and
then
Proof. For the case that d( -
the theorem follows by combining Lemma 3
with Lemma 5. If
there are two cases, either -
A " S n\Thetan
A " S n\Thetan
In the former case, we have hence the error
bound holds, see Section 2. In the latter case, we have that X
the error bound follows from Lemma 3. Q.E.D.
An LMI is said to be weakly infeasible if
1. there is no solution to the LMI, i.e. (B +A) " S n\Thetan
2. dist(B +A;S n\Thetan
For weakly infeasible LMIs, there exist approximate solutions with arbitrarily
small constraint violations. However, the following theorem provides a lower
bound on the size of such approximate solutions to weakly infeasible LMIs.
Theorem 2 Let -
and suppose that
is such that
small enough, we have X(ffl) 6= 0
Proof. Suppose to the contrary that there exists a sequence ffl 1
Applying Lemma 5, it follows that
Together with Lemma 3, we obtain that (B +A) " S n\Thetan
Q.E.D.
There is an extension of Farkas' lemma from linear inequalities to convex
cones, which states that
where K ae S n\Thetan is a convex cone, and K is the associated dual cone. See
e.g. Lemma 2.5 in [23]. If dist(B +A;S n\Thetan
we say that the LMI is
strongly infeasible. Relation (20) states that strong infeasibility can be demonstrated
by a matrix Z 2 A ? " S n\Thetan
0, and such Z is called a dual
improving direction.
For weakly infeasible LMIs, infeasibility cannot be demonstrated by a dual
improving direction. However, an LMI is infeasible if and only if there exist
approximate dual improving directions with arbitrarily small constraint viola-
tions. See e.g. Lemma 2.6 in [23]. The next theorem gives an upper bound for
the minimal norm of such approximate dual improving directions in the case of
infeasibility.
Theorem 3 Let -
there exist
0g such that for all holds that
and
Proof. Let X 2 relint ( -
A " S n\Thetan
suppose without loss of generality
that X is of the form (9). Using the same 2 \Theta 2 partition as in (9), it follows
from Lemma 3 that
dist
S n\Thetan
I
-"
0:
Applying (20), it thus follows that there exists a matrix Y (0) such that
S n\Thetan
I
Partitioning Y (0) , we have
Y (0)
U
We shall now construct fY
A)g such that
Y
1. Remark from (21)-(22) that (23) holds for
construct Y (k) for k 2
in such a way that it satisfies (23),
provided that Y (k\Gamma1) satisfies (23). We can then use induction.
Let
immediately obtain from (23) that
irrespective of t. Furthermore, since Y
positive semi-definite if and only if the Schur-complement
is positive semi-definite. From (18) and the definition of Y t , we have
and hence
positive semi-definite if we choose t as
where we used that kY (k\Gamma1)
Setting Y
The theorem follows by letting
We remark from the proof of Theorem 3 that the matrices Y (0) and Z (k) ,
finite certificate of the infeasibility of the LMI.
Together, these matrices form essentially a solution to the regularized Farkas-
type dual of Ramana [22], see also [10, 15]. Thus, the degree of singularity is
the minimal number of layers that are needed in the perfect dual of Ramana.
As discussed in the introduction, it is easy to calculate the backward error of
an approximate solution. However, the error bound for the forward error of an
LMI, as given in Theorem 1, does not only involve the backward error, but also
the degree of singularity. We will now provide some easily computable upper
bounds on the degree of singularity.
Lemma 6 For the degree of singularity
of a linear subspace -
A ' S n\Thetan ,
it holds that
A ? g:
Proof. If
A " S n\Thetan
by definition of
A). For this case,
we have defined the (d( -
partition (18), where each of
the
diagonal blocks is at least of size 1 \Theta 1. Thus,
Furthermore, Lemma 4 defines a matrix Z
A ? , for each regularization
and it is easily verified that these matrices are
mutually independent. Therefore,
Finally, using the (d( -
partition (18), we claim that
there exists X
A with
Namely, if such X (k) does not exist, then by (6), there must exist \DeltaZ 2 -
A ?
such that
and this contradicts the fact that Z (d( -
of maximal
rank, see its definition in Lemma 4. Again, it is easy to see that the matrices
A), are mutually independent, and hence
A. Q.E.D.
The bounds of Theorem 1 and Theorem 2 quickly become inattractive as
the singularity degree increases. However, the next two examples show that
these bounds can be tight. This means that problems with a large degree of
singularity can be very hard to solve numerically.
Example 2 Consider the LMI
Due to the restriction 'X 22 = 0' and the positive semi-definiteness, we have
which further implies
and inductive argument, we have X
we can construct a sequence fX(ffl) j ffl ? 0g with a constraint violation ffl, but
, viz.
Notice that 'X 22 = 0' is the only constraint that is violated by X(ffl).
To see how unfortunate this example is, consider a backward error
Then, already for
any solution -
X of the LMI.
Example 3 Extending Example 2 with the restriction 'X we obtain a
(weakly) infeasible LMI:
However, we may construct a sequence fX(ffl) violation
ffl and Namely, we let
This example shows that (in)feasibility can be hard to detect. Namely, for
which is not
unusually large; yet, the problem is infeasible.
4 Application to semi-definite programming
bounds for LMIs can be applied to semi-definite optimization models as
well. A standard form semi-definite program is
where B and C are given symmetric matrices. Associated with this optimization
problem is a dual problem, viz.
(D)
An obvious property of the primal-dual pair (P) and (D) is the weak duality
relation. Namely, if X 2 (B +A) " S n\Thetan
Clearly, if X ffl must be optimal solutions to (P) and (D)
respectively; we say then that (X; Z) is a pair of complementary solutions. In
general, such a pair may not exist, even if both (P) and (D) are feasible. (We say
that (P) is feasible if (B+A)"S n\Thetan
and (D) is feasible if (C+A ? )"S n\Thetan
;.) A sufficient condition for the existence of a complementary solution pair is
that (P) and (D) are feasible and satisfy the primal-dual Slater condition, in
which case
Based on (25), we can formulate the set of complementary solutions as the
In principle, we can apply our error bound results for LMIs directly to the above
system. But, tighter bounds can be obtained by exploring its special structure.
Consider a bounded trajectory of approximate primal and dual solutions
C) be a complementary solution pair, i.e.
Such a pair must exist, since in particular any cluster point of f(X(ffl); Z(ffl) j
ffl ? 0g for ffl # 0 is a complementary solution pair. Notice that B
and similarly C +A
, from which we easily derive that
for feasible solutions X and Z, and
for (X(ffl); Z(ffl)) satisfying (26). This means that X(ffl) has an O(ffl) constraint
violation with respect to the LMI
Notice that (27) describes the set of optimal solutions to (P). Letting
A := Img
the Theorems 1 and 2 are applicable to the LMI (27) and hence to the semi-definite
program (P). Specifically, given a bounded trajectory fX(ffl); Z(ffl)
0g satisfying (26), we know that the distance from X(ffl) to the set of optimal
solutions to (P) is O(ffl 2 \Gammad( -
is the degree of singularity of the
linear subspace defined in (28).
0, we can move the parentheses in definition (28) to get
from which we get
Noticing the primal-dual symmetry, we conclude that the distance from Z(ffl)
to the set of optimal solutions to (D) is O(ffl 2 \Gammad( -
A ? ) is the degree
of singularity of -
A ? .
Concluding remarks
Theorem 1 provides a H-olderian error bound for LMIs. For weakly infeasible
LMIs, we have derived relations between backward errors and the size of approximate
solutions, see Theorems 2 and 3. In Section 4, we applied the error
bound of Theorem 1 to semi-definite programming problems (SDPs). If the
SDP has a strictly complementary solution, then its degree of singularity can
be at most 1, and the bound becomes
backward error):
For this case, Luo, Sturm and Zhang [16] obtained a Lipschitzian error bound if
the approximate solutions (X(ffl); Z(ffl)) are restricted to the central path. The
sensitivity of central solutions with respect to perturbations in the right-hand
side was studied by Sturm and Zhang [24].
Acknowledgment
. Tom Luo's comments on an earlier version of this paper
have resulted in substantial improvements in the presentation.
--R
Regularizing the abstract convex program.
Linear matrix inequalities in system and control theory
Computable error bounds for semidefinite program- ming
On parametric semidefinite programming.
An interior-point method for semidefinite programming
On approximate solutions of systems of linear inequalities.
Interior Point Methods for Semidefinite Programming.
Perfect duality in semi-infinite and semidefinite programming
Extensions of Hoffman's error bound to polynomial systems.
bounds for analytic systems and their applications.
bounds for quadratic systems.
Duality results for conic convex programming.
Superlinear convergence of a symmetric primal-dual path following algorithm for semidefinite programming
Interior point polynomial methods in convex programming
bounds in mathematical programming.
On the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues.
An exact duality theory for semidefinite programming and its complexity implications.
On sensitivity of central solutions in semidefinite programming.
programming.
Global error bounds for convex quadratic inequality systems.
--TR
--CTR
Dominique Az , Jean-Baptiste Hiriart-Urruty, Optimal Hoffman-Type Estimates in Eigenvalue and Semidefinite Inequality Constraints, Journal of Global Optimization, v.24 n.2, p.133-147, October 2002 | regularized duality;error bounds;linear matrix inequality;semidefinite programming |
589089 | A Feasible BFGS Interior Point Algorithm for Solving Convex Minimization Problems. | We propose a BFGS primal-dual interior point method for minimizing a convex function on a convex set defined by equality and inequality constraints. The algorithm generates feasible iterates and consists in computing approximate solutions of the optimality conditions perturbed by a sequence of positive parameters $\mu$ converging to zero. We prove that it converges q-superlinearly for each fixed $\mu$. We also show that it is globally convergent to the analytic center of the primal-dual optimal set when $\mu$ tends to 0 and strict complementarity holds. | Introduction
. We consider the problem of minimizing a smooth convex function
on a convex set defined by inequality constraints. The problem is written as
# R is the function to minimize and c(x) # 0 means that each component
m) of c must be nonnegative at the solution. To simplify
the presentation and to avoid complicated notation, the case when linear equality
constraints are present is discussed at the end of the paper. Since we assume that the
components of c are concave, the feasible set of this problem is convex.
The algorithm proposed in this paper and the convergence analysis require that
f and c are di#erentiable and that at least one of the functions f , -c (1) , . , -c (m) is
strongly convex. The reason for this latter hypothesis will be clarified below. Since the
algorithm belongs to the class of interior point (IP) methods, it may be well suited for
problems with many inequality constraints. It is also more e#cient when the number
of variables remains small or medium, say, fewer than 500, because it updates n - n
matrices by a quasi-Newton (qN) formula. For problems with more variables, limited
memory BFGS updates [39] can be used, but we will not consider this issue in this
paper.
Our motivation is based on practical considerations. During the last 15 years
much progress has been realized on IP methods for solving linear or convex minimization
problems (see the monographs [29, 10, 38, 44, 23, 42, 47, 49]). For nonlinear
convex problems, these algorithms assume that the second derivatives of the functions
used to define the problem are available (see [43, 35, 36, 12, 38, 26]). In practice, how-
# Received by the editors September 15, 1998; accepted for publication (in revised form) January
26, 2000; published electronically August 3, 2000.
http://www.siam.org/journals/siopt/11-1/34472.html
des Sciences, 123 av. A. Thomas, 87060 Limoges Cedex, France (Paul.Armand@
unilim.fr).
# INRIA Rocquencourt, BP 105, 78153 Le Chesnay Cedex, France (Jean-Charles.Gilbert@inria.fr).
- MIP, UFR MIG, Universit-e Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 4,
France (jan@mip.ups-tlse.fr).
200 P. ARMAND, J. CH. GILBERT, AND S. JAN-J -
EGOU
ever, it is not uncommon to find situations where this requirement cannot be satis-
fied, in particular for large scale engineering problems (see [27] for an example, which
partly motivates this study and deals with the estimation of parameters in a three
phase flow in a porous medium). Despite the possible use of computational di#eren-
tiation techniques [8, 19, 3, 28], the computing time needed to evaluate Hessians or
Hessian-vector products may be so large that IP algorithms using second derivatives
may be unattractive.
This situation is familiar in unconstrained optimization. In that case, qN tech-
niques, which use first derivatives only, have proved to be e#cient, even when there
are millions of variables (see [32, 20] and [9] for an example in meteorology). This fact
motivates the present paper, in which we explore the possibility of combining the IP
approach and qN techniques. Our ambition remains modest, however, since we confine
ourselves to the question of whether the elegant BFGS theory for unconstrained
convex optimization [41, 6] is still valid when inequality constraints are present. For
the applications, it would be desirable to have a qN-IP algorithm in the case when
f and -c are nonlinear and not necessarily convex. We postpone this more di#cult
subject for future research (see [21, 48] for possible approaches).
Provided the constraints satisfy some qualification assumptions, the Karush-
Kuhn-Tucker (KKT) optimality conditions of problem (1.1) can be written (see [17],
for example) as follows: there exists a vector of multipliers # R m such that
where #f(x) is the gradient of f at x (for the Euclidean scalar product), #c(x) is a
matrix whose columns are the gradients #c (i) (x), and is the
diagonal matrix, whose diagonal elements are the components of c. The Lagrangian
function associated with problem (1.1) is defined on R n
Since f is convex and each component c (i) is concave, for any fixed # 0, #) is
a convex function from R n to R. When f and c are twice di#erentiable, the gradient
and Hessian of # with respect to x are given by
Our primal-dual IP approach is rather standard (see [24, 36, 35, 11, 12, 1, 26, 25,
15, 7, 5]). It computes iteratively approximate solutions of the perturbed optimality
system
for a sequence of parameters - > 0 converging to zero. In (1.2),
is the vector of all ones whose dimension will be clear from the context. The last
inequality means that all the components of both c(x) and # must be positive. By
perturbing the complementarity equation of the KKT conditions with the parameter
A BFGS INTERIOR POINT ALGORITHM 201
-, the combinatorial aspect of the problem, inherent in the determination of the active
constraints or the zero multipliers, is avoided. We use the word inner to qualify those
iterations that are used to find an approximate solution of (1.2) for fixed -, while an
outer iteration is the collection of inner iterations corresponding to the same value
of -.
The Newton step for solving the first two equations in (1.2) with fixed - is the
solution
of the linear system
d #f(x) +#c(x)#
in which
xx #(x, #) and This direction is sometimes
called the primal-dual step, since it is obtained by linearizing the primal-dual system
(1.2), while the primal step is the Newton direction for minimizing in the primal
variable x the barrier function
log c (i) (x)
associated with (1.1) (the algorithms in [16, 33, 4] are in this spirit). The two problems
are related since, after elimination of #, (1.2) represents the optimality conditions of
the unconstrained barrier problem
c(x) > 0.
As a result, an approximate solution of (1.2) is also an approximate minimizer of the
barrier problem (1.4). However, algorithms using the primal-dual direction have been
shown to present a better numerical e#ciency (see, for example, [46]).
In our algorithm for solving (1.2) or (1.4) approximately, a search direction d
is computed as a solution of (1.3) in which M is now a positive definite symmetric
matrix approximating # 2
xx #(x, #) and updated by the BFGS formula (see [14, 17] for
material on qN techniques). By eliminating d # from (1.3) we obtain
Since the iterates will be forced to remain strictly feasible, i.e., (c(x), #) > 0, the
positive definiteness of M implies that d x is a descent direction of # - at x. Therefore,
to force convergence of the inner iterates, a possibility could be to force the decrease
of # - at each iteration. However, since the algorithm also generates dual variables #,
we prefer to add to # - the function (see [45, 1, 18])
log # (i) c (i) (x) #
to control the change in #. This function is also used in [30, 31] as a potential function
for nonlinear complementarity problems. Even though the map (x, # -
V(x, #) is not necessarily convex, we will show that it has a unique minimizer, which is
the solution of (1.2), and that it decreases along the direction
this primal-dual merit function can be used to force the convergence of the pairs
(x, #) to the solution of (1.2), using line-searches. It will be shown that the additional
P. ARMAND, J. CH. GILBERT, AND S. JAN-J -
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function V does not prevent unit step-sizes from being accepted asymptotically, which
is an important point for the e#ciency of the algorithm.
Let us stress the fact that our algorithm is not a standard BFGS algorithm for
solving the barrier problem (1.4), since it is the Hessian of the Lagrangian that is
approximated by the updated matrix M , not the Hessian of # - . This is motivated by
the following arguments. First, the di#erence between # 2
involves first derivatives only. Since these derivatives are considered to be available,
they need not be approximated. Second, the Hessian # 2
xx #, which is approximated by
M , is independent of - and does not become ill-conditioned as - goes to zero. Third,
the approximation of # 2
# obtained at the end of an outer iteration can be used as
the starting matrix for the next outer iteration. If this looks attractive, it has also the
inconvenience of restricting the approach to (strongly) convex functions, as we now
explain.
After the computation of the new iterates
the step-size given by the line-search), the matrix M is updated by the BFGS formula
using two vectors # and #. Since we want the new matrix M+ to be an approximation
of # 2
it satisfies the qN equation M+ # (a property of the
BFGS formula), it makes sense to define # and # by
The formula is well defined and generates stable positive definite matrices provided
these vectors satisfy # > 0. This inequality, known as the curvature condition,
expresses the strict monotonicity of the gradient of the Lagrangian between two successive
iterates. In unconstrained optimization, it can always be satisfied by using
the Wolfe line-search, provided the function to minimize is bounded below. If this is
a reasonable assumption in unconstrained optimization, it is no longer the case when
constraints are present, since the optimization problem may be perfectly well defined
even when # is unbounded below. Now, assuming this hypothesis on the boundedness
of # would have been less restrictive than assuming its strong convexity, but it is not
satisfactory. Indeed, with a bounded below Lagrangian, the curvature condition can
be satisfied by the Wolfe line-search as in unconstrained optimization, but near the
solution the information on # 2
collected in the matrix M could come from a region
far from the optimal point, which would prevent q-superlinear convergence of the it-
erates. Because of this observation, we assume that f or one of the functions -c (i)
is strongly convex, so that the Lagrangian becomes a strongly convex function of x
for any fixed # > 0. With this assumption, the curvature condition will be satisfied
independently of the kind of line-search techniques actually used in the algorithm.
The question whether the present theory can be adapted to convex problems, hence
including linear programming, is puzzling. We will come back to this issue in the
discussion section.
A large part of the paper is devoted to the analysis of the qN algorithm for solving
the perturbed KKT conditions (1.2) with fixed -. The algorithm is detailed in the
next section, while its convergence speed is analyzed in sections 3 and 4. In particular,
it is shown that, for fixed - > 0, the primal-dual pairs (x, #) converge q-superlinearly
toward a solution of (1.2). The tools used to prove convergence are essentially those of
A BFGS INTERIOR POINT ALGORITHM 203
the BFGS theory [6, 13, 40]. In section 5, the overall algorithm is presented and it is
shown that the sequence of outer iterates is globally convergent, in the sense that it is
bounded and that its accumulation points are primal-dual solutions of problem (1.1).
If, in addition, strict complementarity holds, the whole sequence of outer iterates
converges to the analytic center of the primal-dual optimal set.
2. The algorithm for solving the barrier problem. The Euclidean or # 2
norm is denoted by #. We recall that a function # : R n
# R is said to be
strongly convex with modulus # > 0, if for all (x, y) # R n
equivalent definitions, see, for example,
[22, Chapter IV]). Our minimal assumptions are the following.
Assumption 2.1. (i) The functions f and -c (i) (1 # i # m) are convex and
di#erentiable from R n to R and at least one of the functions f , -c (1) , . , -c (m) is
strongly convex. (ii) The set of strictly feasible points for problem (1.1) is nonempty,
i.e., there exists x # R n such that c(x) > 0.
Assumption 2.1(i) was motivated in section 1. Assumption 2.1(ii), also called
the (strong) Slater condition, is necessary for the well-posedness of a feasible interior
point method. With the convexity assumption, it is equivalent to the fact that the
set of multipliers associated with a given solution is nonempty and compact (see
[22, Theorem VII.2.3.2], for example). These assumptions have the following clear
consequence.
Lemma 2.2. Suppose that Assumption 2.1 holds. Then, the solution set of problem
(1.1) is nonempty and bounded.
By Lemma 2.2, the level sets of the logarithmic barrier function # - are compact,
a fact that will be used frequently. It is a consequence of [16, Lemma 12], which we
recall for completeness.
Lemma 2.3. Let f : R n
# R be a convex continuous function and c : R n
be a continuous function having concave components. Suppose that the set {x # R
c(x) > 0} is nonempty and that the solution set of problem (1.1) is nonempty and
bounded. Then, for any # R and - > 0, the set
log c (i) (x) #
is compact (and possibly empty).
Let x 1 be the first iterate of our feasible IP algorithm, hence satisfying c(x 1 ) > 0,
and define the level set
Lemma 2.4. Suppose that Assumption 2.1 holds. Then, the barrier problem (1.4)
has a unique solution, which is denoted by - x - .
Proof. By Assumption 2.1, Lemma 2.2, and Lemma 2.3, L P
1 is nonempty and
compact, so that the barrier problem (1.4) has at least one solution. This solution
is also unique, since # - is strictly convex on {x # R Indeed, by
Assumption 2.1(i), # 2 # - (x) given by (1.6) is positive definite.
To simplify the notation we denote by
z := (x, #)
a typical pair of primal-dual variables and by Z the set of strictly feasible z's:
204 P. ARMAND, J. CH. GILBERT, AND S. JAN-J -
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The algorithm generates a sequence of pairs (z, M ), where z # Z and M is a
positive definite symmetric matrix. Given a pair (z, M ), the next one (z
obtained as follows. First
is a step-size and is the unique solution of (1.3). The
uniqueness comes from the positivity of c(x) and from the positive definiteness of M
(for the unicity of d x , use (1.5)). Next, the matrix M is updated into M+ by the
where # and # are given by
This formula gives a symmetric positive definite matrix M+ , provided M is symmetric
positive definite and # > 0 (see [14, 17]). This latter condition is satisfied because
of the strong convexity assumption. Indeed, since at least one of the functions f or
-c (i) is strongly convex, for any fixed # > 0, the function x #(x, #) is strongly
convex, that is, there exists a constant # > 0 such that
Since # sizes the displacement in x and #, the merit function used to estimate
the progress to the solution must depend on both x and #. We follow an idea of
Anstreicher and Vial [1] and add to # - a function forcing # to take the value -C(x)
The merit function is defined for z = (x, # Z by
where
log # (i) c (i) (x) # .
Note that
# .
Using # - as a merit function is reasonable provided the problem
z # Z
A BFGS INTERIOR POINT ALGORITHM 205
has for unique solution the solution of (1.2) and the direction descent
direction of # - . This is what we check in Lemmas 2.5 and 2.6 below.
Lemma 2.5. Suppose that Assumption 2.1 holds. Then, problem (2.4) has a
unique solution -
z -x - ), where -
x - is the unique solution of the barrier problem
# - has its ith component defined by ( - ) (i) := -/c (i) (-x - ). Furthermore,
- has no other stationary point than - z - .
Proof. By optimality of the unique solution -
x - of the barrier problem (1.4)
x such that c(x) > 0.
On the other hand, since log t is minimized at
c (i) (- x - ) (i) for all index i, we have
z # Z.
Adding up the preceding two inequalities gives # -z - (z) for all z # Z. Hence
z - is a solution of (2.4).
It remains to show that - z - is the unique stationary point of # - . If z is stationary,
it satisfies
Canceling # from the first equality gives #f(x) -#c(x)C(x) thus
x - is the unique minimizer of the convex function # - . Now, by the
second equation of the system above.
Lemma 2.6. Suppose that z # Z and that M is symmetric positive definite. Let
be the solution of (1.3). Then
so that d is a descent direction of # - at a point z
z - , meaning that # - (z) # d < 0.
Proof. We have # - (z) # d. Using (1.5),
which is nonpositive. On the other hand, when d satisfies the second equation of (1.3),
one has (see [1])
which is also nonpositive. The formula for # - (z) # d given in the statement of the
lemma follows from this calculation. Furthermore, # - (z) # d < 0, if z #= -
z - .
We can now state precisely one iteration of the algorithm used to solve the perturbed
KKT system (1.2). The constants # ]0, 1[ and 0 < # < 1 are given
independently of the iteration index.
206 P. ARMAND, J. CH. GILBERT, AND S. JAN-J -
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Algorithm A - (for solving (1.2); one iteration).
At the beginning of the iteration, the current iterate
supposed available, as well as a positive definite matrix M approximating
the Hessian of the Lagrangian # 2
xx #(x, #).
1. Compute d := (d x , d # ), the solution of the linear system (1.3).
2. Compute a step-size # by means of a backtracking line search.
2.0.
2.1. Test the su#cient decrease condition:
2.2. If (2.5) is not satisfied, choose a new trial step-size # in [#] and
go to Step 2.1. If (2.5) is satisfied, set z
3. Update M by the BFGS formula (2.1) where # and # are given by (2.2).
By Lemma 2.6, d is a descent direction of # - at z, so that a step-size # > 0 satisfying
(2.5) can be found. In the line-search, it is implicitly assumed that (2.5) is not
satisfied if z holds for the new iterate z + .
We conclude this section with a result that gives the contribution of the line-search
to the convergence of the sequence generated by Algorithm A - . It is in the
spirit of a similar result given by Zoutendijk [50] (for a proof, see [6]). We say that
a function is C 1,1 if it has Lipschitz continuous first derivatives. We denote the level
set of # - determined by the first iterate z
Lemma 2.7. If - is C 1,1 on an open convex neighborhood of the level set L PD
there is a positive constant K such that for any z # L PD
determined by the
line-search in Step 2 of Algorithm A - , one of the following two inequalities holds:
It is important to mention here that this result holds even though - may not be
defined for all positive step-sizes along d, so that the line-search may have to reduce
the step-size in a first stage to enforce feasibility.
3. The global and r-linear convergence of Algorithm A- . In the convergence
analysis of BFGS, the path to q-superlinear convergence traditionally leads
through r-linear convergence (see [41, 6]). In this section, we show that the iterates
generated by Algorithm A - converge to -
z -x - ), the solution of (1.2), with that
convergence speed. We use the notation
Our first result shows that, because the iterates (x, #) remain in the level set L PD
the sequence {(c(x), #)} is bounded and bounded away from zero.
A BFGS INTERIOR POINT ALGORITHM 207
Lemma 3.1. Suppose that Assumption 2.1 holds. Then, the level set L PD
1 is
compact and there exist positive constants K 1 and K 2 such that
1 .
Proof. Since # c(x) -
log(# (i) c (i) (x)) is bounded below by m-(1 - log -),
there is a constant K
1 . By
Assumption 2.1 and Lemma 2.3, the level set L #
1 } is
compact. By continuity, c(L # ) is also compact, so that c(x) is bounded and bounded
away from zero for all z # L PD
1 .
What we have just proven implies that {# -
1 } is bounded
below, so that there is a constant K #
1 . Hence the #-components of the z's in L PD
are bounded and
bounded away from zero.
We have shown that L PD
1 is included in a compact set. Now, it is itself compact
by continuity of # - .
The next proposition is crucial for the technique we use to prove global convergence
(see [6]). It claims that the proximity of a point z to the unique solution of (2.4)
can be measured by the value of # - (z) or the norm of its gradient # - (z). In unconstrained
optimization, the corresponding result is a direct consequence of strong
convexity. Here, # - is not necessarily convex, but the result can still be established
by using Lemma 2.5 and Lemma 3.1. The function # - is nonconvex, for example,
when is minimized on the half-line of nonnegative real numbers.
Proposition 3.2. Suppose that Assumption 2.1 holds. Then, there is a constant
a > 0 such that for any z # L PDa#z - z - # 2
Proof. Let us show that # - is strongly convex in a neighborhood of - z - . Using
(2.3) and the fact that -
# -e, the Hessian of # - at - z - can be written as
# .
From Assumption 2.1, for fixed # > 0, the Lagrangian is a strongly convex function
in the variable x. It follows that its Hessian with respect to x is positive definite at
us show that the above matrix is also positive definite. Multiplying the
matrix on both sides by a vector (u, v) # R n
positive definite and c(-x - ) > 0, this quantity is nonnegative. If
it vanishes, one deduces that next that
positive
definite.
Let us now prove a local version of the proposition: there exist a constant a # > 0
and an open neighborhood N # Z of - z - such that
a #z - z - # 2
208 P. ARMAND, J. CH. GILBERT, AND S. JAN-J -
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The inequality on the left comes from the fact that # -z - and the strong
convexity of # - near - z - . For the inequality on the right, we first use the local convexity
of for an arbitrary z near -
z -z - (z) z). With the
Cauchy-Schwarz inequality and the inequality on the left of (3.2), one gets
a #
# 1Simplifying and squaring give the inequality on the right of (3.2).
To extend the validity of (3.2) for all z # L PD
su#ces to note that, by virtue
of Lemma 2.5, the ratios
and - (z) -z - )
are well defined and continuous on the compact set L PD
z - is the unique
minimizer of - on L PD
2.5), the ratios are respectively bounded away from
zero and bounded above on L PD
1 \ N , by some positive constants K # 1 and K # 2 . The
conclusion of the proposition now follows by taking
The proof of the r-linear convergence rests on the following lemma, which is part
of the theory of BFGS updates. It can be stated independently of the present context
(see Byrd and Nocedal [6]). We denote by # k the angle between M k # k and
and by # the roundup operator:
Lemma 3.3. Let {M k } be positive definite matrices generated by the BFGS formula
using pairs of vectors {(# k , # k )} k#1 , satisfying for all k # 1
where a 1 > 0 and a 2 > 0 are independent of k. Then, for any r # ]0, 1[, there exist
positive constants b 1 , b 2 , and b 3 , such that for any index k # 1,
for at least #rk# indices j in {1, . , k}.
The assumptions (3.3) made on # k and # k in the above lemma are satisfied in
our context. The first one is due to the strong convexity of one of the functions f ,
-c (1) , . , -c (m) , and to the fact that # is bounded away from zero (Lemma 3.1).
When f and c are C 1,1 , the second one can be deduced from the Lipschitz inequality,
the boundedness of # (Lemma 3.1), and the first inequality in (3.3).
Theorem 3.4. Suppose that Assumption 2.1 holds and that f and c are C 1,1
functions. Then, Algorithm A - generates a sequence {z k } converging to -
z - r-linearly,
meaning that lim sup k#z k -
z - # 1/k < 1. In particular,
z - #.
Proof. We denote by K #
positive constants (independent of the iteration
index). We also use the notation
A BFGS INTERIOR POINT ALGORITHM 209
The bounds on (c(x), #) given by Lemma 3.1 and the fact that f and c are C 1,1
imply that # - is C 1,1 on some open convex neighborhood of the level set L PD
1 , for
example, on
where O is an open bounded convex set containing L PD
1 (this set O is used to have #c
bounded on the given neighborhood).
Therefore, by the line-search and Lemma 2.7, there is a positive constant K #
1 such
that either
or
Let us now apply Lemma 3.3: fix r # ]0, 1[ and denote by J the set of indices j
for which (3.4) holds. Using Lemma 2.6 and the bounds from Lemma 3.1, one has for
# .
Let us denote by K #
4 a positive constant such that #c(x)# K # 4 for all x # L PD
1 . By
using (2.3), (1.5), and the inequality (a
and also, by (1.3),
#d x
P. ARMAND, J. CH. GILBERT, AND S. JAN-J -
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Combining these inequalities with (3.5) or (3.6) gives for some positive constant K #and for any j # J
The end of the proof is standard (see [41, 6]). Using Proposition 3.2, for j # J ,
# [0, 1[. On the other hand, by the line-search, # - (z k+1
# 1. By Lemma 3.3, |[1, k] # J | #rk# rk, so
that the last inequality gives for any k # 1
where K #
7 is the positive constant using the inequality on
the left in (3.1), one has for all k # 1
from which the r-linear convergence of {z k } follows.
4. The q-superlinear convergence of Algorithm A- . With the r-linear convergence
result of the previous section, we are now ready to establish the q-superlinear
convergence of the sequence {z k } generated by Algorithm A - . By definition, {z k }
converges q-superlinearly to - z - if the following estimate holds:
z
z - #),
which means that #z k+1 - z - #z k - z - # 0 (assuming z k #= -
z - ). To get this result,
f and c have to be a little bit smoother, namely twice continuously di#erentiable near
x - . We use the notation
We start by showing that the unit step-size is accepted asymptotically by the line-search
condition (2.5), provided the updated matrix M k becomes good (or su#ciently
large) in a sense specified by inequality (4.1) below and provided the iterate z k is
su#ciently close to the solution -
z - .
Given two sequences of vectors {u k } and {v k } in some normed spaces and a
positive number #, we write u k # o(#v k # ), if there exists a sequence of {# k } # R
such that # k # 0 and u k # k #v k # for all k.
Proposition 4.1. Suppose that Assumption 2.1 holds and that f and c are twice
continuously di#erentiable near - x - . Suppose also that the sequence {z k } generated by
Algorithm A - converges to - z - and that the positive definite matrices M k satisfy the
estimate
(d x
when k #. Then the su#cient decrease condition (2.5) is satisfied with #
k su#ciently large provided that # < 1
A BFGS INTERIOR POINT ALGORITHM 211
Proof. Observe first that the positive definiteness of -
- with (4.1) implies that
(d x
for some positive constant K # and su#ciently large k. Observe also that d k # 0
(for d x
use (1.5), (4.2), and # - Therefore, for k large enough, z k
and z k are near - z - and one can expand - (z k . A second order
expansion gives for the left-hand side of (2.5)
We want to show that this quantity is negative for k large.
Our first aim is to show that # - (z k smaller than a
term of order o(#d k # 2 ). For this purpose, one computes
. On the other hand, using
one gets from Lemma 2.6
k .
With these estimates, (4.1), and the fact that # 2
with Lemma 3.1 and the boundedness of {#c k }, (4.3) becomes
(d x
(d #
clear that the result will be proven if we show that, for some
positive constant K and k large, # - (z k ) # d k # -K#d k # 2 . To show this, we use the
212 P. ARMAND, J. CH. GILBERT, AND S. JAN-J -
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last expression of # - (z k ) # d k and an upper bound of |(d x
k |, obtained by the
Cauchy-Schwartz inequality:
(d x
(d #
k .
It follows that
(d x
(d #
k .
Therefore, using (4.2) and Lemma 3.1, one gets
for some positive constant K and k large.
Proposition 4.1 shows in particular that the function V, which was added to # -
to get the merit function - , has the right curvature around - z - , so that the unit
step-size in both x and # is accepted by the line-search.
In the following proposition, we establish a necessary and su#cient condition of
q-superlinear convergence of the Dennis and Mor-e [13] type. The analysis assumes
that the unit step-size is taken and that the updated matrix M k is su#ciently good
asymptotically in a manner given by the estimate (4.5), which is slightly di#erent
from (4.1).
Proposition 4.2. Suppose that Assumption 2.1 holds and that f and c are
twice di#erentiable at - x - . Suppose that the sequence {z k } generated by Algorithm A -
converges to -
z - and that, for k su#ciently large, the unit step-size # accepted
by the line-search. Then {z k } converges q-superlinearly towards -
z - if and only if
Proof. Let us denote by M the nonsingular Jacobian matrix of the perturbed
KKT conditions (1.2) at the solution -
z -x -
# .
A first order expansion of the right-hand side of (1.3) about - z - and the identities
# -e give
Subtracting Md k from both sides and assuming a unit step-size, we obtain
z - #).
A BFGS INTERIOR POINT ALGORITHM 213
Suppose now that {z k } converges q-superlinearly. Then, the right-hand side
of (4.6) is of order o(#z k -
z - #), so that
Then (4.5) follows from the fact that, by the q-superlinear convergence of {z k }, z k -
Let us now prove the converse. By (4.5), the left-hand side of (4.6) is an o(#d k #)
and due to the nonsingularity of M, (4.6) gives z k+1 - z
With a unit step-size, d
z - (z k - z - ), so that we finally get z k+1 - z
For proving the q-superlinear convergence of the sequence {z k }, we need the
following result from the BFGS theory (see [40, Theorem 3] and [6]).
Lemma 4.3. Let {M k } be a sequence of matrices generated by the BFGS formula
from a given symmetric positive definite matrix M 1 and pairs (# k , # k ) of vectors
verifying
< #,
where M is a symmetric positive definite matrix. Then, the sequences {M k } and
k } are bounded and
By using this lemma, we will see that the BFGS formula gives the estimate
#).
Note that the above estimate implies (4.5), from which the q-superlinear convergence
of {z k } will follow.
A function #, twice di#erentiable in a neighborhood of a point x # R n , is said to
have a locally radially Lipschitzian Hessian at x, if there exists a positive constant L
such that for x # near x, one has
Theorem 4.4. Suppose that Assumption 2.1 holds and that f and c are C 1,1
functions, twice continuously di#erentiable near -
x - with locally radially Lipschitzian
Hessians at -
x - . Suppose that the line-search in Algorithm A - uses the constant # <2 . Then the sequence {z k generated by this algorithm converges to
z -x - ) q-superlinearly and, for k su#ciently large, the unit step-size #
accepted by the line-search.
Proof. Let us start by showing that Lemma 4.3 with
M - can be applied.
First, # k # k > 0, as this was already discussed after Lemma 3.3. For the convergence
of the series in (4.7), we use a Taylor expansion, assuming that k is large enough (f
and c are C 2 near -
214 P. ARMAND, J. CH. GILBERT, AND S. JAN-J -
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With the local radial Lipschitz continuity of # 2 f and # 2 c at - x - and the boundedness
of {# k+1 }, there exist positive constants K # and K # such that
z - # .
Hence the series in (4.7) converges by Theorem 3.4. Therefore, by (4.8) with
and the fact that # k is parallel to d x
#).
By the estimate (4.9) and Proposition 4.1, the unit step-size is accepted when
k is large enough. The q-superlinear convergence of {z k } follows from Proposition
4.2.
5. The overall primal-dual algorithm. In this section, we consider an overall
algorithm for solving problem (1.1). Recall from Lemma 2.2 that the set of primal
solutions of this problem is nonempty and bounded. By the Slater condition (As-
sumption 2.1(ii)), the set of dual solutions is also nonempty and bounded. Let us
denote by -
primal-dual solution of problem (1.1), which is also a solution
of the necessary and su#cient conditions of optimality
Our overall algorithm for solving (1.1) or (5.1), called Algorithm A, consists in
computing approximate solutions of the perturbed optimality conditions (1.2), for a
sequence of -'s converging to zero. For each -, the primal-dual Algorithm A - is used
to find an approximate solution of (1.2). This is done by so-called inner iterations.
Next - is decreased and the process of solving (1.2) for the new value of - is repeated.
We call an outer iteration the collection of inner iterations for solving (1.2) for a fixed
value of -. We index the outer iterations by superscripts j # N\{0}.
Algorithm A (for solving problem (1.1); one outer iteration).
At the beginning of the jth outer iteration, an approximation z j
# Z of the solution -
z of (5.1) is supposed available, as well as a positive
1 approximating the Hessian of the Lagrangian. A value
is given, as well as a precision threshold # j > 0.
1. Starting from z j
use Algorithm A - until z j :=
2. Choose a new starting iterate z j+1
for the next outer iteration, as
well as a positive definite matrix M j+1
1 . Set the new parameters - j+1 > 0
and # j+1 > 0, such that {- j
} and {# j
} converge to zero when j #.
A BFGS INTERIOR POINT ALGORITHM 215
To start the (j+1)th outer iteration, a possibility is to take z j+1
, the updated matrix obtained at the end of the jth outer iteration.
As far as the global convergence is concerned, how z are determined
is not important. Therefore, on that point, Algorithm A leaves the user much freedom
to maneuver, while Theorem 5.1 gives us a global convergence result for such a general
algorithm.
Theorem 5.1. Suppose that Assumption 2.1 holds and that f and c are C 1,1
functions. Then Algorithm A generates a bounded sequence {z j
} and any limit point
of {z j
} is a primal-dual solution of problem (1.1).
Proof. By Theorem 3.4, any outer iteration of Algorithm A terminates with an
iterate z j satisfying the stopping criteria in Step 1. Therefore Algorithm A generates
a sequence {z j
}. Since the sequences {- j
} and {# j
} converge to zero, any limit point
of {z j
} is a solution of problem (1.1). It remains to show that {z j
} is bounded.
Let us first prove the boundedness of {x j
}. The convexity of the Lagrangian
implies that
Using the positivity of # j and c(x 1 ) and next the stopping criteria of Algorithm A, it
follows that
#).
If {x j
} is unbounded, setting t j := #x j
# and y j := x j
one can choose a
subsequence J such that
lim
From the last inequality we deduce that
Moreover, since c(x j ) > 0, we have (-c (i) It follows that
(see, for example, [22, Proposition IV.3.2.5] or
[2, Formula (1)]). Therefore, the solution set of problem (1.1) would be unbounded,
which is in contradiction with what is claimed in Lemma 2.2.
To prove the boundedness of the multipliers, suppose that the algorithm generates
an unbounded sequence of positive vectors {# j
subsequence J # . The
sequence
#)} j#J # is bounded and thus has at least one limit point, say,
Dividing the two inequalities in (5.2) by # j
# and taking limits when j #,
we deduce that # 0, #c(x Using the concavity
of the components c (i) , one has
where the inequality on the right follows from the strict feasibility of the first iterate.
Multiplying by # , we deduce that (# c(x 1 contradiction
with
P. ARMAND, J. CH. GILBERT, AND S. JAN-J -
EGOU
In the rest of this section, we give conditions under which the whole sequence
} converges to a particular point called the analytic center of the primal-dual
optimal set. This actually occurs when the following two conditions hold: strict
complementarity and a proper choice of the forcing sequence # j in Algorithm A,
which has to satisfy the estimate
meaning that # j /- j
Let us first recall the notion of analytic center of the optimal sets, which under
Assumption 2.1 is uniquely defined (see Monteiro and Zhou [37], for related results).
We denote by opt(P ) and opt(D) the sets of primal and dual solutions of problem
(1.1). The analytic center of opt(P ) is defined as follows. If opt(P ) is reduced to
a single point, its analytic center is precisely that point. Otherwise, opt(P ) is a
convex set with more than one point. In that case, f is not strongly convex and,
by Assumption 2.1(i), at least one of the constraint functions, -c (i 0 ) say, is strongly
convex. It follows that the index set
is nonempty (it contains i 0 ). The analytic center of opt(P ) is then defined as the
unique solution of the following problem:
log c (i) (-x) # .
The fact that this problem is well defined and has a unique solution is the matter of
Lemma 5.2 below. Similarly, if opt(D) is reduced to a single point, its analytic center
is that point. In case of multiple dual solutions, the index set
is nonempty (otherwise opt(D) would be reduced to {0}). The analytic center of
opt(D) is then defined as the unique solution of the following problem:
log - # (i)
# .
Lemma 5.2. Suppose that Assumption 2.1 holds. If opt(P ) (resp., opt(D)) is not
reduced to a singleton, then problem (5.3) (resp., (5.4)) has a unique solution.
Proof. Consider first problem (5.3) and suppose that opt(P ) is not a singleton. We
have seen that B is nonempty. By the convexity of the set opt(P ) and the concavity
of the functions c (i) , there exists -
Therefore the
feasible set in (5.3) is nonempty. On the other hand, let -
x 0 be a point satisfying the
constraints in (5.3). Then the set
log c i (-x) # i#B
log c i (-x
is nonempty, bounded (Lemma 2.2), and closed. Therefore, problem (5.3) has a
solution. Finally, by Assumption 2.1(i), we know that there is an index i
A BFGS INTERIOR POINT ALGORITHM 217
that -c (i 0 ) is strongly convex. It follows that the objective in (5.3) is strongly concave
and that problem (5.3) has a unique solution.
By similar arguments and the fact that the objective function in (5.4) is strictly
concave, it follows that problem (5.4) has a unique solution.
By complementarity (i.e., C(-x) - and convexity of problem (1.1), the index
sets B and N do not intersect, but there may be indices that are neither in B nor
in N . It is said that problem (1.1) has the strict complementarity property if
{1, . , n}. This is equivalent to the existence of a primal-dual solution satisfying strict
complementarity.
Theorem 5.3. Suppose that Assumption 2.1 holds and that f and c are C 1,1
functions. Suppose also that problem (1.1) has the strict complementarity property
and that the sequence {# j
} in Algorithm A satisfies the estimate # Then the
sequence {z j
} generated by Algorithm A converges to the point -
z
x 0 is the analytic center of the primal optimal set and -
# 0 is the analytic center of the
dual optimal set.
Proof. Let (- x, -
#) be an arbitrary primal-dual solution of (1.1). Then -
x minimizes
#) and - # so that
Using the convexity of # j ) and the stopping criterion (5.2) of the inner iterations
in Algorithm A, one has
x#,
because
. By Theorem 5.1, there
is a constant C 1 such that m 1
adding the corresponding sides
of the two inequalities above leads to
We pursue this by adapting an idea used by McLinden [34] to give properties of
the limit points of the path -x - ). Let us define #
has for all indices i
c (i)
c (i)
Substituting this in (5.5) and dividing by - j give
c (i) (-x)
c (i)
By assumptions, #
Now supposing that (-x 0 , - # 0 ) is a
limit point of {(x j , # j )} and taking the limit in the preceding estimate provide
c (i) (-x)
P. ARMAND, J. CH. GILBERT, AND S. JAN-J -
EGOU
Necessarily, now that, by strict complementarity,
there are exactly m terms on the left-hand side of the preceding inequality. Hence,
by the arithmetic-geometric mean inequality
c (i) (-x)
or
c (i) (-x) #
One can take -
in this inequality, so that
c (i) (-x) #
c (i) (-x 0 ) and #
This shows that - x 0 is a solution of (5.3) and that - # 0 is a solution of (5.4). Since the
problems in (5.3) and (5.4) have unique solutions, all the sequence {x j
} converges to
x 0 and all the sequence {# j
} converges to - # 0 .
6. Discussion. By way of conclusion, we discuss the results obtained in this
paper, give some remarks, and raise some open questions.
Problems with linear constraints. The algorithm is presented with convex
inequality constraints only, but it can also be used when linear constraints are present.
Consider the problem
min f(x),
obtained by adding linear constraints to problem (1.1). In (6.1), A is a p - n matrix
with p < n and b # R p is given in the range space of A.
Problem (6.1) can be reduced to problem (1.1) by using a basis of the null space
of the matrix A. Indeed, let x 1 be the first iterate, which is supposed to be strictly
feasible in the sense that
Let us denote by Z an n - q matrix whose columns form a basis of the null space
of A. Then, any point satisfying the linear constraints of (6.1) can be written
With this notation, problem (6.1) can be rewritten as the problem in u # R
which has the form (1.1).
Thanks to this transformation, we can deduce from Assumption 2.1 what are
the minimal assumptions under which our algorithm for solving problem (6.2) or,
equivalently, problem (6.1) will converge.
A BFGS INTERIOR POINT ALGORITHM 219
Assumption 6.1. (i) The real-valued functions f and -c (i) (1 # i # m) are convex
and di#erentiable on the a#ne subspace X b} and at least one of the
functions f , -c (1) , . , -c (m) is strongly convex on X. (ii) There exists an x # R n
such that
With these assumptions, all the previous results apply. In particular, Algorithm
A - converges r-linearly (if f and c are also C 1,1 ) and q-superlinearly (if f
and c are also C 1,1 , twice continuously di#erentiable near - x - with locally radially
Lipschitzian Hessian at -
Similarly, the conclusions of Theorem 5.1 apply if f and
c are also C 1,1 .
Feasible algorithms and qN techniques. In the framework of qN methods,
the property of having to generate feasible iterates should not be only viewed as a
restriction limiting the applicability of a feasible algorithm. Indeed, in the case of
problem (6.2), if it is sometimes di#cult to find a strictly feasible initial iterate, the
matrix to update for solving this problem is of order q only, instead of order n for
an infeasible algorithm solving problem (6.1) directly. When q # n, the qN updates
will approach the reduced Hessian of the Lagrangian Z # 2 #)Z more rapidly than
the full Hessian # 2 #, so that a feasible algorithm is likely to converge more rapidly.
About the strong convexity hypothesis. Another issue concerns the extension
of the present theory to convex problems, without the strong convexity assumption
(Assumption 2.1(i)).
this hypothesis, the class of problems to consider encompasses linear
programming (f and c are a#ne). It is clear that for dealing properly with linear
programs, our algorithm needs modifications, since then # and the BFGS formula
is no longer defined. Of course, it would be very ine#ective to solve linear programs
with the qN techniques proposed in this paper (M is the desired matrix), but
problems that are almost linear near the solution may be encountered, so that a
technique for dealing with a situation where # k # k # can be of interest.
To accept # can look at the limit of the BFGS formula (2.1) when
possible update formula could be
The updated matrix satisfies M k+1 # positive semidefinite, provided
is already positive semidefinite. The fact that M k+1 may be singular raises some
di#culties, however. For example, the search direction d x may no longer be defined
(see formula (1.5), in which the matrix M +#c(x)C(x) -1 #c(x) # can be singular).
Therefore, the present theory cannot be extended in a straightforward manner.
On the other hand, the strong convexity assumption may not be viewed as an important
restriction, because a fictive strongly convex constraint can always be added.
An obvious example of fictive constraint is "x # x # K." If the constant K is large
enough, the constraint is inactive at the solution, so that the solution of the original
problem is not altered by this new constraint and the present theory applies.
Better control of the outer iterations. Last but not least, the global convergence
result of section 5 is independent of the update rule of the parameters # j and
In practice, however, the choice of the decreasing values # j and - j is essential for
the e#ciency of the algorithm and would deserve a detailed numerical study.
From a theoretical viewpoint, it would be highly desirable to have an update rule
that would allow the outer iterates of Algorithm A to converge q-superlinearly. Along
P. ARMAND, J. CH. GILBERT, AND S. JAN-J -
EGOU
the same lines, an interesting problem is to design an algorithm in which the barrier
parameter would be updated at every step, while having q-superlinear convergence of
the iterates. Such extensions would involve more di#cult issues.
The global convergence result proved in this paper gives us some reasons to believe
that it is not unreasonable to tackle these open questions.
Acknowledgments
. We would like to thank the referees for their valuable com-
ments. One of them has shown us a direct argument for the last part of the proof of
Proposition 3.2, which is the one we have finally chosen to give in the paper. The other
referee has brought McLinden's paper to our attention, which led us to Theorem 5.3.
--R
On the convergence of an infeasible primal-dual interior-point method for convex programming
Asymptotic analysis for penalty and barrier methods in convex and linear programming
Computational Di
A trust region interior point algorithm for linearly constrained optimization
A. Trust Region Method Based on Interior Point Techniques for
A tool for the analysis of quasi-Newton methods with application to unconstrained minimization
A Primal-Dual Algorithm for Minimizing a Non-convex Function Subject to Bound and Linear Equality Constraints
Principles and Techniques of Algorithmic Di
Interior Point Approach to Linear
A Potential Reduction Method for a Class of Smooth Convex Programming Problems
On the classical logarithmic barrier function method for a class of smooth convex programming problems
Numerical Methods for Unconstrained Optimization and Nonlinear Equations
On the formulation and theory of the Newton interior-point method for nonlinear programming
Sequential Unconstrained Minimization Techniques
Practical Methods of Optimization
Feasible direction interior-point technique for nonlinear optimization
Interior Point Techniques in Optimization-Complementarity
On the method of analytic centers for solving smooth convex problems
A practical interior-point method for convex programming
A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems
A new continuation method for complementarity problems with uniform P
Limiting behavior of trajectories generated by a continuation method for monotone complementarity problems
On the limited memory BFGS method for large scale optimization
The projective SUMT method for convex programming problems
An analogue of Moreau's proximation theorem
An interior point algorithm for solving smooth convex programs based on Newton's method
An extension of Karmarkar-type algorithms to a class of convex separable programming problems with global linear rate of convergence
On the existence and convergence of the central path for convex programming and some duality results
Updating quasi-Newton matrices with limited storage
On the convergence of the variable metric algorithm
Some global convergence properties of a variable metric algorithm for minimization without exact line searches
Theory and Algorithms for Linear Optimization-An Interior Point Approach
"analytical center"
Interior Point Methods of Mathematical Programming
Computational experience with a primal-dual interior-point method for smooth convex programming
Why a pure primal Newton barrier step may be infeasible
Superlinear and quadratic convergence of some primal-dual interior point methods for constrained optimization
Interior Point Algorithms-Theory and Analysis
in Integer Nonlinear Pro- gramming
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--CTR
Richard H. Byrd , Jorge Nocedal , Richard A. Waltz, Feasible Interior Methods Using Slacks for Nonlinear Optimization, Computational Optimization and Applications, v.26 n.1, p.35-61, October
Paul Armand, A Quasi-Newton Penalty Barrier Method for Convex Minimization Problems, Computational Optimization and Applications, v.26 n.1, p.5-34, October
Dingguo Pu , Weiwen Tian, The revised DFP algorithm without exact line search, Journal of Computational and Applied Mathematics, v.154 n.2, p.319-339, 15 May | convex programming;superlinear convergence;constrained optimization;interior point algorithm;analytic center;BFGS quasi-Newton approximations;primal-dual method;line-search |
589092 | Auction Algorithms for Shortest Hyperpath Problems. | The auction-reduction algorithm is a strongly polynomial version of the auction method for the shortest path problem. In this paper we extend the auction-reduction algorithm to different types of shortest hyperpath problems in directed hypergraphs. The results of preliminary computational experiences show that the auction-reduction method is comparable to other known methods for specific classes of hypergraphs. | Introduction
The shortest hyperpath problem is the extension to directed hypergraphs [11] of the
classical shortest path problem (SPT) in directed graphs. Though not as pervasive
as SPT, shortest hyperpaths have several relevant applications. In particular,
they are at the core of traffic assignment algorithms for transit networks [13, 14, 18].
Shortest hyperpath models have been constructed for the SPT problem in stochastic
and time-dependent networks [17] and for production planning in assembly lines [12].
Moreover, shortest hyperpath algorithms are used as building blocks of enumerative
algorithms for hard combinatorial problems [10]. As a consequence, there is a growing
interest for efficient shortest hyperpath algorithms. This provides motivations
for further investigating known methods [11, 15], both from a theoretical and a
practical point of view, and for developing new ones.
Auction algorithms were first proposed by Bertsekas [1, 2] for the assignment
problem and later extended to general transportation problems [5, 6]. A survey of
the auction algorithms for network optimization problems is contained in [4, Chapter
4]. Auction algorithms for shortest path problems on graphs were proposed in [3].
For the single-origin single-destination case the method can be viewed as an application
of the auction method (with to a specifically constructed assignment
problem, and finite termination of the procedure can be established. Furthermore,
the algorithm is a dual coordinate ascent method. Strongly polynomial version of the
auction method were proposed by Pallottino and Scutell'a [16], who define a pruning
procedure that reduces the graph to the shortest path tree. Further improvements
to this method are given in [7], where the pruning method is strengthened, and the
structure of the reduced graph is exploited to obtain a better time complexity. A
variant of the auction algorithm with pruning is proposed in [8].
In this paper, we devise an auction method for shortest hyperpaths with nonnegative
hyperarc weights, by slightly modifying the SPT algorithm given in [7]. Our
method can be tailored to solve several types of shortest hyperpath problems; for
the various cases, we provide a worst case complexity bound. Finally, we report the
results of a preliminary computational experience.
In Section 2 we give the basic definitions on hypergraphs and shortest hyper-
paths. The proposed auction method for shortest hyperpaths is presented in Section
3. Computational results and conclusions are presented in Sections 4 and 5,
respectively.
Shortest Hyperpaths in Directed Hypergraphs
A directed hypergraph H is a pair (V; E), where V is a set of nodes and E is a set
of directed hyperarcs; a hyperarc is a pair
the tail of e, and h(e) 2 V n T (e) is its head. A detailed introduction to directed
hypergraphs can be found in [11], where a more general definition of hypergraphs is
introduced; the particular class of hypergraphs considered in this paper are called
B-graphs in [11]. The size jej of hyperarc e is the number of nodes it contains in its
tail and head:
The hyperarc e is an arc if 2, and a proper hyperarc if jej ? 2. Denote by m a and
m h the number of arcs and proper hyperarcs,
j. The size of H is the sum of the cardinalities of its hyperarcs:
Given a node u, the Forward Star of u, FS(u), is the set of hyperarcs e such that
and the Backward Star of u, BS(u), is the set of hyperarcs e such that
A path P st , of length q, in the hypergraph E) is a sequence:
st
nodes s and t are
the origin and the destination of the path P st , respectively. We say that node t is
connected to node s in H if a path P st exists in H. If t 2 T (e 1 ), then the path P st
is a cycle. A path is cycle-free if it does not contain any subpath which is a cycle,
Given a hypergraph E) and two nodes s; t 2 V, a hyperpath - st is a
minimal hypergraph (with respect to deletion of nodes and hyperarcs) H
such
2. s;
3. connected to s in H - by a cycle-free path.
Observe that for each there exists a unique hyperarc e 2 E - such that
is the predecessor hyperarc of u in -, and is denoted by e - (u). We say
that node t is hyperconnected to s in H if there exists a hyperpath from s to t in H.
Given a hyperarc a, we say that a hyperarc a r is contained in a, or is a reduction
of a, if h(a r (a). Note that a is contained in itself, and a r
is strictly contained in a if T (a r Given a and u 2 T (a), we denote by anu
the reduction of a obtained deleting u from T (a). We say that a hypergraph is full
when it contains all the possible reductions of each of its proper hyperarcs. A full
hypergraph can be represented by its support hypergraph H s , obtained by deleting
all the strictly contained hyperarcs. Conversely, given any hypergraph H, we can
obtain the corresponding full hypergraph H f by adding all the strictly contained
hyperarcs.
2.1 Shortest Hyperpaths
A weighted hypergraph is such that each hyperarc e is assigned a nonnegative real
weight w(e). The weight of a hyperpath in a weighted hypergraph can be defined in
several ways. It is known that some definitions lead to intractable shortest hyperpath
problems [9]. Here we restrict to definitions that are known to be tractable [11].
A weighting function is a node function that, given a hyperpath - st
assigns a value W - (u) to its nodes, depending on the weights of its hyperarcs. The
value W - (t) is the weight of - under the chosen weighting function. An additive
weighting function satisfies the properties that W -
a function of the predecessor e - (u) only. Formally, an additive weighting function
can be defined by means of the following recursive equations:
where F (e) is a non-decreasing function of the weights of the nodes in T (e). Clearly,
many different additive weighting functions can be defined (see [11]); consider first
the value function, which is obtained by defining F (e) in (1) as follows:
The minimum value (shortest hyperpath) problem consists of finding a set of minimum
value hyperpaths from the origin node s to each node u 6= s hyperconnected
to s. We denote by V (u) the minimum value of a hyperpath - su in H; we assume
if u is not hyperconnected to s.
The distance function is obtained from (1) defining F (e) as follows:
The minimum distance problem asks for the minimum distance hyperpaths from
origin s. Denote by D the vector of minimum distances, where again
if u is not hyperconnected to s.
Since arc weights are nonnegative, the minimum value and minimum distance
problems can be solved efficiently by procedure SBT [11]. A computational analysis
of several variants of SBT can be found in [15].
2.1.1 Minimum Time Problems in Transit Hypergraphs
The problem of finding the passenger's expected travel time is at the core of several
urban transit networks models. This problem has been formulated in terms of hyper-
paths in transit networks [13] and in F-graphs [11, 14]. Here, introducing the time
weighting function in transit hypergraphs, we define a particular shortest hyperpath
problem that, though using a different (and slightly more general) terminology, is
equivalent to the formulations found in [13, 14].
A transit hypergraph is a weighted support hypergraph E) where a
positive parameter OE u is associated with each node u 2 V. Let H
the full hypergraph represented by H. Consider an e contained in a proper
hyperarc a 2 H: the time weighting function is obtained from (1) by defining the
weight w(e) and the function F (e) as follows:
OE u
In practice, F (e) is the weighted average (with weights OE:) of the values W - (\Delta) in
T (e), while w(e) is the inverse of the sum of the weights OE: in T (e). For an arc
corresponding to an arc a 2 E the function F (e) is defined in the
same way, which gives F however, in this case w(e) = w(a) can be any
nonnegative value.
The minimum time problem consists of finding the minimum time hyperpaths,
from a given origin s, in the full hypergraph H f . Note that H f may be considerably
larger than its support H: in practice, solving the minimum time problem efficiently
requires to work directly on H. This is the aim of the following observations.
Denote by E the vector of minimum times, where
is not hyperconnected to s in H f . For each e 2
OE u
Consider a proper hyperarc a 2 E , and let R(a) be the set of the reductions of a;
denote by e(a) the reduction of a yielding the minimum time t(e):
The following relations hold ([13], Proposition 6):
Consider the nodes in T (a) in increasing order of E, i.e. let T
with be the reduction of a such
that T (a g. It follows from (2) that e(a) = a h , where
In practice, working on the support hypergraph requires finding the hyperarc
e(a) without considering the whole set of reductions of a. Indeed, according to the
previous observations, this can be done by processing the nodes in T (a) in the order
. For each u compute the value t(a i
This technique has been used to compute expected
travel times efficiently [13, 14], and will be adopted in our auction algorithm for
the minimum time problem.
2.1.2 Reductions and Shortest Hyperpaths
A hyperarc reduction operation on a proper hyperarc a consists of replacing a by a
hyperarc a r contained in a, returning a reduced hypergraph. Clearly, if the hyper-graph
is weighted, a nonnegative weight w(a r ) must be assigned to a r . The following
propositions show that by suitably choosing the weight on a r a reduction operation
does not modify the optimal solution of the shortest hyperpath problem. Proofs are
rather straightforward and are omitted.
Suppose we are given a weighted hypergraph H, and the corresponding vectors
of optimal solutions V , D for the value and distance weighting functions. Given a
proper hyperarc a and u 2 T (a), consider replacing a by a r = a n u.
Proposition 2.1 If w(a r is the vector of optimal values in the
reduced hypergraph.
Proposition 2.2 If D(u) - is the
vector of optimal distances in the reduced hypergraph.
Now suppose we are given a transit hypergraph H, the corresponding full hypergraph
and the optimum times E. Let e be a proper hyperarc in H f , with t(e) ? E(u)
for each u 2 T (e), and consider replacing e by an arc e
Proposition 2.3 If w(e r is the vector of optimal times in the
reduced full hypergraph.
Auction Algorithms for Shortest Hyperpaths
In this section we propose an auction method for the minimum value problem and
discuss the adaptation to other weighting functions. Before introducing our ap-
proach, we briefly recall some relevant features of the auction algorithms for SPT;
the reader is referred to the cited literature for further details.
The auction algorithm for shortest path problems on graphs maintains a path P
(the candidate path) starting at the origin s and a set of dual node prices p satisfying
the following complementary slackness (CS) conditions:
is the cost of arc (i; j). The algorithm consists of three basic operations:
path extension, path contraction and dual price raise. At each iteration, the candidate
path P is possibly extended, by adding a new node at the end of the path
without violating (3). When no extensions are possible, the dual price of the terminal
node i in P is raised, and if i 6= s the path is contracted by deleting node i. For the
single-origin, single-destination case the algorithm terminates when the destination
node is reached; several variants have been devised, also for the multiple-destination
case.
Consider the case of nonnegative costs and dual prices initially set to zero. At
the first scan of a node i (i.e. when node i becomes the last node in P for the first
time) the optimal distance of node i is determined, indeed, it is equal to p(s). As
a consequence, since p(s) is never decreased during the algorithm, the sequences of
first scan operations ranks the nodes in increasing order of distance from the origin
s. Based on the above property, the auction-reduction method [16] introduces the
following reduction operation: at the first scan of node i, delete each arc entering
i, except the last arc in P . By means of these reduction operations, the graph is
transformed into the shortest path tree, and a strongly polynomial time complexity
can be obtained. Further reduction operations, including deletion of nodes, have
been proposed in [7], improving the complexity bound.
The following observation is at the core of our auction shortest hyperpath method:
Observation 3.1 According to the definition of value, distance, and time weighting
functions, for an arc a = (fug; v) we have F (a)
In other words, the weighting functions above define a standard SPT problem if the
hypergraph is a directed graph. This suggests the following technique:
ffl apply the auction-reduction SPT algorithm to the arcs of the hypergraph;
ffl at the first scan of node i, apply hyperarc reduction operations according to
Properties (2.1),. ,(2.3), possibly generating new arcs.
Note that the hypergraph is modified during the algorithm; each step is applied
to the current hypergraph, as returned by the previous reduction operations. In
practice, during the execution the proper hyperarcs lay in the not yet explored part
of the hypergraph, and they are not considered until they are replaced by arcs as a
result of successive reductions.
Our auction algorithm for the minimum value problem is described in procedure
MinValue. Remark that procedure MinValue applied to a graph becomes the auction
algorithm with graph reduction described in [7].
For each node i, the predecessor pred(i) gives the last arc in the best s-i path
determined so far; for notational convenience, we consider pred(i) as a set; initially,
;. The label l(i) is the minimum s-i hyperpath value determined so far,
which becomes the optimum s-i hyperpath value V (i) at the first scan of node i. We
denote by FSA (i) and FSH (i) the arcs and proper hyperarcs in FS(i), respectively;
thus Replacing hyperarc a by its reduction ani is denoted
by a := an i; note that an i may be an arc. The last node in P is denoted by last(P ).
During the execution of the algorithm, the contained graph is the
directed graph defined by the nodes and arcs in the current hypergraph
Proposition 3.1 At each step of the algorithm, for each node i 2 VA such that
gives the shortest s-i path length in the current contained
graph HA .
Proof: The property follows from the correctness of the auction algorithm for SPT,
observing that a new arc (i; j) is created only before the first scan of node i.
Theorem 3.1 The vector V determined by the algorithm gives the minimum hy-
perpath values in the original hypergraph.
Proof: The theorem can be proved by induction considering nodes in order of first
scan, that is, in increasing order of value V (\Delta). The claim is clearly true at the
beginning because to node s is assigned V Assume that all the previously
assigned V are correct at the first scan of node i. It follows from Proposition 3.1
that l(i) is a lower bound on the length of any path in HA from node s to each
node j such that Therefore, in the current hypergraph, the value of any
hyperpath containing a proper hyperarc cannot be less than l(i). This implies that
is correct; as a consequence, Step 1(c) does not change the optimal
solution (Proposition 2.1).
3.1 Other Weighting Functions
The auction algorithm for minimum value can be easily adapted to the minimum
distance problem. To this aim, it suffices to skip the weight update w(a) := w(a)
V (i) in Step 1(c). This follows from Property (2.2) since, when a is replaced by an i,
for each node j 2 T (a n i). At the end of the algorithm,
the vector of optimal distances. The proof of correctness for the distance function
is similar to the one of Theorem 3.1.
Procedure MinValue(H; s)
for each u 2 V: pred(u) := ;, p(u) := 0, V (u) := l(u) := +1;
steps (a). (d) (first scan of i)
(a) (set value) V (i) := l(i);
(b) (delete
(c) (reduce hyperarcs) for each a 2 FSH (i):
a := a n i, w(a) := w(a)
(d) (update labels, delete arcs) for each a = (i;
otherwise,
deletion, contraction, or expansion)
if FSA
(i)g: go to Step 4;
a=(i;j)2FSA (i)
go to Step 1;
Step 4 (expansion) expand P by node j i , where:
a=(i;j)2FSA (i)
go to Step 1.
The situation is slightly more complex for travel times. Recall that our goal is to
work with the support transit hypergraph, thus we must deal with the corresponding
full hypergraph implicitly. To this aim, we replace hyperarc reductions by arc
insertion operations, as described below.
Consider a proper hyperarc a in the support, with T and
We know that it suffices to consider the k reductions
a (see Section 2.1.1). At the first scan of
node u we compute the value t(a i ); if t(a generate an arc
according to Proposition 2.3. Otherwise, i.e.,
that a i\Gamma1 is the reduction e(a) of a yielding minimum
time, and we delete a. If necessary, a is removed at the first scan of node u k . In
conclusion, for a proper hyperarc a, up to jT (a)j arcs can be generated.
In order to compute each t(a i ) efficiently, for each proper hyperarc a in the
support hypergraph we keep two values, initially set to 0:
u2T (a)
u2T (a)
At first scan of u i , it is t(a We also keep a counter k(a) of visited
nodes in T (a). We rewrite Step 1(c) as follows:
Step 1(c) (reduce hyperarcs) for each a 2 FSH (i):
oe(a)
k(a)
Observe that a new arc is added only if it can be used to improve the label of h(a). In
this case, the current predecessor pred(h(a)) will be deleted in Step 1(d); therefore,
at most one arc generated from a belongs to the contained graph at the end of Step
1.
Lemma 3.1 If the values V (\Delta) assigned before the first scan of node i are correct
then the value V (i) assigned at the first scan of node i is correct.
Proof: Let S be the set of nodes in the current hypergraph whose first scan occurred
before first scan of node i. We know that, for gives the SPT distance
from s in the contained graph; moreover, V (i) is a lower bound on the SPT distance
for each node u 62 S. Consider a proper hyperarc a in the current support hyper-
graph. The reductions of a containing nodes in S only have been already considered
by the algorithm, possibly adding new arcs. Moreover, it is
from (2) that V (i) is a lower bound on t(e(a)). Therefore V (i) is a lower bound
on the minimum time for each node u 62 S in the current hypergraph. The thesis
follows.
Using Lemma 3.1 we can proof the correctness of the auction algorithm for
minimum times by induction, as we did for Theorem 3.1.
3.2 Computational Complexity
The auction-reduction algorithm presented in [7] solves the SPT problem on a graph
E) in O(jV j minfjEj; jV j log jV jg) time. It is easy to see that the maximum
number of arcs generated during the execution of MinValue is m, for the value and
distance weighting functions, and O(size(H)) for the time function. Moreover, the
total time spent in first scans (Step 1) is O(size(H)). Therefore, we can state the
following proposition:
Proposition 3.2 The running time of the auction shortest hyperpaths algorithm is
log ng), for value and distance, while for the time function
it is O(size(H)
Two techniques for improving the running time of the auction-reduction method
are presented in [7]: path scanning ant multipath restructuring. The resulting complexity
is O(jV In fact, the total computation time between two successive
first scan operations is O(jV j), and clearly there are at most jV j first scans. The
above techniques can be easily applied within our shortest hyperpath algorithm; the
next proposition follows:
Proposition 3.3 The auction shortest hyperpath algorithm with path scanning or
multipath restructuring takes O(size(H)+n 2 ) time, for the value, distance, and time
functions.
Computational Results
In this section we present the preliminary computational results for auction methods
for shortest hyperpath problems. Our main goal here is to compare a few variants of
the basic method; a complete experimental evaluation of auction shortest hyperpath
methods would require a much larger effort.
Our basic shortest hyperpath algorithm, denoted by HAR, is an implementation
of procedure MinValue where we used the last data structure [7]. A variant
of this algorithm, denoted by HAR2, makes use of the "second best" device [4,
Chapter 4] too. We implemented a third version, denoted by HARn, where the
"second best" device is not used, and a node contraction operation is introduced.
A node contraction deletes a node k with indegree and outdegree equal to one: the
arcs incident with node k, say (i; are replaced by an arc (i; j), where
contraction operations simplify the current graph,
and may help keeping the current path shorter.
We compared our auction algorithms to an implementation of procedure SBT-
heap [11], denoted by SBTh. All algorithms were coded in C language, and run
on an IBM RISC-6000 P43 workstation, with 64M RAM, using the AIX operating
system.
In general, devising a reasonable experimental setup for shortest hyperpaths is
not a trivial task, since hypergraphs show many more degrees of freedom than graphs
(see e.g. [15]). Here, we restricted ourselves to one weighting function, namely the
distance, and we considered two different hypergraph topologies: random and grid.
Random hypergraphs do not show any special structures, except that the origin
s is a distinguished node, and FS(s) contains only arcs. The size of proper hyperarcs
is chosen randomly in the interval [d min ; d max ]. In our experiments, we set d
and jF and we defined five classes of random hypergraphs with different
values of d a . For proper hyperarcs, and for arcs exiting the
root, weights were generated randomly in the interval [0; 1
for the remaining arcs,
weights belong to [ 1
This choice has been motivated by the attempt to increase
the relevance of hyperarcs.
The results for random hypergraphs are shown in Table 1. For each class, the
value ffi is the expected size of FS(u) for u 6= s. Execution times are given in
milliseconds; each entry is the (rounded) average of 20 instances.
m a 2n 25n 2n 25n 50n
HARn 176 123 135 150 277
Table
1. Random Hypergraphs
In a grid hypergraph nodes are arranged in a b \Theta h grid; a node is identified by its
cohordinates Nodes with the same x cohordinate form a
level; for each pair (x; y) and (x; y 0 ), with y h, there are two vertical arcs
and
. Hyperarcs connect nodes in successive levels; for
each (x; y) with there exists a hyperarc:
h, and y h, In addition, there is an origin node
s, and arcs
s; (1; y)
for each 1 - y - h.
We generated three classes of grid hypergraphs: square, where h, long, where
b AE h, and high, where h AE b. Parameters b and h where choosen in order to have
the same number of nodes in the three classes. Hyperarc weights lay in the interval
vertical arcs weights lay in [1; 2]; weights of arcs leaving s lay in [0; 1
Execution times are reported in Table 2. Each entry is the (rounded) average of
5 instances; times are given in seconds.
high square long
HAR 44 88 86 172 370 785
HARn 44 85 86 176 370 783
SBT 2:22 3:85 1:18 2:06 :65 1:15
Table
2. Grid Hypergraphs
Though clearly incomplete, the above results allow us to draw some conclu-
sions. For what concerns random hypergraphs, our auction algorithms are comparable
to standard label-setting methods, that are the most efficient for this class
of hypergraphs[15]. Auction methods become more and more competitive as the
density increases; in one case, HAR2 gives the best results. On the other hand,
auction methods do not seem to be suitable for large grid hypergraphs. This result
(that matches the computational results for auction methods for long grid graphs)
is not surprising, since the auction algorithm must maintain a long current path P
in order to connect nodes in the last layers.
The "second best" data structure gives the best results for random hypergraphs,
and for high grids, but it is not suitable for square and long grids. Again, this result
is not surprising, since in a grid hypergraph there exist at most two hyperarcs (plus
two vertical arcs) leaving each node; it is conceivable that the good results for high
grids are due to savings obtained when scanning the origin node.
On the contrary, the node contraction operation is almost useless, also for grid
hypergraphs. This result is rather disappointing, since in some preliminary experiments
this operation proved to be very effective on some classes of grid graphs. A
possible explanation may be the following: if a node has the highest distance in the
tail of a hyperarc, it is likely to have the highest distance also in the tail of the other
hyperarc it belongs to; in this case, hyperarc reduction may create two arcs leaving
the node, so that node contraction cannot be applied. This observation may suggest
some guidelines for improving our algorithms.
Conclusions
In this paper, we proposed an auction method for shortest hyperpath problems, that
can be adapted to several types of weighting functions. Our method is derived, with
minor changes, from the auction-reduction SPT algorithm. Indeed, an appealing
feature of our approach is that several techniques originally developed for graphs
could be easily exported to hypergraphs.
From a practical point of view, auction shortest hyperpath methods are comparable
to other known methods, at least in favourable cases. As one would expect,
their behaviour can be dramatically affected by the structure of the underlying hy-
pergraph; however, this seems to resemble closely what happens for graphs.
We can conclude that auction shortest hyperpath methods deserve more inves-
tigation, both on the theoretical and the practical side. A possible direction could
be adapting some of the variants proposed in the literature, such as the price raise
technique devised in [8], and the forward-reverse approach for the single-origin
single-destination case [4, Chapter 4].
--R
A distributed algorithm for the assignment problem.
The auction algorithm: A distributed relaxation method for the assignment problems.
An auction algorithm for shortest paths.
Linear Network Optimization: Algorithms and Codes.
The auction algorithm for the minimum cost network flow problem.
The auction algorithm for transportation problems.
Polynomial auction algorithms for shortest paths.
A modified auction algorithm for the shortest path problem.
Dynamic maintenance of directed hypergraphs.
Max Horn SAT and the minimum cut problem in directed hypergraphs.
Directed hypergraphs and applications.
Hypergraph models and algorithms for the assembly problem.
Equilibrium traffic assignment for large scale transit networks.
Implicit enumeration of hyperpaths in logit models for transit networks.
A computational study of shortest hyperpath algo- rithms
Strongly polynomial auction algorithms for shortest paths.
A hypergraph model for stochastic time dependent shortest paths.
A simplicial decomposition method for the transit equilibrium assignment problem.
--TR | auction algorithms;shortest paths;directed hypergraphs;hyperpaths |
589094 | Constraint Qualifications for Semi-Infinite Systems of Convex Inequalities. | We introduce and study the Abadie constraint qualification, the weak Pshenichnyi--Levin--Valadier property, and related constraint qualifications for semi-infinite systems of convex inequalities and linear inequalities. Our main results are new characterizations of various constraint qualifications in terms of upper semicontinuity of certain multifunctions. Also, we give some applications of constraint qualifications to linear representations of convex inequality systems, to convex Farkas--Minkowski systems, and to formulas for the distance to the solution set. Some of our concepts and results are new even in the particular case of finite inequality systems. | Introduction
Let I) be a family of convex functions, where I is an arbitrary (but
nonempty) index set, and let us consider the system of \convex inequalities"
Throughout this paper we shall consider only the above framework, which is sucient for many
applications. However, let us mention that some of our results and proofs can be extended to
arbitrary (nite or innite dimensional) normed linear spaces X and to inequality systems (1) with
convex functions
In the sequel we shall assume, without any special mention, that the solution set S of the system
(1) is nonempty, i.e.,
We shall often consider the important particular case when each g i is ane, say
where a denotes the dot product of vectors in IR n . In this case, (1)
becomes a system of linear inequalities,
and (2) becomes
Note that one can formally convert (1) to one convex inequality:
where G() is the sup-function [7] of (1), dened as
i2I
The system (1) is said to be a system with nite-valued sup-function, if
In this paper, we always assume that (8) holds.
For the inequality system (1) and for any x in IR n , we shall denote by I(x) the set of \active
indices" at x, i.e.,
Note that if is the classical
denition of active indices.
One of the reasons of the diculty of extending the results from nite inequality systems to
semi-innite inequality systems is that in the semi-innite case for x 2 bd S the set I(x) may be
empty or may be innite. As we shall see in the sequel, some other reasons, which explain why
many results cannot be extended at all, or can be extended only under some additional assumptions
(and sometimes only with dierent proofs), are the following: while in the nite case the index set
I is compact, in our main results on the general semi-innite case we shall assume no topology on
I; also, while in the nite case for each x 2 IR n the set A x := fg i (x)j i 2 Ig is closed in IR; and
the sup-function G(x) := sup i2I g i (x) is always nite-valued on IR n ; in the general semi-innite case
these are no longer true. Furthermore, it is well-known that for a linear inequality system (4) with
a nite index set I we have
where N S (x) and bd S denote the normal cone of S at x and the boundary of S; respectively. In
general, (10) does not hold for a linear inequality system (4) if I is innite. Given a convex system
(1), another important well-known property for a nite I is (with the convention [ i2; A
where co(A) denotes the convex hull of a set A and @g(x) denotes the subdierential of a convex
function g at x :
In general, (11) does not hold if I is innite.
In the present paper we shall give a detailed discussion of constraint qualications for semi-
innite systems of convex inequalities and linear inequalities and the relations among them. We
shall introduce and study the Abadie constraint qualication, the weak Pshenichnyi-Levin-Valadier
property and related constraint qualications. Our main results are new characterizations of various
constraint qualications in terms of upper semicontinuity of certain multifunctions. Also, we
shall give some applications of constraint qualications to linear representations of convex inequality
systems, convex Farkas-Minkowski systems, and formulas for the distance to the solution set.
Moreover, some of our concepts and results on semi-innite convex inequality systems will yield new
contributions even when applied to the particular case of nite inequality systems (such as Corollary
2).
Let us describe now, brie
y, the sections of our paper.
It is well-known (see e.g. [7, pp. 307-309]) that in the theory of convex minimization over the
solution set of a nite system of convex inequalities the so-called basic constraint qualication, or
brie
y, the BCQ, which requires that the normal cone at each point of the boundary of the solution
set should coincide with \the cone of the active constraints" at that point, plays an important
role; for example, it is satised if and only if the Karush-Kuhn-Tucker (KKT) sucient optimality
conditions are also necessary for optimality (see e.g. [7, Proposition 2.2.1, page 308]). Recently,
the BCQ has been extended to semi-innite linear inequality systems by Puente and Vera de Serio
[14], who have used the term \locally Farkas-Minkowski systems", or brie
y, LFM systems, and
further extended to semi-innite systems of convex inequalities by Goberna and Lopez [5, page
162], who have used the term \convex locally FM systems", or brie
y, CLFM systems. In section
2 we shall introduce a weaker constraint qualication than the BCQ, which is dierent from the
BCQ even in the particular case of nite convex inequality systems, and which we shall call the
Abadie CQ, requiring only that the normal cone at each point of the boundary of the solution set
should coincide with the closure of the cone of the active constraints at that point. We shall give
new characterizations of the Abadie CQ and the BCQ in terms of upper semicontinuity of certain
associated convex cone-valued multifunctions.
In section 3 we shall introduce and study the PLV (Pshenichnyi-Levin-Valadier) property and
the weak PLV property of a semi-innite convex inequality system at a point x, requiring that the
subdierential of the sup-function G() at x should coincide with (respectively, with the closure
of) the convex hull of the subdierentials of constraints corresponding to the active indices at
that point; when this property holds for all points in the boundary of the solution set, we shall
simply use the terms PLV property or weak PLV property, respectively. In the particular case of
nite linear inequality systems the BCQ (and hence the Abadie CQ) is always satised, and for
nite convex inequality systems so is the PLV property (whence also the weak PLV property) at
all points of IR n , but for semi-innite inequality systems the situation is dierent. We shall give
new characterizations of the PLV and weak PLV properties in terms of upper semicontinuity of
certain associated multifunctions. We shall also show some connections among the PLV, weak PLV
properties, the BCQ, and Abadie CQ.
In section 4 we shall be concerned with Slater conditions. In contrast with the case of nite
systems of convex inequalities, in the semi-innite case two dierent Slater conditions appear in
a natural way: the usual one, requiring the existence of a point in the solution set, at which all
inequalities of the system are satised as strict inequalities, and the so-called strong Slater condition
(following the terminology of [5, page 128]), in which the inequalities of the system are required to
be satised uniformly strictly, that is, in which the sup-function of the system is required to satisfy
the usual Slater condition. We shall study the connections between the Slater conditions and the
constraint qualications discussed in sections 2 and 3; it will turn out that the situation concerning
these connections is dierent from that occurring in the case of nite inequality systems. Also, we
shall see that in the general semi-innite case the usual Slater condition is too weak.
The nal section 5 is devoted to some applications of constraint qualications.
Given a system of convex inequalities, we recall that any equivalent system of linear inequalities
(i.e., a system of linear inequalities with the same solution set) is called a linear representation
of the given convex system. It is well known that linear representations and, especially a certain
simple one, which we shall call the \standard" linear representation, are useful tools in the study of
convex inequality systems (see e.g. [5] and the references therein). In subsection 5.1 we shall give a
new linear representation of (nite or semi-innite) convex inequality systems satisfying the Abadie
CQ, which uses a much smaller subset of inequalities of the standard linear representation. Also,
we shall show the connections between some properties of the initial convex inequality system and
its representation.
In subsection 5.2 we shall extend from semi-innite linear inequality systems to semi-innite
convex inequality systems the concepts of consequence relations and FM (Farkas-Minkowski) systems
and, using the standard linear representation, we shall extend a known relation between linear
FM systems and the BCQ, given in [14] and [5], to the case of convex FM systems. We shall also
give a direct proof of this result, which, in contrast with the known proof for the linear case, does
not use any subset of
The exact formulas for the distance of a point to the solution set of a convex inequality system are
important, among other reasons, for their connection with \asymptotic constraint qualications"
and for obtaining results on error bounds for such a system (see [11]). The well-known general
formulas for the distance of a point to a closed convex set are not suciently useful for this purpose,
since they do not exploit the special structure of the constraints of the inequality system. Up to
the present, only a formula for the distance to the solution set of a semi-innite linear inequality
system has been essentially known (for a dual version, see [4] and [19], and for the nite case see
[2]). In subsection 5.3, assuming the Abadie CQ or the BCQ, we shall give the rst formulas for
the distance of a point to the solution set of a semi-innite system of convex inequalities, which are
new even in the nite case. Also, using this result, we shall show that the distance of a point to the
solution set of a convex inequality system (1) satisfying the BCQ is equal to the distance of that
point to some nite subsystem of (1).
We conclude this section by introducing some notations which we shall use in this paper.
We shall consider IR n endowed with the usual scalar product h; i; the Euclidean norm kk ; and
the topology induced by this norm. For an index set J , jJ j denotes the cardinality of J . Let A be
a subset of IR n . Then
A; int(A) and bd(A) denote the closure, the interior, and the boundary of
are the convex hull and the closed convex hull of A; respectively;
are the convex cone and the closed convex cone, generated by vectors in A;
is the polar of A, i.e.,
and A is the bipolar of A; in the particular case when A is a cone, A 0 coincides with the
\negative polar of A", i.e., we have
The results of the present paper and of [11] have been presented at the "Workshop on Error
Bounds and Applications in Mathematical Programming" in Hong Kong, December 8-14, 1998.
We wish to thank A. Auslender and A. M. Rubinov for the references [13] and [8], in connection
with Theorem 3. We also thank M. A. Lopez for sending us the manuscript [3], which we have
received after the present paper had been completed; in order to compare our results with those of
[3], we have inserted Remarks 2, 3(a) and 5(a).
Finally, we wish to express our gratitude to M. A. Lopez and M. D. Fajardo for their careful
reading of the manuscript of the present paper and for their valuable remarks which contributed to
its improvement.
Constraint Qualication and Basic Constraint
Qualication
For characterizations of constraint qualications for (1) we consider the convex cone generated by
the subdierentials of the active members of G() at x:
We use N 0 (x) to denote the closure of N 0 (x).
If every g i is an ane function as dened in (3), then
is the cone generated by the \active constraints" at x. Thus, in [5], N 0 (x) is called \the cone of
active constraints at x".
Let T S (x) be the tangent cone of S at x, i.e., T S x). The normal cone of S at
x 2 S is dened as
It is well-known that N S e.g. [7, Proposition 5.2.4, page 137]).
Denition 1 We shall say that the convex inequality system (1) satises
(a) the Abadie constraint qualication, or brie
y, the Abadie CQ, at a point x 2 bd
or equivalently, N S
(b) the basic constraint qualication, or brie
y, the BCQ, at a point x 2 bd S, if
(c) the Abadie CQ (respectively, the BCQ), if it satises the Abadie CQ (respectively, the BCQ) at
all points x 2 bd S.
Remark 1 (a) The equivalence of the two formulas in (18) follows from the bipolar theorem. In
fact, if N S since T S (x) is a closed convex cone, by the bipolar theorem and by
e.g. [7, Proposition 5.2.4, page 137]) we have
On the other hand, if and the bipolar theorem, and since
is a cone, we obtain
(b) Since we always have (see e.g. [7, Lemma 4.4.1, page 267] and [7, Lemma 2.1.3, page 305])
the system (1) satises the Abadie CQ at x 2 bd S if and only if
and it satises the BCQ at x 2 bd S if and only if
Moreover, for x 2 bd S, by (20), (1) satises the Abadie CQ at x if and only if N 0
and (6) satises the Abadie CQ at x. Similarly, by (21), (1) satises the BCQ at x if and only if
the BCQ at x. Clearly, (1) satises the BCQ at x if and
only if it satises the Abadie CQ at x and N 0 (x) is closed. Thus, (1) satises the BCQ at x if and
only if T S closed. The points x 2 bdS with the latter property have
been called \Lagrangian regular points" in [12, Denition 3.3].
(c) When I is nite, Denition 1 is the classical denition of the Abadie CQ introduced by Abadie
(see [1], also [10]) and, respectively, of the basic constraint qualication (see [7, page 207]). If I is
nite and each g i is a dierentiable convex function, then @g i is the
gradient of g i at x) and the cone N 0 (x) of (15) is closed (see e.g. [7, Lemma 4.3.3, page 130]). In
this case, the Abadie CQ and the BCQ coincide.
(d) The BCQ, in an equivalent form, has been introduced for semi-innite linear inequality systems
(4) in [14] (called locally Farkas-Minkowski systems, or brie
y, LFM systems) and extended to
convex inequality systems (1) in [5, pp. 162-163] (called convex LFM constraint qualication).
When I is nite and each g i is an ane function, the BCQ, and hence also the Abadie CQ, are
satised (see e.g. [7, Example 5.2.6(b), page 138]). When each g i is ane, for semi-innite systems
of linear inequalities, the Abadie CQ may not hold and the Abadie CQ is not the same as the BCQ,
as shown by the following example.
Example 1 Let 2:
(a) Semi-Innite Linear Systems Without the Abadie CQ:
x 2i
0g. Also,
Thus the family (24) does not satisfy the Abadie CQ.
(b) Semi-Innite Linear Systems With the Abadie CQ but Without BCQ:
Also,
Thus the family (25) satises the Abadie CQ, but not the BCQ.
Next we give new characterizations of the Abadie CQ and the BCQ for (1), in terms of the
upper semicontinuity of the multifunctions N 0 (); cone(@G()), N 0 () and cone(@G()). We recall
(see e.g. [16, page 55]) that a multifunction (i.e., a set-valued
(the
collection of subsets of IR n ) is said to be upper semicontinuous in the sense of Kuratowski, or brie
y,
upper semicontinuous, at x 2 IR n , if the relations lim k!+1 x
Clearly, the graph of Q (i.e., the set f(x; y)j x 2
is closed if and only if Q is upper semicontinuous at all x 2 IR n .
We shall rst prove a lemma, in which we shall use the convex hull of the subdierentials of the
active members of G(x), that is, the set
be a multifunction such that, for all z in a neighborhood
of x; Q(z) is a convex set, I(z) 6= ;, and D 0 (z) Q(z). If Q is upper semicontinuous at x,
then @G(x) Q(x).
Proof. Assume the contrary that there exists y 2 @G(x) n Q(x): Since Q is upper semicontinuous
at x and Q(x) is a convex set, Q(x) is a closed convex set. Then, by the strict separation theorem
[7, Theorem 4.1.1, page 121], there exists u 2 IR n nf0g such that
Let G 0 (x; u) be the directional derivative of G at x in the direction u. By y 2 @G(x) and a
well-known formula for G 0 (x; u) (see e.g. [7, page 240]), we get
z in a neighborhood of x, without loss of generality
we can assume I(x+t k u) 6= ;. Let
Then
whence, by t k
is bounded, the set fy
is bounded as well (see e.g. [7, Proposition 6.2.2, page 282]). Hence, we may assume, without loss
of generality, that y
y. Then, letting k ! +1 in (28) and using (27), we obtain
hb
But, since
lim
and since Q is upper semicontinuous at x, we have b
which contradicts (29). 2
Theorem 1 Let x 2 bd S and I(z) 6= ; for z in a neighborhood of x. Then the following two
statements are equivalent.
(a) (1) satises the Abadie CQ at x.
(b) Both cone(@G()) and N 0 () are upper semicontinuous at x.
Proof. (a))(b): Let x y. We claim that
To prove the claim, let
We consider two cases:
Case 1: (6) does not satisfy the Slater condition, that is, G(z) 0 for all z 2 IR n . Then
. Thus, (31) implies
Letting in the above inequality we obtain
That is, y 2 N S (x).
Case 2: (6) satises the Slater condition, that is, G(^x) < 0 for some ^
S. Then there is k 0 > 0 such that z 2 S k for k k 0
By (31), we have
Since (32) holds for any z 2 int S, it also holds for z 2 int
If (1) satises the Abadie CQ at x, we get y 2 N S
Therefore, both N 0 () and cone(@G()) are upper semicontinuous at x.
y. Then x 0 62 S (since x
would imply hy; xi hy; x which is impossible) and
hence
and assume k large enough so that 1
be such that
k. Then, since x 2 S k , we have
Thus, fx
k g is a bounded sequence. Without loss of generality we may assume that x
in (34) we obtain
On the other hand, since x 0 x is the projection of x 0 onto S. Hence,
imply that Consequently, x
y.
We claim that x 0 x
k k we have x
and x 0 x
satises the Slater condition and x
e.g. [7, Theorem 1.3.5, page 245]). Thus, x 0 x
which proves the claim.
by the upper semicontinuity of
cone(@G()) at x we get y 2 cone(@G(x)). Hence, since y was an arbitrary nonzero element in
On the other hand, by the upper semicontinuity of N 0 () at x and
(applied to we have @G(x) N 0 (x), which implies cone(@G(x)) N 0 (x).
Therefore, we have (22), and thus (1) satises the Abadie CQ at x. 2
Theorem 2 Let x 2 bd S and I(z) 6= ; for z in a neighborhood of x. Then the following two
statements are equivalent.
(a) (1) satises the BCQ at x.
(b) Both cone(@G()) and N 0 () are upper semicontinuous at x.
Proof. (a))(b): By Theorem 1, both cone(@G()) and N 0 () are upper semicontinuous at x. Since
closed, we know that N 0 (x) and cone(@G(x)) must be
closed. Thus, both cone(@G()) and N 0 () are upper semicontinuous at x.
(b))(a): By Theorem 1, (1) satises the Abadie CQ at x. But (b) also implies that N 0 (x) is a
closed set. So N 0 satises the BCQ. 2
From Theorems 1 and 2 we obtain the following characterizations of the Abadie CQ and the
BCQ.
Corollary 1 Suppose that I(z) 6= ; for z in a neighborhood of bdS. Then the following two
statements are true.
(a) (1) satises the Abadie CQ if and only if both cone(@G()) and N 0 () are upper semicontinuous
at every x in bdS.
(b) (1) satises the BCQ if and only if both cone(@G()) and N 0 () are upper semicontinuous at
every x in bd S.
Remark 2 Fajardo and Lopez [3, Theorem 3.1(i)] proved that if (1) satises the BCQ, then the
multifunction A(x) := fy 2 N 0 (x)j kyk 1g (B)-upper semicontinuous on S (which is
equivalent to the upper semicontinuity of N 0 () on S) and that if (6) satises the Slater condition,
then the converse is also true [3, Theorem 3.1(iib)].
Here a mapping
is said to be (B)-upper semicontinuous at x 2 IR n , if for every
open set W in IR n containing Q(x) there exists a neighbourhood V (x) of x such that Q(z) W
for each z 2 V (x); furthermore, Q is said to be (B)-upper semicontinuous on a set M IR n if
it is (B)-upper semicontinuous at each x 2 M: It is well-known (see e,g, [5, p. 128]) that if Q is
(B)-upper semicontinuous (at x) then it is upper semicontinuous (at x); but the converse is not
true. One can show that if the set fzj z 2 Q(y) for y in some neighborhood of xg is bounded, then
the (B)-upper semicontinuity of Q() at x is equivalent to the upper semicontinuity of Q() at x.
3 Subdierentials of the Sup-Function and Its Active Member
In this section, we study the relations between the subdierential of the sup-function G and the
subdierentials of its active member functions fg I(x)g.
We shall use the set D 0 (x) dened by (26). Note that
We shall denote by D 0 (x) the closure of the set D 0 (x):
One important property of (1) when I is nite is the equality (11) (see e.g. [13, Theorem 1.4]),
which can be rewritten as
The above equality means that the subdierential of the sup-function G() is the convex hull of
subdierentials of its active members.
In general, (37) does not hold if I is innite. The following sucient (but not necessary)
condition that guarantees , and is satised when I is nite, has been
given by Levin [8, Theorem 2] (and, at the same time, Valadier [18, Theorem 2] has obtained, in
an arbitrary topological linear space instead of IR n ; the weaker conclusion in which D 0 is replaced
by D 0
Theorem 3 (Pshenichnyi-Levin-Valadier Theorem [13, 8, 18]) If I is a compact set (in some
metric space), and is a family of convex functions such that for each xed
upper semicontinuous on I, then (37) holds at all x 2 IR n .
Here a real-valued function f(t) is said to be upper semicontinuous at
lim
i.e.,
lim
sup
Note that (39) is equivalent to the upper semicontinuity of the multifunction F fy 2 IRj y
f(t)g at t 0 .
For convenience, we shall introduce the following denition related to (37) (where PLV stands
for \Pshenichnyi-Levin-Valadier").
Denition 2 Let (1) be a convex inequality system with a nite-valued sup-function. We shall say
that the family fg or the system (1), has
(a) the weak PLV property at a point x 2
(b) the PLV property at a point x 2
(c) the weak PLV property (respectively, the PLV property), if it has the weak PLV property (re-
spectively, the PLV property ) at all x 2 bd S:
One major problem with an innite I is the possibility of (which can
not happen when I is nite). For example, given a family of convex functions
holds if and only if
However, the active index set for (42) is empty at every x 2 bd S and so there is no (weak) PLV
property for (42).
One can avoid the inconvenience of having by requiring the closedness of fg i (x)j i 2 Ig.
Proposition 1 Let x 2 bd S: If the set A x := fg i (x)j i 2 Ig is closed in IR, then I(x) 6= ;:
Proof. Since x 2bd S; we have (by the continuity of G): Taking any sequence fi k g I
such that g i k
by the closedness of A x we obtain 0 2 A x ;
so I(x) 6= ;: 2
However, even if I(x) 6= ;, (4) (and hence, in general, (1)) might not have the weak PLV
property. Also, in general, the weak PLV property is not the same as the PLV property.
(a) A Semi-Innite Linear System Without Weak PLV Property:
Then
k1
4x if x 0:
f3g. Therefore, the system does not have the weak PLV property at
(b) A Semi-Innite Linear System With Weak PLV Property, But Without PLV Property:
Then
k1
x if x < 0;
5x if x 0:
5). Therefore, the system has the weak PLV property at but not the PLV
property.
There is no relation of implication between the CQs and the PLV properties. Indeed, in Example
2(a), the system actually satises the BCQ at but the PLV property does not hold at
while for any I with one index the PLV property is trivially true, but (1) may not satisfy
the Abadie CQ. However, when (1) satises the PLV property (in particular, when I is nite),
and the characterizations for the Abadie CQ and the BCQ can be simplied.
Corollary 2 Suppose that (1) satises the PLV property and I(x) 6= ; for x in a neighborhood of
bd S (e.g., this happens when I is nite). Let x 2 bd S. Then the following two statements are
true.
(a) (1) satises the Abadie CQ at x if and only if N 0 () is upper semicontinuous at x.
(b) (1) satises the BCQ at x if and only if N 0 () is upper semicontinuous at x.
Proof. By the assumption, D 0 all x, which implies N 0
x. Thus Corollary 2 follows from Corollary 1. 2
The following proposition shows that if (1) satises the (weak) PLV property, then (1) and (6)
are \equivalent" in terms of CQ's.
Proposition 2 Let x 2 bdS.
(a) Suppose that (1) satises the weak PLV property at x. Then (1) satises the Abadie CQ at x
if and only if (6) satises the Abadie CQ at x.
(b) Suppose that (1) satises the PLV property at x. Then (1) satises the BCQ at x if and only
if (6) satises the BCQ at x.
Proof. (a) If (1) satises the weak PLV property at x; then, by (40) and (36),
only if N S
(b) If (1) satises the PLV property at x; then, by (41) and (36),
only if N S
It turns out the the weak PLV property or the PLV property at a point x 2 IR n can be characterized
by the upper semicontinuity at x of the multifunction D 0 () or D 0 (), respectively.
Theorem 4 Let x 2 IR n and I(z) 6= ; for z in a neighborhood of x. Then
only if D 0 () is upper semicontinuous at x.
Proof. If D 0 () is upper semicontinuous at x, then, by Lemma 1, @G(x) D 0 (x). Since @G(x)
holds, we have
Next we assume that prove the upper semicontinuity of D 0 () at x.
Let lim k!+1 x
we have y k 2 @G(x k G is a nite convex function, @G(z) 6= ; for all z 2 IR n and @G
is upper semicontinuous (see e.g. [7, Proposition 6.2.1, page 282]). Thus, y 2 @G(x). Since
upper semicontinuous at x. 2
Theorem 5 Let x 2 IR n and I(z) 6= ; for z in a neighborhood of x. Then
only if D 0 () is upper semicontinuous at x.
Proof. Note that D 0 () is upper semicontinuous at x if and only if D 0 () is upper semicontinuous at
x and D 0 (x) is a closed set.
If D 0 () is upper semicontinuous at x, then, by Theorem 4, On the
other hand, if then by the upper semicontinuity of @G(), D 0 (x) is a closed set
and Hence, by Theorem 4, D 0 () is upper semicontinuous at x. Therefore, D 0 () is
upper semicontinuous at x. 2
Using Theorems 4 and 5 we obtain the following characterizations of the weak PLV and PLV
properties at all x 2 IR n .
Theorem 6 The following statements are true.
(a)
semicontinuous on
only if D 0 (x) 6= ; for all x 2 IR n and D 0 () is upper
semicontinuous on
Proof. Since G() is a nite convex function, @G(x) 6= ; for any x. Therefore, every condition in the
above theorem implies I(x) 6= ; for all x 2 IR n . Consequently, Theorem 6 follows from Theorems 4
and 5. 2
Remark 3 (a) Fajardo and Lopez [3, Theorem 4.1(i)] proved that if D 0 (x) 6= ; for all x 2 IR n and
(B)-upper semicontinuous on IR n (see Remark 2 above), then
Since the (B)-upper semicontinuity of D 0 () is equivalent to the upper semicontinuity of D 0 ()
(see Remark 2), the \if" part of Theorem 6(a) is equivalent to [3, Theorem 4.1(i)].
(b) Using Theorem 6, we can give a new proof of Theorem 3, which seems simpler and more natural
than the proofs known in the literature (see e.g. the proof of Theorem 4.4.2 [7, page 267 ]. To this
end, let us rst prove the following fact:
If I is a compact metric space and if is a family of convex functions
such that for each x 2 IR n the function upper semicontinuous on I, then
the set-valued mapping x ! I(x) is upper semicontinuous on IR n and the set-valued
mapping upper semicontinuous on W; where W := f(x; i)j i 2 I(x)g.
i.e., if lim k!+1 x
y with
First we prove by contradiction. In fact, if by the upper semicontinuity
of g i (x) with respect to i, there exist a positive constant and a neighborhood O( ^ i) of ^ i in I such
that
(x). By the assumptions, ^
are
continuous convex functions. Thus,
But (45) implies that ^
G(^x) G(^x) , a contradiction to (46). This proves that
Now, for any z 2 IR n , we have
Thus, letting k ! +1 in (47) and using lim k!+1 g i k
(by the assumption of upper
semicontinuity of the mapping
Since (49) holds for any z, we have ^
Finally, in order to prove Theorem 3 it will be sucient, by Theorem 5, to prove that under the
assumptions of Theorem 3 the mapping D 0 () has closed graph and I(x) 6= ; for all x 2 IR n . Let
By the denition of D 0
there exist
theorem (see e.g. [7,
Theorem 1.3.6, page 98]) we may assume, without loss of generality, that m k n + 1. Since G()
is nite-valued and x
x, the set fy 2 @G(x k )j bounded (see e.g. [7, Proposition
6.2.2, p. 282]). By y j;k 2 @g i j
we know that fy j;k
a bounded set. Since I is compact and fm k g, fy j;k are all bounded with respect to k, by
repeatedly selecting subsequences we may assume, without loss of generality, that m
+1. By the fact proved above, we know that
and y
(x). Thus,
which proves that D 0 () has closed graph. Finally,
since I is compact and i ! g i (x) is upper semicontinuous, we have I(x) 6= ; for all x 2 IR n . This
provides an alternative proof of Theorem 3.
4 Slater Conditions
If there exists x 2 IR n such that
then (1) is said to satisfy the Slater condition. Let us recall the following well-known result, which
gives a sucient condition for the BCQ of (1) or (6).
Proposition 3 ([7, Theorem 1.3.5, page 245] and [7, Remark 1.3.6, page 246]) If I is nite and
(1) satises the Slater condition, then (1) satises the BCQ. In particular, if (6) satises the Slater
condition, then (6) satises the BCQ.
Remark 4 (a) If (6) satises the Slater condition, then (1) also satises the Slater condition. But
the converse is not true. The Slater condition for (6) is sometimes called the strong Slater condition
for the convex system (1) (see e.g. [5, page 128]). However, the term \strong Slater condition"
is also used in the literature in other senses (see e.g. Lewis and Pang [9], where \strong Slater
condition" means that 0 does not belong to the closure of the set @G(G 1 (0)); and, for a dierent
sense, see [7, Denition 2.3.1, page 311]).
(b) When I is nite, (1) satises the Slater condition if and only if (6) satises the Slater condition.
Proposition 4 Suppose that (6) satises the Slater condition.
(a) If (1) has the weak PLV property, then (1) satises the Abadie CQ.
(b) If (1) has the PLV property, then (1) satises the BCQ.
Proof. (a) By Proposition 3, the Slater condition for (6) implies the BCQ for (6). Hence, by
Proposition 2(a), we have the Abadie CQ at all x 2 bd S.
(b) The proof is similar, using Proposition 2(b). 2
Remark 5 (a) Fajardo and Lopez [3, Theorem 4.1(ii)] proved that if (6) satises the Slater condi-
tion, D 0 () is (B)-upper semicontinuous (see Remark 2), D 0 (x) closed
for each x in S, then (1) satises the BCQ. But the (B)-upper continuity of D 0 () is equivalent to
the upper continuity of D 0 () (see Remark 2), so this result also follows from Theorem 6(a) and
Proposition 4(a).
(b) The assumptions in (a) and (b) of Proposition 4 cannot be omitted, as shown by Example 1(a),
in which the Slater condition for (1) or (6) is satised (in fact, for
but the Abadie CQ is not satised. When I is a nite set, the PLV property always holds. In this
case, the Slater condition, the BCQ, and the Abadie CQ are all dierent.
We recall that for a convex system (1) a solution
x 2 S is called a Slater point if we have (50).
Proposition 5 If for each x 2 S the active index set I(x) 6= ;, then every Slater point of (1) is a
Slater point of (6) (and hence, in this case, the Slater condition for (1) and the Slater condition for
are equivalent).
Proof. Let
x be a Slater point of (1), i.e., let
x be a point such that (50) holds. If x were not a
Slater point of (6), i.e., if we had sup i2I g i since I(x) 6= ;, there would exist i 0 2 I
such that g i 0
a contradiction to the assumption (50). 2
Using Theorem 3, one can give a stronger condition which ensures the BCQ. Indeed, combining
Theorem 3 and Proposition 4, we obtain the following result, which has been proved with a more
complicated method by Lopez and Vercher [12, Theorem 3.8].
Corollary 3 If I is a compact set (in some metric space), is a family of
convex functions such that for each xed x 2 IR n the function upper semicontinuous on
I, and (1) satises the Slater condition, then the BCQ holds for (1).
Proof. Since I is compact and i ! g i (x) is upper semicontinuous, I(x) 6= ; for any x. By Proposition
5, (6) satises the Slater condition. By Theorem 3, the PLV property holds. Thus, the corollary
follows from Proposition 4. 2
Even though we stated Corollary 3 in terms of the Slater condition of (1), obviously the Slater
condition of (6) is also satised. In general, the Slater condition for (1) is not very meaningful if
(6) does not satisfy the Slater condition. One might wonder whether we should use the following
stronger version of (50):
where is a positive constant. In the case that G(x) < +1 for x 2 IR n , (51) is nothing more than
the Slater condition for (6). If one allows G(x) to be +1, then (51) does not provide any useful
information about the system as shown in the following example.
Example 3 (a) For fg
Then (51) holds with g i and being replaced by g i and 1, respectively. Note that x 2 S if and
only if g i (x) 0 (i 2 I). Also the sets I(x) and N 0 (x) remain unchanged for x 2 bd S. Thus, (1)
satises the Abadie CQ (respectively, the BCQ) if and only if so does the system
This example shows that replacing (50) by (51) without requiring a nite-valued G does not give
any new information about the underlying system.
(b) Or we could make the situation worse. For example,
Then we always have
It is easy to see that x 2 S if and only if 1. In this case, the active index
set is always empty for any x 2 bdS. Thus, it is not possible to study S by using N 0 (x). This
example shows that (51) without requiring a nite-valued G(x) could be a meaningless condition;
while (51) with a nite-valued G(x) means that (6) satises the Slater condition, which is useful
for constraint qualication properties of (1) (see Proposition 4).
Applications
5.1 Linear Representation of Convex Systems
Given a semi-innite convex inequality system (1), we recall that a semi-innite linear inequality
system,
a is said to be a linear representation of the system (1), provided that x
is a solution of (1) if and only if x is a solution of (54) (i.e., provided that the systems of inequalities
(1) and (54) are equivalent). Each linear representation (54) of (1) is also called a linear system
associated to the convex system (1).
It is well-known (see e.g. the proof of Theorem 5.2 in [6]) that the system,
is a linear representation of the convex system (1), which we shall call the standard linear representation
of (1). For the sake of completeness, we include here the proof. If x 2
so x satises (55). Conversely, if x satises (55), then
Taking here which completes the proof.
A natural question is whether we can use a smaller subsystem of (55) to get a linear representation
of (1). In particular, we study when the following semi-innite linear system,
is a linear representation of (1). Note that (56) is indeed a subsystem of (55), since for z 2 bd S,
Theorem 7 (a) If the convex system (1) satises the Abadie CQ, then the system (56) is a linear
representation of (1).
(b) If the convex system (1) satises the Slater condition (50) and I(x) 6= ; for all x 2 S, then
the system (56) is a linear representation of (1).
Proof. Obviously, if x 2 S, then, since @g i (z) N S (z) for i 2 I(z) and z 2 bd S, (56) follows from
(z). Thus, every solution of (1) is a solution of (56). In order to prove that (56) is a linear
representation of (1), it is sucient to show that if x 62 S, then (56) does not hold.
(a) First we prove the following more general result:
Suppose that for any x 2 bd S and y 2 N S (x) n f0g, there is a vector ^
that linear representation of (1).
Let x 62 S and let z be the projection of x onto S. Then z 2 bdS, x z 6= 0, and x z 2 N S (z)
[7, Theorem 3.1.1, page 117]. If hy
for all y 2 N 0 (z), a contradiction to the assumption there is ^
Hence, x does not satisfy (56). This proves that x 2 S if and only if x satises (56), i.e., (56) is a
linear representation of (1).
Now, if the convex system (1) satises the Abadie CQ, i.e., N S
then it is trivially true that for any x 2 bd S and y 2 N S (x) n f0g, there is a vector
that
(indeed, it is enough to take ^
suciently close to y): So (56) is a linear
representation of (1).
(b) By Proposition 5, (6) satises the Slater condition. Let x 62 S, and let x 2 S be such that
contains exactly one point, say z. Then there is a positive constant
such that
by the denition of @G(z), we have
xi G(z)
Since > 0, it follows from (57) and (58) that 1
Thus, for any x 62 does not hold. So (56) is a linear representation of (1).Remark 6 (a) In [12, the proof of Theorem 4.5], it has been observed that if (1) satises the
assumptions of Corollary 3, then the linear inequality system,
is a linear representation of (1). Let us observe that this follows also from Corollary 3 and Theorem
under the assumption of Corollary 3,
hence (59) is equivalent to (56).
(b) In the particular case when all are convex and dierentiable, Theorem 7(b) (even
with a smaller subsystem of (56), obtained by choosing for each x 2bd S; with the aid of the axiom
of choice, an index i(x) 2 I(x) and an y i(x) 2 @g i (x)) has been shown, essentially, in the proof of
Theorem 5.4 in [6].
Some connections between the inequality systems (1) and (56) are given in the following proposition
Proposition 6 Let (56) be a linear representation of the convex inequality system (1). Then
(a) (1) satises the Abadie CQ (respectively, the BCQ) if and only if so does (56).
(b) Denoting by G and G 0 the sup-functions of (1) and (56) respectively, we have
Proof. (a) Let (56) be a linear representation of (1) and let x 2 bd S: Since (1) and (56) have the
same solution set and hence the same normal cone N S (x) at x; it will be enough to show that
Since the linear system (56) is a subset of the linear system (55), we have
Furthermore, by [5, proof of Theorem 10.7], there holds
Finally, from the denitions it is obvious that
which, together with (62) and (63), yields (61).
(b) By the denitions of the sup-function and of @g i (z) and I(z); we have, for any x 2
z2bd
sup
i2I
Remark 7 (a) The inequality in Proposition 6(b) may be strict.
(b) From Proposition 6(b) above it follows that if (1) satises the Slater condition, then so does
also, if (6) satises the Slater condition, then G 0 (x) < 0 for some x 2 However, the
converse statements are not true.
5.2 Convex Farkas-Minkowski Systems
Related to the BCQ are the convex Farkas-Minkowski systems, dened as follows.
Denition 3 (a) A linear inequality
where a will be called a consequence relation of the convex inequality
system (1), if every x 2 S satises (65):
(b) The system (1) will be called a convex Farkas-Minkowski (or brie
y, a convex FM) system, if
every linear consequence relation of system (1) is also a consequence relation of some nite
subsystem of (1).
Remark 8 In the particular case of a linear inequality system (4), the above denition reduces to
the usual denition of consequence relations and FM systems [5].
One can extend some results on linear FM systems to convex FM systems. For example, the
fact that a linear inequality system (4) satisfying the BCQ and with bounded solution set S is an
FM system (see [5, Exercise 5.7]), admits the following extension.
Proposition 7 A convex inequality system (1) satisfying the BCQ and with bounded solution set
S is a convex FM system.
Proof. By the above proof of Proposition 6(a), (1) satises the BCQ (if and) only if so does its
standard linear representation (55). Furthermore, since (55) is a linear inequality system having
the same bounded solution set it is an FM system (see e.g. [5, Exercise 5.6]). Finally, let us
show that if (55) is an FM system, then so is (1). Indeed, let
where a since (55) is an FM system and has the same solution set
there exists a nite subsystem of (55), say
where jJ j < +1; such that (65) is a consequence relation of (67). Let S J be the solution set of the
nite subsytem
of (1) and let x
so x is a solution of (67). Hence, since (65) is a consequence relation of (67), we obtain ha
which, since x 2 S J was arbitrary, proves that (1) is an FM system. 2
Remark 9 The proofs of some results on linear systems, given in [5], use certain cones of IR n+1
associated to (4). However, let us observe that one can give, directly for the extensions of those
results to convex systems, proofs which are new even for the case of linear inequality systems, and
do not use any subsets of us give such a proof of Proposition 7. Assume that S
is bounded and (1) satises the BCQ, and let (65) be a consequence relation of (1). Let c be the
smallest number such that ha 0 ; xi c is still a consequence relation of (1) (such a number exists,
since otherwise S fx 2 a contradiction to the general assumption made
in this paper).
We claim that there exists z 2 S such that ha Indeed, by the denition of c; we have
cg and for each
Then, since S is bounded and closed, hence compact, fx k g has a subsequence converging to some
which proves our claim.
By the above, we have S fx 2 Hence, by the BCQ,
there exists a nite subset J of I(z); such that
a
Let
Then, by (70) and J I(z), we have a 0 2co([ i2J @g i (z)) N S J
(z); that is, ha
Thus, the inequality ha 0 ; xi c; whence also (65), is a consequence relation of the nite
subsystem of (1), which completes the proof.
Combining Proposition 7 and Corollary 3, there results the following corollary, which has been
proved with more complicated methods in [12, Theorem 4.5].
Corollary 4 If I is a compact set, is a family of convex functions such that
for each xed x 2 IR n the function upper semicontinuous on I, (1) satises the Slater
condition, and the solution set S is bounded, then (1) is an FM system.
The denition of a convex FM system given in [12] referees to its standard linear
representation being a FM system. Under the assumptions of Corollary 4, both denitions are
equivalent.
5.3 The Distance to the Solution Set of a Convex Inequality System
Theorem 8 Let x 2 IR n nS and ^
x be the projection of x onto S.
(a) If (1) satises the Abadie CQ, then
I 0 I(^x)
sup
xi: (72)
(b) If (1) satises the BCQ, then
I 0 I(^x)
sup
xi:
Proof. By [17, Remark 8b)], we have
(a) By the Abadie CQ at ^
x, we have y 2 N S (^x) with only if there exist I k
1 such that
y.
Thus, (72) is equivalent to (74).
(b) By the BCQ at ^
x, we have y 2 N S (^x) with only if there exist I 0 I(^x),
1 such that
y. Thus,
(73) is equivalent to (74). 2
Remark 11 The assumption of Abadie CQ may be too strong, but at least the assumption I(b x)
(which is implied by the Abadie CQ) is necessary in order to have (72). Indeed, if I(b
the right hand side of (72) is meaningless.
When applied to the semi-innite linear system (4), @g j means that
and thus Theorem 8 reduces to the following form.
Corollary 5 Let x 2 IR n nS and ^ x be the projection of x onto S.
(a) If (4) satises the Abadie CQ, then
I 0
(b) If (4) satises the BCQ, then
Remark 12 (a) By a well-known theorem of Caratheodory (see e.g. [15, Corollary 7.1(i), page
94]), in each positive combination
may assume that fy j jj 2 I 0 g is linearly independent.
That is,
fy is linearly independent; i 0 (i 2 I 0 )
fy j jj 2 I 0 g is linearly independent: (78)
and hence we also have the following representation of N 0 (x):
One could rewrite the distance formulas (72), (73), (75), and (76), based on either (77) or (79).
(b) In the particular case when I is nite, (4) satises the BCQ (see the observation before Example
1), and hence Corollary 5(b) reduces to [17, Remark 8(a)].
The following theorem shows that if the BCQ holds, then the distance of a point to the solution
set S of an arbitrary convex inequality system (1) is equal to the distance of that point to the
solution set of some nite subsystem of (1).
Theorem 9 If (1) satises the BCQ and b
x is the projection of x onto S, then there exists
with
Jj < +1; such that
where
J)g:
Proof. Choose any
J I(b x) for which the rst max in (73) is attained. Then, applying Theorem 8
to the inequality system
and to its solution set S
J (of (81)), we obtain
dist (x; S
which, since S S
J (by (2) and (81)), yields (80). 2
Remark 13 In the particular case when I is nite and each g i is an ane function, (1) satises
the BCQ, and hence Theorem 9 yields [2, Corollary 1.1].
--R
On the Kuhn-Tucker theorem
The distance to a polyhedron.
On systems of linear inequalities.
Application of a theorem of E.
bounds for convex inequality systems.
Abadie's constraint quali
Asymptotic constraint quali
Convex programming in a normed space
Theory of linear and integer programming.
The theory of best approximation and functional analysis.
Duality for optimization and best approximation over
Generalizations of some fundamental theorems in linear inequalities.
--TR
--CTR
Chong Li, Strong uniqueness of the restricted Chebyshev center with respect to an RS-set in a Banach space, Journal of Approximation Theory, v.135 n.1, p.35-53, July 2005
Mara J. Cnovas , Marco A. Lpez , Juan Parra, Stability in the Discretization of a Parametric Semi-Infinite Convex Inequality System, Mathematics of Operations Research, v.27 n.4, p.755-774, November 2002 | convex Farkas-Minkowski systems;distance formulas;constraint qualifications;semi-infinite inequality systems |
589096 | Differential Stability of Two-Stage Stochastic Programs. | Two-stage stochastic programs with random right-hand side are considered. Optimal values and solution sets are regarded as mappings of the expected recourse functions and their perturbations, respectively. Conditions are identified implying that these mappings are directionally differentiable and semidifferentiable on appropriate functional spaces. Explicit formulas for the derivatives are derived. Special attention is paid to the role of a Lipschitz condition for solution sets as well as of a quadratic growth condition of the objective function. | Introduction
Two-stage stochastic programming is concerned with problems that require a here-
and-now decision on the basis of given probabilistic information on the random data
without making further observations. The costs to be minimized consist of the direct
costs of the here-and-now (or first stage) decision as well as the costs generated by the
need of taking a recourse (or second stage) decision in response to the random environ-
ment. Recourse costs are often formulated by means of expected values with respect
to the probability distribution of the involved random data. In this way, two-stage
models and their solutions depend on the underlying probability distribution. Since
this distribution is often incompletely known in applied models, or it has to be approximated
for computational purposes, the stability behaviour of stochastic programming
models when changing the probability measure is important. This problem is studied
in a number of papers. We only mention here the surveys [13], [37] and the papers [1],
This research is supported by the Deutsche Forschungsgemeinschaft
[12], [17], [24], [25], [31] and [32]. The paper [1] contains general results on continuity
properties of optimal values and solutions when perturbing the probability measures
with respect to the topology of weak convergence. Quantitative continuity results of
solution sets to two-stage stochastic programs with respect to suitable distances of
probability measures are derived in [24] and [25]. Asymptotic properties of statistical
estimators of values and solutions to stochastic programs are derived in [17], [31], [32].
They are based on directional differentiability properties of the underlying optimization
problems with respect to the parameter that carries the randomness ([17], [32]) or the
probability measure ([31]). These directional differentiability results for values (in [32])
and solutions (in [17], [31]) lead to asymptotic results via the so-called delta-method .
For a description of the delta-method we refer to Chapter 6 in [26], [32], to [33] for
an up-to-date presentation and to [15] for a set-valued variant. These papers illuminate
the importance of the Hadamard directional differentiability (for single-valued
functions) and of the semidifferentiability (for set-valued mappings) in the context of
asymptotic statistics.
The present paper aims at contributing to this line of differential stability studies. The
results in [17], [31] apply to fairly general stochastic optimization models, but impose
conditions that are rather restrictive in our context. The present paper deals with
special two-stage models and, using structural properties, avoids certain assumptions
that complicate or even prevent the applicability of those general results to two-stage
stochastic programs. Such assumptions are the (local) uniqueness of solutions and
differentiability properties of perturbed problems, which are indispensable in [17], [31].
Before discussing this in more detail, let us introduce the class of two-stage stochastic
programs, we want to consider:
is a nonempty closed convex set, A
is a (s; m)-matrix and Q - is the expected recourse function with respect to the (Borel)
probability measure - on IR s ,
Z
~
~
Here
m are the recourse costs, W is an (s; -
m)-matrix and called the recourse
matrix, and ~
corresponds to the value of the optimal second stage decision
for compensating a possible violation of the (random) constraint To have the
problem (1.1) - (1.3) well-defined, we assume
Z
first moment).
The assumptions (A1) and (A2) imply that ~
Q is finite, convex and polyhedral on
. Due to (A3) also Q - is finite and convex on IR s (cf. [14], [36]). Observe that,
in general, an expected recourse function Q - may be nondifferentiable on a certain
union of hyperplanes in IR s and that, indeed, differentiability properties of Q - depend
on the degree of smoothness induced by the measure - (cf. [14], [19], [35], [36] and
Remark 4.8). Another observation shows that the uniqueness of solutions to (1.1) is
guaranteed only if the constraint set C picks just one element from the relevant level
set of g(\Delta) +Q - \Delta). This set may be large since Q - \Delta) is constant on translates of the
null space of the matrix A (see Example 1.1 in [25]). Proposition 2.1 below provides
some more insight into the structure of the solution set to (1.1) and elucidates the role
of the set-valued mapping oe(y) yg in this respect.
Note that assumption (A1) could be relaxed by introducting the set
+1g. Then (A2) and (A3) imply that K is a closed convex polyhedron and
that Q - is convex and continuous on K (cf. [36]). Now (A1) can be replaced by the
condition K ' A(C) (relatively complete recourse), and much of the work done in this
paper carries over to this more general setting by using spaces of functions defined on
K instead of IR s .
Let K C denote the set of all convex functions on IR s which forms a convex cone in the
space C 0 (IR s ) of all continuous functions on IR s . K C will serve as the set of possible
perturbations of the given expected recourse function Q - 2 K C . We define
and regard ' and / as mappings from K C into the extended reals and the set of all
closed convex subsets of IR m , respectively.
In this paper we develop a sensitivity analysis for the mappings ' and / at some given
function Q - . The stochastic programming origin of the model (1.1) takes a back seat
and our results are stated in terms of general conditions on Q - and its perturbations
Q. We identify conditions such that the value function ' has first- and second-order
directional derivatives and the solution-set mapping / is directionally differentiable at
admissible directions. Here, admissibility means that the direction belongs to
the radial tangent cone to K C at Q - , i.e.,
ensuring that the difference quotients are well-defined. For v belonging to T r (K C
the Gateaux directional derivatives of ' and / at Q - and (Q -
tively, are defined as
if the limits exist. The third limit is understood in the sense of (Painlev'e-Kuratowski)
set convergence (e.g. [2]). Recall that the lower and upper set limits of a family (S t ) t?0
of subsets of a metric space (X; d) are defined as
lim inf
lim sup
Both sets are closed and the lower set limit is contained in the upper limit. If both limits
coincide, the family (S t ) t?0 is said to converge and its limit set is denoted by lim
For sequences of sets (S n ) n2IN the definitions of set limits are modified correspondingly.
We also derive conditions implying that the limits defining the directional derivatives
exist uniformly with respect to directions v belonging to compact subsets of certain
functional spaces. The limits are then called (first- or second-order) Hadamard directional
derivatives and semiderivatives for set-valued maps, respectively. The corresponding
directional derivatives are defined on tangent cones to the cone of convex
functions in certain functional spaces. For more information on concepts of directional
differentiability and multifunction differentiability we refer to [5], [30] and to [2], [4],
[21], [23], respectively.
Let us fix some notations used throughout the paper. k \Delta k and h\Delta; \Deltai denote the norm
and scalar product, respectively, in some Euclidean space IR n ; B(x; r) denotes the open
ball around x 2 IR n with radius r ? 0; d(x; D) denotes the distance of x 2 IR n to the
set D ' IR n ; for a real-valued function f on IR n , rf denotes its gradient in IR n and the
its Hessian; if f is locally Lipschitzian near x 2 IR n , @f(x) denotes
the Clarke subdifferential of f at x; f 0 (x; d) denotes the directional derivative of f at
x in direction d if it exists; for denotes the tangent cone to C at x, i.e.,
cl stands for closure;
for denotes the second order tangent set to C at x in
direction -, i.e., T 2 (C; x;
closed and
convex; see [10] for further properties).
In our paper, we use the following linear metric spaces of real-valued functions on
The space C 0 (IR s ) of continuous functions on IR s equipped with the distance
d1 (f; ~
f) =X
\Gamman kf \Gamma ~
, where
kyk-r
jf(y)j, for f; ~
the space C 0;1 (IR s ) of locally Lipschitzian functions on IR s with the metric
d L (f; ~
f) =X
\Gamman kf \Gamma ~
, where
y
the space C 1 (IR s ) of continuously differentiable functions on IR s with the metric d(f; ~
d1 (f; ~
and the space C 1;1 (IR s ) of functions in C 1 (IR s )
whose gradients are locally Lipschitzian on IR s equipped with the distance d(f; ~
d1 (f; ~
f) for all f 2 C 1;1 (IR s ).
The sensitivity analysis of the mappings ' and / is carried out by exploiting structural
properties of the optimization model (1.1). We obtain novel differentiability properties
of solution sets and extend our earlier results on directional differentiability of optimal
values in [12] considerably. As one might expect, the basic ingredients of our analysis
are a Lipschitz continuity result for solution sets with respect to the distance in
(Theorem 2.3) and a quadratic growth condition near solution sets (Theo-
rem 2.6). Both theorems extend earlier results in [25] to more general situations for
the first stage costs g and constraint set C. All results in the paper apply to the
linear-quadratic case, i.e., to linear or convex quadratic g and polyhedral C. Indeed,
all results are formulated as general as possible and most of them are accompanied by
illustrative examples. The second-order analysis of ' in Section 3 utilizes some ideas
from [28] and [29], but its proof is entirely different and its Gateaux differentiability
part is valid for nondifferentiable directions (Theorem 3.4). It is also elaborated that
the Hadamard directional differentiability properties require the C 0 -topology for the
first-order result and the C 1 -topology for the second-order one (Theorem 3.8), while
the C 1;1 -topology is needed for the semidifferentiability of the solution-set mapping /
(Theorem 4.7). All results on differentiability properties of / in Section 4 are new and
do not follow from recent sensitivity results (as e.g. [3], [6], [7], [16], [29]; see also the
survey [8] for further references).
The results of Sections 3 and 4 have direct implications to asymptotic properties of
values and solution sets of two-stage stochastic programs when applying nonparametric
estimation procedures to approximate Q - . For a discussion of some of the related
aspects we refer to [11], where the delta-method is utilized and a central limit theorem
for all selections belonging to a Castaing representation of the approximate solution
sets is derived. Further applications to asymptotics are beyond the scope of this paper
and will be done elsewhere.
Basic directional properties
The first step in our analysis of directional properties consists in establishing results
on the lower Lipschitz continuity of / and on the directional uniform quadratic growth
of the objective near its solution set. Both results become important for our method
of deriving directional differentiability properties for the optimal value function ' and
the solution set mapping / at some given expected recourse function Q - . Their proofs
are based on a decomposition of the program
with Q belonging to K C , into two auxiliary problems. The first one is a convex program
with decisions taken from A(C) and the second represents a parametric convex program
which does not depend on Q.
Proposition 2.1 Let Q 2 K C and /(Q) be nonempty. Then we have
Moreover, - is convex on A(C) and dom oe is nonempty.
Proof. Let -
For the converse inequality, let " ? 0 and -
be such that
Then there exists a -
"-
is arbitrary, the first statement has been shown. In particular, x 2 oe(Ax)
and Ax 2 Y (Q) for any x 2 /(Q) . Hence, it holds that /(Q) ' oe(Y (Q)). Conversely,
implying
Since the convexity of - is immediate, the proof is complete. 2
In the following, it will turn out that Lipschitzian properties of the solution set mapping
y 7! oe(y) and a quadratic growth property of g near oe(y) are essential. For the linear-quadratic
case we are in a comfortable situation in this respect. Namely, we have the
following
Proposition 2.2 Let g be linear or convex quadratic, C be convex polyhedral and assume
dom oe to be nonempty. Then oe is a polyhedral multifunction which is Hausdorff
Lipschitzian on its domain dom there exists a constant L ? 0 such that
yk; for all
where dH denotes the (extended) Hausdorff distance on subsets of IR m .
Moreover, for each r ? 0 there exists a constant j(r) ? 0 such that
(Here - and oe are defined as in Proposition 2.1).
Proof. The Lipschitz property of oe is shown in [18], Theorem 4.2. To prove the second
statement, let g be of the form positive
semidefinite and c 2 IR m . For each y 2 A(C) we fix some z(y) 2 oe(y). An elementary
characterization of solution sets to convex quadratic programs with linear constraints
yields that
Due to the Lipschitz behaviour of convex polyhedra (cf. [34]), there exists a constant
for all y 2 A(C) and x 2 C with y. Using the decomposition
2 denotes the square root of H, and the representation
one arrives at the estimate
for all y 2 A(C) and x 2 C with y.
us fix some element -
r) and a corresponding
oe(A-x). For each y 2 A(C) we now select z(y) 2 oe(y) such that
Hausdorff Lipschitzian on A(C), this
implies A(C). Hence, there exists a constant
r). Thus our estimate continues
to d(x;
and some constant -
Furthermore, the equation
implies kH 1
y.
Therefore, we finally obtain
for all x
Due to the above proposition, the main results in this section apply to the linear-quadratic
case. Although this case represents the main application of our results, the
assumptions of the following theorems are formulated in terms of general conditions on
the mapping oe in order to gain generality and clarity. The first theorem states (lower)
Lipschitz continuity of / at Q - and supplements Theorem 2.4 in [25].
Theorem 2.3 Let Q nonempty, bounded and Q - be strongly convex
on some open, convex neighbourhood of A/(Q - ). Let -
assume that there
exist a constant L ? 0 and a neighbourhood U of -
y with
Then there exist constants -
Proof. We may assume that U is open, convex and that Q - is strongly convex on U .
Let V be an open, convex, bounded subset of IR m such that /(Q -
It follows from Proposition 2.3 in [25] (where a slightly different terminology is used)
that there exists a constant
cl A(V
chosen such that cl A(V
r). Hence, we have ; 6=
Proposition 2.1 yields the relation
strongly convex
on U , there exists a constant - ? 0 such that
belongs to A(V ) ae U ,
we obtain
and, hence,
yk --
--
The proof can now be completed as follows. Let Q 2 K C be such that
Then
Remark 2.4 The proof shows that a Lipschitz modulus of / can be chosen as the
quotient of a Lipschitz constant to oe and a strong convexity constant to Q - .
From the proof it is immediate that replacing the local Lipschitz condition on oe by
stronger conditions like
sup
leads to corresponding stronger Lipschitz continuity properties of solution sets. Because
of Proposition 2.2, all of this applies to the linear-quadratic case. However, it is worth
mentioning that the theorem also applies to more general problems such that the
corresponding solution sets oe(y) enjoy Lipschitzian properties. Conditions ensuring
Lipschitz behaviour of oe can be derived from stability results for the corresponding
parametric generalized equation
which describes the first order necessary optimality condition. Here L(x; -; y) := g(x)+
is the Lagrangian function, rL(x; -;
, where g is
assumed to be continuously differentiable, and N C \ThetaIR s is the normal cone map of convex
analysis. Such stability results are presently available for broad classes of parametric
generalized equations (e.g. [16], [20], [22]). A typical recent result in this direction,
which applies to our situation for twice continuously differentiable g, is Theorem 5.1
in [20]. It says that the solution set mapping of the parametric generalized equation
(2.2) is pseudo-Lipschitzian around (-x; -
y) if the adjoint generalized equation
has only the trivial solution w
Here D N C \ThetaIR s (-x; -; \GammarL(-x; -
y)) is the Mordukhovich coderivative ([20]) of the normal
cone multifunction at the point (-x; -; \GammarL(-x; -
belonging to the graph of
. Translating this into our framework, we obtain that the mapping oe is pseudo-
Lipschitzian around (-x; -
y) if the following two conditions are satisfied:
(a) There exists an element -
x belonging to the relative interior of C such that
y
(Slater
(b) the equations Aw
have only the trivial solution w is a solution
of (2.2) for
y.)
The next example shows that the theorem also applies to instances of two-stage stochastic
programs with nonpolyhedral convex constraint sets C.
Example 2.5 In (1.1) -
be the uniform distribution on [\Gamma 1; 1] and
x g. Then we have ~
R
jyj otherwise ,
strongly convex on (\Gamma 1; 1). For y we have
and, hence d((0; 0); . Thus Theorem 2.3 applies for -
Example 2.8 shows that Theorem 2.3 gets lost if Q - fails to be strongly convex on some
neighbourhood of A/(Q - ). Our next result establishes a sufficient condition for the
uniform quadratic growth near solution sets.
Theorem 2.6 Let Q nonempty, bounded and Q - be strongly convex
on some open convex neighbourhood U of A/(Q - ). Assume that there exists a constant
yk; for all
and, for each r ? 0 there exists a constant j(r) ? 0 such that
Then, for some open, bounded neighbourhood V of /(Q - ) and each
there exist constants c ? 0 and ffi ? 0 such that the following uniform growth condition
holds:
for all x
Proof. Let be an open, bounded subset of IR m such that
As in Theorem 2.3 we choose ffi ? 0 such that ; 6=
and, in addition, that strongly convex on U for all t 2 [0; ffi)
(with a uniform constant - ? 0). For each t 2 [0; ffi) Proposition 2.1 then yields that
is the unique minimizer of the strongly convex function
- +tv on A(C) and, moreover, we have -ky \Gammay t k 2 -(y)+(Q - +tv)(y)\Gamma'(Q - +tv),
for all y . Now, we choose r ? 0 such that V ' B(0; r) and continue for
each
Putting c
completes the proof. 2
The following examples show that the quadratic growth condition gets lost even for
the original problem, i.e. either the Lipschitz condition for oe or the strong
convexity property for Q - are violated.
Example 2.7 Consider again the set-up of Example 2.5. It holds that dH (oe(y);
oe is not Hausdorff Lipschitzian on A(C).
Supposed there exists a neighbourhood V of /(Q -
such that the growth condition
is satisfied. Since the sequence (( 1
belongs to C " V for sufficiently large n 2 IN ,
this would imply %( 1
for large n, which is a contradiction.
Example 2.8 In (1.1) -
- be the probability distribution on IR having the density
R
jyj otherwise ,
there is no neighbourhood of /(Q - ) where Q - is strongly convex.
It is clear that the quadratic growth condition fails to hold, since the inequality %x 2 -
cannot be true for some % ? 0 and all x belonging to some
neighbourhood of
With the linear function we obtain for all t 2 [0; 1] that /(Q -
f
(cf. Example 3.7). Hence, the lower Lipschitz property of / has got lost, too.
Since the strong convexity and later also the strict convexity of the expected recourse
function Q - (on certain convex subsets of IR s ) form essential conditions in most of our
results, we record a theorem (Theorem 2.2 in [27]) that provides a handy criterion to
check these properties for problem (1.1) - (1.3).
Proposition 2.9 Let V ae IR s be open convex and assume (A1), (A3). Consider the
following conditions:
absolutely continuous on IR s ;
there exist a density f - for - and a constant
Then (A2) and (A4) imply that Q - is strictly convex on V if V is a subset of the
support of -, and (A2) , (A4) imply that Q - is strongly convex on V .
In addition, it is shown in [27] that under (A1) - (A4) the condition (A2) is also
necessary for the strict convexity of Q - . For extended simple recourse models (i.e.
is equivalent to q
(componentwise), where This may be used to check
strict or strong convexity properties in the Examples 2.5 and 2.8.
Directional derivatives of optimal values
In this section, we study first- and second-order directional differentiability properties of
the optimal value function ' on its domain K C . We begin with the first-order analysis
and show that ' as a mapping from K C to the extended reals is Hadamard directionally
differentiable at some given expected recourse function Q - 2 K C . Here K C is regarded
as a subset of C 0 (IR s ). Recall that ' is Hadamard directionally differentiable at Q - on
K C iff for all sequences (v n ) converging to some v in C 0 (IR s ) and all sequences t n ! 0+
such that the elements belong to K C the limit
exists. Since the condition means that v
the limit v belongs to the tangent cone T (K C ; Q - ) to K C at Q - in C 0 (IR s ). In
[32], [33] this property is also called Hadamard directional differentiability tangentially
to K C .
Proposition 3.1 Let Q - 2 K C and assume that /(Q - ) is nonempty, bounded. Then
' is Hadamard directionally differentiable at Q - on K C and it holds for all v 2
If, in addition, Q - is strictly convex on some open convex neighbourhood of A/(Q - ),
we have
Proof. Arguing similarly as in the proof of Propostion 2.1 in [24] there exists a
neighbourhood N of Q - in C 0 (IR s ) such that /(Q) is nonempty for all Q
sequences such that t n ! 0+,
belongs to K C for all n 2 IN . Then sufficiently large n 2 IN .
upper semicontinuous at Q -
([24]), the sequence has an accumulation point x 2 /(Q - ) and we obtain
lim sup
where the last inequality follows from the uniform convergence of (v n ) to v on bounded
subsets of IR s . In order to show the reverse inequality for lim inf, let x 2 /(Q - ). Then
lim inf
This completes the proof of the first part. The second part is an immediate conclusion,
since A/(Q - ) is a singleton whenever Q - is strictly convex on some of its open, convex
neighbourhoods. 2
The preceding result can also be proved by using the methodology of Theorem 6.4.1 in
[26]. There the compactness of the constraint set is assumed and Gateaux directional
differentiability of ' at Q - together with its Lipschitz continuity is shown. Here we
prefer a direct two-sided argument, which will also be used in the subsequent second-order
analysis of '. Namely, we will first derive an upper bound for the second-order
Hadamard directional derivative of ' at some Q - 2 K C , where K C is equipped with
the C 0;1 -topology. Secondly, we identify conditions implying that the upper bound
coincides with the Gateaux directional derivative of ' at Q - for all directions taken
from T r (K C
Lemma 3.2 Let y sequence in K C such that
sequence converging to - in IR s .
Then we have lim sup
(v
Proof. Each function v n is locally Lipschitzian on IR s and, hence, Lebourg's mean
value theorem for Clarke's subdifferential ([9]) implies the existence of elements ~ y n
belonging to the segments
(v
The convergence v n ! v in C 0;1 (IR s ) implies that
n!1holds for any r ? 0. This yields
0:
Here dH denotes the Hausdorff distance and the inequality is a consequence of general
properties of the subdifferential (cf. Lemma 2.1 in [25]). Hence, there exist elements
~
belonging to @v(~y n ) such
(v
and, for some ~
lim sup
(v
Here, the identity follows from the upper semicontinuity of @v(\Delta). This completes the
proof. 2
Proposition 3.3 Let Q - 2 K C and assume that /(Q - ) is nonempty, bounded. Let g
be twice continuously differentiable, Q - be strictly convex on some open convex neighbourhood
of A/(Q - ) and twice continuously differentiable at -
y, where
sequence in K C such that v n := 1
in C 0;1 (IR s ). Then
lim sup
x) is the tangent
cone to C at -
x and T 2 (C; -
x; -) the second order tangent set to C at -
x in direction -.
Proof. Let -). Then there exists a sequence (z n ) such that
Using Proposition 3.1, this allows for
the following estimate
After dividing by t 2
n and using Lemma 3.2 the limes superior as of the right-hand
side can be bounded above by
Taking the infimum on the right-hand side yields the assertion. 2
We notice that the upper second-order Hadamard directional derivative
lim sup
nonpositive, since ' is concave on K C
and, hence, the inequality '(Q - +t n v n
We also note that the upper bound is nonpositve, since (0;
belongs to S(-x) \Theta T 2 (C; - x;
Next we consider particular perturbations Q n of Q - , namely, Q
for some Q 2 K C sufficiently large n 2 IN . Then v
In the following result we give conditions implying that the second-order
directional derivative exists and coincides with the upper bound of the
previous proposition. The result extends those in [12] although its proof parallels in
parts that of Theorem 3.6 in [12].
Theorem 3.4 Let Q - 2 K C and assume that /(Q - ) is nonempty, bounded. Let g be
twice continuously differentiable, Q - be strictly convex on some open convex neighbourhood
of A/(Q - ) and twice continuously differentiable at - y, where
assume that
(ii) the second-order set S 2 (-x; -) := fz 2 T 2 (C; -
is nonempty for each - 2 S(-x).
Then the second-order Gateaux directional derivative of ' at Q - in direction v exists
and it holds that
Moreover, the infimum is attained at some -
having the property that
-).
(Here S(-x) and T 2 (C; -
are defined as in the previous result, v 0 (-y; j) is the directional
derivative of v at -
y in direction j and O(t) denotes a real quantity such that 1
jO(t)j is
bounded as t ! 0+.)
Proof. (i) implies that there exist constants L ? 0,
expanding g and Q - and
using Proposition 3.1 we obtain
xi +2
xi
Moreover, we have that
denotes a real quantity having
the property 1
the optimality of - x implies
for any t 2 (0; ffi), we
Now take a sequence (t n ) tending to 0+ in such a way that
lim inf
and that - n := 1
-. The latter is possible since k 1
sufficiently large. Then -
x) and Proposition 3.1 yields
This implies -
From (3.1) and (3.2) we obtain
lim inf
Here we have used the fact that v is Hadamard directionally differentiable and Clarke
regular ([9]), i.e. v 0 (-y;
ji. From Proposition 3.3 we obtain
lim sup
The latter equality is due to (ii) and to the fact that the necessary optimality condition
for - x yields
Hence, the limit lim
exists and is equal to the
infimum subject to - 2 S(-x). Moreover, this infimum is attained at -
S(-x). For the
remainder of the proof we put a(-) := v 0 (-y; A-) and
Since S(-x) is a (convex) cone, we have
-)); for all - ? 0:
In case of B( -
-) ? 0, the quadratic function f vanishes at with the property
and the final assertion is shown. If B( -
the fact that
-) holds for any - ? 0, implies a( -
-) and the proof is complete. 2
The proof shows that the previous theorem remains true when replacing condition
(ii) by the condition that both infima in (3.3) coincide. Next we state a more handy
criterion implying that ' 00 (Q - ; v) exists for any direction v 2 T r (K C
Corollary 3.5 Let Q - 2 K C and assume that /(Q - ) is nonempty, bounded. Let g be
twice continuously differentiable, Q - be strongly convex on some open convex neighbourhood
of A/(Q - ) and twice continuously differentiable at -
y where
assume that
(i) 0 there exist a constant L ? 0 and a neighbourhood U of - y such that
(ii) the second-order set S 2 (-x;
is nonempty for each - 2 S(-x).
Then the second-order Gateaux directional derivative of ' at Q - exists for any direction
the formula for ' 00 (Q - ; v) in Theorem 3.4 holds true.
Moreover, condition (ii) is satisfied if C is polyhedral and (i) 0 is satisfied for any - x 2
in addition to the polyhedrality of C, g is linear or (convex) quadratic.
Proof. Let Theorem 2.3 then says that there exist constants -
Hence, the strong convexity of Q - and condition (i) 0 imply that condition (i) of the
previous theorem is satisfied and that the first part of the assertion is shown. If C
is polyhedral, we have T 2 (C; -
x). Hence, (ii) is satisfied. If C is polyhedral and g is linear or (convex)
quadratic, Proposition 2.2 implies (i) 0 to hold for any -
Let us consider two illustrative examples to provide some insight into the benefit and
limits of the previous results.
Example 3.6 We revisit Example 2.5 and know that condition (i) 0 is satisfied for
Furthermore, it holds that T (C; -
Hence, (ii) and the general assumptions of Corollary 3.5 are satisfied and ' 00 (Q - ; v)
exists for any v 2 T r (K C ; Q - ). It holds that ' 00 (Q -
IRg. Let us finally replace the function g(x) j 0
by
IR, and condition (ii) is violated. But, since we have
both infima in (3.3) coincide, the result holds true and we have
Example 3.7 Here we revisit Example 2.8, and have
For the function
2 . Hence, '
has no second-order directional derivative at Q - in direction v. Note that there is no
neighbourhood of -
strongly convex.
Finally, we aim at showing that ' is even second-order Hadamard directionally differentiable
at equipping K C with a suitable topology. To this end we need a
certain counterpart of Lemma 3.2 for the corresponding limes inferior. Since this is not
available for nonsmooth functions, it is a natural idea to consider the space C 1 (IR s ),
to restrict ' to the subset K C " C 1 and to equip K C " C 1 with the C 1 -topology. Then
we are able to show that the assumptions of Corollary 3.5 even imply the second-order
Hadamard directional differentiability of ' at Q - .
Theorem 3.8 Let Q assume that /(Q - ) is nonempty, bounded. Let g
be twice continuously differentiable, Q - be strongly convex on some open convex neighbourhood
of A/(Q - ) and twice continuously differentiable at -
y where
assume the conditions (i) 0 and (ii) of Corollary 3.5 to hold.
Then the second-order Hadamard directional derivative of ' at Q - exists in any direction
v belonging to the tangent cone T (K any such
v, and all sequences t n ! 0+ and (Q n ) in K C such that v n := 1
exists, and it holds
0g.
Proof. Let sequence in K C such that
together with Theorem 2.3 then
imply that there exist constants L ?
such that
Since the sequence (v n ) converges in C 1 (IR s ), the norms kv n k L;r are uniformly bounded
and we have
Expanding g and Q - as in the proof of Theorem 3.4
we obtain analogously to (3.1), for all n - n 0 :t 2
Putting - n := 1
and using the mean value theorem for v n we may continue
with some -
Arguing as in the proof of Theorem 3.4 and using v n ! v in C 1 (IR s ) we arrive at the
estimate
lim inf
for some element -
Furthermore, we conclude from (ii) and Proposition 3.3 that
lim sup
Hence, the desired limit exists and the proof is complete. 2
Let us finally note that all minimization problems appearing as bounds or formulas for
second-order directional derivatives represent convex programs. Those in the results
3.4, 3.5 and 3.8 have convex cone constraints, which are polyhedral if C is polyhedral.
Moreover, the solution sets of the convex minimization problems in 3.4, 3.5 and 3.8 are
nonempty. Indeed, we show next that these solution sets represent certain derivatives
of the set-valued mapping / at the pair (Q - x).
4 Differentiability of solution sets
It is well-known that second-order differentiability properties of optimal values in perturbed
optimization are intrinsic for establishing the differentiability of solutions (see
e.g. [8]). We also pursue this approach and derive conditions implying directional
differentiability properties of the solution set mapping by exploiting the results of the
previous section. Our first results in this direction concern Gateaux directional differ-
entiability, and complement Theorem 3.4 and its corollary.
Theorem 4.1 Assume that the general conditions on g, Q - and C of Theorem 3.4 are
suppose the conditions (i) and (ii) of
Theorem 3.4 to be satisfied. In addition, assume that
(iii) there exist a neighbourhood V of such that the
uniform growth condition
for all x
Then the Gateaux directional derivative of / at the pair (Q - x) into direction v exists
and it holds that
Proof. Let M(-x; v) denote the solution set in the assertion. First we show that
lim sup
x). Then there exists a sequence (t converging to
(0+; -) such that - n 2 1
Hence, analogously to the proof of Theorem 3.4 we deduce that - belongs to S(-x).
In view of Theorem 3.4 it remains to show that 1hr 2 g(-x)-i
expanding g and Q - as in the proof of Theorem 3.4, we
obtain analogously to (3.1):
After dividing by t 2
n and taking the lim
on both sides of the inequality, we obtain the
desired estimate. In a second step we show that
or, equivalently, that it holds for any - 2 M(-x; v),
lim t!0t
sequence with t n ! 0+. We have to show that
lim
there exists an element z
and a sequence (z n ) converging to z with -
suffices to show that
lim
Condition (iii) implies the following estimate for all sufficiently large n
By expanding g and Q - as in the proof of Theorem 3.4 and using the fact that - belongs
to S(-x), we may continue
After dividing by t 2
n and taking the lim sup
on both sides of the latter inequality, we
obtain
lim sup
c
where we made use of z Theorem 3.4. This completes the
proof. 2
Complementing Corollary 3.5 we provide a result on the directional differentiability of
/ at Q - into any direction v
Theorem 4.2 Assume that the general conditions on g, Q - and C of Corollary 3.5
are satisfied. Let -
assume that
(i) 00 there exists a constant L ? 0 such that
and, for each r ? 0, there exists a constant j(r) ? 0 such that
(ii) the second-order set S 2 (-x;
is nonempty for each - 2 S(-x).
Then the Gateaux directional derivative / 0 (Q -
x; v) of / at the pair (Q -
exists for
any direction v the formula in Theorem 4.1.
Moreover, condition (ii) is satisfied if C is polyhedral, and (i) 00 is satisfied if C is
polyhedral and g is linear or (convex) quadratic.
Proof. Let strongly convex on some open convex neighbourhood
of A/(Q - ), we infer from condition (i) 00 and Theorem 2.6 that condition (iii)
of Theorem 4.1 is satisfied. Moreover, condition (i) 00 implies (i) 0 and, thus, Corollary 3.5
says that the second-order directional derivative ' 00 (Q - ; v) exists. Hence, the first part
of the assertion follows from the proof of the previous theorem. If C is polyhedral,
we have 0 2 S 2 (-x; -) for any - 2 S(-x), and if, in addition, g is convex quadratic,
Proposition 2.2 implies condition (i) 00 to hold. 2
We note that Example 3.7 shows that, in general, the directional differentiability property
of / gets lost at those pairs (Q -
is not strongly convex
on some neighbourhood of A/(Q - ).
Finally, we turn to directional differentiability properties of / where the derivatives
exist uniformly with respect to directions taken from compact sets of certain functional
spaces. For our first result we consider the space C 1 (IR s ) and equip the set K
with the C 1 -topology.
Proposition 4.3 Let Q assume that the general conditions on g,
and C in Proposition 3.3 are satisfied. In addition, we suppose condition (ii) of
Theorem 3.4 to be satisfied. Let -
sequence in K C
such that v n := 1
Then the upper set limit of the sequence ( 1
of closed convex subsets
in IR m , i.e., lim sup
x)), is contained in the closed convex set
argmin
Proof. Let D n := 1
x) for all n 2 IN and let -
- belong to the upper set
limit lim sup
. Then there exist a subsequence (again denoted by (D n )) and elements
-. Since -
we have that -
As in the proof of Theorem 3.4 we deduce that hrg(-x); -
thus, -
expanding g and Q - as in the proof of Theorem 3.4, we also obtain
analogously to (3.1):
After dividing by t 2
n and taking the lim sup
on both sides of the inequality, we obtain
as in the proof of Theorem 3.8
lim sup
-i:
Hence, we may conclude from (ii) and Proposition 3.3 that -
- belongs to the set
and we are done.Remark 4.4 The upper limit of the sequence ( 1
in Proposition 4.3
is nonempty if the mapping d(-x; /(\Delta)) from K C into the extended reals has the Lipschitzian
property of Theorem 2.3 at Q - . Indeed, we may select x
for large n 2 IN , such that for some constants -
. Hence, the sequence ( 1
is bounded and
has a convergent subsequence whose limit belongs to lim sup
x). If
the Lipschitz property of d(-x; /(\Delta)) is violated, the upper set limit may be empty. This
is illustrated by Example 3.7, in which we have -
g.
In order to establish the semidifferentiability of / at a pair (Q -
x) belonging to the
graph of /, it remains to show, according to Proposition 4.3, that the solution set
argmin
is contained in the lower set limit lim inf
converges to v. To this end, a uniform quadratic
growth condition of the objective functions g(\Delta)
is significant. In view of Theorem 2.6, the uniform strong convexity of Q - and its
approximations Q n , for large n 2 IN , is decisive for the growth condition. The next
example and the following result show that the approximations Q n do not maintain
the strong convexity property of Q - in general if the sequence (Q n ) converges to Q -
in C 1 (IR s ), but that the situation is much more advantageous when considering the
C 1;1 -topology.
Example 4.5 Let Q be the following differentiable
convex functions
\Gammay \Gamman
0; y \Gamman
Note that Q
and Q n is not strongly convex for each n 2 IN ,
but (Q n ) converges to Q - in C 1 (IR s ).
strongly convex on some bounded convex set
(with some constant - ? 0).
Then there exists a neighbourhood N of Q - in C 1;1 (IR s ) such that each function Q
belonging to N is strongly convex on U with constant -Proof. The strong convexity of Q - on U (with constant - ? 0) is equivalent to the
condition
such that cl U ' B(0; r) and let N be a neighbourhood of Q - in C 1;1 (IR s ) having the
property
y. Then we
obtain for any Q 2 N ,
and, hence
This means that Q is strongly convex on U with constant -. 2
Now we are able to show that the solution set mapping / is semidifferentiable on
at some pairs (Q -
direction v from the tangent cone
in C 1;1 (IR s ). The assumptions are essentially
the same as in Theorem 4.2.
Theorem 4.7 Let Q assume that /(Q - ) is nonempty, bounded.
Let g be twice continuously differentiable, Q - be strongly convex on some open convex
neighbourhood U of A/(Q - ) and twice continuously differentiable at -
y, where
Assume that, for each r ? 0, there exist constants L ? 0 and j(r) ? 0 such
that the following condition (i) 00 is satisfied for
Then the solution set mapping / from K C " C 1;1 into IR m is semidifferentiable at any
such that S 2 (-x; -) is nonempty for each - 2 S(-x), and into
any direction v 2 T (K any such -
x and v, t n ! 0+, and (Q n ) in
exists. The semiderivative D/(Q -
x; v) is equal to the set
argmin
Moreover, / is semidifferentiable at any pair (Q -
direction
polyhedral. Condition (i) 00 is satisfied if C is polyhedral
and g is linear or (convex) quadratic.
Proof. Let be such that S 2 (-x; -) is nonempty for each - 2 S(-x),
and (Q n ) is a sequence in K C " C 1;1 . We may assume that U is bounded. Since
converges to Q - in C 1;1 (IR s ), we obtain from Lemma 4.6 that there exists an
, such that Q n is strongly convex on U for each n - n 0 with a uniform constant
sufficiently large such that /(Q n ) is nonempty, for each
Arguing as in the proof of Theorem 2.6, we obtain a constant c ? 0 and a
neighbourhood V of /(Q n ) such that the growth condition
holds for all x
be a minimizer of the function 1hr 2 g(-x)-i
subject to - 2 S(-x). Because of Proposition 4.3 it remains to show that -
belongs to the lower limit lim inf
there exists an element z
-) and a sequence (z n ) converging to z with
As in the proof of Theorem 4.1 it suffices to show
that
lim
By using the above growth condition and by expanding the function g and Q - , we
obtain as in the proof of Theorem 4.1
and
lim sup
c
This implies -
x) and the semidifferentiability of / at (Q -
in
direction v is shown. The remaining part of the assertion follows as in the proof of
Theorem 4.2. 2
For the linear-quadratic case, the essential assumptions in Theorem 4.7 are the strong
convexity of Q - , and the smoothness properties of Q - and its perturbations Q, respec-
tively. While criteria for strong convexity were already discussed in Section 2, we now
close this section by adding some comments on C 1;1 - and C 2 -properties of expected
recourse functions.
Remark 4.8 Assume (A1) - (A3) and - to have a density with respect to the Lebesgue
measure on IR s . Then the function Q - in (1.2) is continuously differentiable on IR s and
its gradient is of the form rQ -
)), for all y
are certain basis submatrices of the recourse matrix W such that the
simplicial cones B i (IR s
are linearity regions of ~
Q and d i is the gradient
of ~
Denoting by F - the distribution
function of - and using the formula
-(y +B(IR s
for any nonsingular (s; s)-matrix B, C 1;1 - and C 2 -properties of Q - may thus be formulated
in terms of Lipschitz and differentiability properties of the distribution functions
to the linear transforms - of the measure -.
The distribution function F - of a probability measure - on IR s is locally Lipschitzian
if all one-dimensional marginal distribution functions of - are locally Lipschitzian (cf.
[24], [35]). F - is continuously differentiable if - has a continuous density function and
all one-dimensional marginal distribution functions of - are continuously differentiable
(cf. [19], [35]). If - has a continuous density function, then - ffi B has a continuous density
for any nonsingular (s; s)-matrix B, too. Hence, we may conclude, for instance,
that Q - belongs to C 1;1 (IR s has a (continuous) density and the
above-mentioned conditions on the one-dimensional marginal distribution functions
for F -ffiB belonging to C 0;1 (IR s are satisfied for any nonsingular
(s; s)-matrix B. This criterion is particularly useful for probability distributions
which have the property that all one-dimensional marginal distributions of - and
all linear transforms - ffi B, for all nonsingular matrices B, belong to the same class of
measures. For instance, all multivariate normal and all logarithmic concave probability
measures (e.g. [14]) form classes having this property.
Acknowledgement
: The authors wish to thank Alexander Shapiro (Georgia Institute
of Technology, Atlanta) and Ren'e Henrion (WIAS Berlin) for beneficial discussions
--R
Stability results for stochastic programs and sen- sors
First and second order sensitivity analysis of nonlinear programs under directional constraint qualification conditions
A comparative study of multifunction differentiability with applications in mathematical programming
Directional derivatives in nonsmooth optimization
Perturbed optimization in Banach spaces I: A general theory based on a weak directional constraint qualification
Quadratic growth and stability in convex programming problems with multiple solutions
Optimization problems with perturbations
Optimization and Nonsmooth Analysis
tangent sets and second-order optimality condi- tions
Differentiable selections of set-valued mappings with application in stochastic programming
Strong convexity and directional differentiability of marginal values in two-stage stochastic programming
Stability and sensitivity analysis for stochastic programming
Generalized delta theorems for multivalued mappings and measurable selections
Sensitivity analysis for nonsmooth generalized equations
Asymptotic theory for solutions in statistical estimation and stochastic programming
bounds for solutions of linear equations and inequalities
Approximationen der Entscheidungsprobleme mit linearer Ergebnis- funktion und positiv homogener
Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis
Differentiability of relations and differential stability of perturbed optimization problems
Strongly regular generalized equations
Stability of solutions for stochastic programs with complete recourse
Lipschitz stability for stochastic programs with complete recourse
Discrete Event Systems.
Strong convexity in stochastic programs with complete recourse
Second order directional derivatives in parametric optimization prob- lems
Sensitivity analysis of nonlinear programs and differentiability properties of metric projections
On concepts of directional differentiability
On differential stability in stochastic programming
Asymptotic analysis of stochastic programs
Weak Convergence and Empirical Pro- cesses
A Lipschitzian characterization of convex polyhe- dra
Distribution sensitivity analysis for stochastic programs with complete recourse
Stochastic programs with fixed recourse: the equivalent deterministic program
in: Handbooks in Operations Research and Management Science.
--TR
--CTR
Svetlozar T. Rachev , Werner Rmisch, Quantitative Stability in Stochastic Programming: The Method of Probability Metrics, Mathematics of Operations Research, v.27 n.4, p.792-818, November 2002 | semide-rivatives;sensitivity analysis;two-stage stochastic programs;directional derivatives;solution sets |
589099 | Multiple Cuts in the Analytic Center Cutting Plane Method. | We analyze the multiple cut generation scheme in the analytic center cutting plane method. We propose an optimal primal and dual updating direction when the cuts are central. The direction is optimal in the sense that it maximizes the product of the new dual slacks and of the new primal variables within the trust regions defined by Dikin's primal and dual ellipsoids. The new primal and dual directions use the variance-covariance matrix of the normals to the new cuts in the metric given by Dikin's ellipsoid.We prove that the recovery of a new analytic center from the optimal restoration direction can be done in O(plog (p+1)) damped Newton steps, where p is the number of new cuts added by the oracle, which may vary with the iteration. The results and the proofs are independent of the specific scaling matrix---primal, dual, or primal-dual---that is used in the computations.The computation of the optimal direction uses Newton's method applied to a self-concordant function of p variables.The convergence result of [Ye, Math. Programming, 78 (1997), pp. 85--104] holds here also: the algorithm stops after $O^*(\frac{\bar p^2n^2}{\varepsilon^2})$ cutting planes have been generated, where $\bar p$ is the maximum number of cuts generated at any given iteration. | Introduction
The analytic center cutting plane (ACCPM) algorithm [5, 19] is an efficient
algorithm in practice [2, 4]. The complexity of related algorithms was given in
[1, 13], and subsequently in [6]. Extensions to deep cuts were given in [7] and
to very deep cuts in [8]. The method studied in [8] corresponds to the practical
implementation of ACCPM [11] with a single cut.
In practice, it often occurs that the oracle in the cutting plane scheme generates
multiple cuts. The papers [12, 20, 17] show that it is possible to handle
several cuts at a time provided they are central [20] or moderately shallow [12].
Although these analyses show how one can recover feasibility after introducing
multiple cuts, there is no clear argument as to the choice of a feasibility restoration
direction. Intuitive, but well justified, arguments about how to introduce
multiple cuts were given in [2] in the context of a primal projective algorithm
and two cuts (one shallow, one deep) and in [10] with an infeasible primal-dual
approach to the introduction of several cuts in general position.
The case of two central cuts was analyzed in [9]. It was shown that there exist
explicit primal and dual directions which allow a best move towards primal and
dual feasibility. An argument using the primal, dual and primal-dual potentials
at this new optimal primal and dual point proves that O(1) damped Newton
steps are enough to recover centrality. The updating direction depends on the
cosine in the metric of Dikin's ellipsoid of the normals to the cuts.
In this paper, we analyze the multiple cut generation scheme in the analytic
center cutting plane method. We propose an optimal updating direction when
the cuts are central. The direction is optimal in the sense that it maximizes the
product of the new slacks within the trust region defined by Dikin's ellipsoid.
The new primal and dual directions use the variance-covariance matrix of the
normals to the new cuts in the metric given by Dikin's ellipsoid.
We prove that the recovery of a new analytic center from the optimal restoration
point can be done in O(p log(p+1)) damped Newton steps, where p is the number
of new cuts added by the oracle. The results and the proofs are independent of
the specific scaling matrix -primal, dual or primal-dual- that is used in the
computations.
The computation of the optimal direction uses Newton's method applied to a
self-concordant function of p variables. This could be very advantageous in
practice if the number of cuts p is a small multiple of n, the dimension of the
space.
The convergence result of [20] holds here also: the algorithm stops after O
cutting planes have been generated.
Analytic center cutting plane method
2.1 Cutting planes
The problem of interest is that of finding a point in a convex set C ae IR n . We
make the following assumptions.
Assumption 2.1 The set C is convex, contains a ball of radius " ? 0 and is
contained in the cube 0 - y - e.
Assumption 2.2 The set C is described by an oracle. That is, the oracle either
confirms that y 2 C, or answers at least one cutting plane that contains C and
does not contain y in its interior.
A cut at -
takes the form
a T y - a T -
fl:
the cut is deep; if -
the cut is shallow; if - the cut passes
through -
y, and is thus a central cut.
The algorithm may generate multiple cuts at a time. They take the form
a T
We define the matrix B by
Assumption 2.3 All the cutting planes generated have been scaled so that
We also assume that the cuts are central, that is -
A cutting plane algorithm constructs a sequence of query points fy k g. The
answers of the oracle to the queries, together with the cube 0 - y - e, define a
polyhedral outer approximation
of C. Since A contains the identity matrix associated with the cube, A has
full row rank. Therefore there is a one-to-one correspondence between points
and the slack y, leading to the equivalent definition of FD
The number of columns in A is denoted as m and is equal to 2n plus the number
of cutting planes generated until the k th iteration.
The analytic center cutting plane method chooses as a query point an approximate
analytic center of FD .
2.2 Analytic center
The analytic center of FD is the unique point maximizing the dual potential
log s i
with formally introduce the optimization problem
and the associated first order optimality conditions
where x is a vector in R m . The notation xs indicates the Hadamard or componentwise
product of the two vectors x and s.
The analytic center can alternatively be defined as the optimal solution of
where
log x i
denotes the primal potential. One easily checks that problem (2) shares with
(1) the same first order optimality conditions.
At this stage it is convenient to introduce the primal-dual potential
and an associated duality relationship.
Lemma 2.4 Let x 2 intF P and s 2 intF D . Then 'PD (x; s) - \Gammam; with
equality if and only if
Proof
Consider the simple inequality
log
with equality if and only if Apply (3) with
By summing the resulting inequalities, one gets
log
log
with equality if and only if xs = e. Therefore,
with equality if and only if
Finally, we define approximate centers by relaxing the condition in the
first order optimality conditions. Formally, any solution (x; s) of
defines a pair of '-approximate centers, or '-centers in short.
2.3 Analytic center cutting plane method
ACCPM can be shortly stated as follows.
Initialization Let F 0
fy eg be the unit cube and y
e be its
center. The centering parameter is
Basic Step y k is a '-center of F k
the total number
hyperplanes describing F k
D .
1. The oracle returns the cuts amk+j , k , at y k .
2. Update
g.
3. Compute a '-center of F k+1
D .
The computation of a new '-center after adding new cuts will be discussed in a
further section.
3 Some useful properties
The literature on interior point methods essentially proposes three approaches
for computing analytic centers. All of them are based on Newton's method. The
primal (resp. dual) Newton direction is initiated at an interior primal (resp.
dual) feasible point; it involves the scaling matrix
(We recall the standard notation X which denotes the diagonal matrix diag(x).)
The primal-dual direction is initiated at an interior primal-dual feasible pair,
involves the scaling matrix
Let us shortly recall the formulas. The primal direction is given by
xp(x) with
The dual direction is given by \Deltas = sq(s) with
Finally the primal dual direction is
\Deltas and
3.1 Properties of the Newton step
There are two basic properties, a local one in the vicinity of the analytic center,
and a global one. Since the results are well-known we state them without proofs.
Missing proofs can be found in the books [18] or [21].
Let us start with the local properties. Proximity to analytic center is measured
with the quantity sxk. In this definition, either
Note that if
and case). The Newton step defines a pair
Theorem 3.1 Assume
3 . Let be the point resulting
from a Newton step (primal, dual, or primal-dual). Then,
intF D \Theta intF P .
In the primal and dual cases, the theorem holds with any 0 ! ' ! 1.
One can derive from the above theorem a useful corollary that yields lower
bounds on the potentials near the analytic center. Let be the pair of
exact analytic centers. Denote ' c
Corollary 3.2 Assume (5)-(7) at (x; s). Then
1. ' c
2. ' c
3. \Gammam - 'PD (x; s) - \Gammam
Let us now consider the global properties of a damped Newton step. The properties
are consequences of the well-known inequality on the logarithm function
Lemma 3.3 Let h be any point in R m such that khk ! 1. Then,
The main result bounds the variation of the potentials after a damped Newton
step.
Theorem 3.4 Assume
s+ff\Deltas. (\Deltax and \Deltas may be the primal, dual or primal-dual directions.) Then,
there exists a step size ff ? 0 and absolute constants oe P , oe D and oe PD such that
1. 'P
2. 'D (x(ff)) - 'D
3. 'PD (x(ff);
In the primal and dual cases the constants are oe '). The
above result allows to design a potential increase algorithm based on damped
Newton steps. The convergence estimate is given by the following theorem.
Theorem 3.5 Let x potential increase algorithm
(primal, dual, primal-dual) produces an interior feasible pair such that
a number of iterations not greater than
oe
with or oe PD , depending on which approach (primal, dual or primal-
dual) is taken.
3.2 Dikin's ellipsoids
. From the observation that x \Deltax such that
1, we can define an ellipsoidal neighborhood of x that is entirely
contained in FP . Formally,
We shall be particularly concerned with ellipsoids around a '-center.
We can extend the definition of Dikin ellipsoid to include a different scaling.
Lemma 3.6 Let (x; s) be a pair of '-centers.
1. If
2. If
Proof
For the dual scaling the proof follows from
and
For the primal-dual scaling the proof follows from
and
We can similarly define Dikin's ellipsoids in the dual. Let s 2 intF D . The dual
ellipsoid is
The extension of Dikin's ellipsoid to a different scaling at a '-center is given by
Lemma 3.7 1. If
2. If
The proof is the same as for Lemma 3.6.
It is well-known that an homothety of Dikin's ellipsoid contains the feasible set.
We shall use this property in the restricted context of the set FD .
Lemma 3.8 Let (x; s) be a '-centered feasible pair. Then
ae
oe
Proof
s) be a '-centered feasible pair and ~
point of FD . Since are orthogonal,
Since ~
one has x T s -
thus obtain the weak bound
Finally, from kD(~s \Gamma s)k -
Hence,
ae
oe
4 Multiple central cuts
We assume now that a '-center (x; s; y) has been computed, i.e.,
The cuts are
a T
We define
The new cuts lead to two new sets:
e
or
e
and
e
We shall use the notation
so
After adding the cuts, one has
~
FD
and
~
Let us introduce the notation
The primal and dual potentials at the new points (-x; fi) and (-s; fl) are:
~
log -
log
log
and
~
log -
log
log
The points lie on the boundary of the primal and dual sets respectively. To
recover the new analytic center, one has to increase the components fi and fl.
Since the terms
log fi i and
log fl i are dominant near
maximizing those terms while limiting the variation on 'P and 'D is likely to
produce a good step towards the solution.
This approach requires the knowledge of the level sets of the potential, something
that we don't have, but that can be approximated by Dikin's ellipsoids.
Therefore, we are interested in solving the following problems
log
and
log
\Deltay
Here D is one of the scaling matrices X
depending whether the
computations are done with the primal, the dual or the primal-dual algorithm.
Let show here that the above problems are well-defined and have a finite optimum
Lemma 4.1 Under Assumptions 2.1 and 2.2, Problems (11) and (12) are well
defined and have a finite optimum that is uniquely defined by the first order
optimality conditions.
Proof Both problems have a strictly concave objective. Their optimum, if it
exists, is unique in fi (resp., fl).
By Assumptions 2.1 and 2.2, there exists a -
\Deltay such that B T -
Problem (12) is well-defined. Since \Deltay is bounded, fl is bounded and the
feasible set is compact. Since the objective tends to \Gamma1 close to the boundary,
the problem has a finite solution that is uniquely defined by the set of first order
optimality conditions.
To show that Problem (11) is also well-defined, we note that the equation A\Deltax+
has a solution for any fi ? 0 since A has full row rank. Let us show
that the feasible set is bounded. Indeed, let fi - 0 and
\Deltay
Recalling that A has full row rank,
we conclude from A\Deltax
fi is bounded, since \Deltax is bounded by
Problem (11) is thus
well-defined and has a finite optimum.
The solutions of Problems (11) and (12) define the primal dual pair of rays
fffiA
and
\Deltay
ffflA
~
FP and ~ s(ff) 2 int ~
The following positive semidefinite matrix
plays a fundamental role in the analysis. V can be interpreted as variance-covariance
matrix between the vectors (a m+j in the metric induced
by the matrix (AD 2 A
Theorem 4.2 The solution of Problems (11) and (12) is given by
and
with fi defined as the unique solution of
log
and
Proof Let - 2 R n and oe 2 be the multipliers associated with the constraints of
Problem (11). The optimality conditions are
?From the definition of \Deltax, one immediately sees that A\Deltax Letting
This proves the second relation. To prove the first relation, we shall use the
optimality condition for Problem (13). However, we must check first that (13)
has a bounded optimum. In Lemma 4.1 we proved that
nonegative solution. Thus, for all fi - 0, fi 6= 0, one has
This proves that the objective \Gamma p
log fi i is bounded above and Problem
has a unique optimum.
The optimality condition for Problem (13) is
Replacing fi \Gamma1 by pV fi we get the identity
It remains to check that
and
Let us now consider Problem (12). The optimality conditions are
\Deltay
are the multipliers associated with the two constraints
We want to show that are the optimal multipliers, where fi
is the optimal solution of Problem (13). Solving for \Deltay, one gets:
Now
Remembering the optimality condition for fi, one may replace
thus check that the first optimality condition holds.
Finally,
\Deltay
proves that with our choice of multipliers, the last optimality condition also
holds.
Remark 4.1 If V is nonsingular, fl is also the unique solution of
log
We can now give an explicit formula for the restoration direction. Noting that
we have the new primal-dual pair
~
and
~
Remark 4.2 We note a significant dissymmetry between the primal and dual
directions:
1. any positive value of fi, say primal feasible direction
2. but fi ? 0 does not guarantee
then taking a feasible dual direction.
Different stepsizes (ff could be used in the primal and dual space.
Note that, by construction,
D\Deltas. At the optimum direction, one has
The computation of fi requires solving the nonlinear optimization problem (13).
Since the function F
log
self-concordant, it can easily
be minimized by classical Newton schemes. We postpone to a later section the
discussion on the complexity estimate for getting approximate solutions.
For the sake of a simpler presentation we shall assume in our analysis of ACCPM
that the minimizers are exact. However, this is not the case in practice and we
must be concerned with the impact of errors on fi and fl on the performance
of ACCPM. This discussion is also postponed to a later section. Below, we
sketch the result that enables an easy extension of our analysis of ACCPM with
multiple cuts in the case of inexact computations of fi and fl.
The convergence analysis of section 5 relies on the following properties:
e:
If we can guarantee that the solutions satisfy pfifl - e and 1
then the convergence result on ACCPM is essentially unaffected, while
the proofs need only minor adjustments.
We give here a theorem that stipulates the condition that must be met by fi and
fl to carry the analysis with inexact minimizers. In a later section we shall show
that classical interior point schemes make it possible to meet the condition.
Theorem 4.3 Assume fi ? 0 and kpfi(V fi) \Gamma ek - j. Let
and
In particular,
Proof
The first set of inequalities follows directly from the assumption and the definition
of fl. These inequalities also imply that
Multiplying these inequalities by e T one gets
5 Convergence analysis
We now assume that (x; s) is a pair of '-centers and that \Deltax and \Deltas are
computed as in Section 4 with fi and fl being the exact minimizers of problems
and (14). We assume that the computations are done with either the
primal, the dual of the primal-dual scaling.
Lemma 5.1 Independently of the specific scaling matrix D (primal, dual or
primal-dual), one has, for any ff
Proof
By construction
1. From Lemma 3.6, for any primal,
dual or primal-dual scaling D, we have
The proof is the same in the dual case.
Remark 5.1 The above result can be sharpened by considering separately the
three different scaling matrices D. However, we prefer the weaker result since
it allows a single formulation for the three cases.
Lemma 5.2 The following inequalities hold:
and
je
Proof
?From \GammaA\Deltax, one has
Thus,
To prove the second statement, we note that x T
je
In view of the above lemmas, we can bound the potentials e
'P and e
'D at the
new pair of points (~x(ff); ~ s(ff)).
Lemma 5.3 For any 0 ! ff the new potentials satisfy
e
log
e
log
and
e
Proof
Let us prove first the inequality on the primal potential. At the updated point
~
x(ff) the potential is
e
log x
log
log x
log
log
1. We can apply Lemma 3.3 to get
Then, by Lemma 5.2
ffe
decreasing, we can bound khP k
e
log
Let us prove now the dual case. We have
e
log s
log
log
\Deltas. By Lemma 5.1 khD k ! 1. We can apply Lemma 3.3 to get
Since by Lemma 5.2
we obtain, by putting the inequalities together, the same result as in the primal
case
e
log
To conclude the proof of the theorem, we just sum the inequalities on ~
'P and
~
'D and use
e to get
e
5.1 Recovering the new analytic center
Theorem 5.4 The number of Newton steps to compute the updated '-analytic
center is bounded by
oe
where
and, depending on the Newton scheme,
Proof
To bound the number of Newton steps, we compute the optimality gap
for the sum of the primal and dual potentials. On the one hand,
~
On the other hand, we can write
e
Finally,
Hence
e
Thus
Using theorem 3.4 and the above bound on the potential variation we conclude
the proof of the theorem.
5.2 Convergence of ACCPM with multiple cuts
The next lemma is a first step on bounding the number of calls to the oracle.
Theorem 5.5 For all
~
log
with
Proof
The first inequality uses ~
'P (~x(ff)), the duality on potential and Lemma
2.4 to yield
log
We now need to deal with the contribution of the new variables
log
Since fi solves (13), we have fi T V
log
f
log
log
for any arbitrary fi 0 .
Let us define the vector - by
a T
Note that - while the off-diagonal terms of V are
The off-diagonal elements
Those properties are typical of a variance-covariance matrix. Let us choose
Then
The correlation matrix: all its coefficient
are bounded in absolute value by 1, and
Thus
log
log
log
log
Using corollary 3.2 we have
Putting together (21), (22) and (23) yields
~
ff
ff
log
The bound
ff
can be analyzed by selecting, somewhat arbitrarily,
guaranteeing
ff
which is exactly the same result as in [20], but with a rather different derivation,
as we show that this inequality is actually achieved at the iterate obtained by
the restoration step.
Remark 5.2 If the p cuts generated are identical, then the correlation matrix
R is the rank-one matrix ee T . Otherwise for the optimal fi
log fi
log
may be significantly greater than 0 and speed the convergence in practice, even
though this does not appear to affect the worst case complexity bound.
5.3 Convergence of ACCPM
The convergence analysis uses the proof given in [20], for the case of multiple
cuts.
Denote
and let P k be the same value after k calls to the oracle, that is, after adding
denotes the number of cuts added at iteration
j. By Theorem 5.5 and the observation (24) the following inequality holds
log
Theorem 10 of [20] can be used here, with p - n denotes the maximum number
of cuts generated by any call to the oracle.
Theorem 5.6 The algorithm stops with a solution as soon as k satisfies:
Furthermore the number of damped Newton steps per call to the oracle is O(p log(p+
1)). The number of cutting planes generated is at most O
The assumption that p - n is not required in the proof of [20], and in fact
would still lead to O
cutting planes (this would only impact
the constant).
6 Computing the optimal direction of restora-
tion
The restoration direction requires the solution of the concave problem
log
We note that in the computation of the restoration direction a significant absence
of symmetry occurs: it is easy to give a feasible value for fi, say
or
that gives a feasible solution to the problem of finding a feasible
direction, but, in general, this is not the case for the dual side.
If V is invertible, then the dual direction could also be computed by maximizing
log
1 The notation O indicates that lower order terms are ignored.
A good starting value for fl could also be given, say
or
, with - 2
D being the diagonal of V \Gamma1 .
The following bounds on F (fi) will be useful in the computation of complexity
estimate of a Newton method to solve (13).
Theorem 6.1 For
log
and
Proof
The inequality on F (fi 0 ) was derived in the proof of theorem 5.5. (See (22).)
Let us construct an upper bound on F (fi ) where fi denotes the optimal solution
of problem (13), and fl . From the optimality condition
log
log(fi
Hence,
F (fi
log
By Lemma 3.8, an homothety of Dikin's ellipsoid contains the current set of
localization, i.e.,
ae
oe
By assumption (2.1) and the fact that the algorithm has not terminated, a
sphere of radius " is contained in FD . Hence,
contains a sphere of radius "(1\Gamma')
(m+1)(1+') . Denoting by y c the center of this sphere,
and selecting one has
log
with
And thus
If V is invertible, one can derive alternative upper bounds on F (fi ) as follows.
Using
we have
log
(setting
If instead of
log(- D
The bounds on F (fi 0 ) and F (fi ) are used to derive a complexity estimate for
the computation of an approximate optimal solution. Using the fact that the
function F is self-concordant [15], we can resort to a potential increase scheme.
The scheme uses the Newton direction
\Gamma[F
Let us denote kuk
the norm of an arbitrary vector u in the metric
induced by the positive definite matrix H . The norm
a critical role in the analysis. The potential increase scheme is based on an
extension of lemma 3.3. The proof can be found in the unpublished lecture
notes [14]. (The proof is also made available in [16].)
Lemma 6.2 Let \Deltafi be such that k\Deltafik [F 00 (fi)]
with
Assume now
and satisfies the condition of the above
lemma. Thus,
easily shows that
is bounded from below by an absolute constant.
The complexity estimate for the potential increase scheme follows directly from
the above analysis and a bound on the achievable potential increase
Theorem 6.3 Let fi
. The potential increase algorithm applied
to the maximization of F produces a point fi such that
in a number of iterations not greater than6 6 6
Remark 6.1 The total number of Newton steps involved in the computation
of all the approximate optimum directions can easily bounded by m k log(1="),
using theorem (6.3), where k is the number of calls to the oracle at termination,
long step argument similar to the one given in [20] could
most likely be used to reduce this bound.
Looking at every iteration individually, and using the fact that A T y - c contains
the cutting planes 0 - y - e, we can assert that
and hence
This indicates that, in practice, the number of iterations needed at each iteration
to compute the optimal fi should not increase with the number of cutting planes.
It remains to prove that the potential increase scheme yields a solution fi that
meets the proximity condition kpfi(V used in Theorem 4.3. In other
words, we must show that for j small enough the condition
implies To this end, we adapt some results and proofs of [3]
developed for quadratic programming.
We then relate a few critical norms.
Lemma 6.4 Let
The following inequality holds
Proof
Since
This proves the left-hand side inequality.
As V is positive semidefinite, one has
Therefore
We can now prove the main result of the section.
Lemma 6.5 Assume
Proof
Since
over,
we get
We conclude that
The above lemma shows that once the condition
met, one more Newton step is enough to generate point satisfying
and thus, by Theorem 4.3, a point
7 Conclusion
In this paper, we defined an efficient direction to restore primal and dual feasibility
and centrality after adding p new central cuts simultaneously. The direction
is efficient in the sense that it maximizes the the product of the new variables
brought into the primal or the dual potentials, under the constraints that the
other variables remains within the Dikin ellipsoid. The computation of the optimal
direction takes place in a space of dimension p equal to the number of cuts
added at a given iteration. If p is sufficiently smaller than n, then significant
gains in efficiency can be expected.
The analysis has been derived under the assumption that the cuts are central. If
deep cuts are present, which is to be expected in practice, primal feasibility can
always be recovered but dual feasibility appears difficult to achieve in general,
except by the use of a primal Newton method. One could then extend the long
step argument of [8] in the case of one deep cut to multiple deep cuts.
The implementation of ACCPM [11] uses
e. Other choices using the
variance-covariance matrix V , if it is invertible, have been proposed in [10], and
the analysis of this paper actually strengthens that line of thinking.
Both the heuristic and optimal choices for fi and fl need to be tested in practice,
and rigorous extensions to multiple deep cuts deserve a more thorough study.
--R
"A cutting plane algorithm that uses analytic centers"
"A Cutting Plane Method from Analytic Centers for Stochastic Programming"
Interior Point Approach to Linear
"Solving Non-linear Multicommodity Flows Problems by the Analytic Center Cutting Plane Method"
"Decomposition and non-differentiable optimization with the projective algorithm"
"Complexity analysis of an interior cutting plane for convex feasibility problems"
"Using the Primal Dual Infeasible Newton Method in the Analytic Center Method for Problems Defined by Deep Cutting Planes"
"Shallow, deep and very deep cuts in the analytic center cutting plane method"
"A two-cut approach in the analytic center cutting plane method"
"Warm start of the Primal-Dual Method Applied in the Cutting Plane Scheme"
"ACCPM - A Library for Convex Optimization Based on an Analytic Center Cutting Plane Method"
"Analysis of a Cutting Plane Method That Uses Weighted Analytic Center and Multiple Cuts"
"Cutting plane algorithms from analytic centers: efficiency estimates"
Introductory lectures on Convex Optimization.
Interior Point Polynomial Algorithms in Convex Programming
Homogeneous Analytic Center Cutting Plane Methods for Convex Problems and Variational Inequalities
"On Updating the Analytic Center after the Addition of Multiple Cuts,"
Theory and Algorithms for Linear Optimization: An Interior point Approach
"A potential reduction algorithm allowing column genera- tion"
"Complexity Analysis of the Analytic Center Cutting Plane Method That Uses Multiple Cuts"
Interior Point Algorithms: Theory and Analysis.
--TR
--CTR
Olivier Pton , Jean-Philippe Vial, Multiple Cuts with a Homogeneous Analytic Center Cutting Plane Method, Computational Optimization and Applications, v.24 n.1, p.37-61, January
Fernanda Raupp , Clvis Gonzaga, A Center Cutting Plane Algorithm for a Likelihood Estimate Problem, Computational Optimization and Applications, v.21 n.3, p.277-300, March 2002
Shu-Cherng Fang , Soon-Yi Wu , Jie Sun, An Analytic Center Cutting Plane Method for Solving Semi-Infinite Variational Inequality Problems, Journal of Global Optimization, v.28 n.2, p.141-152, February 2004 | cutting plane method;self-concordance;primal Newton algorithm;interior-point methods;analytic center;multiple cuts |
589112 | On the Accurate Identification of Active Constraints. | We consider nonlinear programs with inequality constraints, and we focus on the problem of identifying those constraints which will be active at an isolated local solution. The correct identification of active constraints is important from both a theoretical and a practical point of view. Such an identification removes the combinatorial aspect of the problem and locally reduces the inequality constrained minimization problem to an equality constrained problem which can be more easily dealt with. We present a new technique which identifies active constraints in a neighborhood of a solution and which requires neither complementary slackness nor uniqueness of the multipliers. We also present extensions to variational inequalities and numerical examples illustrating the identification technique. | Introduction
In this paper we consider the problem of identifying the constraints which are active
at an isolated stationary point -
x of the nonlinear program
where it is assumed that the functions f are at least
continuously differentiable. More specifically, we are interested in the following
question: Given an (x; -) 2 IR n+m belonging to a sufficiently small neighborhood
of a Karush-Kuhn-Tucker (KKT) point of Problem (P), is it possible to correctly
estimate, on the basis of the problem data in x, the set of indices
I 0 := fij g i
of the active constraints? The correct identification of active constraints is important
from both a theoretical and a practical point of view. Such an identification,
by removing the difficult combinatorial aspect of the problem, locally reduces the
inequality constrained minimization problem to an equality constrained one which
is much easier to deal with. In particular, the study of the local convergence rate
of most algorithms for Problem (P) implicitly or explicitly depends on the fact that
I 0 is eventually identified.
The identification of the active constraints is not difficult if strict complementarity
holds at the solution, see the discussion in the next section. However, as far
as we are aware of, to date no technique can successfully deal with the case in which
the complementary slackness assumption is violated, except in the case of linear
programs, see [10]. In this paper we present a new technique which, under mild as-
sumptions, correctly identifies active constraints in a neighborhood of a KKT point.
This technique appears to improve on existing techniques. In particular, it enjoys
the following properties:
(i) It is simple and independent of the algorithm used to generate the point (x; -).
(ii) It does not require complementary slackness.
(iii) It does not require uniqueness of the multipliers.
(iv) It does not rely on any convexity assumption.
(v) In the case of unique multipliers it also permits the correct identification of
strongly active constraints.
(vi) The identification technique can be applied also to the Karush-Kuhn-Tucker
system arising from variational inequalities.
IDENTIFICATION OF ACTIVE CONSTRAINTS 3
Strategies for identifying active constraints are part of the optimization folklore
[2, 13, 15], however, they almost invariably lack many of the good characteristics
listed above. In the last ten years a special attention has been devoted to this
problem in the field of interior point methods for linear programs; we refer the reader
to the survey paper [10]. Recent works on the nonlinear case include [9, 11, 23], where
the case of box constraints is considered, and [12, 36], where the general nonlinear
case is studied. Related material can also be found in [4, 5, 6], where the problem
of establishing whether or not a sequence fx k g, converging to a solution -
x, in some
way identifies the set I 0 is dealt with. Note, however, that in these latter papers
no explicit rule is given in order to identify the active constraints from a close, but
arbitrary point.
We remark that, in order to identify the active set, we suppose we are given a
pair (x; -) of primal and dual variables. If we think of algorithmic applications of
the results in this paper, we stress that most algorithms will produce a sequence
of primal and dual variables. Even in the rare cases in which this does not occur,
it is usually possible, under reasonable assumptions, to generate a continuous dual
estimate by using a multiplier function, see, e.g., [12] and references therein, as well
as Section 4.
This paper is organized as follows. In the next section we introduce the identification
technique and prove its main properties. The identification technique critically
depends on the definition of what we call identification function. Therefore, the
more technical Section 3 is devoted to the definition of identification functions under
different sets of assumptions. In Section 4 we use the results of the previous
sections in order to define a local active set Newton-type algorithm for the solution of
inequality constrained optimization problems for which a Q-quadratic convergence
rate of the primal variables can be proved under very weak conditions. In Section 5
we give some final comments.
We conclude this section by providing a list of the notation employed. Through-out
the paper, k \Delta k indicates the Euclidean vector norm. The symbol B ffl denotes
the open Euclidean ball with radius ffl ? 0 and center at the origin; the dimension of
the space will be clear from the context. The Euclidean distance of a point y from a
nonempty set S is abbreviated by dist[y; S]. We write x+ for the vector maxf0; xg;
where the maximum is taken componentwise. We set I := use
of the notation x J for J ' I in order to represent the jJ j-dimensional vector with
components Finally, the transposed Jacobian of the vector-valued mapping
g at a point x will be denoted by rg(x); i.e., the ith column of this matrix is the
gradient rg i (x):
Active Constraints
Following the usual terminology in constrained optimization, we call a vector -
a stationary point of (P) if there exists a vector - 2 IR m such that (-x; -
-) solves the
4 F. FACCHINEI, A. FISCHER AND C. KANZOW
Karush-Kuhn-Tucker system
(1)
The pair (-x; -) is called a KKT point of Problem (P). In the sequel -
x will always
denote a fixed, isolated stationary point, so that there is a neighborhood of -
x which
does not contain any further stationary point of (P). Moreover, we shall indicate
by the set of all Lagrange multipliers -
- associated with - x and by K the set of all
KKT points associated with -
x, that is,
-) solves (1)g; K := f(-x; -
The set is closed and convex and therefore, so is the set K. Gauvin [14] showed
that is bounded (and hence compact) if and only if the Mangasarian-Fromovitz
constraint qualification (MFCQ) is satisfied, i.e., if and only if
On the other hand, Kyparisis [22] showed that reduces to a singleton if and only
if the strict Mangasarian-Fromovitz constraint qualification (SMFCQ) holds, i.e., if
and only if
denotes the index set
I
In particular, the linear independence constraint qualification (LICQ), i.e., the linear
independence of the gradients of the active constraints, implies that is a singleton.
Our basic aim is to construct a rule which is able to assign to every point (x; -)
an estimate A(x; -) ' I so that A(x; lies in a suitably small
neighborhood of a point (-x; -) 2 K.
Usually estimates of this kind are obtained by comparing the value of g i (x) to
the value of - i . For example, it can easily be shown that the set
I \Phi (x; -) :=
coincides with the set I 0 for all (x; -) in a sufficiently small neighborhood of a KKT
point (-x; -) which satisfies the strict complementarity condition. If this condition is
violated, then only the inclusion
I \Phi (x; -) ' I 0 (2)
IDENTIFICATION OF ACTIVE CONSTRAINTS 5
holds. Furthermore, if is a singleton, then we also have, in a sufficiently small
neighborhood of (-x; -
-),
I
This relation was exploited to construct locally superlinearly convergent QP-free
optimization algorithms when the unique multiplier - does not satisfy the strict
complementarity condition, see, e.g., [12, 20, 35].
We refer the reader to [2, 12] and references therein for a more complete discussion
of these kind of results. An analysis of results established in the literature
shows that this conclusion holds in general: if strict complementarity is satisfied, it
is usually possible to correctly identify the active constraint set, otherwise a relation
like (3) is the best result that has been established in the general nonlinear case.
To overcome this situation we propose to compare g i (x) to a quantity which goes
to 0 at a known rate if (x; -) converges to a point in the KKT set K. To this end,
we introduce the notion of identification function.
called identification function for
(a) ae is continuous on K,
(b) (-x; -) 2 K implies ae(-x; -
(c) if (-x; -
-) belongs to K, then
lim
ae(x; -)
dist [(x; -); K]
In the next section we shall give examples of how to build, under appropriate as-
sumptions, identification functions. Basically, Definition 2.1 says that a function is
an identification function if it goes to 0 when approaching the set K at a "slower"
rate than the distance from the set K. We note that dist [(x; -); K] ? 0 whenever
since K is a closed set; hence the denominator in (4) is always nonzero.
Using Definition 2.1 it is easy to prove that the index set
correctly identifies all active constraints if (x; -) is sufficiently close to the KKT set
K.
Theorem 2.2 Let ae be an identification function for K. Then, for any - 2 , an
exists such that
6 F. FACCHINEI, A. FISCHER AND C. KANZOW
Proof. Since g is continuously differentiable, g is locally Lipschitz-continuous.
Hence there exists a constant c ? 0 such that, for all x sufficiently close to -
Suppose now that g i using (4) and (7), we obviously have, for
in a sufficiently small neighborhood of (-x; -),
dist [(x; -); K] - ae(x; -);
so that, by (5), i 2 A(x; -).
If, instead, (x; -) 2 K, then we have
x by the local uniqueness of -
x. From
the definition of an identification function, we also have ae(x; so that
and also in this case i 2 A(x; -).
On the other hand, if by continuity, that i 62 A(x; -) if (x; -)
is sufficiently close to (-x; -
-). Therefore, for any - 2 , we can find
such that (6) is satisfied. 2
From the previous theorem it is obvious that there exists an open set containing
K where the identification of the active constraints is correct. Using the MFCQ
condition we can obtain a somewhat stronger result.
Theorem 2.3 Let ae be an identification function for K. If the MFCQ condition
holds, then there is an ffl ? 0 such that
Proof. By the previous Theorem, for every (-x; -) 2 K, there exists a neighborhood
-) such that A(x; -)). The
collection of open
obviously forms an open cover of K. Since the
set K is compact in view of the MFCQ condition, we can extract from the infinite
cover
- such that (-x; -) 2 K a finite subcover \Omega\Gamma ffl( -
s: Then it is easy to see that the Theorem holds with ffl := min j=1;:::;s fffl( -
the SMFCQ holds, it is even possible to identify the set of strongly active constraints
at -
x, i.e., the set of constraints whose multipliers are positive. To this end,
let be defined by
The following theorem holds.
IDENTIFICATION OF ACTIVE CONSTRAINTS 7
Theorem 2.4 Let ae be an identification function for K. If the SMFCQ holds at - x,
then there is an ffl ? 0 such that
Proof. We first recall that the SMFCQ implies that reduces to a singleton,
i.e., -g. Theorem 2.2 shows that A+ (x; -) ' I 0 for all (x; -) in a certain
neighborhood of (-x; -). Now, consider an index . By continuity, this implies
in a sufficiently small neighborhood of (-x; -). On the other hand, let
in a sufficiently small neighborhood of
-), we have
dist [(x; -); K] - ae(x; -)=2 ! ae(x; -):
This means i 62 sufficiently close to
Until now we made reference to the Karush-Kuhn-Tucker system (1) which expresses
first order necessary optimality conditions for the minimization Problem (P). We
showed how the active constraints associated to an isolated stationary point -
x can be
identified. However, the fact that the Karush-Kuhn-Tucker system (1) derives from
an optimization problem plays no role in the theory developed. What we actually
proved is the following: Given a solution (-x; -
-) of a system with the structure of
system (1) and with an isolated x-part, we can identify, in a suitable neighborhood
of this solution, those inequalities which hold as equalities at the solution (-x; -).
Therefore, if we consider the KKT system
continuous function, the theory of this section goes
through without any change. This is an important observation, since it allows us to
extend the theory developed so far to the identification of active constraints for the
variational inequality problem:
Find - x 2 X such that F (-x) T
is continuous and
is continuously differentiable. It is well known that, under a standard regularity
assumption [17], a necessary condition for -
to be a solution of the variational
inequality problem is that - 2 IR m exists such that (-x; -) solves system (8). There-
fore, if we have a sequence f(x converging to a solution of system (8) which
has an isolated primal part, we can apply the techniques described in this section
in order to identify which of the constraints g i (x) - 0 will be active at -
8 F. FACCHINEI, A. FISCHER AND C. KANZOW
3 Defining Identification Functions
From what exposed in the previous section we see that the crucial point in the
identification of active constraints is the definition of an identification function. In
this section we show how it is possible to define such a function for Problem (P):
We consider three cases. In the first one we assume that the functions f and g
are analytic, in the second case we require them to be LC 1 , but then we also need
that the MFCQ condition and a second order sufficient condition for optimality are
satisfied. Finally, in the third case, the functions are required to be C 2 and the
KKT point is assumed to satisfy a regularity condition related to (but weaker than)
Robinson's strong regularity [33] and which we call quasi-regularity. Extensions of
these results to the KKT system (8) are possible. We shall point out the relevant
changes in corresponding remarks.
The cases considered here do not cover all the situations in which an identification
function can be defined and computed, but they certainly show that the definition
and computation of an identification function is possible in most of the cases of
interest.
3.1 The Analytic Case
Let f and each g i (i 2 I) be analytic around a point x. We recall that this means
that f and each g i (i 2 I) possess derivatives of all orders and that they agree with
their Taylor expansions around x. We say that f and each g i (i 2 I) are analytic
on an open set X ' IR n if they are analytic around each x 2 X: We shall make use
of the following result due to Lojasiewicz, Luo and Pang [25, 27].
Theorem 3.1 Let S denote the set of points in IR r satisfying
are analytic functions defined on an open set
Suppose that S 6= ;. Then, for each compact
ae X, there exist
Using this result, it is possible to define an identification function for Problem (P):
Theorem 3.2 Suppose that f and g are analytic in a neighborhood of a stationary
point - x. Then, the function ae defined by
log(r(x;-))
if r(x; -) 2 (0; 0:9);
IDENTIFICATION OF ACTIVE CONSTRAINTS 9
where
is an identification function for K.
Proof. It is obvious, by definition, that ae 1 is a nonnegative function such that
lim
so that ae 1 is also continuous on K. Hence we only have to check the limit
lim
dist [(x; -); K]
To this end we recall that, for arbitrary - ? 0 and fl ? 0, the limit
lim
holds, see, e.g., [28, p. 328]. We can now apply Theorem 3.1 by considering the
system (1) which defines KKT points. It is then easy to see that (9) yields, for
every given compact
containing (-x; -) in its interior,
where - and fl are fixed positive constants. Therefore we can write
lim
dist [(x; -); K]
from which (11) follows taking into account (12), recalling the definition of ae 1 and
noting that r(x; -) is a continuous function that goes to 0 from the right as (x; -)
tends to (-x; -). 2
We stress that Theorem 3.2 holds under the mere assumption that f and g are
analytic.
Remark 3.3 If we want to define an identification function for the solutions of the
KKT system (8), we only have to substitute the definition of the residual (10) by
the following one:
Obviously, also in this case, we have to assume that F and each g i (i 2 I) are
analytic in a neighborhood of the KKT point under consideration.
F. FACCHINEI, A. FISCHER AND C. KANZOW
3.2 The Second Order Condition Case
In this subsection we assume that f and g are LC 1 , i.e., that they are differentiable
with Lipschitz-continuous derivatives. We denote the Lagrangian of problem (P) by
and write r x L(x; -) for the gradient of L with respect to the x-variables. We
further assume that the MFCQ holds along with the following second order sufficient
condition for optimality:
Assumption 3.4 There is
Here, W ( -) denotes the cone
and @ x r x L(-x; -
-) denotes Clarke's [8] generalized Jacobian with respect to x of the
gradient r x L, calculated at (-x; -
-).
We remark that, if the functions f and g are twice continuously differentiable and
only one multiplier exists, then the previous definition reduces to the classical KKT
second-order sufficient condition for optimality.
We shall show that these two conditions allow us to define an identification
function for K which, because the MFCQ holds, is a compact set. To this end
consider the perturbed nonlinear program
denotes the perturbation parameter. In what follows
we will assign to any vector (y; -) 2 IR n \Theta IR m a particular perturbation vector
For this purpose we first define the function
componentwise by
where, we recall, I \Phi (y; g. We can now introduce the function
\Gammag i (y) if i 2 I \Phi (y; -);
Although, in general, the functions - \Phi and - are not everywhere continuous, the
following properties can be proved.
IDENTIFICATION OF ACTIVE CONSTRAINTS 11
Lemma 3.5 (a) If - 2 , then - \Phi (-x; -.
(b) The function - \Phi is continuous on K.
(c) If - 2 , then -x;
(d) The function - is continuous on K.
Proof. (a) Since (-x; -) is a KKT point, it readily follows that I \Phi (-x;
the definition of the function - \Phi yields - \Phi (-x; - for all - 2 .
(b) Let (-x; -
-) belong to K. According to assertion (a), in order to show continuity
of - \Phi in (-x; -
-), we have to show that, for every i 2 I,
lim
If -
easily follows from the definition of - \Phi
sufficiently large, continuity. Using the
definition of - \Phi
again, we have - \Phi
i for all k large enough. Thus, (14)
follows also in this case.
(c) Taking into account assertion (a), the KKT conditions (1) for problem (P)
yield
On the other hand, analogously to point (a), we have I \Phi (-x;
readily follows from the definition of - g .
(d) The continuity of the function - f on K follows by its definition and assertion
(b). In order to prove the continuity of - g on K; let (-x; -
be given and let
be any sequence converging to (-x; -
-). In view of part (c), we have to show
that
lim
To this end, first consider an index i 2 I Hence
(15) follows from the definition of - On the other hand, if
sufficiently large. Hence - g (y k
all these indices, i.e., (15) holds also for i 62 I
Using the particular perturbation vector -), we can prove the following
result.
Lemma 3.6 Let (y; -) 2 IR n \Theta IR m be arbitrarily chosen. Then, (y; - \Phi (y; -)) is a
KKT point for problem (P(t)); where
Proof. The KKT system for the perturbed program (P(t)) reads as follows:
12 F. FACCHINEI, A. FISCHER AND C. KANZOW
Let (y; -) be arbitrary but fixed. Obviously, since we find that (x; -) :=
solves (16) and (17). Now, we will show that (y; - \Phi (y; -)) also satisfies
(18) and (19). For i 2 I \Phi (y; -) the definition of - g (y; -) yields (g(y)
that both (18) and (19) are fulfilled. If, instead, i 2 I n I \Phi (y; -), it follows from
the definition of - \Phi (y; -) that - \Phi
and (19) is satisfied. Moreover, the
definition of - g (y; -) implies
Thus, (18) is also valid for i 2 I n I \Phi (y; -). We therefore conclude that (y; - \Phi (y; -))
is a KKT point of (P(t)) when
The next result can easily be derived from Theorem 4.5 b) and formula (3.2 f) in
Klatte [18]. If the functions f and g of the program (P) are twice continuously
differentiable it can also be obtained from a corresponding result in Robinson [34,
Corollary 4.3]. We further note that Assumption 3:4 can be weakened by using
generalized directional derivatives, see [18] for more details and references.
Theorem 3.7 Let the MFCQ and Assumption 3.4 be satisfied. Then, there are
for every t 2 B ffi and for every KKT point (-x(t); -
-(t)) of problem (P(t)) for which
Putting together the last two results, we can easily prove the following theorem.
Theorem 3.8 Let the MFCQ and Assumption 3.4 be satisfied. Then, there are
Proof. By Lemma 3.5 (c), we have -x;
Lemma 3.5 (d) and by the compactness of K, we have that, for ffi from Theorem 3.7,
we can find an " ? 0 such that, if Therefore,
since ffl - j (with j from Theorem 3.7) can be assumed without loss of generality,
Theorem 3.7 together with Lemma 3.6 yields the desired result. 2
We are now in the position to show that the function ae defined
by
can be used as an identification function.
Theorem 3.9 Let the MFCQ and Assumption 3.4 be satisfied. Then ae 2 is an identification
function for K.
IDENTIFICATION OF ACTIVE CONSTRAINTS 13
Proof. By Lemma 3.5 we easily obtain that ae 2 is continuous on K and that
any sequence with
lim
Using Theorem 3.8 we get, for k sufficiently large,
Let z 1 2 K and z 2 2 K be the projections of
tively, on the closed convex set K. Then, using the triangle inequality, we get
Combining relations (21) and (22), we obtain, for k sufficiently large,
ae 2 is continuous on the compact set K and since its value is 0 on K it follows
from (20), (21) and Lemma 3.5 (b) that the quantity dist[(x
goes to 0 for k ! 1. But then the right hand side of (23), and thus
also the left hand side, tends to infinity. Therefore, we have shown that ae 2 possesses
all properties of an identification function. 2
If, instead of the upper Lipschitz-continuity as stated in Theorem 3.7, the multi-function
t 7! K(t) is upper H-older-continuous at with a known rate - 2 (0; 1],
that is, if, for some
dist [(-x(t); -(t)); K] - cktk -
for every t 2 B ffi and for every KKT point (-x(t); -
-(t)) of Problem (P(t)) for which
then the technique presented in this subsection can easily be
extended if we define ae
In particular, Theorems 3.8 and 3.9 remain valid for this ae 2 if Assumption 3.4 is
replaced by the upper H-older-continuity.
An interesting case in which it is possible to prove, under an assumption weaker
than Assumption 3.4, the upper H-older-continuity at of the multifunction
14 F. FACCHINEI, A. FISCHER AND C. KANZOW
t 7! K(t) is the case of convex problems. Assume that f is convex and each g i (i 2 I)
is concave, that the MFCQ holds and that the following growth condition holds (in
place of Assumption 3.4): positive -
exist such that
Under these assumptions and using the results in [19], it is possible to show (we
omit the details) that ffi ? exist such that
dist [(-x(t); -(t)); K] - c
ktk
for every t 2 B ffi and for every KKT point (-x(t); -(t)) of Problem (P(t)) for which
It may be interesting to note that the growth condition holds, in
particular, if Assumption 3.4 is fulfilled.
Remark 3.10 The extension of the results of this section to general KKT systems
is not straightforward, since the sensitivity analysis of perturbed KKT systems re-
quires, to date, stronger assumptions. The key point is to establish a result analogous
to Theorem 3.7. Once this has been done, we can easily prove theorems analogous
to Theorem 3.9 by substituting F to rf in every relevant formula. As an example
of the kind of the results that can be obtained we cite the following one. Suppose
that F is C 1 and g is C 2 . Assume also that the SMFCQ holds at -
x along with
Assumption 3.4. Then, according to [16, Corollary 8 (c)], Theorem 3.7 holds and
therefore ae 2 is a regular identification function for the KKT system (8).
3.3 The Quasi-Regular Case
In this subsection we assume that the functions f and g are C 2 . We shall introduce a
condition which we call quasi-regularity. As will be clear later, this quasi-regularity
is related to , but weaker than Robinson's strong regularity [33]. In order to motivate
the definition of a quasi-regular KKT point we will first recall a condition which is
equivalent to the notion of a strongly regular KKT point. To this end we shall use
the index set I 00 := I 0 n I + of all those indices for which the strict complementarity
condition does not hold at the KKT point (-x; -). For any J ' I 00 (empty set
included) introduce the matrix
xx L rg+ rg J
\Gammarg T
\Gammarg T
xx L, rg+ and rg J are abbreviations for the matrices r 2
-), rg I+ (-x)
and rg J (-x), respectively. The following result is due to Kojima et al. [21].
Theorem 3.11 The following statements are equivalent:
IDENTIFICATION OF ACTIVE CONSTRAINTS 15
(a) (-x; -) is a strongly regular KKT point.
(b) For any J ' I 00 (empty set included), the determinants of the matrices M(J)
all have the same nonzero sign.
Motivated by point (b) in Theorem 3.11, we introduce the following definition.
Definition 3.12 The KKT point (-x; -
-) is a quasi-regular point if the matrices M(J)
are nonsingular for every J ' I 00 (empty set included).
Note that, in view of Theorem 3.11, quasi-regularity is implied by Robinson's strong
regularity condition, but the converse is not true. In fact, consider the following
example:
It is easy to check that -
is a global minimizer and that the Lagrange
multipliers of the two constraints are both zero, so that I
Therefore (-x; 0; 0) is a quasi-regular KKT point, but not a strongly regular one. Note
that in this example the KKT point is an isolated KKT point. This is not a chance.
In fact we shall show in this section that quasi-regularity of a KKT point implies
its local uniqueness. It is also worth pointing out that quasi-regularity implies the
linear independence of the active constraints. This easily follows from the fact that
Now let us introduce the operator
r x L(x; -)
Note that the KKT conditions are equivalent to the nonlinear system of equations
By the differentiability assumption we have that \Phi is locally Lipschitzian. Hence,
by Rademacher's Theorem, \Phi is differentiable almost everywhere. Denote by D \Phi
the set of points where \Phi is differentiable. Then we can define the B-subdifferential
(see, e.g., [31]) of \Phi at (x; -) as
F. FACCHINEI, A. FISCHER AND C. KANZOW
Note that the B-subdifferential is a subset of Clarke's generalized Jacobian [8, 31].
The next lemma illustrates the structure of the B-subdifferential of \Phi: Before stating
this lemma, however, we introduce three index sets:
Lemma 3.13 Let (x; -) 2 IR n+m be arbitrary. Then
xx L(x; -) rg(x)D a (x; -)
where
D a (x; -) := diag (a 1
are diagonal matrices with
a
and D b (x;
Proof. This follows immediately from the definition of the operator \Phi: 2
We are now in the position to prove the following result.
Lemma 3.14 Let (-x; -
n+m be a quasi-regular KKT point. Then all matrices
are nonsingular.
Proof. Let In view of Lemma 3.13, there exists an index set
-) such that
\Gammarg T
\Gammarg T
\Gammarg T
\Gammarg T
denotes the complement of J in the set fi(-x; -): Obviously,
this matrix is nonsingular if and only if the matrixB @
xx L rg ff rg J
\Gammarg T
\Gammarg T
IDENTIFICATION OF ACTIVE CONSTRAINTS 17
is nonsingular. In turn, this matrix is nonsingular if and only if the matrix M(J) is
nonsingular. Hence the thesis follows immediately from Definition 3.12. 2
We are now able to prove the main result of this subsection.
Theorem 3.15 Let (-x; -
n+m be a quasi-regular KKT point of problem (P).
Then,
(a) (-x; -) is an isolated KKT point,
(b) the function ae 3 : defined by
ae 3 (x; -) :=
is an identification function for
-)g:
Proof. Obviously, ae 3 is a continuous and nonnegative function with ae 3 (-x; -
Furthermore, since f and g have locally Lipschitzian gradients and the min operator
is semismooth (see [29, 32] for the definition of semismoothness and [29] for the proof
that the min operator is semismooth) it follows that also \Phi, which is the composite
of semismooth functions, is semismooth [29, 32]. Hence it follows from Lemma 3.14
and [30, Proposition 3] that there exists a constant c ? 0 such that
for all (x; -) in a neighborhood of (-x; -
only if (x; -) is a
KKT point, part (a) follows immediately.
From (24) we also get
ae 3 (x; -)
c
and therefore
lim
ae 3 (x; -)
i.e., ae 3 is an identification function. 2
Remark 3.16 In the case of the KKT system (8) everything goes through. It is
sufficient to assume that F is continuously differentiable and to substitute everywhere
the gradient r x L(x; -) by the function F Also in this case
the definition of quasi-regularity is related to and weaker than that of a strongly
regular KKT point since Theorem 3.11 carries over to the KKT system (8), see
Actually, the case of KKT systems of variational inequalities
is probably the main case in which quasi-regularity can be applied. In fact, it is
F. FACCHINEI, A. FISCHER AND C. KANZOW
not difficult to see that, if strict complementarity holds and -
x is a local minimum
point of Problem (P), quasi-regularity implies the conditions of the previous sub-
section. However, these conditions and quasi-regularity are fairly distinct if one
considers variational inequalities. For example, it can easily be checked that, given
the variational inequality defined by the function F
and the
set 0g, the point (0; is a quasi-regular solution but does not
satisfy the conditions stated in Remark 3.10 of the previous subsection.
4 An Application
In this section we apply the results obtained in the previous sections to a local
active-set Newton algorithm for the solution of Problem (P). The algorithm to be
introduced here is a simple variation of the one presented in [12]. However, using the
new results obtained in this work, we are able to relax the assumptions used in [12].
The result is an algorithm which, by solving only linear systems at each iteration,
guarantees Q-quadratic convergence of the sequence fx k g to the solution under very
mild assumptions and without requiring strict complementarity. We remark that,
as far as we are aware of, there exist only two other algorithms which ensure Q-
quadratic convergence of the primal variables. The first one is due to Bonnans [3]
and requires, at each iteration, the solution of a quadratic and possibly nonconvex
subproblem. Furthermore the algorithm of Bonnans also requires the selection of a
suitable solution of the quadratic subproblem, which appears to be a difficult task in
practice. The other algorithm that guarantees Q-quadratic convergence is the one
discussed in [12] which, as already said, requires stronger assumptions. We refer the
interested reader to [12] for a more detailed discussion of these issues.
Consider problem (P) and assume that f and g are twice continuously differen-
tiable. The algorithm we consider generates a sequence fx k g as
with d k being obtained by solving the linear system
z k
In the previous system,
where N(x) is the m \Theta m matrix defined by:
diag
i2I
while
IDENTIFICATION OF ACTIVE CONSTRAINTS 19
We shall assume that the LICQ holds at - x, along with the following weak second
order assumption.
Assumption 4.1 It holds that
We note that Assumption 4.1 is extremely weak if compared to second order assumptions
usually used in the local analysis of algorithms for the solution of inequality
constrained optimization problems. In particular, even if coupled with the linear
independence assumption, it does not even imply that - x is an isolated local solution
of problem (P). This can be checked on the following example, where a is a
nonnegative constant:
t. x 2 - 0:
It is easy to see that -
is a stationary point satisfying both the LICQ
assumption and Assumption 4.1. However, if a ? 0, -
x is not a local solution, while,
if we have that -
x is indeed a local solution but not isolated.
We recall a result which illustrates the properties of the multiplier function defined
by (27).
Theorem 4.2 (see [26]) Let -
x be a KKT point where the linear independence of the
active constraints holds. Then,
- and there exists ffl 1 ? 0 such that, for all
(a) -(x) is well defined;
(b) -(x) is continuously differentiable.
We now pass to the proof that the algorithm (25)-(26) is Q-quadratically convergent
in the primal variables. To this end we first need a simple Lemma.
Lemma 4.3 Let (-x; -
-) be a KKT pair for Problem (P) which satisfies the LICQ and
Assumption 4.1. Then there exist
the matrix
xx L(x; -(x)) \Gammarg I 0
rg I 0
is nonsingular and kM(x)
Proof. Assumption 4.1 and well known properties of quadratic forms (see, e.g., [1,
p. 78]) imply that there exists a constant oe ? 0 such that the matrix
(-x)rg I 0
F. FACCHINEI, A. FISCHER AND C. KANZOW
is positive definite. Then, by continuity, the matrix
is positive definite for all x 2 f-xg +B ffl 2
sufficiently small. This implies
(see, e.g., [1, p. 78]) that, for all x 2 f-xg +B ffl 2
Therefore (29) and the linear independence assumption imply that ffl 2 ? 0 can be
chosen as small as necessary so that, for all x 2 f-xg +B ffl 2
and
rg I 0
using (30) and (31), it is easy to show that the matrix M(x) is nonsingular for
. Hence, the remaining result follows by the continuity of M(x). 2
Theorem 4.4 Let f and g i (i 2 I) be twice continuously differentiable with locally
Lipschitz-continuous Hessian matrices r 2 f and r 2 g i (i 2 I). Let (-x; -
-) be an
isolated KKT pair for Problem (P) which satisfies the LICQ and Assumption 4.1,
and suppose that an identification function ae for known. Then there
exists
the system (26) is nonsingular and the
sequence fx k g produced by (25) satisfies converges to - x, and
the rate of convergence is Q-quadratic.
Proof. For us consider the linear system
with
I 0
is arbitrary but fixed. Recalling that r x L(-x;
I 0
0: (33)
Taking into account (33) and the differentiability assumptions on f and each g i
(i 2 I), it is possible to show (see [12] for the details), by repeated use of Taylor's
formula, that positive numbers ffl 3 , C 1 , and C 2 exist such that, for all x 2 f-xg +B ffl 3
IDENTIFICATION OF ACTIVE CONSTRAINTS 21
By Lemma 4.3, (32) and (33), we have, for all x 2 f-xg +B ffl 3
Moreover, if ffl 3 ? 0 is small enough, it follows from Theorem 4.2 that C 3 ? 0 exists
such that, for all x 2 f-xg +B ffl 3
Assume now that
and suppose that ffl 3 2 (0; 1) is chosen small enough so that, according to Theorem
2.2,
Therefore, setting
the linear systems (26) and (32) are equivalent as long as x k
. By
(34)-(36), we also have, if x k 2 f-xg +B ffl 3
xk:
From these relations, all the assertions of the theorem easily follow by induction. 2
We stress that the example in Section 3.3 satisfies the assumptions of the previous
theorem, but not those of the corresponding Theorem 4.1 in [12].
Final Remarks
In this paper we introduced a technique to accurately identify active constraints in
inequality constrained optimization and variational inequality problems. The most
remarkable feature of the new identification technique is that it identifies all active
constraints even if strict complementarity does not hold. Furthermore, as discussed
in the introduction, it also enjoys several other favorable characteristics. In particu-
lar, the identification technique can be used in combination with any algorithm for
the solution of inequality constrained optimization or variational inequality prob-
lems. In Section 4 we gave an example of an application of the results of this paper
to an active set Newton-type method; however, we believe that the techniques introduced
in this paper can be useful in many other cases, especially in the theoretical
analysis and in the design of optimization methods.
From a practical point of view, the following questions may also be of interest:
22 F. FACCHINEI, A. FISCHER AND C. KANZOW
(a) How large is the region where exact identification occurs?
(b) Can we build identification functions which are scale invariant?
(c) Can we relax the assumption that -
x is an isolated stationary point and still
obtain useful results?
It is difficult to answer to these questions at the level of generality adopted in this
paper. We think that an answer can come from practical experiments and from an
analysis of structured classes of problems, e.g., linear or quadratic problems, box or
linearly constrained problems etc.
Acknowledgment
. We would like to thank Professor D. Klatte for helpful discussions
on the stability of KKT-systems.
--R
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Constrained Optimization and Lagrange Multiplier Meth- ods
Rates of convergence of Newton type methods for variational inequalities and nonlinear programming.
On the identification of active constraints II: The nonconvex case.
Optimization and Nonsmooth Analysis.
Global convergence for a class of trust region algorithms for optimization problems with simple bounds.
A study of indicators for identifying zero variables in interior-point methods
"La Sapienza"
Quadratically and superlinearly convergent algorithms for the solution of inequality constrained minimization problems.
Practical Methods of Optimization.
A necessary and sufficient regularity condition to have bounded multipliers in nonconvex programming.
Practical Optimization.
Stability analysis of variational inequalities and nonlinear complementarity problems
Nonlinear optimization problems under data perturbations.
On quantitative stability for C 1
Strongly stable stationary solutions in nonlinear programs.
On uniqueness of Kuhn Tucker multipliers in nonlinear pro- gramming
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Strong stability in variational inequalities.
Sur la probl'em de la division.
New results on a continuously differentiable exact penalty function.
bounds for analytic systems and their applications.
Calculus I.
Semismooth and semiconvex functions in constrained optimiza- tion
Nonsmooth equations: motivation and algorithms.
Convergence analysis of some algorithms for solving nonsmooth equa- tions
A nonsmooth version of Newton's method.
Strongly regular generalized equations.
Generalized equations and their solution
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--TR
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Andreas Fischer , Houyuan Jiang, Merit Functions for Complementarity and Related Problems: A Survey, Computational Optimization and Applications, v.17 n.2-3, p.159-182, December 2000
Naihua Xiu , Jianzhong Zhang, Some recent advances in projection-type methods for variational inequalities, Journal of Computational and Applied Mathematics, v.152 n.1-2, p.559-585, 1 March | active constraints;constrained optimization;degeneracy;variational inequalities;identification of active constraints |
589114 | Robust Solutions to Uncertain Semidefinite Programs. | In this paper we consider semidefinite programs (SDPs) whose data depend on some unknown but bounded perturbation parameters. We seek "robust" solutions to such programs, that is, solutions which minimize the (worst-case) objective while satisfying the constraints for every possible value of parameters within the given bounds. Assuming the data matrices are rational functions of the perturbation parameters, we show how to formulate sufficient conditions for a robust solution to exist as SDPs. When the perturbation is "full," our conditions are necessary and sufficient. In this case, we provide sufficient conditions which guarantee that the robust solution is unique and continuous (Hlder-stable) with respect to the unperturbed problem's data. The approach can thus be used to regularize ill-conditioned SDPs. We illustrate our results with examples taken from linear programming, maximum norm minimization, polynomial interpolation, and integer programming. | Introduction
. A semidefinite program (SDP) consists of minimizing a linear
objective under a linear matrix inequality (LMI) constraint; precisely,
subject to F
(1)
- {0} and the symmetric matrices F
are given. SDPs are convex optimization problems and can be solved in polynomial
time with, e.g., primal-dual interior-point methods [24, 35, 26, 19, 2]. SDPs include
linear programs and convex quadratically constrained quadratic programs, and arise
in a wide range of engineering applications; see, e.g., [12, 1, 35, 22].
In the SDP (1), the "data" consist of the objective vector c and the matrices
In practice, these data are subject to uncertainty. An extensive body of
work has concentrated on the sensitivity issue, in which the perturbations are assumed
to be infinitesimal, and regularity of optimal values and solution(s), as functions of
the data matrices, is studied. Recent references on sensitivity analysis include [30,
31, 10] for general nonlinear programs, [33] for semi-infinite programs, and [32] for
semidefinite programs.
When the perturbation a#ecting the data of the problem is not necessarily small,
a sensitivity analysis is not su#cient. For general optimization problems, a whole field
# Received by the editors June 21, 1996; accepted for publication (in revised form) September 22,
1997; published electronically October 30, 1998.
http://www.siam.org/journals/siopt/9-1/30571.html
Ecole Nationale Sup-erieure de Techniques Avanc-ees, 32, Bd. Victor, 75739 Paris, France
(elghaoui@ensta.fr, oustry@ensta.fr, lebret@ensta.fr).
34 L. EL GHAOUI, F. OUSTRY, AND H. LEBRET
of study (stochastic programming) concentrates on the case where the perturbation
is stochastic with known statistics. One object of this field is to study the impact of,
say, a random objective on the distribution of optimal values (this problem is called
the "distribution problem"). References relevant to this approach to the perturbation
problem include [15, 9, 29]. We are not aware of special references for general SDPs
with randomly perturbed data except for the last section of [30], some exercises in
the course notes of [13], and section 2.6 in [23].
The main objective of this paper is to quantify the e#ect of unknown but bounded
deterministic perturbation of problem data on solutions. In our framework, the perturbation
is not necessarily small, and we seek a solution that is "robust," that is,
remains feasible despite the allowable, not necessarily small, perturbation. Our aim
is to obtain (approximate) robust solutions via SDP. Links between regularity of solutions
and robustness are, of course, expected. One of our side objectives is to
clarify these links to some extent. This paper extends results given in [16] for the
least-squares problem.
The approach developed here can be viewed as a special case of stochastic programming
in which the distribution of the perturbation is uniform.
The ideas developed in this paper draw mainly from two sources: control theory,
in which we have found the tools for robustness analysis [36, 17, 12] and some recent
work on sensitivity analysis of optimization problems by Shapiro [31] and Bonnans,
Cominetti, and Shapiro [10].
Shortly after completion of our manuscript, we became aware of the ongoing
work of Ben-Tal and Nemirovski on the same subject. In [7], they apply similar ideas
to a truss topology design problem and derive very e#cient algorithms for solving
the corresponding robustness problem. In [8], the general problem of tractability of
obtaining a robust solution is studied, and "tractable counterparts" of a large class of
uncertain SDPs are given. The case of robust linear programming, under quite general
assumptions on the perturbation bounds, is studied in detail in [6]. Our paper can
be seen as a complement of [8], giving ways to cope with (not necessarily) tractable
robust SDPs by means of upper bounds. (In particular, our paper handles the case
of nonlinear dependence of the data on the uncertainties.) A unified treatment, and
more results, will appear in [4].
The paper is divided as follows. Our problem is defined in section 2. In section
3, we show how to compute upper bounds on our problem via SDP. We give
special attention to the so-called full perturbations case, for which our results are
nonconservative. In section 4, we examine sensitivity of the robust solutions in the
full perturbations case. We provide conditions which guarantee that the robust solution
is unique and a regular function of the data matrices. We then consider several
interesting examples in section 5, such as robust linear programming, robust norm
minimization, and error-in-variables SDPs.
2. Problem definition.
2.1. Robust SDPs. Let F(x, #) be a symmetric matrix-valued function of two
variables x # R m
. In the following, we consider x to be the decision
variable, and we think of # as a perturbation. We assume that # is unknown but
bounded. Precisely, we assume that # is known to belong to a given linear subspace
D of R p-q , and in addition, #, where # 0 is given.
In section 2.2, we will be more precise about the dependence of F on #.
We define the robust feasible set by
for every # D, #,
F(x, #) is well defined and F(x, # 0
# .
(2)
Now let c(#) be a vector-valued rational function of the perturbation #, such that
We consider the following min-max problem:
c(#) T x subject to x # X # .
From now on, we assume that the function c(#) is independent of # (in other
words, the objective vector c is not subject to perturbation). This is done with no
loss of generality: introduce a slack variable # and define
Problem (3) can be formulated as
x subject to -
# is the robust feasible set corresponding to the function -
F.
In the following, we thus consider a problem of the form
subject to x # X #
and refer to it as a robust semidefinite problem (RSDP). In general, although X #
is convex, P # is not a tractable problem-in particular, it is not an SDP. Our aim
is to find a convex, inner approximation of X # that is described by a linear matrix
inequality constraint. This inner approximation is then used to find an upper bound
on the optimal value of P # by solving an SDP. In some cases, we can prove our results
are nonconservative, that is, as in the so-called "full perturbation" case.
We refer to the set X 0 (resp., problem P 0 , i.e., (1)) as the nominal feasible set
nominal SDP). We shall assume that the nominal SDP is feasible, that is,
course, the robust feasible set X # may become empty for some # > 0; we
return to this question later.
2.2. Linear-fractional representation. In this paper, we restrict our attention
to functions F that are given by a "linear-fractional representation" (LFR):
where F (x) is defined in (1), R(-) is an a#ne mapping taking values in R q-n , and
L # R n-p and D # R q-p are given matrices. Together, the mappings F (-), R(-), the
matrices L, D, the subspace D, and the scalar # constitute our perturbation model for
the nominal SDP (1).
The above class of models seems quite specialized. In fact, these models can
be used in a wide variety of situations, for example, in the case where the (matrix)
coe#cients F i in P 0 are rational functions of the perturbation. The representation
lemma, given below, and the examples of section 5 illustrate this point.
A constructive proof of the following result can be found in [37].
Lemma 2.1. For any rational matrix function M : R k
singularities
at the origin, there exist nonnegative integers r 1 , . , r k , and matrices M # R n-c ,
36 L. EL GHAOUI, F. OUSTRY, AND H. LEBRET
, such that M has the
following linear-fractional representation (LFR): For all # where M is defined,
Using the LFR lemma, we may devise LFR models for SDPs, where a perturbation
vector
enters rationally in the coe#cient matrices. The resulting set D of
perturbation matrices # is then a set of diagonal matrices of repeated elements, as
in (6). Componentwise bounds on the vector #, such as |#| i #,
into a norm-bound # on the corresponding matrix #.
2.3. A special case. We distinguish a special case for which exact (nonconser-
vative) results can be obtained via SDP. This is when F(x, #) is block diagonal, each
block being independently perturbed-precisely, when
where each F i (x, # i ) assumes the form shown in section 2.2 for appropriate L i , R i , D i ,
with consists of block-diagonal matrices of the form
# .
We refer to this situation as the block-full perturbation case. When
of the full perturbation case. As will be seen later, all results given for can be
generalized to the case L > 1.
3. Robust solutions for SDPs. Unless otherwise specified, we fix # > 0.
3.1. Full perturbations case. In this section, we consider the full perturbations
case, that is,
. We assume #D# -1 , which is a necessary and
su#cient condition for F(x, #) to be well defined for every x
#.
The following lemma is a simple corollary of a classic result on quadratic inequal-
ities, referred to as the S-procedure [12]. Its proof is detailed in [16].
Lemma 3.1. Let real matrices of appropriate size. We have
det(I -D#= 0 and
for every # 1, if and only if #D# < 1 and there exists a scalar # such that
A direct application of the above lemma shows that, in the full perturbations
case, the RSDP (4) is an SDP.
Theorem 3.1. When the RSDP (4) and a corresponding solution x
can be computed by solving the SDP in variables
subject to # F (x) - #LL T R(x) T
ROBUST SOLUTIONS TO UNCERTAIN SEMIDEFINITE PROGRAMS 37
Special barrier functions adapted to a conic formulation of the problem can be
devised and yield an interior-point algorithm that has the same complexity as the
nominal problem; see [24].
We may define the maximum allowable perturbation level, which is the largest
number # max such that X # for every #, 0 #
Computing # max is a (quasi-convex) generalized eigenvalue minimization
problem [24, 11]:
minimize # subject to # F (x) - #LL T R(x) T
Remark. The above exact results are readily generalized to the block-full perturbation
case (L > 1) as defined in section 2.2.
3.2. Structured case. We now turn to the general case (D is now an arbitrary
linear subspace). In this section, we associate with D the following linear subspace:
-R q-q
-R q-p
S#T , G# T G T for every # D # .
As shown in [16], a general instance of problem (4) is NP-hard. Therefore, we
look for upper bounds on its optimal value. The following lemma is a generalization
of Lemma 3.1 that traces back to [17]. Its proof is detailed in [16].
Lemma 3.2. Let real matrices of appropriate size. Let D be
a subspace of R p-q , and denote by B the set of matrices associated with D as in (11).
We have det(I -D#= 0 and
for every # D, # 1 if there exist a triple (S, T,
and
Using Lemma 3.2, we obtain the following result.
Theorem 3.2. An upper bound on the RSDP (4) and a corresponding solution
x can be computed by solving the SDP in variables x, S, T
subject to (S, T,
Note that when the perturbation is full, the variable G is zero and S, T are of the
form #I p , #I q , resp., for some # 0. We then recover the exact results of section 3.1.
As before, we may define the maximum allowable perturbation level, which is the
largest number # max such that X # for every #, 0 # max . Computing a
lower bound on this number is a (quasi-convex) generalized eigenvalue minimization
problem:
subject to (S, T,
38 L. EL GHAOUI, F. OUSTRY, AND H. LEBRET
4. Uniqueness and regularity of robust solutions. In this section, we derive
uniqueness and regularity results for the RSDP in the case of full perturbations. As
before, we first take . The results of this section
remain valid in the general case L > 1 (several blocks).
We fix #, 0 < # max . For simplicity of notation (and without loss of generality)
we take 1). For well-posedness reasons, we must assume
#D# < 1. We make the further assumption that #) is
a#ne in #). In section 4.5, we show how the case D #= 0 can be treated.
For full perturbations and the RSDP is the SDP
subject to # F (x) - #LL T R(x) T
4.1. Hypotheses. We assume that the SDP (15) (with satisfies the
following hypotheses:
H1. The Slater condition holds, that is, the problem is strictly feasible.
H2. The problem is inf-compact, meaning that any unbounded sequence
feasible points (if any) produces an unbounded sequence of objectives. An
equivalent condition is that the Slater condition holds for the dual problem
[28, p. 317, Thm. 30.4].
H3. (a) The nullspace of the matrix
x
is independent of (#, x) #= (0, 0) and not equal to the whole space.
(b) For every x,
# is full column-rank.
Hypotheses H1 and H2 ensure, in particular, the existence of optimal points for problem
(15) and its dual. Hypotheses H3(a) and (b) are di#cult to check in general, but
sometimes can be easily tested in practical examples, as seen in section 5. We note
that H3(a) implies that R(x) #= 0 for every x.
Hypothesis H1 is equivalent to Robinson's condition [27], which can be expressed
in terms of
# .
Robinson's condition is stated in [27] as the existence of x 0
R such that
where dF is the di#erential of F , and S
n+q is the set of positive semidefinite matrices
of q. The equivalence between H1 and Robinson's assumption is not true,
in general. Here, this equivalence stems from the fact that the problem is convex and
that the cone S
n+q has a nonempty interior.
Remark. Hypothesis H1 holds if and only if it holds for the nominal problem (1)
(recall our assumption # max > 1). Also, hypothesis H2 implies L #= 0 (otherwise, we
can let # without a#ecting the objective value). If H2 holds for the nominal
problem and L #= 0, then H2 holds for the RSDP (15).
4.2. An equivalent nonlinear program. Let x opt , # opt be optimal for (15).
Hypothesis H3(a) ensures that any # that is feasible for (15) is nonzero (otherwise,
R(x) would be zero for some x). We thus have # opt > 0.
We introduce some notation. For x
Using Schur complements and # opt > 0, we obtain that problem (15) can be
rewritten as
minimize d T y subject to G(y) # 0
and that y
Our aim is first to prove that the
so-called quadratic growth condition [10] holds at y opt for problem (16). Then, we
will apply the results of [10] to obtain uniqueness and regularity theorems.
4.3. Checking the quadratic growth condition. Following [10], we say that
the quadratic growth condition (QGC) holds at y opt if there exists a scalar # > 0 such
that, for every feasible y,
Roughly speaking, this condition guarantees that y opt is not on a facet on the boundary
of the feasible set.
Define the set of dual variables associated with y opt by
The following result is a direct consequence of a general result by Bonnans,
Cominetti, and Shapiro [10]. Roughly speaking, this result states that, if an optimization
problem satisfies Robinson's condition and has an optimal point, and if a
certain "curvature" condition is satisfied, then the QGC holds at that point.
Theorem 4.1. With the notation above, if H1 and H2 hold, and if
then problem (16) satisfies the QGC.
The following theorem is proven in appendix A.
Theorem 4.2. If H1-H3 hold, problem (15) satisfies the quadratic growth condition
at every optimal point y opt . Consequently, there exists a unique solution to the
SDP (15).
Remark. Note that the QGC is satisfied independent of the objective vector. This
means that the boundary of the feasible set is strictly convex (it contains no facets).
4.4. Regularity results. In problem (15), the data consist of the matrices L,
We seek to examine the sensitivity of the problem with
respect to small variations in F i , L i , and R i .
In this section, we consider matrices
are functions of class C 1 of a (small) parameter vector u. Define
x
We denote by P(u) the corresponding problem (15), where F (-), R(-), and L are
replaced by F (-, u), R(-, u), and L(u). We assume that F (-,
and L, so that P(0) is (15).
We first note that, in the vicinity of problem P(u) satisfies the hypotheses
H1 and H2 if P(0) does. In this case, for every # > 0 we may define the set S # (u) of
#-suboptimal points of P(u):
where v(u) is the optimal value of P(u).
Recall that, if P 0 satisfies hypotheses H1 and H2, the optimal value v(u) is con-
tinuous, and even directionally di#erentiable, at Thm. 5.1]. With the QGC
in force, and using [31, Thm. 4.1], we can give quite complete regularity results for
the robust solutions.
Theorem 4.3. If hypotheses H1-H3 hold for P(0), then for every
there exists a # > 0 and a neighborhood V of such that for every u # V and
When H1-H3 hold for P(0), the above theorem states that every (su#ciently)
suboptimal solution to P(0) is H-older-stable (with coe#cient 1/2). This is true, in
particular, for any optimal solution of P(u) (that is, for 0). The fact that the
theorem remains true for # > 0 guarantees regularity of numerical solutions to the
RSDP. The main consequence is that even if the nominal SDP is ill conditioned (with
respect to variations in the F i 's), the RSDP becomes well conditioned for every # > 0.
Now assume #= 1. We seek to examine the behavior of problem (10) (with
when the uncertainty level # for 0 < # max varies. This is a special case of
the problem examined above, with
Corollary 4.1. For every #, 0 < # max , the solution to (10) (with
is unique and satisfies the regularity results (written with #) of Theorem 4.3.
Remark. The results of this section are all valid in the block-full perturbation
case (L > 1), as defined in section 2.2. Of course, the conditions given in H3 should
be understood blockwise.
4.5. Case D #= 0. When D #= 0, we can get back to the case
Recall that we have #D# < 1 in order to ensure that F(x, #) is defined everywhere
on D. With this assumption, we can define, for x
Using Schur complements, we have, for every x and # > 0,
Hypothesis H3 holds for -
L, -
R(-) if and only if it holds for L, R(-). We can then follow
the steps detailed previously.
Corollary 4.2. If the SDP (10) (with
then the results of Theorem 4.3 hold.
5. Examples.
5.1. Unstructured perturbations. Assume
This case corresponds to the representation in section 5, with
x
Using Lemma 3.2, we obtain that problem (4) is equivalent to the SDP
subject to # F (x) - #I # 1 x T
#I
T#I #I
It turns out that we may get rid of the variable # and get back to a convex problem
of the same size as that of the unperturbed problem (1). To see this, first note that
every feasible variable # in problem (20) is strictly positive. Use Schur complements
to rewrite the matrix inequality in (20) as
Minimizing (over variable #) the scalar in the left-hand side of the above inequality
shows that the RSDP (1) is equivalent to
subject to F (x) #
Formulation (21) is more advantageous than (20), since (21) involves a (convex) matrix
inequality constraint of the same size as the original problem. As noted before, special
barrier functions can be devised for this problem and yield an interior-point algorithm
that has the same complexity as the original problem; see [24].
We note that, with the above choice for L, R, hypothesis H3 holds, which yields
the following result.
Theorem 5.1. The optimal value of the RSDP (20) can be computed by solving
the convex problem (21). If (21) satisfies hypotheses H1 and H2, then for every # > 0,
the solution is unique and satisfies the regularity conditions of Theorem 4.3.
Remark. A su#cient condition for hypotheses H1 and H2 to hold for (21) is that
they hold for the nominal problem. A more restrictive su#cient condition is that the
nominal feasible set X 0 is nonempty and bounded, and # max .
42 L. EL GHAOUI, F. OUSTRY, AND H. LEBRET
.3
x
Fig. 1. Nominal and robust solutions of an SDP, with a 5 - 5 matrix F (x). Here
5.2. Robust center of a linear matrix inequality. In this section, for # > 0,
we consider the SDP (21) and corresponding feasible (convex) set X # . We assume
that X 0 is nonempty and bounded, and that P 0 is strictly feasible. Then, for every #,
is nonempty and bounded, and we can define a (unique) solution
x(#) to the strictly convex problem (21).
In view of Corollary 4.1, x(#) is a continuous function of # in ]0 # max [. Since (X # )
is a decreasing family of bounded sets, we may define
x(#).
Note that x # is independent on the objective vector c.
Thus, to the matrix inequality F (x) # 0, we may associate the robust center,
defined by (22). The robust center has the property of being the most tolerant (with
respect to unstructured perturbation) among the feasible points.
An example is depicted in Fig. 1. The nominal feasible set X 0 is described by a
linear matrix inequality F (x) # 0, where F is a 5 - 5 matrix. For various values of #,
we seek to minimize x 2 . The dashed lines correspond to the optimal objectives. As #
increases, we observe that the robust feasible sets shrink. A crucial property of these
robust sets is that they do not possess any straight faces, as observed in the figure.
For the robust feasible set is a singleton (in this example, x
the optimal solution is not unique and not continuous with respect
to changes in the coe#cient matrices F i , (although the optimal value is
continuous). Since the sets X # become strictly convex as soon as # > 0, the resulting
robust solutions are continuous.
5.3. Robust linear programs. An interesting special case arises with linear
programming (LP). Consider the LP
subject to a T
Assume that the a i 's and b i 's are subject to unstructured perturbations. The perturbed
value of [a T
We seek a
ROBUST SOLUTIONS TO UNCERTAIN SEMIDEFINITE PROGRAMS 43
robust solution to our problem, which is a special case of the block-full perturbation
case referred to in section 2.2, with F given by (7), and
and D is the set of diagonal, L - L matrices. The robust LP is
subject to a T
The above program is readily written as an SDP by introducing slack variables. In
fact, the robust LP is a second-order cone program (SOCP) for which e#cient special-purpose
interior-point methods are available [24, 20, 23].
We note that hypothesis H3 holds blockwise. This yields the following result.
Theorem 5.2. The optimal value of the robust LP can be computed by solving
the convex problem (23). If the latter satisfies hypotheses H1 and H2, then for every
#, 0 < # max , the solution is unique and satisfies the regularity conditions of
Theorem 4.3.
In [6], robust linear programming is studied in detail. For a wide class of perturbation
models, where the data of every linear constraint vary in an ellipsoid, explicit
robust solutions are constructed using convex SOCPs. Reference [23] also provides
examples of robust linear programs solved via SOCP.
5.4. Robust eigenvalue minimization. Consider the case where the nominal
problem consists of minimizing the largest eigenvalue of a matrix-valued function
When F (-) is subject to unstructured perturbations (as defined in section 5.1), the
robust version of the problem is
subject to #I # F (x),
or equivalently
When written in an SDP form, the above problem satisfies the
hypotheses H1-H3. From Theorem 4.3 we obtain that the solution is unique. If we
consider that the data of the above problem consist of the matrices F i ,
then we know that the corresponding solution is H-older-stable (with coe#cient 1/2).
Since the problem is unconstrained, we can use a result of Shapiro [31, Thm. 3.1], by
which we conclude that the solution is actually Lipschitz stable (inequality (18) holds
with the exponent 1/2 replaced by 1). Finally, using the results from Attouch [3], we
can show that computing the solution for # 0 amounts to selecting the minimum
norm solution among the solutions of the nominal problem.
Theorem 5.3. The optimal value of the min-max problem (24) can be computed
by solving the convex problem (25). For every # > 0, the solution is unique and is
Lipschitz stable with respect to perturbations in F i , the
solution converges to the minimum norm solution of the nominal problem (24).
Remark. In this case, the RSDP is a regularized version of the nominal SDP, which
belongs to the class of Tikhonov regularizations [34]. The regularization parameter
44 L. EL GHAOUI, F. OUSTRY, AND H. LEBRET
is 2# and is chosen according to some a priori information on uncertainty associated
with the nominal problem's data. Taking # close to zero can be used as a selection
procedure for choosing a particular (minimum norm, regular) solution among the (not
necessarily unique and/or regular) solutions of the nominal problem.
Problem (25) is particularly suitable to the recent so-called U-Newton algorithms
for solving problem (24). These methods, described in [21, 25], require that the Hessian
of the "smooth part" (the so-called U-Hessian) of the objective of (24) be positive
definite. For general mappings F (-), this property is not guaranteed. However, when
looking at the robust problem (25), we see that the modified U-Hessian is guaranteed
to be positive definite for every x and # > 0. This indicates that the RSDP approach
may be used to devise robust algorithms for solving SDPs.
5.5. Robust SOCPs. An SOCP is a problem of the form
subject to #C
can be
formulated as SDPs, but special-purpose, more e#cient algorithms can be devised for
them; see [24, 5, 23].
Assuming that C i , d i , e i , f i are subject to linear-or even rational-uncertainty,
we may formulate the corresponding RSDP as an SDP. This SDP can be written as
an SOCP if the uncertainty is unstructured and a#ects each constraint independently.
The subject of robust SOCPs is explored in [5] in detail. Explicit SDPs that
yield robust counterparts to SOCPs nonconservatively are given for a wide class of
uncertainty structures. In some cases, albeit not all, the robust counterpart is itself
an SOCP. In [16, 14], the special case of least-squares problems with uncertainty in
the data is studied at length.
5.6. Robust maximum norm minimization. Several engineering problems
take the form
minimize #H(x)#,
where
are given p-q matrices. A well-known instance of this problem is
the linear least-squares problem, with Another example is a minimal
norm extension problem for a Hankel operator studied in [18], in which H 0 is a given
Hankel matrix and H i , is the n - n Hankel matrix
associated with the polynomial 1/z i . In practice, the matrices H i , are
subject to perturbation, which motivates a study of the robust version of problem (27).
Note that the least-squares case is extensively studied in [16].
Consider the full perturbation case, which occurs when each H i is perturbed independently
in a linear manner. Precisely, consider the matrix-valued function
For a given # > 0, we address the min-max problem
min
x
#H(x, #.
This problem is an RSDP for which we can get exact results using SDP. Indeed, for
every x # R m and # 0, the property
is equivalent to F(x, # 0 for every #, where
where
F (x, #I H(x)
x
I # .
We thus write problem (28) as (4), where the perturbation set D is R p-q .
Applying Theorem 3.2, we obtain that the RSDP above is equivalent to the
As in section 5.1, we may get rid of the variable # and
obtain the equivalent formulation
This RSDP satisfies hypotheses H1-H3, so we conclude that the results of Theorem 4.3
hold. As in robust eigenvalue minimization, we can get improved results using [31,
section 3, Thm. 3.1].
Theorem 5.4. The optimal value of the min-max problem (28) can be computed
by solving the convex problem (29). For every # > 0, the solution is unique and
Lipschitz stable with respect to perturbations in H i , the
solution converges to the minimum norm solution of the nominal problem (27).
Remark. As for the RSDP arising in robust eigenvalue minimization, the robust
minimum norm minimization problem is a regularized version of the nominal problem,
which belongs to the class of Tikhonov regularizations.
We now consider the general case where each matrix H i in (27) is perturbed in
a structured manner. To be specific, we concentrate on the minimal norm extension
problem mentioned above.
In practice, the matrix H 0 is obtained from measurement and is thus subject to
error. We may assume that this matrix is constructed from an n - 1 vector h 0
is unknown but bounded. The perturbed matrix H 0 is of the form
where L, R are given matrices (the exact form of which we do not detail), and
(In the above, each # i corresponds to the uncertainty associated with the ith antidiagonal
of H 0 .) We address the min-max problem
L. EL GHAOUI, F. OUSTRY, AND H. LEBRET
This problem is amenable to the robustness analysis technique. Defining
we obtain the following result.
Theorem 5.5. An upper bound on the objective value of the min-max problem
(30) can be computed by solving the SDP in variables x, S, G:
subject to
5.7. Polynomial interpolation. This example, taken from [16], can be formulated
as an RSDP with rational dependence. For given integers n # 1, k, we seek a
polynomial of degree that interpolates given points
(a
If we assume that (a i , b i ) are known exactly, we obtain a linear equation in the unknown
x, with a Vandermonde structure:
1 a 1 . a n-11 a k . a n-1
which can be solved via standard least-squares techniques.
Now assume that the interpolation points are not known exactly. For instance,
we may assume that the b i 's are known, while the a i 's are parameter dependent:
a
where the # i 's are unknown but bounded: |# i | #,
We seek a robust interpolant, that is, a solution x that minimizes
where
1 a k (#) . a k (#) n-1
# .
The above problem is an RSDP. Indeed, it can be shown that
where
. a n-2
and, for each i,
. a n-2
. a i
. a n-3
. a i
(Note that det(I - D#= 0, since D is strictly upper triangular.) With the above
notation, if we define F(x, #) as in section 5, then problem (31) can be formulated as
the RSDP (4).
With the approach described in this paper, one can compute an upper bound for
the minimizing value of (31), and a corresponding suboptimal x. We do not know if
the problem can be solved exactly in polynomial time, e.g., using SDP. We conjecture
(as the reviewers of this paper did) that the answer is no. To motivate this claim,
note that the solution to the problem of computing (31) for arbitrary a#ne functions
A is already NP-hard [16].
5.8. Error-in-variables RSDPs. In many SDPs that arise in engineering, the
variable x represents physical parameters that can be implemented with finite absolute
precision only. A typical example is integer programming, where integer solutions
to (linear) programs are sought. These problems (which are equivalent to integer
programming) are NP-hard. We now show that we may find upper bounds on these
problems using robustness analysis.
Consider, for instance, the problem of finding a solution x to the feasibility SDP
find an integer vector x such that F (x) # 0.
Now, consider the robust SDP
maximize # subject to
Assume there exists a feasible pair (x feas , #) to the above problem, with # 0. By
construction, x feas satisfies F Furthermore, any vector x chosen such that
is guaranteed to satisfy F (x) # 0. This is true, in particular,
for x int , the integer closest to x feas . Thus, if we know a positive lower bound #,
and corresponding feasible point for problem (33), then we can compute an integer
solution to our original problem.
Finding a lower bound for (33) and an associated feasible point can be done as
follows. For
Let
Rm
# , and
48 L. EL GHAOUI, F. OUSTRY, AND H. LEBRET
Problem (33) can be formulated as
maximize # subject to #I # F
for every # D, # 1/2.
The above is a special instance of the structured problem examined in section 3.2.
Theorem 5.6. A su#cient condition for an integer solution to the feasibility
SDP (32) is that the constraints
are feasible. If x feas is feasible for the above constraints, then any integer vector closest
to x feas (in the maximum norm sense) is feasible for (32).
6. Conclusions. In this paper, we considered semidefinite programs subject to
uncertainty. Assuming the latter is unknown but bounded, we have provided su#cient
conditions that guarantee "robust" solutions to exist via SDPs. Under some conditions
(detailed in section 4), the robust solution is unique, and not surprisingly, stable. The
method can then be used to regularize possibly ill-conditioned problems. For some
perturbation structures (as for unstructured perturbations), the conditions are also
necessary. That is, there is no conservatism induced by the method.
The paper raises several open questions.
In our description, we have considered the problem of making the primal SDP
robust, thereby obtaining upper bounds on an SDP subject to uncertainty. The
dual point of view should be very interesting. One might be interested in applying
the approach to the dual problem instead. Does this lead to lower bounds on the
perturbed problem? Also, in some cases, the RSDP approach leads to a unique (and
stable) primal solution. May we obtain a unique solution to the dual problem by
making the latter robust? (This would lead to analyticity of the primal solution;
see [32].)
As seen in section 5.2 the notion of robust center has, certainly, connections with
the well-known analytic center; is the latter related to some robustness characterization
It seems that the RSDP method could be useful for deriving fast and robust
(stable) algorithms for solving SDPs (see section 5.4), especially in connection with
maximum eigenvalue minimization.
Finally, as said in section 2.2 (Lemma 2.1), an SDP with coe#cient matrices
depending rationally on a perturbation vector can always be represented by an LFR
model. Now, this LFR model is not unique. However, the results given here (for
example, Theorem 3.2) hinge on a particular linear-fractional representation for a
perturbed SDP. Hence we have the question: are our results independent of the chosen
representation? We partially answer this di#cult question in Appendix B.
Appendix
A. Proof of Theorem 4.2. We take the notation of section 4.
diag(Z, -) be dual variables associated with are optimal
(their existence is guaranteed by H1 and H2). Then, Y # Y(y opt ). Let us show that
condition (17) holds for this choice of Y .
Since the problem satisfies H1 and H2, the complementarity conditions hold;
therefore, the (optimal) dual variable - associated with the constraint # opt is
zero. Consequently the variable Z is nonzero (recall c #= 0). Using
TrY
we obtain
From # opt #= 0 (implied by H3(a)), and using hypothesis H3(b), we can show that
are impossible for Z # 0, Z #= 0. This yields TrR(x opt ) T R(x opt )Z > 0.
Now let # R m and # R, and define
We have, for every i,
(R(x)
(R T
By summation, we have
(R(#)
opt
opt
R T
R+# opt
R(x opt ). We obtain finally,
opt
0 with means that every column of Z 1/2 belongs
to the nullspace of R(#) -R(0). Now assume #= 0. By hypothesis H3(a), we obtain
that every column of Z 1/2 also belongs to the nullspace of R(x opt ), which contradicts
50 L. EL GHAOUI, F. OUSTRY, AND H. LEBRET
We conclude that # 2
yy L is positive definite at (y opt , Y ).
Thus, problem (15) satisfies the QCG.
Appendix
B. Invariance with respect to the LFR model. In this section,
we show that the su#cient conditions obtained in this paper are, in some sense,
independent of the LFR model used to describe the perturbation structure.
Consider a function F taking values in the set of symmetric matrices having an
LFR such as that in section 5. This function can be written in a more symmetric
#)
where we have dropped the dependence on x for convenience, and
# .
It is easy to check that, if an invertible matrix Z satisfies the relation Z -
# for
every # D, then
#)
LZ) T .
In other words, the "scaled" triple
DZ)) can be used to represent F
instead of F, -
L, -
D in (35). By spanning valid scaling matrices Z, we span a subset of
all LFR models that describe F.
A valid scaling matrix Z can be constructed as follows. Let (S, T, G) # B, and
define
I
# .
It turns out that such a Z satisfies the relation Z -
# for every # D.
Using the above facts, we can show that if condition (13) is true for the original
LFR model F, L, R, D with appropriate S, T, G, then it is also true for the scaled LFR
obtained using any scaling matrix Z such as that above, for appropriate matrices -
T . That is, the condition is independent of the scaling Z.
In this sense, the conditions we obtained are independent of the LFR used to
represent the perturbation structure.
Acknowledgments
. This paper has benefitted from many stimulating discussions
with several colleagues, including Aharon Ben-Tal, Stephen Boyd, Arkadii Ne-
mirovski, Michael Overton, and particularly, Lieven Vandenberghe (who pointed out
a mistake just before the final version was sent). Last but not least, the authors would
like to thank the editor and reviewers for their very helpful comments and revisions.
--R
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Linear Matrix Inequalities in System and Control Theory
Introduction to convex optimization with engineering applications
New York
Robust solutions to least-squares problems with uncertain data
Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics
Minimal norm extensions and eigenstructures
Global and Local Convergence of Predictor-Corrector- Interior-Point Algorithm for Semidefinite Programming
Interior Point Polynomial Methods in Convex Program- ming: Theory and Applications
On homogeneous interior-point algorithms for semidefinite pro- gramming
Stability theorems for systems of inequalities
Convex Analysis
Discrete Event Systems
Perturbation theory of nonlinear programs when the set of optimal solutions is not a singleton
Perturbation analysis of optimization problems in Banach spaces
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Frank Lutgens , Jos Sturm , Antoon Kolen, Robust One-Period Option Hedging, Operations Research, v.54 n.6, p.1051-1062, November 2006
Budi Santosa , Theodore B. Trafalis, Robust multiclass kernel-based classifiers, Computational Optimization and Applications, v.38 n.2, p.261-279, November 2007
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Mung Chiang, Geometric programming for communication systems, Communications and Information Theory, v.2 n.1/2, p.1-154, July 2005 | uncertainty;robustness;regularization;semidefinite programming;convex optimization |
589118 | Towards a Practical Volumetric Cutting Plane Method for Convex Programming. | We consider the volumetric cutting plane method for finding a point in a convex set ${\cal C}\subset\Re^n$ that is characterized by a separation oracle. We prove polynomiality of the algorithm with each added cut placed directly through the current point and show that this "central cut" version of the method can be implemented using no more than 25n constraints at any time. | Introduction
Let C ae ! n be a convex set. Given a point -
separation oracle for C either reports
that -
returns a separating hyperplane a 2 ! n such that a T x ? a T - x for every x 2 C.
The convex feasibility problem is to use such an oracle to find a point in C, or prove that
the volume of C must be less than that of an n-dimensional sphere of radius 2 \GammaL , for given
It is well known [9] that a variety of convex optimization problems can be cast as instances
of the convex feasibility problem, and moreover the problem plays a fundamental role the
complexity analysis of many combinatorial optimization problems. For many years the
standard approach to the convex feasibility problem has been the ellipsoid algorithm; see for
example [5] or [9]. In [14], Vaidya proposed an alternative algorithm for the convex feasibilty
problem based on a new barrier for a polyhedral set, the volumetric barrier. On each iteration
algorithm has a point x a polyhedral set P
where A k is a m k \Theta n matrix. For each k the set P k is bounded, is an
approximation of the volumetric center of P k , the minimizer of the volumetric barrier (see
Section 2). The algorithm then either deletes one constraint that defines P k , or calls the
separation oracle to see if x k 2 C. If not, the oracle returns a separating hyperplane which is
used to add a constraint to P k . After the addition or deletion of a constraint, the algorithm
takes a number of Newton, or Newton-like, steps for the volumetric barrier to obtain a new
point x k+1 which is an approximation of the volumetric center of the new polyhedron P k+1 .
Let T represent the cost, in numerical operations, of a call to the separation oracle.
fundamental result is that the complexity of his volumetric cutting plane algorithm
for the convex feasibility problem is O(nLT +n 4 L) operations, compared to O(n 2 LT +n 4 L)
operations for the ellipsoid algorithm. (In theory, the complexity of Vaidya's method can be
further reduced through the use of "fast matrix multiplication," which cannot be applied to
the ellipsoid algorithm.) Although Vaidya's result is theoretically significant, the algorithm
of [14] does not appear to very practical. In particular, the analysis of [14] requires that the
polyhedral sets P k have up to 10 7 n constraints, and the algorithm might require thousands
of Newton-like steps following the addition or deletion of a constraint.
In [3], Anstreicher describes a strengthened version of Vaidya's volumetric cutting plane
algorithm for the convex feasibility problem. The algorithm of [3] reduces the maximum
number of constraints to 200n, while requiring no more than 5 Newton steps following a
constraint addition or deletion. Although these figures represent a substantial improvement
over [14], the algorithm of [3] is still not fully practical. In particular:
i. For reasonable n, 200n constraints is still quite large, given that least-squares systems
with this number of rows must be repeatedly solved on each iteration.
ii. The algorithm of [3] uses true Newton steps, which in practice are expensive to compute
compared to the Newton-like steps used in [14].
iii. As in [14], the algorithm of [3] cannot place a new constraint directly through the
current point, but must rather "back off" each separating hyperplane to generate a
shallow cut.
Ramaswamy and Mitchell [13] describe a "central cut" version of the volumetric cutting
plane algorithm that allows for the placement of each new constraint through the current
point, and uses Newton-like steps following constraint additions and deletions. (The algorithm
of [13] actually solves the problem of minimizing a linear function over a convex set C
using a separation oracle, but most of the analysis is very similar to that required to solve the
convex feasibility problem.) Unfortunately [13], which uses many results from [14], requires
that the algorithm maintain up to 10 8 n constraints.
The purpose of this paper is to develop a central cut volumetric cutting plane algorithm
that also improves on the 200n constraints required by the algorithm of [3]. As in [13],
the algorithm uses an "affine" step to move off of a cut placed through the current point.
The use of such a step in the context of a cutting plane method based on analytic centers
is well known [7]. (See [4], [8], [10], [11], [12], and [15] for other results on analytic center
cutting plane methods.) In fact the affine step we use is based on that of [7], rather than the
step used in [13]. Our analysis uses a number of results from [2] and [3], and an improved
second-order expansion of the volumetric barrier, to improve upon the analysis of [13]. As
in [13], the method described here requires O(
n) Newton-like steps following the addition
or deletion of a constraint, compared to O(1) Newton, or Newton-like, steps in [14] and [3].
Although this is certainly a disadvantage from the standpoint of theoretical complexity, the
fact that the O(
n) bound arises from a worst-case analysis of descent in the volumetric
barrier suggests that in practice far fewer steps would likely be required. Our final result is
a central cut volumetric cutting plane method that requires no more than 25n constraints
at any time.
In
Table
1 we summarize important attributes of four papers (including this paper) on
volumetric cutting plane methods. These features are the placement of added cuts (shallow
or central), the number of Newton or Newton-like steps required after a constraint addition
or deletion, the maximum number of constraints required, and the value of a scalar \DeltaV ,
defined as the difference between the mimimal increase in the volumetric barrier following a
constraint addition, and the maximal decrease following a constraint deletion (see Section 3).
For all four algorithms the number of oracle calls is O(nL), with a constant that is inversely
proportional to \DeltaV (see for example the proof of [3, Theorem 3.2]).
Reference Placement Steps after Number of \DeltaV
of cut addition/deletion constraints
Anstreicher [3] Shallow O(1) 200n 3:7 \Theta 10 \Gamma4
Ramaswamy and Mitchell [13] Central O(
This Paper Central O(
n) 25n 1:4 \Theta 10 \Gamma3
Table
1: Attributes of volumetric cutting plane methods
2 The Volumetric Barrier
In this section we collect a number of properties of the volumetric barrier V (\Delta) which will be
used in the subsequent analysis. To start, let A is an m \Theta n
. Whenever we refer to P, we are implicitly refering to the constraint
system [A; b] which defines it. The volumetric barrier for P is the function
x be a point having
denote the vector equal to the diagonal of the projection
In other words oe m. It is easy to show (see for example the Appendix of
[1]) that 0 - oe - e, e T is the vector with each component equal to one.
The gradient and Hessian of V (\Delta) at x are then given by
denotes the Schur, or Hadamard product of P with itself:
ij . Let good approximation of H(x), in that
where - denotes the ordering for positive semi-definite matrices defined by A - B ()
A is positive semidefinite. See for example the Appendix of [1] for a derivation of (1)
and a proof of (2); these and other properties of V (\Delta) are originally due to Vaidya [14].
In the sequel we will often be interested in the behavior of V (\Delta) for a step of the form
1. For such an -
Q(-x). The proof of the following is very straightforward; see for example [14, Lemma
5] or [1, Lemma 2.2].
Proposition 2.1 Let -
It follows immediately from Proposition 2.1 that if -
Using a Taylor series expansion, (2), and (3), it is easy to show that
The bounds in (4) have been used in [1], [13], and [14]. The following theorem provides a
strengthening of (4) that will be used throughout the paper.
Theorem 2.2 Suppose that -
Proof: We have
dff
To prove the lemma we will obtain lower and upper bounds on the final term in (5). We
begin with the lower bound. Using (2) and (3), we have
and therefore
But it is straightforward to compute that
d
and therefore
Substituting (8) into (7), we obtain
Z 1ff
An integration by parts shows that
Z 1ff
and substituting (10) into (9) produces the lower bound of the lemma. The proof of the
upper bound is similar. Again using (2) and (3), we have
and therefore
But
d
and therefore
Substituting (12) into (11), we obtain
Z 1ff
Another integration by parts shows that
Z 1ff
and substituting (14) into (13) produces the upper bound of the lemma. 2
Since the bounds in Lemma 2.2 involve both k-k
to consider how these two quantities are related. For x with
g. Note that oe min ? 0 under the trivial assumption that A contains no zero
row. Define
In the following lemma we give two bounds for kS \Gamma1 A-k1 in terms of k-k Q ; one involving
-, and therefore oe min , and the other independent of oe.
Theorem 2.3 Let x have
2. kS
m)=2] 1=2 k-k Q .
Proof: See [1, Theorem 3.3] for the proof of 1, and [3, Lemma 2.3] for the proof of 2. 2
Motivated by Theorem 2.3, we define
It then follows from Theorem 2.3 that
The fundamental proximity criterion that we employ throughout the paper is -
this quantity is "large" (that
is,\Omega\Gamma,/23 we will take a damped Newton-like step in an effort to
reduce V (\Delta), and thus move closer to !, the volumetric center of P. When -
we will be close enough to ! to adequately control the effect of adding or deleting a constraint,
as required. The following theorem and corollary obtain a simple condition on -
suffices to demonstrate the boundedness of P.
Theorem 2.4 Let where the columns of A are independent. Let x have
g. Suppose that kS
Proof: P is bounded if and only if
6 9x 6= 0, Ax - 0. Since the columns of A are independent,
ve T
where the third equivalence uses a standard "Theorem of the Alternative" for systems of
linear inequalities. But is exactly A T S \Gamma2
can be written
A
It follows that if kS
is bounded. 2
Corollary 2.5 Let where the columns of A are independent. Let x have
suppose that -
Proof: This follows from Theorem 2.4, (17), and the fact that show that if -
sufficiently small, then we can bound the possible
remaining decrease in V (\Delta). The proximity allowed in Theorem 2.6, -
weaker than in previous, similar results in [14], [2], and [3].
Theorem 2.6 Let x have
where ! is the minimizer of V (\Delta), and
Proof: Assume that -
1. Then there is a -,
has -
from (17), we have
d
dff
where the inequality uses (6), and the final equality uses (8), both with 1. Using the
fact that jg T
we then obtain
d
dff
where the last inequality uses and the assumption that -
V (\Delta) is strictly convex, it follows that V (!) ? V (-x), which is a contradiction. Therefore
1, from (17), so Theorem 2.2 implies that
where the second inequality uses jg T -j - kgk Q and the final inequality uses -
1, and the assumption that -
In the corollary below we use Theorem 2.6 to establish bounds on V for two
values of the parameter fl which are useful in the sequel.
Corollary 2.7 Let x have
Proof: For is straightforward to show that the minimum in (20), for 0 - ff - 1,
occurs at
2. Substituting this value of ff into (20), and simplifying, then implies
that
It is also easy to show that for fl - 4=27, the right-hand side in (20) is monotonically
decreasing for 0 - ff - 1. Substituting
final topic that we consider in this section is that of reducing V (\Delta) when -
\Omega\Gamma/54 To accomplish this we use a Newton-like step of the form
for some ff ? 0. When -
=\Omega\Gamma338 it can be shown that ff may be
chosen in (21) so that an
m) reduction is obtained in V (\Delta). In the following lemma
we give a result for a particular value of fl used in the sequel.
Lemma 2.8 Suppose that x has
x(ff) be as in (21).
m.
Proof: Let construction we have -
by (17). Applying Theorem 2.2, we obtain
Substituting using -
. Finally from (16), -
m)=2 -
m. 2
3 The Algorithm and its Complexity
In this section we describe the central-cut volumetric cutting plane method, and establish its
complexity using results from the two following sections. At the start of each iteration k - 0,
we have an interior point x k of a bounded polyhedron P k oe C, where
and A k is an m k \Theta n matrix with independent columns. We assume that C is contained in
the hypercube kxk1 - 1, and set P is straightforward to
show that x 0 is the volumetric center of P 0 .) The algorithm to be analyzed is as follows:
Central-Cut Volumetric Cutting Plane Algorithm
1. Go to Step 1.
Step 1. If V k
Else go to Step 2.
Step 2. If oe k
min - ffl, go to Step 3. Else go to Step 4.
Step 3. (Constraint Addition) Call the oracle to see if x k 2 C. If so, STOP. Otherwise
the oracle returns a vector a 2 ! n such that a T x ? a T x k for all x 2 C. Let
be an augmented constraint system having a k+1
ff ? 0. Go to Step 5.
Step 4. (Constraint Deletion) Suppose that oe k
be the reduced
constraint system obtained by removing the j'th row of
Go to Step 5.
Step 5. (Centering Steps) Take a sequence of damped Newton-like steps of the form -
-x J ). Let x
to Step 1.
In Step 1 of the algorithm, the value of V k
max is such that V
proves that the
volume of P k , and therefore also C ae P k , is less than that of an n-dimensional sphere of
radius 2 \GammaL . An explicit value for
is given in Lemma 3.1, below. A suitable steplength
ff in Step 3 is given in Theorem 4.6. Note that by construction each P k is bounded, by
Corollary 2.5, since -
each k. In addition, the fact that a constraint
is only added if oe k
For the Newton-like steps in Step 5, we assume that the steplengths ff are chosen so that
each step produces an
n) decrease in V k (\Delta). That this is always possible follows from
\Omega\Gamma17 (see for example Lemma 2.8), and the fact that m k -
O(n).
Lemma 3.1 Consider the volumetric cutting plane algorithm with fl - :03. Assume that
termination in Step 1 proves that the volume
of C is less than that of an n-dimensional sphere of radius 2 \GammaL .
Proof: See [3, Lemma 3.1]. 2
Next we consider the issue of how many iterations might be required for the algorithm
to terminate. Assume that each time a constraint is added the algorithm achieves
while each time a constraint is deleted it is assured that
. The following theorem provides a complexity result for the
algorithm under simple assumptions regarding \DeltaV and the number of
Newton-like steps taken in Step 5.
Theorem 3.2 Assume that the iterates of the volumetric cutting plane algorithm, using fl -
:03, satisfy (23) and (24) on iterations where a constraint is added or deleted, respectively.
Assume further that \DeltaV + is O(1),
is\Omega\Gamma/4 , and the number of
Newton-like steps in Step 5 of the algorithm is always O(
n). Then for L
=\Omega\Gamma47 (n)) the
algorithm terminates in O(nL) iterations, using a total of O(nLT +n 4:5 L) operations, where
T is the cost of a call to the separation oracle.
Proof: See the proof of [3, Theorem 3.2]. 2
Compared to the algorithms of [14] and [3], Theorem 3.2 demonstrates that the central cut
method of this paper has the same order number of oracle calls, O(nL), but performs more
non-oracle work; O(n 4:5 L) versus O(n 4 L) operations. The reason for this is the larger number
of centering steps, O(
n) versus O(1), required after a constraint addition or deletion. Using
results from the next two sections, we now show that the assumptions of Theorem 3.2 hold
for certain choices of the parameters ffl and fl.
Theorem 3.3 Let :01. Then the central-cut volumetric cutting plane method
satisfies the assumptions of Theorem 3.2, with
Proof: First consider an iteration where a cut is added. In Theorem 4.6, it is shown that
for a particular choice of ff in Step 3 it is assured that
where we are using the fact that kg k k (Q
In addition, in Theorem
4.5 it is shown that for kg k k (Q
is the volumetric center of P k+1 . Combining (25) and (26) we obtain
Next, in Lemma 2.8 it is shown that if -
in Step 5, then a steplength
ff may be chosen so that
together imply that after
n)
steps we must obtain -
x J having -
Finally, V k+1
implies that
Next consider an iteration where a constraint is deleted. In Lemma 5.2 it is shown that
dropping constraint i to obtain a new
polyhedron P k+1 results in V k+1 Arguing exactly as above, it
follows that after
n) steps in Step 5, we must obtain - x J with -
In addition, in Lemma 5.2 it is shown that V k+1 (! k+1 must
have
The assumptions of Theorem 3.2 thus hold with
The value demonstrated in Theorem 3.3 may seem relatively small, but it
should be noted that this is the largest value of \DeltaV to date for a volumetric cutting plane
algorithm; see Table 1 in Section 1.
4 Adding a Central Cut
Let x be an interior point of P. In this section we consider augmenting the constraint system
defining P by imposing a central cut through x, to obtain a new polyhedron ~
b; a T ~
x - a T xg. Let ~
V (\Delta) be the volumetric barrier for ~
! the volumetric center. Note
that for any -
x with -
x ? a T x, we have
~
(a T -
I
(a T -
(a T -
We will first use (29) to establish a lower bound on ~
a cut is added through
x. We will obtain two versions of this result. The first, using produces a relatively
simple bound for the fundamental quantity ~
(!). Although this bound may be
of some independent interest, in practice it cannot be used since
Therefore we will also obtain a lower bound using x in a certain neighborhood of !. We
begin with a series of lemmas. Throughout we let
Lemma 4.1 Assume that -
x ? a T x, and k -
a T
(a T -
Proof: Consider the problem
a T -
Letting -
can be written as
a T -
the solution of which is
a T
with solution value equal to ae
a T
It follows that
a T
(a T -
- a T
ae 2 a T
:Lemma 4.2 Assume that -
and -r - 1. Then
a T
(a T -
immediately implies that
oe min
and also, from Lemma 2.3, that
From (32) and Proposition 2.1 it follows that
Then (31) and (33) together imply that
oe min
and the lemma follows from Lemma 4.1. 2
Lemma 4.3 Assume that -
Proof: This follows immediately from (32), and the lower bound of Lemma 2.2. 2
Now let the volumetric center of P. We will use Lemmas 4.2 and 4.3 to establish
a lower bound on ~
P is obtained by placing a central cut through !.
Theorem 4.4 Suppose that ! is the volumetric center of P,
~
!g. Let ~
V (\Delta) be the volumetric barrier for ~
! the volumetric
center. Then ~
assume for the moment that
-(!). Using (29), Lemmas 4.2 and 4.3, and the fact that
~
Next we use the fact that ln(1 to obtain
where the second inequality uses - 1, and
and the final inequality uses Substituting (35) into (34) then gives
~
A straightforward differentiation shows that the minimum of the right-hand side of (36), for
occurs at From (36) we then have
Next assume that r ? 1=-. Then there is an ff 2 (0; 1) so that -
1=-. From the convexity of V (\Delta), and Lemma 4.3, we obtain
and (29) certainly implies that ~
It follows that ~
oe min )=8. 2
It is worthwhile to mention that the analysis in [13, Section 4.1] actually shows that
~
oe min ), although the authors of [13] do not note this fact. In practice
the added cut a T ~ x - a T x cannot be passed through as in Theorem 4.4, but rather
through a point x which is close to ! in some sense. As a result the lower bound of Theorem
4.4 must be modified to account for the use of x 6= !. In the next theorem we give a result
based on particular parameter choices used throughout the paper.
Theorem 4.5 Let x have
~
V (\Delta) be the volumetric barrier for ~
! the volumetric
center. Then ~
assume for the moment that
Proceeding as in the proof of Theorem 4.4, but including the effect of
~
where the second inequality uses the fact that jg T We distinguish
two cases.
Case 1: oe min - :04725. Note that 1=-
monotonically increasing
in oe min , so oe min - :04 implies that 1=- 2 - 2
In addition, - 2 oe
oe min =(2
monotonically increasing in oe min , so oe min - :04 also implies that
Finally oe min - :04725 implies that - (2
1:606. Using these facts in (37), and the assumption that kgk Q
~
It can be verified numerically that the minimum of the right-hand side in (38), for
occurs at approximately with value greater than :0340. (See Figure 1, Case 1 for a
plot of the right-hand side of (38), for
Case 1
Case 2
Figure
1: Lower bound on ~
Case 2: oe min - :04725. In this case we have 1=-
Using these facts in (37), with - 1
and the assumption that kgk Q
~
It can be verified numerically that the minimum of the right-hand side in (39), for
occurs at approximately with value greater than :0340. (See Figure 1, Case 2 for a
plot of the right-hand side of (38), for
This completes the proof under the assumption that r - 1=-. However, arguing as at
the end of Theorem 4.4, it is easy to show that if k~! \Gamma xkQ ? 1=-, then
~
:For the final topic of the section, we consider moving off of the cut a T ~
x - a T x to a
new point -
x having a T -
x ? a T x. Our goal is to obtain an upper bound for the quantity
~
Consider a point of the form
a T
1. Note that -
x in (40) is based on A T S \Gamma2 A, the Hessian of the logarithmic
barrier at x, and not Q as used in [13, Section 4.1.2].
Theorem 4.6 Suppose that x has
x - a T xg,
and let ~
V (\Delta) be the volumetric barrier for ~
P. Then using
x having
~
Proof: By construction we have a T - x \Gamma a T
a T
It follows that
a T
(a T -
where the last inequality uses (41) and Proposition 2.1. Let -
x. Then from (41) and
Lemma 2.2 we have
where the last inequality uses the facts that Q - A T S \Gamma2 A, and - T A T S \Gamma2 Combining
(29), (42), and (43), we obtain
~
The proof is completed by substituting
5 Dropping a Constraint
In this section we consider the effect of dropping a constraint, as in Step 4 of the algorithm.
For simplicity we assume that oe
P be the new constraint system obtained
by deleting the mth constraint in the original system [A; b] defining P. Throughout we use
the tilde (~) notation to denote quantities related to the reduced constraint system [ ~
Theorem 5.1 Suppose that x has
P is obtained by deleting
the mth constraint defining P. Then
1. ~
2. oe i - ~
3. k~gk ~
Proof: See [3, Lemma 5.1, Lemma 5.2, and Theorem 5.3]. 2
We will use Theorem 5.1 to bound the change in our fundamental proximity measure
following the deletion of a constraint. We use - ~
-(x) to denote the value of -
with respect to the reduced constraint system [ ~
Theorem 5.2 Assume that x has
Let ~
P be obtained by deleting the mth constraint defining P. Then ~
P is bounded, ~
~
! is the volumetric center of ~
P.
Proof: Note that - ~
-, from part 2 of Theorem 5.1, and the fact that ~
Applying part 3 of Theorem 5.1, we obtain
-k~gk ~
min
where the second inequality uses the assumption that -
It is clear that the
right-hand side of (45) is increasing in oe min , and substituting oe results in
-k~gk ~
Assume for the moment that - ~
- :833-. Then (46) implies that - ~
-k~gk ~
~
from Corollary 2.7. Alternatively assume that - ~
- :833-. Since in any case - ~
implies that - ~
-k~gk ~
In addition, oe min - :04 implies - (2
From Corollary 2.7 we then have
~
so in all cases ~
as claimed. In addition, we have
~
and part 1 of Theorem 5.1 gives
~
Then ~
:0326 follows from (47), (48), and ~
6 Conclusion
From a practical standpoint, this paper gives the best result to date for a cutting plane
method for the convex feasibility problem based on the volumetric barrier. From the stand-point
of theoretical complexity, the most interesting open problem is the use of central cuts
with the volumetric barrier, while requiring only O(1) Newton (or Newton-like) steps following
the introduction of a cut, as is possible when shallow cuts are employed ([14], [3]).
Although the affine step (40) is sufficient to obtain an O(1) bound on ~
as in Theorem
4.6, this bound is too weak relative to - oe min to show that O(1) steps suffice to return to
a suitable proximity of the new volumetric center ~ !. As a result it becomes necessary to use
a proximity measure based on - in place of -, leading to a worst-case decrease of
n)
instead of
\Omega\Gamma/1 in the steps on Step 5 of the algorithm. In practice the algorithm might
of course do much better than these worst-case bounds indicate, but serious computational
work using the volumetric barrier has not yet been conducted.
For the analytic center cutting plane method it is relatively easy to show that O(1) steps
suffice to return to a suitable proximity of the new analytic center following the addition of
a central cut [7]. (The basic analytic center cutting plane method is not a polynomial time
algorithm, however. To date the only polynomial cutting plane algorithm based on analytic
centers, due to Atkinson and Vaidya [4], uses shallow cuts.) The complexity analysis for
the analytic center cutting plane method can also be extended to multiple cuts ([11], [15]),
and deep cuts ([6], [8]). Similar results for the volumetric cutting plane method would
be desirable. In [13] a result allowing multiple cuts is developed, but in addition to the
very small constants required throughout [13], the multiple cut result requires a "Selective
Orthonormalization" procedure that weakens the original cuts in the interest of constructing
a feasible affine step.
--R
"Large step volumetric potential reduction algorithms for linear programming,"
"Volumetric path following algorithms for linear program- ming,"
"On Vaidya's volumetric cutting plane method for convex programming,"
"A cutting plane algorithm for convex programming that uses analytic centers,"
"The ellipsoid method: a survey,"
"Using the primal dual infeasible Newton method in the analytic center method for problems defined by deep cutting planes,"
"Complexity analysis of an interior point cutting plane method for convex feasibility problems,"
"Shallow, deep, and very deep cuts in the analytic center cutting plane method,"
Geometric Algorithms and Combinatorial Optimization
"Complexity of some cutting plane methods that use analytic cen- ters,"
"Analysis of a cutting plane method that uses weighted analytic centers and multiple cuts,"
"Complexity estimates of some cutting plane methods based on analytical barrier,"
"A long step cutting plane algorithm that uses the volumetric barrier,"
"A new algorithm for minimizing convex functions over convex sets,"
"Complexity analysis of the analytic center cutting plane method that uses multiple cuts,"
--TR | convex programming;cutting plane method;volumetric barrier |
589127 | An Interior-Point Approach to Sensitivity Analysis in Degenerate Linear Programs. | We consider an interior-point approach to sensitivity analysis in linear programming developed by the authors. We investigate the quality of the interior-point bounds under degeneracy. In the case of a special type of degeneracy, we show that these bounds have the same nice asymptotic relationship with the optimal partition bounds as in the nondegenerate case. We prove a weaker relationship for general degenerate linear programs. | Introduction
Sensitivity analysis (or post-optimality analysis) is the study of how the optimal solution
of an optimization problem changes with respect to the changes in the problem
data. The possible presence of errors in the problem data often makes sensitivity
analysis as important as solving the original problem itself.
In the context of linear programming (LP), sensitivity analysis can be performed
using an optimal basis approach (as in the simplex method) or an optimal partition
approach, where the optimal partition refers to knowing, for each index, whether the
corresponding component of an optimal primal solution or of an optimal dual slack
vector can be positive. The latter approach has close connections with interior-point
methods since such methods, when properly terminated, provide an optimal solution
in the relative interior of the optimal face, from which the optimal partition is readily
available. In fact, as will shortly be discussed in detail, the optimal partition approach
has been developed by Adler and Monteiro [1] and Jansen, de Jong, Roos and
[7] as a promising alternative in order to circumvent the drawbacks of the classical
optimal basis approach in the presence of degeneracy. Later, Monteiro and Mehrotra
[9] extended this approach by relaxing the requirement that the optimal partition be
known. They also provided two methods to estimate the range of perturbations, each
of which can be performed at any optimal solution, regardless of where it lies on the
optimal face. More recently, Greenberg, Holder, Roos and Terlaky [5] related the
dimension of the optimal set to the dimension of the set of objective perturbations for
which the optimal partition is invariant. Greenberg [4] considered the simultaneous
perturbations of the right-hand side and the cost vectors from an optimal partition
perspective.
Recently, the authors studied perturbations of the right-hand side and the cost
parameters in linear programming [12], motivated by how interior-point methods from
a near-optimal pair of strictly feasible solutions for a problem and its dual would
compare with the optimal basis approach obtained from a nondegenerate optimal basic
solution for such perturbations. The proposed interior-point perspective stems from the
objectives of regaining feasibility and maintaining near-optimality in a single iteration
of the interior-point method. This requires the setup of the "right" Newton system
among many possible choices in order to achieve both objectives simultaneously. Such
a perspective provides a basis for the comparison of the interior-point and the simplex
approaches to sensitivity analysis.
Under the assumption of a unique, nondegenerate optimal solution, the authors
showed that the Newton system proposed in [12] is the "right" one in the sense that it
yields asymptotically the same bounds on perturbations as those that keep the current
basis optimal (after symmetrization with respect to the origin). Similar results, but
changing only one of the primal or dual near-optimal solutions, were obtained by Kim,
Park and Park [8].
However, most LPs arising from real-life problems are degenerate. Our goal in this
paper is to investigate the quality of the bounds from the interior-point perspective in
the absence of the strong assumption of nondegeneracy. This will lead to a complete
analysis of the interior-point perspective proposed in [12]. In doing so, we need something
to compare our interior-point bounds with. In contrast to the nondegenerate
case, the presence of multiple optimal bases makes a simplex-based approach unsuit-
able, as will be explained shortly. We therefore compare our bounds to those obtained
from considering how much the right-hand side or the cost vector can change while
maintaining the same optimal partition. Consequently, we use completely different
tools for our analysis in this paper.
The next section is devoted to the preliminaries including the introduction of the
tools relevant for the analysis as well as the restatement of our interior-point approach.
Section 3 discusses the equivalence between the primal and dual formulations and shows
that it suffices to consider perturbations of the right-hand side only. We analyze the
interior-point bounds under a special case of degeneracy in Section 4 and extend the
analysis to the general degenerate case in Section 5. We present and discuss some
computational results in Section 6 and Section 7 concludes the paper.
Preliminaries
We consider the LP in the following standard form:
c T x; subject to
The associated dual LP is given by
subject to A T y
constitute the data, and (x;
are the decision variables. Throughout this paper, the coefficient matrix A will be
fixed and we will consider one-dimensional perturbations of the right-hand side vector
b and the cost vector c, i.e., b will be replaced by b + t\Deltab and c by c
and \Deltac will be fixed in IR m and IR n , respectively, and t 2 IR will be the parameter.
This is also called parametric analysis in the literature.
We will make the following assumptions:
1. The coefficient matrix A has full row rank.
2. Both (P) and (D) have strictly feasible solutions, i.e., there exist x ?
and y such that
The classical approach to sensitivity analysis has been based on the simplex method.
Assuming that an optimal solution exists, the simplex method terminates with a basic
optimal solution along with a corresponding basis. A natural criterion for the allowable
perturbations in the data is then given by the following: how much perturbation in the
data can one allow so that the current basis remains optimal for the perturbed LP?
Let us consider the parametric right-hand side (RHS) problem, i.e., let b be replaced
0g. It is well-known that v
is a convex, piecewise linear, continuous function of t. The parametric RHS problem
includes finding out all the "breakpoints" of v(t).
Fixing a value of t, say at 0 for the purposes of this paper, the classical approach to
sensitivity analysis then provides the set of values of t for which an optimal basis for
remains optimal for the resulting LPs parametrized by t. This is called the optimality
interval associated with an optimal basis. Note that the optimal basis approach
indeed yields the breakpoints of v(t) around 0 under primal and dual nondegeneracy
(which holds only if 0 itself is not a breakpoint of v(t)). However, the presence of
primal and/or dual degeneracies is a shortcoming for this approach since, for example,
multiple optimal bases might yield different optimality intervals. This shortcoming has
been observed by several researchers. Adler and Monteiro [1], and Jansen, de Jong,
Roos and Terlaky [7] developed an optimal partition approach to sensitivity analysis
and showed that the optimality intervals associated with the optimal partitions
uniquely and unambiguously identify the breakpoints of v(t) and the intervals between
the consecutive breakpoints. By the symmetry between (P) and (D), which will be
treated in more detail in Section 3, the same conclusions also hold for the parametric
analysis of the cost vector c.
The idea of the optimal partition is based on a well-known result of Goldman and
Tucker [2]. The optimality conditions for (P) and (D) are given by primal and dual
feasibility and complementary slackness, that is, a triple (x; y; s) is optimal for (P) and
(D) if and only if it satisfies
where x i and s i denote the ith components of x and s, respectively.
and\Omega D
denote the set of optimal solutions for (P) and (D), respectively. Then, we can define
two index sets as
ng
2\Omega D g: (2.2)
The optimality conditions (2.1) imply that B " ;. The Goldman-Tucker result
indicates that B and N actually partition the index set
ng. Therefore, there exist at least one primal solution x
2\Omega P and one dual
solution
2\Omega D such that x Such a solution will be called strictly
complementary and B and N will be called the optimal partition. In contrast to the
possibility of multiple optimal bases, the optimal partition is unique for a given LP
instance.
We will denote by B and N the columns of A corresponding to the indices in B
and N , respectively, and we will also partition the cost vector c as c B and c N , and the
variables x and s as xB and xN , and s B and s N accordingly. Note that if (x; y; s) is a
strictly complementary solution, then we have xB ? 0,
Let us again restrict our attention to one-dimensional perturbations of the right-hand
side vector b. The optimal partition approach is based on maintaining the whole
dual optimal set invariant rather than an optimal basis as in the classical simplex
approach. Note that perturbations of b do not affect the dual feasible region. Conse-
quently, the range of t is given by solving two auxiliary LPs. More precisely, if b is
replaced by b + t\Deltab, and
if\Omega D denotes the dual optimal set for (D) (i.e.,
the lower and upper bounds on t are given by the optimal values of
subject to
We will call the resulting bounds the optimal partition bounds. Note that both problems
are always feasible since together with any x
2\Omega P satisfy all the constraints.
Fixing the dual optimal
set\Omega D is equivalent to fixing the optimal partition B and N
by the Goldman-Tucker result. Therefore, the (possibly infinite) last constraint set in
(AUX1) can be replaced by the equivalent single constraint x T s
point in the relative interior
of\Omega D (hence s
This condition, in turn, is the same
as setting Consequently, (AUX1) can be written in the following simplified
subject to
The analogous derivation for the one-dimensional perturbations of the cost vector c
leads to the following auxiliary problems, whose optimal values give the optimal partition
bounds for t when c is replaced by c
subject to
Here, \Deltac B and \Deltac N constitute the corresponding partition of \Deltac.
Before getting into the symmetrized bounds we would like to recall an important
result about the dimensions of the optimal solution
and\Omega D . In what follows,
dim(\Delta) denotes the dimension and j \Delta j denotes the cardinality of a set. The reader is
referred to Lemma IV.44 in [10] for a proof.
Proposition 2.1
dim(\Omega
dim(\Omega
2.1 Symmetrized Bounds
The auxiliary problems (AUX1) and (AUX2) can be reformulated in the following
way. Let us consider (AUX1) and let x
2\Omega P . Then, the equality constraint can be
rewritten as
Therefore, by a change of variable, if we let
then (AUX1) is equivalent
to
subject to
Next, we will tighten the constraints in the above formulation by putting upper bounds
on u as well, and our choice for the upper bound will be x
B , which will give the largest
L1 -box around the origin which is contained in the feasible region:
subject to
We will call (SA1) the symmetrized LP and the resulting optimal solutions the symmetrized
bounds. The formulation of (SA1) reveals that if (u ; - ) solves the maximization
problem, then (\Gammau ; \Gamma- ) solves the minimization problem. Therefore, it suffices
to solve one LP as opposed to solving two LPs to obtain the optimal partition bounds
from (AUX1). A similar treatment of (AUX2) gives rise to the following symmetrized
LP:
subject to
\Gammas
which is obtained by replacing y \Gamma y by v and s
N by w, where (y ; s )
2\Omega D .
Finally, a similar symmetrization has been applied to w.
Next, we would like to discuss the relationship between the auxiliary and the symmetrized
LPs. First of all, let us assume that both (P) and (D) have unique and
nondegenerate solutions. Then, Proposition 2.1 implies that B is actually a square
and nonsingular matrix, hence invertible. In fact, B is the optimal basis. Conse-
quently, (AUX1) and (AUX2) are trivial to solve and their optimal solutions coincide
with the optimal basis bounds arising from the simplex method. With this observation,
the constraints of (AUX1) reduce to
-B
B or -(X
B is the diagonal matrix whose components are given by x
B and e denotes the
vector of ones in the appropriate dimension. Similarly, the constraints of (SA1) can be
rewritten as
\Gammae -(X
is the L1 -norm. A similar treatment reveals that the constraints of
are equivalent to
-(S
N is defined similarly, and that those of (SA2) to
The derivations (2.3)-(2.6) imply the following relationship between the auxiliary
and the symmetrized LPs: let the optimal partition bounds given
by the optimal solutions of the auxiliary LPs (including possibly \Sigma1). Then, the
symmetrized bounds for t are (\Gamma- s
Therefore, the symmetrized bounds are indeed equal to the "symmetrization" of the
optimal partition bounds.
Next, let us assume that (P) has a unique but degenerate solution. Then, by
Proposition 2.1, B is nonsquare but it has full column rank. Therefore, (AUX1) is still
easy to solve. If \Deltab does not lie in the range space of B, then the optimal solutions
of (AUX1) and (SA1) are all zero (which implies that is a breakpoint of v(t)).
Otherwise, there exists a unique vector v such that \Deltab, and hence, the constraints
of (AUX1) are equivalent to
-(X
Similarly, the constraints of (SA1) can be stated as
Once again, we conclude that a similar symmetry as in (2.7) continues to hold between
(SA1) and (AUX1). In a similar manner, one can show that such a relationship holds
between (SA2) and (AUX2) if (D) has a unique but degenerate solution.
The preceding discussion shows that the optimal solutions of the auxiliary and
the symmetrized LPs have the nice relationship (2.7) as long as there is a unique
optimal solution that one can use to symmetrize the constraints of the auxiliary LPs
to obtain the symmetrized LPs. An interesting question then is whether the same
nice relationship continues to hold between the auxiliary and the symmetrized LPs
if there are multiple optimal solutions, that is whether the symmetrized bounds are
independent of the choice of the optimal solution used to symmetrize the constraints.
Unfortunately, the answer is no as shown by the following example. Let (P) be given
by 0g. Then (P) has multiple optimal
solutions given by with an optimal value
of 0. If the right-hand side is perturbed to (0; then the reader can
easily verify that (AUX1) yields (\Gamma1=3; +1) as the optimal partition bounds, whereas
the symmetrized bounds are (\Gammafi; +fi) if one uses the optimal solutions with
to symmetrize the constraints, and (\Gamma1=3; 1=3) if those with fi - 1=3 are used. This
example illustrates that in case of multiple optimal solutions, the symmetrized bounds
are dependent on the optimal solution used in the formulation of the symmetrized
LPs. Therefore, the relationship (2.7) no longer holds between the symmetrized and
the auxiliary LPs.
However, we will keep using the symmetrized LPs for two reasons. First of all, at
least in the unique solution case, they bear a nice relationship to the auxiliary LPs.
For our analysis, we will always choose an optimal solution in the relative interior of
the optimal set; therefore the symmetrization will hopefully allow more room for the
decision variables of the symmetrized LPs. Secondly, the symmetrized LPs are easier
to deal with than the auxiliary LPs and the symmetrized bounds will provide a good
comparison basis for our interior-point approach proposed in [12], as will be analyzed
in the subsequent sections.
2.2 Interior-Point Approach and Central Path Neighborhood
We will start with a brief review of the primal-dual path-following interior-point meth-
ods. The reader is referred to [11] for an extensive treatment. The central path is a
path of strictly feasible points (x(-); y(-); s(-)) parametrized by a positive scalar -.
Each point on the central path satisfies the following system for some - ? 0:
Under the two assumptions in Section 2, such a solution exists
and is unique for each positive -. Interior-point methods are iterative algorithms
that generate iterates which "follow" the central path in the direction of decreasing
- towards the primal-dual optimal
\Theta\Omega D . The iterates generated typically lie
in some neighborhood of the central path. For any given feasible iterate (x; the
duality gap is given by c T and we define the duality measure - as
denote the set of feasible and strictly feasible
primal-dual points respectively, that is,
One of the commonly used neighborhoods in interior-point methods is the so-called
wide neighborhood, denoted by N
where
At each iteration, given (x; (fl), the algorithm determines a search
direction (\Deltax; \Deltay; \Deltas). This direction is usually obtained by seeking an approximation
to the point on the central path corresponding to some parameter -, and then
applying Newton's method to the nonlinear system of equations (2.10). Finally, a
(damped) step is taken in this direction in such a way that the resulting iterate still
lies in N \Gamma1 (fl).
However, as in the context of target-following methods, one might seek an approximation
to a point other than the one on the central path. We will say that a
Newton step from (x; targeting the feasible pair of points
is the direction (\Deltax; \Deltay; \Deltas) obtained from the Newton's method applied
to (2.10) with -e replaced by X e:
A T \Deltay
Next, we describe the interior-point approach proposed by the authors in [12].
Given a primal-dual pair of LPs (P) and (D), let us assume that b or c is perturbed
in some fixed direction. Assuming strictly primal-dual feasible for (P) and
(D), a full Newton step is taken from (x; targeting "a feasible point"
of the perturbed LPs which satisfies X is possible that there is no
such feasible point for the perturbed LPs, however, the Newton step as given above
is still well-defined.) We state the results formally, referring the reader to [12] for the
proofs. Note, in particular, that the duality gap of the resulting feasible iterate for the
perturbed LPs is no greater than that of the original iterate.
Proposition 2.2 Assume that (x; y; s) is a strictly feasible point for (P) and (D) and
the right-hand side vector b is replaced by b+t\Deltab, where t 2 IR and \Deltab 2 IR m . Suppose
a Newton step is taken from (x; targeting the feasible pair of points
the perturbed pair of LPs that satisfies X a full Newton step will
yield a feasible iterate for the new problem if and only if
. Moreover, in this case the new iterate will have duality gap at
most x T s.
Proposition 2.3 Assume that (x; y; s) is a strictly feasible point for (P) and (D) and
the cost vector c is replaced by c Suppose a Newton
step is taken from (x; targeting the feasible pair of points of the perturbed
pair of LPs that satisfies X a full Newton step will yield a feasible
iterate for the new problem if and only if
. Moreover, in this case the new iterate will have duality gap at
most x T s.
Under primal-dual nondegeneracy, the bounds arising from Propositions 2.2 and
2.3 computed at near-optimal solutions for (P) and (D) asymptotically equal the symmetrized
bounds arising from (SA1) and (SA2) [12]. The goal of this paper is to
investigate the quality of these bounds in the absence of the nondegeneracy assumption
We first present a nice characterization of the distance of the strictly feasible primal-dual
points strictly complementary optimal solutions in terms of the
duality gap -n. Using this characterization, we derive some bounds on the components
of such points. In what follows, xB , xN , s B and s N denote the partitions of x and
s according to the optimal partition B and N as before. Furthermore, we will use
the bounds O(-), \Omega\Gamma -) and \Theta(-) interchangeably for scalars as well as vectors and
matrices by which we mean each entry satisfies the stated bounds. O(-) will indicate
that the quantity (in absolute value) is bounded above by some positive multiple of
-, where the multiple depends on the primal-dual instance (P) and (D) but does not
depend on the particular strictly feasible point or on -. Similarly, \Omega\Gamma -) will indicate
a lower bound by some positive multiple of - and \Theta(-) will mean a lower and upper
bound by some positive multiples of -.
The following proposition will be useful for the analysis that follows. Actually, the
proposition continues to hold for any feasible solutions and even for a point where
feasibility is violated by O(-). The statement below suffices for the purposes of this
paper.
Proposition 2.4 Let (x; y; s) be a strictly feasible point for (P) and (D) with duality
gap -n. Then, there exists a strictly complementary optimal solution
and (D) such that
Proof:
Optimal solutions of (P) and (D) satisfy the linear system
strictly feasible point satisfies the same
linear system with the third equality replaced by c T
[6] indicates that there exists a solution (-x; -
of the first system such that (-x; -
+O(-). The result follows immediately if (-x; -
s) is strictly complementary. If
not, there exists an arbitrarily small perturbation of (-x; -
s) which leads to a strictly
complementary solution and (2.17) follows since - ? 0.
The following corollary immediately follows from Proposition 2.4 since x
s
optimal solution of (P) and (D).
Corollary 2.1 Let (x; y; s) be a strictly feasible point for (P) and (D) with duality gap
-n. Then,
Note that both Proposition 2.4 and Corollary 2.1 hold for any primal-dual strictly
feasible (x; s). Next, we derive some more bounds by restricting the iterates to lie
in a wide neighborhood given by (2.13).
Proposition 2.5 Let (x; duality gap -n for (P) and (D). Then,
Proof:
similar argument shows
\Omega\Gamma126 Finally, together with s N
imply
O(-). The
proof of SB X \Gamma1
Equivalence
In this section, we show that the interior-point bounds are independent of the problem
formulation. It is well-known that although (P) and (D) do not look symmetric, they
can easily be reformulated so that (D) is in the form of (P) and vice versa. We briefly
review this reformulation. Let (-x; -
s) be such that
us
consider (D) first. The objective function can be rewritten as
where we used and the fact that every feasible pair (y; s) for (D) satisfies
Note that the first term is a constant: therefore maximizing b T y is the
same as minimizing - x T s. Let K 2 IR (n\Gammam)\Thetan be such that its rows form a basis for the
null space of A. Then, premultiplying the equality constraints in (D) by K yields
Moreover, if s satisfies (3.2), then c \Gamma s lies in the null space of K, for which the columns
of A T form a basis by definition of K. Therefore, there exists y such that A T
Consequently, (D) is equivalent to
x T s; subject to
Note in particular that K has full row rank by its definition. If we take the dual of
(D'), we obtain
subject to K T
It is not hard to see that (P) and (P') are also equivalent by a similar argument.
Therefore, the roles of (P) and (D) can be interchanged via this reformulation.
Let us now focus on perturbations of c, i.e., let c be replaced by c+t\Deltac. By the above
reformulation, this is the same as replacing the right-hand side of (D') by - c
Therefore, Proposition 2.2 can be used to evaluate the interior-point bound at a strictly
feasible primal-dual pair (s; x) (note that the roles of x and s are interchanged). We
need to compute
On the other hand, one can also use Proposition 2.3 to compute the interior-point
bound directly at (x; s), which requires the evaluation of
A simple manipulation of (3.3) gives rise to another equivalent formula:
where \Psi is the orthogonal projection matrix onto the range space of X \Gamma1=2 S 1=2 K T .
Similarly, (3.4) is equivalent to
where \Xi is the orthogonal projection matrix onto the null space of AX 1=2 S \Gamma1=2 . There-
fore, in order to prove that (3.3) and (3.4) are equivalent, it suffices to show that \Psi
and \Xi project onto the same subspace, or that the null space of AX 1=2 S \Gamma1=2 equals
the range space of X \Gamma1=2 S 1=2 K T . This is easily proven by an inclusion argument: if w
satisfies AX 1=2 S \Gamma1=2 Thus, w is in
the range space of X \Gamma1=2 S 1=2 K T . Conversely, if
AX This proves the equivalence of the interior-point bounds.
We next argue that the range of t resulting from the optimal partition bounds is
also independent of the formulation. If the two LPs are formulated in the form of (P)
and (D), then (AUX2) yields the range of t for perturbations of c. Premultiplying the
equality constraints of (AUX2) by leads to (AUX1') given by
min
w;-
w;-
which exactly yields the range of t for perturbations of the right-hand side of (D') if
one uses the form (D') and (P'). Similarly, if (w; -) is feasible for (AUX1'), then
lies in the null space of K. Then, by our previous observation, there exists v such
that which is exactly the constraints of (AUX2),
completing the proof of the claim.
Using this observation, we will carry out our analysis for perturbations of b only
in the subsequent sections, and state the corresponding results for changes in c as
corollaries. We begin with a special case of degeneracy first and then consider the
most general case.
4 Unique Primal Solution
Throughout this section, we assume that (P) has a unique but degenerate optimal
solution x . Note that by Proposition 2.1, we have linearly
independent columns. In this particular case, Proposition 2.4 provides another useful
bound on xB for a strictly feasible primal-dual point (x;
Corollary 4.1 Assume that (P) has a unique optimal solution x . Let (x; y; s) be
primal-dual strictly feasible for (P) and (D) with duality gap -n. Then,
An analogous corollary follows if (D) has a unique solution.
Corollary 4.2 Assume that (D) has a unique optimal solution (y
be primal-dual strictly feasible for (P) and (D) with duality gap -n. Then,
Next, we will analyze one-dimensional perturbations of b.
4.1 Perturbations of b
In this subsection, we assume that the right-hand side vector b is replaced by b
We also assume that (x;
strictly feasible point for (P) and (D) for some fl 2 (0; 1]. We will compare the interior-point
bounds arising from Proposition 2.2 at with the optimal values of (SA1),
i.e., the symmetrized bounds. The interior-point bounds are given by the L1-norm of
where
Let us now consider (SA1). Since B has full column rank, \Deltab either does not lie in
the range space of B, in which case the optimal values of (SA1) as well as (AUX1) are
all 0, or there exists a unique v 2 IR jBj such that \Deltab, in which case the constraints
of (SA1) reduce to (2.9), from which the symmetrized bounds can be obtained easily.
We will consider both situations in turn.
Let us start with the second case. Without loss of generality, we can assume that
\Deltab has unit L 2 -norm, which implies a bound on v. Then, we need to compute
in order to obtain (4.3). However, (4.4) is equivalent to
where B and N are the partitions of the coefficient matrix A with respect to B and
N as before. Since B has linearly independent columns, there exists a matrix C 2
IR m\Theta(m\GammajBj) such that the augmented matrix [B C] is square and nonsingular: let W
be its inverse. Therefore, premultiplying the second equality in (4.5) by W , we obtain
I#
~
I#
where ~
I is a jBj \Theta jBj identity matrix. Therefore, if we
partition ~
N and ~
accordingly as
~
~
~
~
can then be decomposed in the following way:
~
~
~
~
~
~
v#
where DB and DN are the corresponding partitions of D. By (4.3), we need to compute
For notational convenience, let us define
F := ~
Note that G has full row rank since A does. The bottom equality in (4.7) can be
rewritten as
Substituting (4.9) in the top equality in (4.7) gives
~
~
where PG is the orthogonal projection matrix onto the range space of G T . Therefore,
I \Gamma PG is the orthogonal projection matrix onto the null space of G.
We briefly review the Neumann lemma now [3]. Let U be an invertible matrix and
being used does not really matter:
we will always use k \Delta k for the Euclidean norm or the operator norm arising from it.)
Then, I +U \Gamma1 V is invertible with kI +U 2. Moreover U
given by
Now, we apply this result to (4.10) with U := D 2
2.5 implies that both U \Gamma1 and V are O(-) since I \Gamma PG is a projection matrix and has
unit Euclidean norm. Therefore, assuming the duality gap -n is small,
~
I +D \Gamma2
It then follows that
I +D \Gamma2
However, by Proposition 2.5, F is O(- 1=2 ), D \Gamma2
B is O(-) and X \Gamma1
B is O(1). Consequently,
the second term on the right hand side of (4.13) is O(- 2 ) since kI \Gamma PG k - 1. Finally,
Corollary 4.1 implies X \Gamma1
We have thus obtained the top part of (4.8). For the lower part, we get
where we substituted (4.9) for ~
Proposition 2.5 implies (XN SN
Combining these bounds with
leads to
Using (4.8), we conclude that the L1-norm of the quantity (4.3) we need to evaluate
is given by
O(-)
The reciprocal of (4.17) gives the desired interior-point bound. Consequently, if the duality
gap -n is small, we conclude by comparing (4.17) with (2.9) that the interior-point
approach yields exactly the same bound as the optimal solution to (SA1) asymptotically
in -.
Next, we address the situation where \Deltab does not lie in the range space of B. In this
case, the optimal values of both (AUX1) and (SA1) are clearly 0. \Deltab can be uniquely
written as
where [B C] is nonsingular as before and v C is a nonzero vector. Once again, we need
to compute (4.3). We follow a similar treatment as before, and corresponding to (4.7)
we
~
~
~
~
~
~
The bottom part can be expanded as
~
~
~
However, (4.8) implies that the term in the brackets is exactly the bottom part of the
quantity (4.3) we seek. Let us denote that term by p and let XN
is equivalent to ~
nonzero, the norm of q is bounded below,
that is, kqk - ff ? 0 where ff is the norm of the least squares solution. Therefore,
e.g. [3]). (Note that jBj ! n since
can happen only if case \Deltab is always in the range of B.) However,
k1 kpk1 since XN This implies
kXN k1
where the last equality follows from Corollary 2.1. Therefore, as - tends to 0, kpk1
tends to 1, which implies that the interior-point bound given by its reciprocal tends
to 0 as desired.
We remark that if is the only optimal solution of (P), which can
happen only if In this case, the top part of (4.8) disappears. The interior-point
bound is then given by the reciprocal of kpk1 , where p is as defined after (4.20). By
the preceding argument, the interior-point bound tends to 0 as - approaches 0. This
is still in agreement with the optimal partition bounds since any nonzero perturbation
of b leads to a change in the optimal partition and hence, the optimal partition bounds
in this case are also equal to 0. Therefore, we have proved the following theorem:
Theorem 4.1 Let (x; be a primal-dual strictly feasible point for (P)
and (D). Assume that (P) has a unique but degenerate optimal solution and that b
is replaced by b . Then the interior-point bound
evaluated at yields exactly the same value as the optimal solution of (SA1)
asymptotically in -, where
The following corollary of Theorem 4.1 is an immediate consequence of the equivalence
between (P) and (D) as discussed in Section 3. One uses Corollary 4.2 in place
of Corollary 4.1 in the preceding analysis.
Corollary 4.3 Let (x; be a primal-dual strictly feasible point for (P)
and (D). Assume that (D) has a unique but degenerate optimal solution and that c is
replaced by c+t\Deltac where t 2 IR and \Deltac 2 IR n . Then the interior-point bound evaluated
at yields exactly the same value as the optimal solution of (SA2) asymptotically
in -, where
It does not appear that we can obtain better results for perturbations of c in the
case of a unique primal optimal solution (but not dual optimal solution) than those
arising from the analysis of the general case in the next section. A similar remark holds
for perturbations of b in the case of a unique dual optimal solution (but not primal
optimal solution).
5 General Case
In this section, we turn our attention to the most general case where both (P) and (D)
may have multiple optimal solutions. As the small example given at the end of Section
2.1 reveals, some complications arise in the presence of multiple optimal solutions. For
instance, unlike the previous case, the symmetrized bounds become dependent on the
optimal solution of (P) used in the formulation of (SA1) if (P) has multiple optimal
solutions. Furthermore, they do not necessarily coincide with the "symmetrizations"
of the optimal partition bounds arising from (AUX1). Similar remarks hold for the
relationship between (SA2) and (AUX2) if (D) has multiple optimal solutions.
Despite this complication arising from the presence of multiple optimal solutions,
we aim to be able to say something about the quality of the interior-point bounds at
least in comparison with the symmetrized bounds. In the next subsection, we analyze
perturbations of b in this general setting.
5.1 Perturbations of b
Let (P) have multiple optimal solutions and let b be replaced by b
and \Deltab 2 IR m . Suppose that (x; strictly feasible where
For such a point, Proposition 2.4 guarantees the existence of a strictly
complementary solution whose distance from (x; y; s) is bounded above by
the duality gap n-. We will compare the interior-point bounds evaluated at
with the optimal values of (SA1). Among other optimal solutions of (P), the x above
will be the particular choice of the primal optimal solution to be used in the formulation
of (SA1). The use of such an optimal solution in the relative interior of the primal
optimal set is likely to leave more room for the decision variables of (SA1) since x
Let us first consider (SA1). The constraints of (SA1) are
k. Clearly we have r - m and r ! k since Proposition 2.1
implies
dim(\Omega which is positive by our assumption. This, in turn, implies
that r ? 0 since (assuming no columns of A are identically
zero). A QR factorization of B yields orthogonal and
R 2 IR m\Thetak is upper triangular with
R 1#
rows. Note that R 1 has full row rank.
Premultiplying the equality constraints in (5.1) by Q T yields
R 1#
f
with f
that (SA1) has a nontrivial optimal solution
- if and only if f
First, we consider the nontrivial case. (Since f
\Deltab is nonzero, this implies that k ? 0.)
Let (- ; u ) be an optimal solution to the maximization problem with - 6= 0. Note
that - is finite since u is bounded (this follows since B 6= ;). Then, we have
f
The interior-point approach, on the other hand, requires the evaluation of (4.3) at
s). By (5.4), we then need to evaluate the L1-norm of
Let
Premultiplying the second equality in (5.6) by Q T gives
~
R T
~
~
~
R 1#
where ~
are the appropriate partitions of ~
are those
of ~
us define
F := ~
can then be decomposed into two equations as
Note, in particular, that both G and H have full row rank since R 1 and A do. From
the second equation in (5.9), we obtain
~
Substituting (5.10) in the first equation of (5.9) leads to
where I \Gamma PH is the orthogonal projection matrix onto the null space of H. Proposition
2.5 implies that the second term in parentheses in the second equation above is O(-)
since kI \Gamma PH k - 1. In order to apply Neumann's lemma, we need to show that
Lemma 5.1 (GG
Proof:
We use the "thin" QR factorization of G
columns and Z is upper triangular and nonsingular. Then, (GG
Therefore, it suffices to find an upper bound on Z \Gamma1 . We have
Therefore, (R
(R
Y . However, by
Proposition 2.5, D \Gamma1
which implies that Z completing the
proof.
We can now apply Neumann's lemma to (5.11). Using the same notation as in
(4.11) we have U := GG T and V := F . Note that both U \Gamma1 and V are
O(-). We obtain
~
where we used R
By (5.5) and (5.6), we need
Let us define
For the top part of (5.14) we need to evaluate
where we used (5.13), (5.15) and where PG is the orthogonal projection matrix onto
the range space of G T . Consider the second term in the right hand side of the second
equality. By Proposition 2.5, (SB XB ) \Gamma1=2 is O(- \Gamma1=2 ), V is O(-) and D \Gamma1
B is O(- 1=2 ).
Lemma 5.1 implies that
fore, the whole expression is O(- 2 ). We conclude that the top part of (5.14) is
Let us next consider the lower part of (5.14). We need to compute
~
~
By (5.13) the first term in (5.18) is given by
Note that by the preceding discussion. As for the second term in brack-
ets, we have both (GG are O(-), which implies the whole second term is
O(- 5=2 ). Thus, the expression in brackets is O(- 1=2 ). By Proposition 2.5, (SN XN ) \Gamma1=2
is O(- \Gamma1=2 ), whereas both F T and D \Gamma1
are O(- 1=2 ). We therefore conclude that (5.19)
is O(-).
For the second term in (5.18), we use (5.10) together with (5.13):
\Gamma(S
Note that ~
O(-) by the preceding arguments. The fact that kPH k - 1 together
with (SN XN ) \Gamma1=2 being O(- \Gamma1=2 ) and F T being O(- 1=2 ) implies (5.20) is O(-).
Therefore, we conclude that the lower part of (5.14) is O(-). Combining this result
with (5.17) yields the following:
r := (1=- )
O(-)
Consequently, we need to evaluate the L1-norm of (5.21) and take its reciprocal.
Observe that X \Gamma1
Proposition 2.4. Using this, we derive an
upper bound on the L1-norm of (5.21).
Thus, . Furthermore, since
e.g. [3]), where k. Finally, since u is
optimal for (SA1), k(X
Therefore,
/s
We conclude that the interior-point bound, which is the reciprocal of (5.24), is then
bounded below bykrk 1
Note, in particular, that the lower bound tends to 1=
k, independent of n, as
if s) is on the central path.
We next consider the case where \Deltab is not in the range space of B. Again, in this
case, the symmetrized bounds as well as the optimal partition bounds are all 0. The
QR factorization of B can be rewritten as
use (5.2) and [Q 1 is the appropriate partition of Q. Since Q is orthogonal, \Deltab
can uniquely be expressed as
Arguing similarly to Section 4, we need to evaluate (4.3), which
in turn requires the computation of
Premultiplying (5.27) by Q T leads to
~
R T
~
~
~
which looks like (4.19). Essentially the same arguments as in Section 4 reveal that the
interior-point bound tends to 0 as - approaches 0.
Therefore, we have proved the following theorem.
Theorem 5.1 Let (x; be a primal-dual strictly feasible point for (P)
and (D) with duality gap -n. Assume that (P) has multiple optimal solutions and that
b is replaced by b . If the strictly feasible solution
given by Proposition 2.4 is used for symmetrization in (SA1), then the ratio of the
interior-point bound evaluated at (x; y; s) to the value of the optimal solution of (SA1)
is at least p
Note that the presence of multiple primal optimal solutions implies k ? 0, therefore,
the expression (5.29) is well-defined. As in Section 4, Theorem 5.1 leads to the following
corollary by the discussion in Section 3. Due to the interchange of the roles of the basic
and nonbasic variables in the reformulation given in Section 3, k in the denominator of
is replaced by (n \Gamma k). Under the assumption of multiple dual optimal solutions,
Proposition 2.1 indicates that m ? r, which implies k ! n since A has full row rank.
Corollary 5.1 Let (x; be a primal-dual strictly feasible point for (P)
and (D) with duality gap -n. Assume that (D) has multiple optimal solutions and that
c is replaced by c . If the strictly feasible solution
given by Proposition 2.4 is used for symmetrization in (SA2), then the ratio of the
interior-point bound evaluated at (x; y; s) to the value of the optimal solution of (SA2)
is at least p fl
6 Computational Results
In the previous sections, we have provided a theoretical basis for the behavior of the
interior-point bounds evaluated at the near-optimal solutions. We present some computational
results in this section to shed some light on the performance of the interior-point
bounds in practice.
We have generated random LPs with 400. The input parameters
are the number of basic variables (jBj) and dimension of the primal optimal set
(dim(\Omega P )), which together
determinedim(\Omega D ) and rank(B). This allows us to incorporate
all scenarios of primal and dual degeneracies into the random LPs. We first
generate a suitable matrix A, then a strictly complementary pair of solutions, and
finally set b and c to make these feasible and hence optimal.
Having generated a random LP with the prespecified degeneracies, we obtain a
strictly feasible, near-optimal solution by perturbing the known strictly complementary
optimal solution. Next, random perturbations of b and c are generated in the correct
subspaces so that (AUX1) and (AUX2) have nontrivial optimal solutions. We compute
the interior-point bounds evaluated at those near-optimal solutions and compare
them with the optimal solutions to (AUX1) and (AUX2) as well as the optimal solutions
to the symmetrized LPs (SA1) and (SA2), where the initially generated strictly
complementary optimal solution is used to symmetrize the constraints.
We present our results for various degeneracy scenarios in Table 6. Eight instances
with various levels of primal-dual degeneracies are reported. For each instance, the
interior-point bounds are evaluated at two iterates corresponding to each row. DP and
DD are the dimensions of the primal and dual optimal sets, respectively. - is the duality
gap measure given by x T s=n, and fl is the parameter of the narrowest wide central-
path neighborhood containing the iterate. (AUX1) and (AUX2) are the minimum
of the absolute values of the optimal values of the corresponding minimization and
maximization problems (symmetrizations). (SA1) and (SA2) are the optimal values of
the symmetrized maximization problems. Finally, IPB and IPC are the upper interior-point
bounds for changes in b and c evaluated at the corresponding iterates.
The predicted nice theoretical behavior of the interior-point bounds is exhibited
in Instances 1,2 and 4 for changes in b and in Instances 4,6 and 8 for changes in c.
Observe that the bounds converge to the symmetrized bounds even though fl is very
small, which is typical in practice. For the remaining degeneracy scenarios, the interior-point
bounds lie within a factor of the symmetrized bounds as discussed in the previous
section. It is worth noting, however, that the actual ratio seems to be much better
than the theoretical worst-case ratios (5.29) and (5.30). In our extensive computational
tests, the ratio was never worse than a hundredth although the predicted lower bounds
and (5.30) are on the order of 10 \Gamma5 in most of the instances.
Finally, we note that the condition number of AD 2 A T blew up in all of the degenerate
instances as expected. Therefore, the numerically unstable results have been
discarded. Furthermore, the bound for changes in b seems to be computationally much
more stable than its counterpart for c; however, this is most likely due to the fact
that we use (2.15) and (2.16) to compute the bounds. By the equivalence discussed in
Section 3, this problem can be overcome using (3.3) instead of (3.4) at the extra cost
of computing K, which can easily be obtained by a QR factorization of A T .
7 Conclusion
In this paper, we have studied the quality of the bounds arising from the interior-point
perspective on sensitivity analysis developed by the authors in [12]. By relaxing
the strong assumption of nondegeneracy, we have been able to consider all possible
degeneracy scenarios and to investigate how our bounds compare with those arising
from the optimal partition approach to sensitivity analysis.
If the primal problem has a degenerate but unique optimal solution, then our approach
yields the same bounds as the "symmetrized" optimal partition bounds for
perturbations of b. By the equivalence discussed in Section 3, the same relationship
holds for perturbations of c if the dual problem has a degenerate but unique opti-
Table
1:
Computational
Results
(m
200,
Ins
DD
IPB
IPC40160e-5e-65.47264.1633.005e-3e-63.06112080e-5e-614.61327.6653.806e-3e-62.670120140e-5e-52195.607223.65325.957e-3e-525.9572000e-6e-50.6760.008300.00829e-4e-50.00823200100e-5e-54030.46669.4843.086e-3e-53.0992800e-7e-5220.6561.5611.561e-5e-51.547280100e-4e-48512.007115.83438.813e-3e-438.5403600e-6e-4552.0142.6362.638e-4e-4
87.
mal solution. This result directly extends the previous result proved in [12] under the
assumption of a unique and nondegenerate solution.
We then considered general degenerate LPs. In this case, we were able to show that
our interior-point approach would yield bounds that are at least a certain fraction of
the symmetrized bounds, where the fraction depends on certain characteristics of the
problem instance and of the iterate at which the bounds are evaluated. Our extensive
computational tests suggest that the ratio in practice is much better than the predicted
worst-case ratio. Although this result is not as strong as the aforementioned results,
our interior-point bounds still yield some useful information on the range of allowable
perturbations. The fact that the cost of the evaluation of our bounds is simply the
same as that of an interior-point iteration makes it more appealing given the cost of
solving two LPs to obtain the range from the optimal partition approach.
--R
A geometric view of parametric linear program- ming
Theory of linear programming.
Matrix Computations.
Simultaneous primal-dual right-hand-side sensitivity analysis from a strictly complementary solution of a linear program
On the dimension of the set of rim pertubations for optimal partition invariance.
On approximate solutions of systems of linear inequalities.
Sensitivity analysis in linear programming
A general parametric analysis approach and its implication to sensitivity analysis in interior point methods.
Theory and Algorithms for Linear Optimization
Sensitivity analysis in linear programming and semidefinite programming using interior-point methods
--TR | degeneracy;sensitivity analysis;linear programming;interior-point methods |
589159 | On Some Properties of Quadratic Programs with a Convex Quadratic Constraint. | In this paper we consider the problem of minimizing a (possibly nonconvex) quadratic function with a quadratic constraint. We point out some new properties of the problem. In particular, in the first part of the paper, we show that (i) given a KKT point that is not a global minimizer, it is easy to find a "better" feasible point; (ii) strict complementarity holds at the local-nonglobal minimizer. In the second part of this paper, we show that the original constrained problem is equivalent to the unconstrained minimization of a piecewise quartic merit function. Using the unconstrained formulation we give, in the nonconvex case, a new second order necessary condition for global minimizers. In the third part of this paper, algorithmic applications of the preceding results are briefly outlined and some preliminary numerical experiments are reported. | Introduction
In this paper we study the problem of minimizing a general quadratic function q :
subject to an ellipsoidal constraint, that is
where H is a symmetric positive definite n \Theta n matrix and a is a positive scalar.
The interest in this problem initially arose in the context of trust region methods for
solving unconstrained optimization problems. In fact, such methods require at each
iteration an approximate solution of Problem (1) where q(x) is a local quadratic model
of the objective function over a restricted ellipsoidal region centered around the current
iterate. However, recently, it has been shown that problems with the same structure of
y This work was partially supported by Agenzia Spaziale Italiana, Roma, Italy
z Universit'a di Roma "La Sapienza" - Dipartimento di Informatica e Sistemistica - via Buonarroti, 12
Italy and Gruppo Nazionale per l'Analisi Funzionale e le sue Applicazioni del Consiglio
Nazionale delle Ricerche.
(1) play an important role not only in the field of unconstrained minimization. In fact,
the solution of Problem (1) is at the basis of algorithms for solving general constrained
nonlinear problems (e.g. [3, 42, 20, 27]), and integer programming problems (e.g.
[21, 41, 22, 31, 19]).
Many papers have been devoted to point out the specific features of Problem (1).
Among the most important results there are the necessary and sufficient conditions
for a point x to be a global minimizer, due to Gay [12] and Sorensen [35], and the
characterization and uniqueness of the local-nonglobal minimizer due to Mart'inez [26].
The particular structure of the Problem (1) has led to the development of algorithms for
finding a global solution. The first algorithms proposed in literature were those of Gay
and Sorensen [12, 35]. Mor'e and Sorensen [29] developed an algorithm that produces
an approximate global minimizer in a finite number of steps. More recently, it has been
proved that an approximation to the global solution can be computed in polynomial
time (see for example [39, 38, 40, 41, 21]). Furthermore, Mor'e [28] has considered a
more general case by allowing in Problem (1) a general quadratic constraint and has
extended the results of [12, 35, 29].
In spite of all these results, there is still interest in studying Problem (1). In fact,
as we mentioned before, there is a growing use of Problem (1) as a tool for tackling
large nonlinear programming problems and combinatorial optimization problems. This
leads to the necessity of solving more and more efficiently large scale problems of
the type (1) and motivates further research on theoretical properties of Problem (1)
and on the definition of efficient methods for locating its global minimizers. Recently
some interesting algorithms for tackling large scale trust region problems have been
proposed in [36, 34, 33]. The basic idea behind these algorithms is to recast the trust
region problem in term of a parametrized eigenvalue problem and then to adjust the
parameter to find an optimal solution.
In this paper we point out further theoretical properties of Problem (1). In par-
ticular, our research develops along two lines: the study of some new properties of
its Karush-Kuhn-Tucker points and its equivalence to an unconstrained minimization
problem. Besides their own theorical interest, these results allows us to define new
classes of algorithms for solving large scale case trust region problems. These algorithms
use only matrix-vector product and do not require the solution of an eigenvalue
problem at each iteration (see [24] for details).
The paper is organized as follows. In Section 2 we recall some preliminary results. In
Section 3 we show that
(i) given a KKT point - x which is not a global minimizer, it is possible to find a
new feasible point -
x such that the objective function is strictly decreased, i.e.
(ii) the strict complementarity condition holds at the local minimizer, hence in the
nonconvex case, strict complementarity holds at local and global minimizers.
In Section 4 we show that there is a one to one correspondence between KKT
points (global minimizers) of Problem (1) and stationary points (global minimizers) of
a piecewise quartic merit function Therefore, Problem (1) is equivalent
to the unconstrained problem of P over IR n . In Section 5, by exploiting some results
of the preceding sections, we give a new second order necessary condition for global
minimizers of Problem (1). Finally, in Section 6, we sketch some possible applications
of the results of Section 3 and Section 4 for defining new classes of algorithms for solving
large scale trust region problems.
In the sequel we will use the following notation. Given a vector x 2 IR n , we
denote by kxk the ' 2 -norm on IR n . The ' 2 -norm of a n \Theta n matrix Q is defined by
1g. Moreover, we denote by - 1 - 2 - n the
eigenvalues of Q.
Preliminaries
Without loss of generality we can assume that the feasible set F is defined by
so that the problem under consideration is
and Q is a n \Theta n symmetric matrix and c 2 IR n .
In fact, since H is positive definite, we can reduce Problem (1) to the form (2) by employing
the transformation
we refer to [25] for the direct treatment
of Problem (1)).
The Lagrangian function associated with Problem (2) is the function
A Karush-Kuhn-Tucker point for Problem (2) is a pair (-x; -) 2 IR n \Theta IR such that:
Furthermore, we say that strict complementarity holds at a KKT pair (-x; -
for
It is well known that it is possible to completely characterize the global solutions
of Problem (2) without requiring any convexity assumption on the objective function.
In fact, the following result due to Gay [12] and Sorensen [35] holds (see also Vavasis
Proposition 2.1 A point x such that kx k 2 - a 2 is a global solution of Problem (2),
if and only if there exists a unique - - 0 such that the pair (x ; - ) satisfies the KKT
conditions
and the matrix (Q positive semidefinite. If (Q positive definite
then Problem (2) has a unique global solution.
Moreover, Mart'inez [26] gave the following characterization of the local-nonglobal minimizer
for Problem (2).
Proposition 2.2 There exists at most one local-nonglobal minimizer -
x of Problem
(2). Moreover we have and the KKT necessary conditions holds with - 2
are the two smallest eigenvalues of Q.
3 Further features of KKT points
In this section we give some new properties of the KKT points for Problem (2). Our
interest in the characterization of KKT points is due to the fact that, in general,
algorithms for the solution of constrained problems, converge towards KKT points.
We show that the number of different values that the objective function can take at
KKT points is bounded from above by the number of negative eigenvalues of the matrix
Q. First we state a preliminary result that extends one given in [38].
Lemma 3.1 Let (b x; -) and (-x; -) be KKT pairs for Problem (2) with the same KKT
multiplier. Then q(b
Proof We observe that the function q(x) can be rewritten at every KKT pair (x; -)
as follows
By using the KKT conditions we obtain
Now, we can state the following proposition whose proof follows from a result of
Forsythe and Golub [11] on the number of stationary values of a second degree polynomial
on the unit sphere. For sake of completeness we give a sketch of the proof.
Proposition 3.2 There exist at most minf2m points with distinct
multipliers -, where m is the number of negative distinct eigenvalues of Q.
Proof First we observe that at every KKT point (x; -) such that kxk 2 ! a 2 the value
of the objective function q is constant. This easily follows from Lemma 3.1 by observing
that all these pairs are characterized by the fact that
Now, we consider the values of the function q(x) at all the points such that kxk
there exists an orthogonal matrix V such that V T
diag are the eigenvalues of Q. By considering
the transformation ff we can write the first equation of the KKT condition (4)
(premultiplied by V T ) as follows:
diag
with recalling that kxk we have that the KKT
multipliers must satisfy the system
where
The function g(-) has poles at \Gamma2- i
and it is convex on the subintervals
\Gamma2-
Thus there exists at most 2 roots of in each subinterval. Moreover, since
a 2 has one root in each exteme subinterval.
If all eigenvalues - i are positive there exists at most one non negative root; if all the
eigenvalues are negative there are at most 2n non negative roots; in the case of m ! n
negative eigenvalues, there are at most 2m negative roots. Hence the number
of the solutions of system (6) is at most minf2m
Finally, by summarizing the two cases, we can conclude that the number of distinct
KKT multipliers is bounded above by minf2m
Recalling Lemma 3.1, we get directly the following corollary.
Corollary 3.3 The number of distinct values of the objective function q(x) at KKT
points is bounded from above by minf2m 1g.
Now we can state the main result of this section. In particular, we show that the
peculiarity of Problem (2) can be exploited to escape from the KKT points that are
not global solutions in the sense that, whenever we have a KKT point - x, either -
x is
a global minimizer of Problem (2), or it is possible to compute the expression of a
feasible point with a strictly lower value of the objective function. This results is very
appealing from a computational point of view, as discussed in Section 6.
Proposition 3.4 Let (-x; -) be a KKT point for Problem (2). Let us define the point
x in the following way
(a) If c T -
x:
(b) If c T - x - 0 and a vector z 2 IR n such that z T (Q
with
z:
with
z
-I)z
Then we have q(-x) ! q(-x) and k-xk 2 - a 2 .
Proof In case (a), the point -
x is still feasible and
Now consider case (b). In case (i) we have by the KKT conditions that -
hence we have that z is a vector of negative curvature for q(x). Therefore, for every
satisfies the inequality
In particular, if we take ff - ~
ff with
~
we have that k-xk 2 - a 2 .
let us consider case (ii). Let -
x be the vector defined as follows
z
and consider the quadratic function
We note that k-xk and that z is a negative curvature direction for the quadratic
function L(x; -). By simple calculation, taking into account that (Q
get
-I)z
and hence L(-x; -) ! L(-x; -
-). Hence, recalling the expression (8) we can write
Hence we get the result for case (ii).
Let us consider the case (iii). Let us define the vector -
0: We can
find a value for ff such that -
s is a negative curvature direction for L(x; -) and - s T - x 6= 0,
so that we can proceed as in case (ii). In fact, by simple calculation we have:
and by using the KKT conditions
xj:
By solving the quadratic equation with respect to ff we get that - s T
ff where
\Gammac
Hence, by proceeding as in case (ii), we get the result by introducing the point
with ff ? -
ff.
Remark We note that the local-nonglobal minimizer can corresponds either to the
case (a) with k-xk or to the case (b)(ii).
The preceding proposition shows that, if the KKT point - x is not a global minimizer,
it is possible to determine a feasible point -
x such that q(-x) ! q(-x) by computing at
most a direction z such that z T (Q The existence of such a direction is
guaranteed by Proposition 2.1 and from the numerical point of view, its computation
is not an expensive task. In fact, we can obtain such a direction by using, for example,
the Bunch-Parlett decomposition [2, 30], modified Cholesky factorizations [10] or, for
large scale problem, methods based on Lanczos algorithms [4].
Now, as last result of this section, we investigate a regularity property of the local
and global minimizers. In particular, we focus our attention on the strict complementarity
property, that, roughly speaking, indicates that these points are "really
constrained". Also this property can be interesting from an algorithmic point of view.
Proposition 3.5 At the local-nonglobal minimizer for Problem (2) the strict complementarity
condition holds.
Proof Since - x is a local minimizer the KKT conditions (4) hold. Moreover the second
order necessary conditions require that
z
By Proposition 2.2 we have that -
there is no local-nonglobal minimizer. Furthermore, if necessarily
restrict ourselves to the case
since in this case 0 2 us assume by contradiction that -
(4) and Proposition 2.2 we have that
z
x is not a global minimizer, by Proposition 2.1 there exists a direction y such
that y T Qy ! 0 and from the second order necessary conditions y T -
x 6= 0. We assume,
without loss of generality, that y T -
us consider the point ffy with
We prove that for sufficiently small values of ff the point x(ff) is feasible and
produces a smaller value of the objective function, thus contradicting the assumption
of local optimality. In fact, we have
and hence for
we obtain kx(ff)k 2 ! a 2 . Moreover,
By this proposition and by Proposition 2.1 we directly obtain the following result.
Proposition 3.6 In the nonconvex case at every local or global minimizer the strict
complementarity holds.
Unconstrained formulation
In this section, we show that Problem (2) is equivalent to an unconstrained minimization
problem of a piecewise quartic merit function. A general constrained optimization
problem can be transformed into an unconstrained problem by defining a continuously
differentiable exact penalty function by following, for example, the approach proposed
in [6, 7]. However, in the special case of minimization of a quadratic function with
box constraints, it has been shown in [16] and [23] that it is possible to define simpler
penalty functions by exploiting the particular structure of the problem. In the same
spirit of these papers we show that also for Problem (2) it is possible to construct a
particular continuously differentiable penalty function. This new penalty function takes
full advantage of the peculiarities of the trust region problem and enjoys distinguishing
features that make its unconstrained minimization significantly simpler in comparison
with the unconstrained minimization of the penalty functions proposed in [6, 7]. The
main properties of the penalty function proposed in this section are:
ffl it is globally exact according to the definition of [7];
ffl it does not require any shifted barrier term hence it is defined on the whole space;
ffl it has a very simple expression (it is piecewise quartic);
ffl it is known, a priori, for which values of the penalty parameter the correspondence
between the constrained problem and the unconstrained one holds.
As a first step for the definition of the exact penalty function, we recall the Hestenes-
Powell-Rockafellar augmented Lagrangian function [32, 18]
L a (x; -;
""
ae
is a given positive parameter.
Now, according to the classical approach, we replace the multiplier vector - in the
function L a (x; -; ") with a multiplier function which yields an estimate
of the multiplier vector associated to Problem (2) as a function of the variables x. In the
literature different multiplier functions have been proposed (see e.g. [9, 13, 6, 7, 23]).
However, all the expression of the multiplier functions given in [9, 13, 6, 7] are not
defined in the origin of the space.
Here we define a new simpler multiplier function that is defined on the whole space IR n
whose expression is the following
Its properties are summarized in the following proposition.
Proposition 4.1
(i) -(x) is continuosly differentiable with gradient
(ii) If (-x; -) is a KKT point for Problem (2) then we have
(iii) For every x 2 IR n we have x
Proof Part (i) easily follows from the definition of the multiplier function (10). As
regards part (ii), from (4) we have that a pair (-x; -
It is easy to see that if k-xk corresponds exactly to the definition of the
multiplier function (10). Otherwise, if k-xk 2 ! a 2 , (4) imply that hence by
comparing (11) and (10) that
Now let us consider part (iii). By simple calculations we have
a 2
On the basis of the previous considerations we can replace the vector - in the function
L a with the multiplier function -(x). Furthermore, as regards the penalty parameter
", we can select, a priori, an interval of suitable values depending on the problem
data Q; c; a. Therefore, we are now ready to define our merit function P
L a (x; -(x); "(Q; c; a)), that is
ae
where -(x) is the quadratic function given by (10) and " is any parameter that satisfies
the following inequality:
a
First, we show some immediate properties of the merit function P .
Proposition 4.2
(i) P (x) is continuosly differentiable with gradient
ae
(ii) P (x) is twice continuosly differentiable except at points where"
(iii) P (x) is twice continuosly differentiable in a neighborhood of a KKT point -
x where
strict complementarity holds;
(iv) for every x such that kxk 2 - a 2 we have that P (x) - q(x);
(v) the penalty function P (x) is coercive and hence it admits a global minimizer.
Proof Part (i), (ii) and (iii) directly follows from the expression of the penalty function
P . As regards Part (iv) it follows from a classical results on penalty functions (see
Theorem 2 of [7]).
As regards part (v), we want to show that as kxk ! 1 the function P (x) goes to
infinity. First, we observe that"
hence for sufficiently large values of kxk the leading term of the preceding inequality is
strictly positive since, recalling that " satisfies (13), we have that " -
sufficiently large values of kxk, we can assume that
ae
oe
By simple calculation, the expression of the penalty function becomes in this case:
and the following inequalities hold:
As " satisfies (13), we have that " -
and hence we get lim
1. The
existence of the global minimizer immediately follows from the continuity of P and the
compactness of its level sets.
Now, we state the first result about the exactness properties of the penalty function
P . Since its proof is technical and lenghty we report it in the Appendix.
Proposition 4.3 A point -
is a stationary point of P (x) if and only if (-x; -x))
is a KKT pair for Problem (2).
Furthermore, at this point we have P
Now we prove that there is a one to one correspondence betweeen global minimizers
of Problem (2) and global minimizers of the penalty function P .
Proposition 4.4 Every global minimizer of Problem (2) is a global minimizer of P (x)
and conversely.
Proof By Proposition 4.3, the penalty function P admits a global minimizer -
which
is obviously a stationary point of P and hence by the preceding proposition we have
On the other hand, if x is a global minimizer of Problem (2), it is also a KKT point and
hence the preceding proposition implies again that P (x proceed
by contradiction. Assume that a global minimizer -
x of P (x) is not a global minimizer
of Problem (2), then there should exists a point x , global minimizer of Problem (2),
such that
that contradicts the assumption that -
x is a global minimizer of P . The converse is true
by analogous considerations.
In order to complete the correspondence between the solution of Problem (2) and the
unconstrained minimization of the penalty function P we prove the following result
that considers the corrispondence between local minimizers.
Proposition 4.5 The function P (x) admits at most a local-nonglobal minimizer - x
which is a local minimizer of Problem (2) and -x) is the associated KKT multiplier.
Proof We first prove that if -
x is a local minimizer of P (x) then the pair (-x; -x)) satisfies
the KKT conditions for Problem (2). Moreover, by Proposition 4.3, we have that
x is a local minimizer of P , there exists a neighbourhood
of - x such that
Thus, by using (iv) of Proposition 4.2, we obtain
and hence -
x is a local minimizer for Problem (2). The proof can be easily completed
by recalling Proposition 2.2.
5 A new second order optimality condition
The results given in Section 3 and Section 4 can be combined to state new theoretical
properties of Problem (2). In this section we introduce a new second order necessary
optimality condition for Problem (2) for the nonconvex case that follows from the
unconstrained formulation.
Proposition 5.1 Assume that Q is not positive semidefinite, if - x is a global (local)
minimizer of Problem (2) then there exists a unique -
such that the KKT conditions
hold and
a 2
a 2
is positive semidefinite for every " satisfying (13).
Proof If - x is a global minimizer of Problem (2), by Proposition 3.6, we have that
. Then, there exists a neighborhood \Omega\Gamma - x) of -
x such that"
Thus, by (ii) of Proposition 4.2, the function P (x) is twice continuously differentiable
in \Omega\Gamma -
x). and the Hessian matrix evaluated at - x is given by:
a 2
By Proposition 4.4, -
x is also a global minimizer of P (x) and therefore - x satisfies the
second order necessary conditions to be a global unconstrained minimizer of P , that is
positive semidefinite. Then the result follows.
Recalling point (a) of Proposition 3.4, we have that in a global minimizer - x, it results
Hence, the matrixa 2
a 2
is not necessarily positive semidefinite. A similar second order necessary condition was
given in [1], where it has been proved, without requiring any assumptions on the matrix
Q, that if the global minimum is on the boundary, the matrix
a 2
is positive semidefinite where again the matrix 1
a 2
x)-x-x T is not
necessarily positive semidefinite.
6 Algorithmic application
Besides their own theorical interest, the results of the preceding sections are appealing
also from a computational point of view. Although the study of a numerical algorithm
for the solution of Problem (2) is out of the aim of this paper, in this section we give
a hint of possible algorithmic applications of the results of Section 3 and Section 4.
We recall that Proposition 3.4 ensures that given a KKT point which is not a global
solution for Problem (2), it is possible to find a new feasible point with a lower value of
the objective function and that Proposition 3.2 states that the number of KKT points
with different value of the objective function is finite.
These results indicate a new possibility to tackle large scale trust region problems.
In fact they show that a global minimum point of Problem (2) could be efficiently
computed by applying a finite number of times a constrained optimization algorithm
that presents the following features:
(i) given a feasible starting point, it is able to locate a KKT point with a lower value
of the objective function;
(ii) it presents a "good" (at least superlinearly) rate of convergence;
(iii) it does not require an heavy computational burden.
A possibility to ensure property (i) is to use any feasible method that forces the decrease
of the objective function, following, for example, the approach of [37, 17]. Another
possibility is to exploit the unconstrained reformulation of Problem (2) described in
Section 4 which allows us to use any unconstrained method for the minimization of the
penalty function P . In fact, starting from a point x 0 , any of this algorithm obtains a
stationary point -
x for P such that
Then, Proposition 4.3 ensures that -
x is a KKT point of Problem (2) and that P
q(-x). On the other hand, if x 0 is a feasible point, part (iv) of Proposition 4.2 yields
that
In conclusion by using an unconstrained algorithm, we get a KKT point of Problem
(2) with a value of the objective function lower than the value at the starting point.
Furthermore, the possibility of transforming the trust region problem into an unconstrained
one, seems to be quite appealing also as regards properties (ii) and (iii).
In fact Proposition 3.6 and (iii) of Proposition 4.2 guarantees that, in the nonconvex
case, the penalty function is twice continuosly differentiable in every local and global
minimizer of the problem. Therefore, in this case, any unconstrained Truncated Newton
algorithm (see for example [5, 37, 15]) can be easily adapted in order to define globally
convergent methods which show a superlinear rate of convergence in a neighbourhood
of every global or local minimizer.
Nevertheless, we can define algorithm with superlinear rate of convergence without
requiring that the penalty function is twice continuosly differentiable in the neighbourhood
of the points of interest, that is without requiring the strict complementarity in
these points. In fact we can drawn our inspiration from the results in [8].
In particular, we can define a search direction d k as follows:
!/
z k
The results of [8] ensure that the algorithm x locally superlinearly
convergent without requiring the strict complementarity. Following the approach of
truncated Newton method (see for example [5, 15]), in [24] it is shown that an approximate
solution ~
d k of (15)(16) is able to preserve the local superlinear rate of convergence
of the algorithmic scheme. Furthermore it is also proved that this direction ~
suitable descent conditions with respect to the penalty function P . This strict connection
between the direction ~
d k and the penalty function P (x) allow us to define globally
and superlinearly convergent algorithms of the type
~
where ff k can be determined by every stabilization technique and ~
d k is computed by
using a conjugate gradient based iterative method for solving approximately the linear
system (15)(16) .
The paper [24] is devoted to a complete description of this approach with the analysis
of its theoretical properties and to the definition of an efficient algorithm. Here, in
order to have only a preliminary idea of the viability of this unconstrained approach for
solving Problem (2), we have performed some numerical experiments with a rough implementation
of algorithm (17) where ff k is determined by the line-search technique of
[14] and ~
d k is computed by a conjugate gradient algorithm similar to the one proposed
in [5].
We coded the algorithm in MATLAB and run the experiments on a IBM/RISC 6000.
We run two sets of problems randomly generated that we take from the collection of
[34]. We solved ten related problems for each of the two classes both with the easy and
the hard case. According to [34], the hard case occurs when the vector c is orthogonal
to the subspace generated by the smallest eigenvalue of the matrix Q. In Table 1 we
report the results in terms of average number of iterations for problems with increasing
dimension
We run also a set of near hard-case problems (with that is with
c nearly orthogonal to the subspace of the smallest eigenvalue of Q. The results are
FIRST SET SECOND SET
100 11.3 21.9 10.7 25.6
Table
1: Average number of iterations
NEAR HARD CASE
mult. - min
Table
2: Average number of iterations
reported in Table 2. We tested the invariance with respect to the multiplicity of the
smallest eigenvalue (mult. of -
The results obtained are encouraging. The number of iteration is almost constant when
the dimension increases. This feature is appealing when solving large scale problems
taking into account that, at each iteration, the main effort is due to the approximate
solution of a linear system of dimension n or n \Gamma 1 that requires only matrix-vector
products. Furthermore the efficiency of the algorithm seems not to be seriously affected
by the occurrence of the hard case, while it is completely insensible to the near-hard
case.
Of course, even if no final conclusion can be drawn by these limited numerical exper-
iments, the results obtained encourage further research in defining new algorithms for
solving large scale trust region problems which use the results described in this paper.
In particular, as we said before, the possibility of defining efficient algorithms based on
the unconstrained reformulation is investigated in [24].
Acknowledgments
We wish to thank S. Santos, D. Sorensen, F. Rendl and H. Wolkowiz, for providing us
their Matlab codes and test problems. Moreover we thank the anonymous referees for
their helpful suggestions which led to improve the paper.
--R
New optimality conditions and algorithms for homogeneous and polynomial optimization over spheres.
Direct methods for solving symmetric indefinite systems of linear equations.
Computing a trust region step for a penalty function.
Lanczos Algorithms for Large Symmetric Eigenvalue Computation.
An exact penalty method with global convergence properties for nonlinear programming problems.
Exact penalty functions in constrained optimization.
Quadratically and superlinear convergent algorithms for the solution of inequality constrained optimization problems.
A class of methods for nonlinear programming with termination and convergence properties.
Computing modified Newton directions using a partial Cholesky factorization.
On the stationary values of a second-degree polynomial on the unit sphere
Computing optimal locally constrained steps.
A multiplier method with automatic limitation of penalty growth.
A nonmonotone line search technique for Newton's method.
A truncated Newton method with nonmonotone linesearch for unconstrained optimization.
A differentiable exact penalty function for bound constrained quadratic programming problems.
On the solution of a two ball trust region subproblem.
Multiplier and gradient methods.
A continuous approach to compute upper bounds in quadratic maximization problems with integer constraints.
Fast algorithms for convex quadratic programming and multicommodity flows.
An interior-point approach to NP-complete problems
An interior point algorithm to solve computationally difficult set covering problems.
A differentiable piecewise quadratic exact penalty functions for quadratic programs with simple bound constraints.
"La Sapienza"
"La Sapienza"
Local minimizers of quadratic functions on Euclidean balls and spheres.
Trust region algorithms on arbitrary domains.
Generalization of the trust region problem.
Computing a trust region step.
On the use of directions of negative curvature in a modified Newton method.
Algorithms for the solution of quadratic knapsack problems.
A method for nonlinear constraints in minimization problem.
A semidefinite framework to trust region subproblems with applications to large scale minimization.
A new matrix-free for the large scale trust region subproblem
Newton's method with a model trust region modification.
Minimization of a large scale quadratic function subject to an ellipsoidal constraint.
Towards an efficient sparsity exploiting Newton method for minimiza- tion
Nonlinear Optimization.
Proving polynomial-time for sphere-constrained quadratic programming
A new complexity result on minimization of a quadratic function with a sphere constraint.
On affine scaling algorithms for nonconvex quadratic programming.
An extension of Karmarkar's projective algorithm for convex quadratic programming.
--TR
--CTR
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589162 | Superlinear Convergence of a Symmetric Primal-Dual Path Following Algorithm for Semidefinite Programming. | This paper establishes the superlinear convergence of a symmetric primal-dual path following algorithm for semidefinite programming (SDP) under the assumptions that the semidefinite program has a strictly complementary primal-dual optimal solution and that the size of the central path neighborhood tends to zero. The interior point algorithm considered here closely resembles the Mizuno--Todd--Ye predictor-corrector method for linear programming which is known to be quadratically convergent. It is shown that when the iterates are well centered, the duality gap is reduced superlinearly after each predictor step. Indeed, if each predictor step is succeeded by r consecutive corrector steps then the predictor reduces the duality gap superlinearly with order 2 The proof relies on a careful analysis of the central path for SDP. It is shown that under the strict complementarity assumption, the primal-dual central path converges to the analytic center of the primal-dual optimal solution set, and the distance from any point on the central path to this analytic center is bounded by the duality gap. | Introduction
Recently, there have been many interior point algorithms developed for semidefinite programming
(SDP), see for example [1, 2, 5, 9, 11, 13, 17]. These algorithms differ in their choices of scaling
matrix, the size of the central path neighborhoods, and stepsize rules, among others. In particular,
the algorithms of Kojima-Shida-Hara [5] and Nesterov-Todd [11] are based on the primal-dual
scaling and they both can be viewed as extensions of the predictor-corrector method for linear
programming [8]. It has been shown [4, 6, 11, 13, 17] that these algorithms for SDP retain many
important properties of the interior point algorithms for linear programming including polynomial
complexity. For an overview of SDP and its applications, we refer to Vanderberghe and Boyd [15].
However, there exists considerable difficulty in extending one key property of the predictor-corrector
method for linear programming to the interior point algorithms for SDP. This is the
property of quadratic convergence of the duality gap (see [16] for a proof of the LCP case). In
some sense, the need for superlinear convergence in solving SDP is more pronounced than that for
the linear programming case. This is because for SDP there cannot exist any finite termination
procedures as in the case of linear programming. Indeed, the recent papers of Kojima-Shida-Shidoh
[4] and Potra-Sheng [12] are both focused on the issue of superlinear convergence for solving SDP.
In particular, the latter reference provided a sufficient condition for the superlinear convergence of
an infeasible path following algorithm, while the former reference [4] established the superlinear
convergence of their algorithm [5] under certain key assumptions. These assumptions are: (1) SDP
is nondegenerate in the sense that the Jacobian matrix of its KKT system is nonsingular; (2) SDP
has a strictly complementary optimal solution; (3) the iterates converge tangentially to the central
path in the sense that the size of the central path neighborhood in which the iterates reside must
tend to zero. Among these three assumptions for superlinear convergence, (2) is inevitable since it
is needed even in the case of LCP (see [16]). Assumption (3) is needed to ensure the duality gap
is reduced superlinearly after each predictor step for all points in the central path neighborhood.
In the reference [4], an example was given which showed that, without the tangential convergence
assumption, the duality gap is reduced only linearly after one predictor step for certain points in
the central path neighborhood.
Our goal in this paper is to establish the superlinear convergence of a symmetric path following
algorithm for SDP under the only assumptions of (2) and (3) (i.e., without the nondegeneracy
assumption). In particular, we consider the primal-dual path following algorithm of Nesterov-
Todd [11] (later discovered independently by Sturm and Zhang [13] using a V -space notion). In
this paper we adopt the framework of [13] since it greatly facilitates the subsequent analysis. We
show that this symmetric primal-dual path following algorithm has an order of convergence that
is asymptotically quadratic (i.e., sub-quadratic). Indeed, for any given constant positive integer r,
the algorithm can be set so that the duality gap decreases superlinearly with order 2
1+2 \Gamma2r after one
predictor (affine scaling) step followed by (at most) r corrector steps. The cornerstone in our bid
to establish this superlinear convergence result is a bound on the distance from any point on the
central path to the optimal solution set (see Section 3). Specifically, it is shown that, under the
strict complementarity assumption, the primal-dual central path converges to the analytic center
of the optimal solution set, and that the distance to this analytic center from any point on the
central path can be bounded above by the duality gap. These properties of the central path are
algorithm-independent and are likely to be useful in the analysis of other interior point algorithms
for SDP.
The organization of this paper is as follows. At the end of this section, we describe some basic
notation to be used in this paper. In Section 2, we will discuss some fundamental background
notions, and we will make two assumptions concerning the solution set of the SDP. In Section 3
we will analyze the limiting behavior of the primal-dual central path. In Section 4, the notion
of V -space for SDP is reviewed and a path following algorithm in the spirit of [13] is introduced.
The superlinear convergence of this algorithm is established in Section 5. Finally, some concluding
remarks are given in Section 6.
Notation. The space of symmetric n \Theta n matrices will be denoted S. Given X and Y in ! n\Thetan ,
the standard inner product is defined by
where tr (\Delta) denotes the trace of a matrix. The notation X ? Y denotes orthogonality in the sense
that The Euclidean norm and its associated operator norm, viz. the spectral norm,
are both denoted by k\Deltak. The Frobenius norm of X is kXk
positive
(semi-) definite, we write (X - The cone of positive semi-definite matrices is denoted
by S+ and the cone of positive definite matrices is S++ . The identity matrix is denoted by I. We
use the standard "big O" and "small o" notation. In particular, w(-) = O(-) with - ? 0 means
that there is a positive constant \Gamma, possibly dependent on problem data but independent of -, such
that w(- \Gamma- for all -; means that lim -!0
O(w(-)). For a positive definite
matrix, we use "O" and "\Theta" to denote the order of all its eigenvalues. Hence, for W (-) 2 S++ ,
the notation W signifies the existence of \Gamma ? 0 such
Problem formulation
A semidefinite programming (SDP) problem is given as
subject to A (i) ffl
. The decision variable is S. The
corresponding dual program can be formulated as
subject to
Denote the feasible sets of (P) and (D) by F P and FD respectively, i.e.
and
We make the following assumptions throughout this paper.
Assumption 1 There exist positive definite solutions X 2 F P and Z 2 FD for (P) and (D)
respectively.
Assumption 2 There exists a pair of strictly complementary primal-dual optimal solutions for (P)
and (D). Specifically, there exists (X ; Z FD such that
0, we can diagonalize X and Z simultaneously. Therefore, by applying
an orthonormal transformation to the problem data if necessary, we can assume without loss of
generality that X , Z are both diagonal and of the form
positive scalars Here the subscripts B and N signify the "basic" and
"nonbasic" subspaces (following the terminology of linear programming). Throughout this paper,
the decomposition of any n \Theta n matrix X is always made with respect to the above partition B
and N . In fact, we shall adhere to the following notation throughout:
U XN
so XU will always denote the off-diagonal block of X with size K \Theta (n \Gamma K), etc.
Notice that X 2 F P is an optimal solution to (P) if and only if XZ Hence, by Assumption
2, the primal optimal solution set can be written as
F
Analogously, the dual optimal solution set is given by
F
Given - 2 !++ , the pair (X; Z) 2 F P \Theta FD is said to be the -center (X(-); Z(-)) if and only
if
We refer to [5, 14] for a proof of the existence and uniqueness of -centers. The central path of the
problem (P) is the curve
To be consistent with the above definition of the central path, we define the analytic center of
F
P as the unique solution X a of the system
X a
P and ZB - 0:
In a similar fashion, we define the analytic center of F
D as the unique solution Z a of the system
XNZ a
A (i)
3 Properties of the central path
The notion of central path plays a fundamental role in the development of interior point methods
for linear programming. In this section, we shall study the analytic properties of the central path
in the context of semidefinite programming. These properties will be used in Section 5 where we
perform convergence analysis of a predictor-corrector algorithm for SDP.
For linear programming (i.e., A (i) 's and C are diagonal), it is known that the central path
curve converges: (X(-); being the analytic center of
the primal and dual optimal solution sets F
P and F
respectively ([7]). It is also known for linear
programming that the central path does not approach (X a ; Z a ) tangentially to the optimal solution
set, viz. it is shown in [10] that
In the following we shall extend these result to the semidefinite programs (P) and (D).
The following lemma shows that the set
is bounded.
Lemma 3.1 For any - ? 0 there holds
Proof. We have
where we used the property (X(-) in the second equality. Since X(1) - 0
and Z(1) - 0, we have
Q.E.D.
It follows from Lemma 3.1 that the central path has a limit point. We will now show that
any limit point of the central path f(X(-); Z(-))g is a strictly complementary optimal primal-dual
Lemma 3.2 For any - 2 (0; 1) there holds
Hence, any limit point of f(X(-); Z(-))g as - ! 0 is a pair of strictly complementary primal-dual
optimal solutions of (P) and (D).
Proof. Let 1. For notational convenience, we will use X and Z to denote the matrices
X(-) and Z(-). Let (X ; Z ) be the pair of strictly complementary primal-dual optimal solutions
postulated by Assumption 2. Since A (i) ffl
Therefore, we have
where the last step follows from (2.1). Since (by the
positive semidefiniteness of X and Z), we obtain
From
Now consider the identities
log det
log det
U
log det XB log det
log det ZN log det
By the estimates (3.1) and using Lemma 3.1, we see that
Therefore each of the four logarithm terms in the preceding equation are bounded from above as
these four terms sum to zero, we must have
Together with (3.1), this implies
This completes the proof of the lemma. Q.E.D.
Lemma 3.2 provides a precise result on the order of the eigenvalues of XB (-); XN (-); ZB (-)
and ZN (-). We will now prove a preliminary result on the order of the off-diagonal blocks XU (-)
and ZU (-).
Lemma 3.3 For - 2 (0; 1), there holds
\GammaX U (-) ffl ZU
Proof.
By the central path definition, we have
Expanding the right-hand side and comparing the upper-right corner of the above identity, we have
or equivalently,
Using XB Lemma 3.2), this implies that
This proves the first part of the lemma.
We now prove (3.2). Let f(X(- k ); Z(- k :::g be an arbitrary convergent sequence
of the central path with - k ! 0. By Lemma 3.2, the limit of this sequence satisfies strict comple-
mentarity. Let (X ; Z ) denote this limit point so that
As before, we assume without loss of generality that X and Z are diagonal. In addition, since
(3.2) holds trivially when kXU (- k
First, we divide both sides of (3.3) by kXU (- k )k and let k !1 to obtain
U Z
U and Z 1
U are defined by
U := lim
U := lim
(If the above limits do not exist, then we define X 1
U and Z 1
U to be any two limit points of the
corresponding sequences.) Since X
B and Z
are both positive diagonal matrices, it follows that
the nonzero entries of the matrices X 1
U must have opposite signs. By kX 1
that
This establishes (3.2) along the sequence f(X(- k ); Z(- k :::g. Since this sequence is
arbitrary, we see (3.2) holds.
It remains to establish the last part of the lemma. Once again, we consider an arbitrary
convergent sequence f(X(- k ); Z(- k on the central path with - k ! 0; we continue
to use the same notation X , Z , X 1
U defined above. Since kZU (- k
only need to show kXU (- k . Assume this is not the case. Using Lemma 3.2 and
passing onto a subsequence if necessary, we have kXU (- k
Dividing both sides of this equation by kXU (- k )k 2 and taking limit yields
U
Therefore, the limit in the preceding equation equals zero, implying
But this contradicts (3.5), so we must have
The proof is complete. Q.E.D.
We now use Lemma 3.2 and Lemma 3.3 to prove that the central path f(X(-); Z(- ? 0g
converges to (X a ; Z a ), and to estimate the rate at which it converges to this limit.
Lemma 3.4 The primal-dual central path f(X(-); Z(- ? 0g converges to the analytic centers
(X a ; Z a ) of F
P and F
D respectively. Moreover, if we let
ffl(-) := kXU (-)k
then
Proof. Suppose 1. By expanding X(-)Z(-I and comparing the upper-left block,
we obtain
Pre-multiplying both sides with
Let J be an index set of minimal cardinality such that
As Z
it follows from the dual feasibility and (3.6) that-
Now consider the following nonlinear system of equations:
A (i)
By (2.3), we know that X a
B is a solution of (3.8) for some - a
. Using the linear independence
of the matrices A (i)
using the fact that X a
B is positive definite, it can be checked that
the Jacobian (with respect to the variables XB and - i , of the nonlinear system (3.8) is
nonsingular at the solution X a
Hence we can apply the classical inverse function
theorem to the above nonlinear system at the point:
By (3.7) we have
and from X(-) 2 F P we obtain
Combining this with (3.9) and (3.10) yields
It can be shown with an analogous argument that
The proof is complete. Q.E.D.
Lemma 3.4 only provides a rough sketch of the convergence behavior of the central path as
Our goal is to characterize this convergence behavior more precisely.
Theorem 3.1 Let - 2 (0; 1). There holds
and
Proof. The estimate (3.11) is already known from Lemma 3.2, so we only need to prove (3.12).
By Lemma 3.3 and Lemma 3.4, it is sufficient to show that
Suppose to the contrary that there exists a sequence
with kXU (- k )k ? 0 for all k and
lim
0:
To simplify notations, we introduce
(By virtue of Lemma 3.4, we can assume the above limit exists because otherwise we can always
pass onto a convergent subsequence.) From Lemma 3.3 it follows that
lim
Since for each Z 2 FD we have
it follows
We know from Lemma 3.2 that ZB (- k so that the above relation implies
lim
0:
Analogously, it can be shown that
lim
0: (3:15)
As (X(- k we have from (3.14) and (3.15) that
which clearly contradicts (3.2). The proof is complete. Q.E.D.
Theorem 3.1 characterizes completely the limiting behavior of the primal-dual central path as
We point out that this limiting behavior was well understood in the context of linear
programming and the monotone horizontal linear complementarity problem, see Megiddo [7] and
Monteiro and Tsuchiya [10] respectively. Notice that under a Nondegeneracy Assumption (i.e.,
the Jacobian of the nonlinear system (2.2) is nonsingular at (X a ; Z a )), the estimates (3.12) follow
immediately from the application of the classical inverse function theorem. Thus, the real contribution
of Theorem 3.1 lies in establishing these estimates in the absence of the nondegeneracy
assumption.
It is known that in the case of linear programming the proof of quadratic convergence of
predictor-corrector interior point algorithms required an error bound result of Hoffman. This error
bound states that the distance from any vector x 2 ! n to a polyhedral set P ag can
be bounded in terms of the "amount of constraint violation" at x, namely
denotes the positive part of a vector. More precisely, Hoffman's error bound ([3]) states that there
exists some constant - ? 0 such that
Unfortunately, this error bound no longer holds for linear systems over the cone of positive semidefinite
matrices (see the example below). In fact, much of the difficulty in the local analysis of interior
point algorithms for SDP can be attributed to this lack of Hoffman's error bound result (see the
analysis of [4, 12]). Specifically, without such error bound result, it is difficult to estimate the
distance from the current iterates to the optimal solution set. In essence, what we have established
in Theorem 3.1 is an error bound result along the central path. In other words, although Hoffman
type error bound cannot hold over the entire feasible set of (P), it nevertheless still holds true on
the restricted region "near the central path". One consequence of this restriction to the central
path is that we will need to require the iterates stay "sufficiently close" to the central path in order
to establish the superlinear convergence of the algorithm. Such a requirement on the iterates was
called "tangential convergence to the optimal solution set" by Kojima et. al. [4]. Notice that
the analysis in this reference required the additional nondegeneracy assumption to establish their
superlinear convergence result. In contrast, this assumption is no longer needed in our analysis
because Theorem 3.1 holds without the nondegeneracy assumption.
Example. Consider the following linear system over the cone of positive semidefinite matrices in
Clearly, there is exactly one solution X to the above linear system, namely
For each ffl ? 0, consider the matrix
Clearly, X(ffl) - 0. The amount of constraint violation is equal to ffl 2 . However, the distance
\Theta(ffl). Thus, there cannot exist some fixed - ? 0 such that
for all ffl ? 0. Instead, we have in this case that is, the error bound
holds with an exponent of 1=2.
4 A polynomial predictor-corrector algorithm
We begin by summarizing some of the results on V -space path following for SDP that were obtained
in [13].
Let (X; Z) 2 F P \Theta FD with X - 0; Z - 0. Then, there exists a unique positive definite matrix
S++ such that ([13, Lemma 2.1])
Let L be such that
and let V := L T ZL. It follows that
The quantity
serves as a centrality measure, with - := X ffl Z=n. It is easy to see that the central path is the set of
solutions (X; Z) with ffi(X; equivalently, those solutions for which
we have
In V -space path following, we want to drive the V -iterates towards the origin by Newton's method,
in such a way that the iterates reside in a cone around the identity matrix. Before stating the
Newton equation, we need to introduce the linear space A(L),
and its orthoplement in S
A Newton direction for obtaining a (fl-center, for some fl 2 [0; 1], is the solution (\DeltaX; \DeltaZ ) of
the following system of linear equations ([13], equation (17)):
\DeltaX
For we denote the solution of (4.4) by (\DeltaX p ; \DeltaZ p ), the predictor direction. For
solution is denoted by (\DeltaX c ; \DeltaZ c ), the corrector direction. If we let
then we can rewrite (4.4) as
It follows from orthogonality that
F
F
The corrector direction does not change the duality gap,
whereas
for any t 2 !, see equation (16) of [13].
Algorithm SDP
Given positive integer r.
REPEAT (main iteration)
Compute the largest step t k such that for all there holds
and
IF ffi(X; Z) - fi k THEN exit loop.
Compute
UNTIL convergence.
Interestingly, each corrector step reduces ffi(\Delta; \Delta) at a quadratic rate as stated in the following
result:
Lemma 4.1 If ffi(X; Z) - 1
Proof. It follows from Lemma 4.5 in [13] that
Hence, the desired result is an immediate consequence of Lemma 4.4 in [13]. Q.E.D.
Also, it follows from (4.6), (4.7) and Lemma 4.1 that for any k ? 1
Furthermore, if fi only one corrector step (i.e., needed to recenter the iterate
(see [13]). In other words, the iterations of Algorithm SDP are identical to that of the primal-dual
predictor-corrector algorithm of [13], for all k with
We can therefore conclude from Theorem 5.2 in [13] that the algorithm yields - k - fflfor -
O(
log(- 0 =ffl)). Thus, we have the following polynomial complexity result.
Theorem 4.1 For each generate an iterate (X
ffl in at most O(
predictor-corrector steps.
In addition to having polynomial complexity, Algorithm SDP also possesses a superlinear rate
of convergence. We prove this in the next section.
5 Convergence analysis
We begin by establishing the global convergence of Algorithm SDP. Notice that Algorithm SDP
chooses the predictor step length t k to be the largest step such that for all there holds
oe
It was shown in [13] (equation (21) therein) that
Combining (5.1) and (5.2), we can easily establish the global convergence of Algorithm SDP.
Theorem 5.1 There holds
lim
i.e. Algorithm SDP is globally convergent.
Proof. Due to (4.7), - . is a monotone decreasing sequence. Hence, the sequence has a
limit. Suppose contrary to the statement of the lemma that
Then, we obtain from (4.5), (5.1) and (5.2) that t Together with (4.7) this implies that
= \Theta(1), which contradicts (5.3). Q.E.D.
Next we proceed to establish the superlinear convergence of Algorithm SDP. In light of (4.7),
we only need to show that the predictor step length t k approaches to 1. Hence we are led to bound
t k from below. For this purpose, we note from (5.2) that, for t 2 (0; 1),
Thus, if we can properly bound
, then we will obtain a lower bound on the predictor
step length t k .
To begin, let us consider L - with
Remark that
Now define the predictor direction starting from the solution (X(-); Z(-)) on the central path as
Z a ) be the analytic center of the optimal solution set in the L -transformed space,
Z a := L T
- Z a
We will show in Lemma 5.1 below that \Delta -
is close to the optimal step -
We will bound the difference between \Delta -
afterwards.
Lemma 5.1 There holds
X a
Z a
Proof. Since
it follows
Z a
Z a
X a
Therefore, the matrix (
Z a ), or equivalently, the matrix
is symmetric. By the property of F-norm, we obtain
Z a )
where the last step follows from Theorem 3.1. Now since -
we have
Z a )
Z a
As
Z a 2 A(L - );
it follows that fl fl
X a
F
Z a
F
F
where last step is due to (5.5). This proves the lemma. Q.E.D.
Lemma 5.1 applies only to (\Delta -
namely the predictor directions for the points
located exactly on the central path. What we need is a similar bound for (\Delta -
at points close to the central path). This leads us to bound the difference \Delta -
our next goal is to show (Lemma 5.5) that
O(
We prove this bound by a sequence of lemmas. Let D be given by (4.1) and define
so that -
Choose L by
and notice that indeed LL stipulated by (4.2).
Lemma 5.2 Suppose ffi(X; Z) - 1. There holds
Proof. Let
Clearly, \Delta x (-) and \Delta z (-) are symmetric and \Delta x (-) ? \Delta z (-). Let ae denote the smallest eigenvalue
of
aeIg:
tr
where the last step follows from (X(-)
Consider
tr
0:
By (4.3), the matrix V is symmetric positive definite and Diagonalize the symmetric
matrix
the diagonal entries of QV Q T must be \Theta(1). Therefore, the preceding
equation implies that the diagonal matrix must have a nonpositive eigenvalue and that its diagonal
entries are of same order of magnitude. In other words, ae - 0 and O(jaej). This further
implies
By the definition of the central path, we have
Now using the fact that the above matrix is symmetric, it follows that
and therefore,
Using (4.3), we obtain
Combining this with (5.6) and using the fact that \Delta x (-) ? \Delta z (-), we have
Q.E.D.
Lemma 5.3 Suppose ffi(X; Z) - 1=2. Then there holds
Proof. Notice that
and
Now using
we have, by pre- and post-multiplying the above two equations with -
D \Gamma1=2 and rearranging terms,
Together with Lemma 5.2, this implies -
The lemma is proved.
Q.E.D.
Notice that
Lemma 5.4 We have
Proof. We have
Now using Lemma 5.3 and (4.5), we see that
It can be shown in an analogous way that
Q.E.D.
Now we are ready to bound the difference between \Delta -
Lemma 5.5 Suppose ffi(X; Z) - 1=2. We have
Proof. By definition of the predictor directions, we have
and
Combining these two relations yields
Now using Lemma 5.4 and using the fact that
we obtain fl
Z p ), the lemma follows from the above relation, after
applying Lemma 5.4 once more. Q.E.D.
Combining (5.5), Lemma 5.1 and Lemma 5.5 we can now estimate the order of
and hence, using (5.4), we can estimate the predictor step length t k .
Lemma 5.6 We have
Proof. Combining Lemma 5.5 with Lemma 5.1, we have
X a
Z a
so that, using (4.5), fl fl
X a
Z a
Moreover,
Z a )
X a
Applying (5.5), (5.7), (5.8) and (4.5) to the above relation yields
Q.E.D.
Theorem 5.2 The iterates (X k ; Z k ) generated by Algorithm SDP converge to (X a ; Z a ) superlinearly
with The duality gap - k converges to zero at the same rate.
Proof. From (5.4) we see that for any t - 0 satisfying
there holds
This implies using (4.8) and Lemma 5.6 that
so that
This shows that the duality gap converges to zero superlinearly with order 2=(1 It remains
to prove that the iterates converge to the analytic center with the same order. Notice that
However, using the definition of F-norm and applying Lemma 5.3,
Recall that L - k
definition, so that using Lemma 3.1,
Combining (5.9) and (5.10) with Lemma 5.2, we have
Hence, we obtain from Theorem 3.1 that
Similarly, it can be shown that
This shows that the iterates converge to the analytic center R-superlinearly, with the same order
as - k converges to zero. Q.E.D.
6 Conclusions
We have shown the global and superlinear convergence of the predictor-corrector algorithm SDP,
assuming only the existence of a strictly complementary solution pair. The local convergence analysis
is based on Theorem 3.1, which states that O(-). By enforcing
the iterates "inherit" this property of the central path. For the generalization of
the Mizuno-Todd-Ye predictor-corrector algorithm in [13], we do not enforce ffi(X
hence we cannot conclude superlinear convergence for it yet. In this respect, it will be interesting to
study the asymptotic behavior of the corrector steps. Finally, it is likely that our line of argument
can be applied to the infeasible primal-dual path following algorithms of Kojima-Shindoh-Hara [5]
and Potra-Sheng [12].
--R
"Interior point methods in semidefinite programming with applications to combinatorial optimization problems,"
"An interior-point method for semidefinite programming,"
"On approximate solutions of systems of linear inequalities"
"Global and local convergence of predictor-correct infeasible-interior-point algorithm for semidefinite programming,"
"Interior-point methods for the monotone linear complementarity problem in symmetric matrices,"
"An infeasible start predictor corrector method for semi-definite linear programming,"
"Pathways to the optimal solution set in linear programming,"
"On Adaptive-Step Primal-Dual Interior-Point Algorithms for Linear Programming,"
"Primal-dual path following algorithms for semidefinite programming,"
"Limiting behavior of the derivatives of certain trajectories associated with a monotone horizontal linear complementarity problem,"
"Primal-dual interior-point methods for self-scaled cones,"
"A superlinearly convergent primal-dual infeasible-interior-point algorithm for semidefinite programming,"
"Symmetric primal-dual path following algorithms for semidefinite programming,"
"A primal-dual potential reduction method for problems involving matrix inequalities,"
"Semidefinite programming,"
"On quadratic and O( p nL) convergence of a predictor-corrector algorithm for LCP,"
"On extending primal-dual interior-point algorithms from linear programming to semidefinite programming,"
--TR | path following;central path;superlinear convergence;semidefinite programming |
589163 | The Sequential Knapsack Polytope. | In this paper we describe the convex hull of all solutions of the integer bounded knapsack problem in the special case when the weights of the items are divisible. The corresponding inequalities are defined via an inductive scheme that can also be used in a more general setting. | Introduction
In this paper we deal with the integer bounded knapsack problem
a
and the numbers
a i are divisible, i.e., a i
a
n. In this case we say that the
knapsack problem has the divisibility property. It is also called the sequential
knapsack problem (see [1]). Whenever we are given a knapsack problem having
the divisibility property, we will assume without loss of generality that a
Our main result is the construction of the system of inequalities that describes
the convex hull of all solutions in this special case.
Since years the knapsack polytope is of particular interest for researchers in
polyhedral combinatorics. This is due to several reasons: one is the increasing
number of applications like in circuit design, telecommunication, vehicle routing
and scheduling that involve the knapsack problem as a subproblem. In order to
apply polyhedral methods to such complex problems, a good understanding of the
knapsack polytope is important. Secondly, the knapsack problem is the "easiest"
case of a number dependent problem. A slight change of the weights of the items
might change the inequalities that describe the polyhedron drastically. There-
fore, it is important to understand "general principles" according to which valid
inequalities are constructed. Examples in this direction are, for instance, Gomory
cutting planes [2], covers [12], (1; k)-configurations [8], the concept of lifting [7],
the weight reduction principle [10] or inequalities based on the Hilbert basis of
a cone of exchange vectors [11]. The knapsack polytope is one of the very interesting
and challenging polyhedra for which beautiful results can be discovered.
We present an inductive scheme to construct valid inequalities for the knapsack
polytope and show, in case that the weights of the items have the divisibility
property, that we obtain the complete description of the associated polyhedron.
The special case of the knapsack problem with the divisibility property has been
studied in the literature by several authors.
Hartmann and Olmstead [4] give an O(n log n) algorithm for optimizing a linear
objective function whose bottleneck operation is sorting the ratios fl i
a i
The case of the sequential knapsack problem when s
considered by Marcotte [6]. He shows that an optimum solution can be found
in linear time and applies his algorithm to the cutting stock problem. Pochet
and Wolsey [9] give an explicit description of the knapsack polyhedron with the
divisibility property when there are no bounds on the variables. They also refer
to applications in local area networking.
In Section 2 we present a transformation of any given sequential knapsack problem
to a special one such that in terms of feasible solutions and optimization both
formulations are equivalent. In Section 3 we outline a decomposition result for
all the optimal solutions of such a transformed sequential knapsack problem.
Our main result is contained in Section 4. Here we present an inductive scheme
to generate valid inequalities for the sequential knapsack problem. Given an
objective function, we construct an inequality via this scheme whose induced
face contains the set of all optimal solutions. This sufficies to show that our
inductive class of inequalities describes the sequential knapsack polyhedron. How
inequalities defined via our inductive scheme can be interpreted combinatorially
is the issue of Section 5. The discussions end in Section 6 with some extensions.
Throughout the paper we use the following notation.
For
vg.
The constraint
is called the knapsack inequality. The number
a is termed the weight of item i and a 0 2 IN is called the knapsack capacity.
We set N := ng and we always assume that 0 ! a 1 - a n - a 0 .
An integer vector that satisfies the knapsack constraint and the lower and upper
bound constraints is called feasible.
We say, F c is a face of some polytope P induced by the valid inequality c T x - fl,
flg. Every x 2 F c is also called a root of c T x - fl. The
inequalities x are called trivial. For real numbers
ng we use
the notation - (I) := P
In this section we present a transformation of the given sequential knapsack problem
to a special sequential knapsack problem that satisfies certain requirements.
We show that in terms of polyhedra and in terms of optimization both formulations
are equivalent. We start by introducing the notion of blocks.
l be a subset of items. B is
called a block if, for every
holds.
Let B be a block. The above definition implies that for every number - 2
there exists a subset W ' B such that
. The number uB :=
a
is called the
multiplicity of block B. We replace block B by a single item B with weight a i 1 and
multiplicity (upper bound)
a
. The objective function coefficient
of B is the number
Bm be a partition of N into blocks and denote by fw , c w , uw the
weight, objective function coefficient, multiplicity of block Bw , respectively,
m. Now consider the knapsack problem where every block is replaced by
a single item:
z
From the construction of the blocks it is clear that (MSKP) is a sequential knapsack
problem (MSKP stands for modified sequential knapsack problem). We now
show that there is a many to one correspondence between the feasible solutions
of the original problem (SKP) and the feasible solutions of its modified version
(MSKP). For ease of notation we assume that f 1 - f 2 - fm , and in case
holds. By P SKP and PMSKP we denote the convex
hull of all feasible vectors of the problem (SKP) and (MSKP), respectively.
Let z 2 IR m be a feasible solution of (MSKP), i.e., 0 - z w - uw , z w integer for
all m. By Definition 2.1, for every w there exist integers
Bw such that P
In fact, for all subsets I w of
items in Bw with
IN, the vector x 2 IR n
defined via x otherwise, is
feasible for problem (SKP).
Conversely, with every vector x 2 IR n that is feasible for problem (SKP) we
associate a vector z 2 IR m by setting z w :=
j2Bw a j x j
It follows that an integer vector z with z w
feasible for (MSKP) if and only if there exist feasible vectors of (SKP) with the
same total weight as z.
Now suppose that
is a valid inequality for the polytope PMSKP .
By setting fi i := b w a i
fw if item i belongs to block Bw , the inequality
is valid for P SKP . This statement follows from the fact that if x is feasible
for (SKP) then defined via z
j2Bw a j x j
is feasible for (MSKP) and satisfies
This shows that valid
inequalities for PMSKP can be transformed into valid inequalities for P SKP .
In the following we focus on a special partition of the set N into blocks
For an item i of (SKP), its gain per unit is defined as fl i
a i
. Let
the different values of gains per unit for all items of (SKP) (clearly, v - n).
We partition each set V g := fi
a i
into blocks
ng such that B g
j is not a block anymore, for
denote the final blocks constructed this way. Each block B i ,
is called a maximal block and, by definition, all items belonging to
the block B i have the same gain per unit.
Example 2.2. Consider the instance of the sequential knapsack problem
with upper bounds s i on the variables x i as follows: s
1. The set of items is partitioned into the
f7g. After transformation we obtain the instance
of the sequential knapsack problem:
with upper bounds on the variables
For a given sequential knapsack problem, the aggregation of items into maximal
blocks is unique. If V l is the set of
all items in N with gain per unit equal to g, then the unique maximal block
containing
g and a i l+1
is defined as
item in this subset fi can belong to some maximal block containing
an item t, because a i j
- a i t+1
. By removing
1 from V g and iteratively using the same argument, the unique partition of V g
into maximal blocks B g
ng , with B g
can be constructed
easily. This argument applies to all numbers g 2 fg g.
From the above discussions follows that, if we define (MSKP) using the unique
partition into maximal blocks, a vector z is feasible for (MSKP) if and only if
the associated vectors x are feasible for (SKP). As each maximal block contains
items in N with the same gain per unit we obtain in addition: a vector x 2 IR n
is optimal with respect to (SKP) if and only if the associated vector z 2 IR m is
optimal with respect to (MSKP) and vice versa, a vector z 2 IR m is optimal with
respect to (MSKP) if and only if all of its associated vectors x 2 IR n are optimal
solutions to (SKP).
To simplify notation, we always assume, when transforming (SKP) to (MSKP)
using maximal blocks, that f 1 - f 2 - fm , and in case
. Moreover, the above arguments for the construction of the unique
partition into maximal blocks show that for the transformed problem (MSKP)
the following property always holds:
fw
This property will be used in the next section to derive a decomposition scheme
of all optimal solutions of (MSKP).
3 Decomposition of optimum solutions
In this section, we characterize the optimal solutions of a problem (MSKP) obtained
by the maximal block transformation of an initial (SKP) problem presented
in the previous section.
Let positive rational numbers c positive integers
fm be given such that
We also assume that for every
For every F 2 IN and we denote by P F (j) the convex hull
of all solutions of the following (MSKP) problem with knapsack capacity F and
restricted to the variables 1 to j.
The optimization problem OP F (j) is the program
Note that in this section we only consider optimization problems OP F (j) with
positive objective coefficients. Using this notation we have that
and (m). By O F (j) we denote the set of all optimal solutions to
OP F (j). Finally, for an item
oe
i.e., \Delta i is the set of all items before i whose gain per unit is strictly better than
the one of i. Let f
be the total weight of items in \Delta i .
For every F and j, we now construct a decomposition tree whose paths from the
root node to the leaves contain all the optimal solutions of OP F (j). The key for
this result is the next lemma showing that for every optimum solution z 2 O F (j)
the component z j can attain at most two different values.
Lemma 3.1. For the optimization problem OP F (j) with positive cost coefficients
the following statement is true:
z 2 O F (j) implies that z j - min
ae
and z j - min
ae
l
Proof. We prove this result by contradiction using standard exchange arguments.
Several cases are distinguished.
the lemma states that z (j). By contra-
diction, suppose that there exists z 2 O F (j) with z j ? 0. As
f l z l
the divisibility of the weights, there exist integers - l 2
all l 2 \Delta j such that P
. We now define a solution z 0 with
z 0
i and the solution z 0 has
strictly better objective value than z by definition of \Delta j . This contradicts
the optimality of z.
because u j is in-
tegral. In this case the lemma states that z (j). By
contradiction, suppose that there exists z 2 O F (j) with z
the new solution z 0 with z 0
z 0
belongs to P F (j) and has strictly better
objective value than z, because c j
for all
contradiction.
Hence, we can assume that By the divisibility of the
weights, there exist integers - l 2
with
The new solution z 0 with z 0
for l 2
belongs to P F (j).
0g. As
(where the last inequality holds by as-
sumption), there exists i 2 W with c i
. Then the solution z 0 has
strictly better objective value than z, again a contradiction.
In the remaining cases we have that and the lemma states
and z j -
l F \Gammaf
suppose, by contradiction, that there exists
z 2 O F (j) with z
similar argument as in case (ii) shows that there exists
a solution z
and with a strictly better
objective value than z, a contradiction.
suppose, by contradiction, that there exists
z 2 O F (j) with z
l F \Gammaf
As
f l z l +f j z j -
l F \Gammaf
similar argument as in case (i) shows that
there exists a solution z
z 0
l F \Gammaf
and with a strictly
better objective value than z, a contradiction.
Lemma 3.1 can be applied inductively to build a binary decomposition tree containing
all potential optimal solutions in O F (j). We illustrate this on an example.
Example 2.2 Continued. The modified sequential knapsack problem P 396 (5)
using the maximal block transformation was defined as
with upper bounds on the variables
We have c 1
0:56, and hence
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
(2) (2) (2) (2)
(1) (1) (1) (1) (1) (1) (1)
Figure
1: Decomposition of Optimal Solutions for Example 2.2
Figure
1 illustrates the decomposition tree that we obtain from applying Lemma
3.1 iteratively. The node labels identify the problems P F (j) to be solved and
the value of z j is fixed on the corresponding branches. For example, Lemma 3.1
applied to P 396 (5) yields z are left with problem
are left with problem P 36 (4). Potential optimal solutions
of problems P 396 (4) and P 36 (4) are further decomposed using Lemma 3.1.
The set S 396 (5) of potential optimal solutions to P 396 (5) is defined by all the paths
from the leaves to the root node in the decomposition tree, that is
and, by Lemma 3.1, O 396 (5) ' S 396 (5).
For a given problem P F (j) and its associated decomposition tree, we define in
the next section valid inequalities that are satisfied at equality by all solutions in
this decomposition tree, and thus by all optimal solutions in O F (j).
4 The convex hull of all solutions to the sequential
knapsack problem
Let (SKP) be a sequential knapsack problem and suppose that C is a class of valid
inequalities for P SKP . The technique that we use in order to show that C describes
P SKP is due to Lovasz [5]: for every objective function fl we prove that the set of
optimal solutions to (SKP) belongs to the face induced by some inequality in C.
This suffices to show that C describes P SKP , because when an objective function
is parallel to a facet defining inequality, then the only inequality satisfied at
equality by all optimal points in (SKP) is this facet defining inequality. Hence,
C contains all the facet defining inequalities.
We first consider the case that all objective function coefficients are positive.
As outlined in Section 2, we partition N into maximal blocks
construct the modified sequential knapsack problem (MSKP). Associated with
the transformed problem (MSKP), we use the notations P F (j), OP F (j), O F (j),
introduced in Section 3.
For every knapsack capacity F 2 IN and for every mg, we now
define an inequality I F (j) satisfying the conditions (i), (ii), (iii) listed below:
(i) The left hand sides of inequalities I F (j) and I F 0 (j) are equal if F modulo f j
holds.
(ii) I F (j) is a valid inequality for P F (j).
(iii) The set of optimal solutions O F (j) is contained in the face induced by the
inequality I F (j).
The inequalities I F (j) are defined inductively on j.
We define the inequality I F (1) as z 1 - minfF; u 1 g. This inequality clearly
satisfies all the properties (i) - (iii).
number r between 0 and f given and assume that for
every number F 2 IN with F modulo f there exists an inequality I F (j \Gamma 1)
that satisfies the properties (i) - (iii).
In particular, property (i) guarantees that this family of inequalities is of the
are the coefficients of the
inequalities I F (j \Gamma 1) for all F with F modulo f r. With the parameter r we
associate a number F r . We set F r := r if f
r is the largest number of residuum class r with respect
to f j not exceeding the sum of weights in that have a better gain
per unit than j.
For every F 2 IN with F modulo f r the left hand side of the inequality I F (j)
is of the form
defined as in I F (j \Gamma 1) and d j defined
by
In order to define the corresponding right hand side - that we denote by g F;j -
we need to distinguish several cases.
First,
is an integer.
We set
Under these assumptions the inequality I F (j) defined via
the three properties (i), (ii) and (iii). These statements are shown below. We
first illustrate this (inductive) construction on the initial example. Then three
technical lemmas are proved and afterwards applied to show that I F (j) satisfies
(i)-(iii).
Example 2.2 continued.
with upper bounds on the variables
The construction of the inequalities is defined for any value of F . If we are only
interested in the inequality I 396 (5), then we need not find I F (j) for all values of
F . The node labels in Figure 1 represent the subproblems we have to solve in
order to obtain an optimum solution for the original problem OP 396 (5). They
also give the F and r values we must consider in order to construct I 396 (5).
I
I F (2) with
I
I 6 (2)
I 1+5s (2)
I 1+5s (2)
I F (3) with
I
I
I
I F (4) with
I
I
I F (5) with
I
I
The inequality I 396 (5) is satisfied at equality by all solutions in S 396 (5) containing
all optimal solutions in O 396 (5).
Lemma 4.1. Let F and G be natural numbers such that F - G and F modulo
holds.
Proof. For the statement is certainly true. So assume, it holds for all
numbers that are less or equal than j \Gamma 1. We show that it is true for j as well.
We G, we know that s - t. We
define
G; if t ! 0;
Checking all cases we notice that F 0 - G 0 , F 0 modulo f
As F 0 modulo by assumption of the induction
and the claim follows.
Lemma 4.2. Let F and G be natural numbers such that F - G and F modulo
holds for every
Proof. g G+oef j Applying Lemma 4.1
yields [g G+oef
Iterating this argument proves Lemma 4.2.
Accordingly, we obtain Lemma 4.3.
Lemma 4.3. Let F and G be natural numbers such that F - G and F modulo
holds for every
Proof. 4.1, we conclude that [g F;j \Gamma
Iterating
this argument proves Lemma 4.3.
Using Lemmas 4.1 - 4.3 we are now able to prove the following theorem.
Theorem 4.4. Given a modified sequential knapsack problem with positive
objective function obtained from the maximal block transformation. If the inequalities
I F (j \Gamma 1) satisfy the three conditions (i), (ii), (iii) with
so do the inequalities I F (j) with
(i) The left hand sides of two inequalities I F (k) and I F 0 (k) are identical whenever
(ii) I F (k) is valid for P F (k);
(iii) Every optimum solution to problem OP F (k) is contained in the face induced
by the inequality I F (k);
Proof. We write I F (j) as
(i) Let F and F 0 be two natural numbers satisfying F modulo f
As \Delta j and F r are uniquely defined by the residuum class r and the
objective function we obtain - per definition - that the left hand sides of the two
inequalities I F (j) and I F 0 (j) are the same.
(ii) The inequality I F (j) is valid for the polyhedron P F (j). Let z 2 P F (j) be
a feasible point, then
G;j \Gamma1 is a valid inequality for all values of G with G modulo f j
Again, we write
cases.
(ii) (a) F - F r . Then s - 0. If z it follows from the definition of g
F;j \Gamma1 that the inequality is valid. Suppose that z j ? 0. Then
, where the
last inequality follows from Lemma 4.3, and the statement follows.
it follows from the definition of
that the inequality is valid. Suppose that z j ! u j . By
applying Lemma 4.2, we obtain:
(ii) (c) What remains is the case where F r
holds and we obtain
If z the inequality is valid by construction. Otherwise,
if z j ? s, then Lemma 4.3 implies that g Fr \Gamma(z j \Gammas)f
together with (?) we have that I F (j) is valid. Finally, if z
implies that g Fr \Gamma(z j \Gammas)f
which again shows that the inequality I F (j) is valid.
(iii) It remains to be shown that the set of optimal solution O F (j) is contained
in the face induced by the inequality I F (j).
By definition of F r , we can always write (j), then by
Lemma 3.1 and by definition of F r we have F \Gammaf
. If r - f
then F r is the unique number such that F r - f
r. In this case Lemma 3.1 implies that
or z
In this case, z
Hence, r ? f implies that z sg.
Summarizing all cases yields z or z
In case s - 0, i.e., F - F r , we have z in every optimum solution. Therefore
by assumption of the induction, every optimum solution to problem OP F (j) is
contained in the face
this case, the claim
follows.
In case s - every element in the set O F (j) satisfies
By assumption of the induction, every optimum solution z to problem
F;j . This proves the claim in this case.
Finally, we have 1 - s - u j . Then every optimum solution z of OP F (j) satisfies
either z By assumption of the induction we obtain that (a)
This yields in case (a):
In case (b) we obtain:
This shows that in both cases the
inequality I F (j) is satisfied at equality by all optimal points.
Let us now present the final theorem describing P SKP as a system of inequalities.
Let W ' N be a subset of items in N , let be a partition of W
into blocks and let - be a permutation of
l g be
the weight of block
a l s l
be the multiplicity of block B i and assume
that f 0
m . We set f
Denote by P b a 0
f 0c (m) the modified knapsack polytope defined with the block partition
B of W , weights f multiplicities um and knapsack capacity
b a 0
f 0c. That is
z
f 0c
If the inequality I b a 0
f 0c (m), written as
f 0c;m , denotes the valid inequality
developed in this section for P b a 0
f 0c (m) using the sets \Delta j induced by the
permutation -, then the inequality K(W;B;-) is defined as
a i
Theorem 4.5. Given an instance of (SKP), the following system of inequalities
describes the polyhedron
K(W;B;-); for all W ' N; all partitions
W into blocks, and all permutations - of
Proof. We first show validity of the inequalities K(W;B;-). Given W ,
and -. It is easy to check that there exists an objective function
is the partition of W into maximal blocks and there exists
- such that
g:
Then, the inequality I b a 0
f 0c (m) is valid for the polyhedron P b a 0
f 0c (m) by Theorem
4.4 (i) and (ii). By the arguments on the transformation of valid inequalities for
PMSKP to valid inequalities for P SKP (see Section 2), the inequality K(W;B;-)
is valid for the polyhedron
a i
This polyhedron is a relaxation of P SKP , because f 0
1 is the smallest weight among
all items in W . As K(W;B;-) is valid for this relaxation of P SKP , it is certainly
valid for P SKP .
Now given any objective function construct an inequality
satisfied at equality by all optimal solutions of (SKP). If
then and we set W := fi 2
Bmg be the partition of W into maximal blocks and let (MSKP)
denote the modified sequential knapsack problem of Section 2. From Theorem
4.4 (iii) we know that I b a 0
f 0c (m) is satisfied at equality by all optimal solutions
of (MSKP). By the arguments on the equivalence of optimal solutions beween
problems (SKP) and (MSKP) (see Section 2), K(W;B;-) is satisfied at equality
by all optimal solutions of
a i
Now if x is an optimal solution of the original problem with K(W;B;-) not
satisfied at equality (because some i 2 N n W has value x i ? 0), then a solution
with strictly better objective function value can be found by setting x
This completes the proof.
5 Explicit Inequalities
In the previous section we have inductively defined a class of inequalities that
depends on the choice and ordering of the blocks. Can we find a more explicit
or combinatorial formulation for those inequalities? This question is addressed
now.
Given a sequential knapsack problem of the form
where u is the vector of upper bounds on the variables and 1
A large class of inequalities for the associated polyhedron P SKP can be described
as follows:
Let r i denote the residuum of the capacity F modulo f i . We choose sets S i ' N i ,
with the following properties:
Setting
the inequality
is valid for P SKP . This statement can be verified by applying our inductive
scheme: we define a modified sequential knapsack problem and, for every item i
in this modified problem, we choose a set \Delta i such that the inequality constructed
via our inductive scheme coincides with (?).
We first consider the case where
2. Here we define the transformation to (MSKP) by
considering k. Thus, the
modified sequential knapsack is of the form
The ordering of blocks is defined by \Delta
F;k denote the
inequality I F (k) for this modified problem (MSKP) with sets
now show that (?) coincides with the inequality
F;k that is
obtained by transforming I F (k) to a valid inequality for (SKP) (see Section 2).
As
we have that, for any j - 2, r
thus
To derive I F (k) using our inductive scheme, we have to compute the numbers
g. As f
Starting from d going through the
inductive scheme (see Section 4) we obtain for each
finally
which shows that the inequality (?) is obtained via our inductive
scheme and thus is valid for P SKP when
When
then we consider
as a single block. Performing this for all j with
generating the corresponding modified knapsack problem, constructing the valid
inequality using our inductive scheme and transforming it to a valid inequality
for (SKP) yields the inequality (?).
The inequalities of the form (?) are already a strong generalization of other known
inequalities:
In case that if u is the vector of all ones (the 0=1 case), then the
inequality (?) is of the form
cl
. The latter class of inequalities plus the trivial inequalities
plus the cover inequality
c describe the polyhedron
g. This result was
shown in [11] and, independently in [3].
As a special case we obtain Padberg's result on (1; k)-configurations [8]: Suppose,
we are given a knapsack problem such that the set of feasible solutions is equal
to
The corresponding polyhedron is described by the lower and upper bound constraints
plus the inequalities
for all subsets
Summarizing our discussions, the inequalities (?) are only a subclass of the inequalities
needed to describe a sequential knapsack polyhedron. Nevertheless, this
subclass is quite large and extends all the explicitly known inequalities for special
cases of the knapsack problem having the divisibility property.
6 Extensions
The previous sections deal exclusively with the sequential knapsack polytope
which is still a restrictive assumption when considering integer programs in gene-
ral. Can we use parts of this polyhedral knowledge presented so far and apply it
within a more general framework? The answer is "yes" and we outline now some
directions.
A first question in using our inductively defined inequalities computationally is
whether we have a combinatorial algorithm for solving the separation problem,
i.e., given a fractional solution y: does there exists an inequality that is violated
by y and if so, then what is the inequality? We did not succeed in solving this
separation problem. "Only" for the subclass of inequalities
cl where ;
Hartmann [3] gives a linear time algorithm for solving the separation problem .
The general problem is still open. However, we can use our inductive scheme as
a separation heuristic. For instance, defining every item i 2 N as a single block,
setting generating an inequality
according to this ordering seems to be a promissing approach to end up with a
violated inequality, if one exists. Other reasonable definitions of \Delta i might be to
set
Whether those ideas work is
certainly not clear, but similar "greedy type" of procedures work pretty well for
the separation of cover- and (1; k)-configuration inequalities.
Given an integer programming problem Ax - b; 0 - x - u; x integer with
. If there exists some row P
such that a subset S of items in fj 2 ng 0g has the divisibility
property, then we can investigate the polyhedron: convfx 2
and generate inequalities for this polyhedron. By
computing lifting coefficients for the items in N n S, we obtain a valid inequality
for the overall polyhedron convfx 2 g. This
approach can always be used to apply knowledge about special integer programs
to more general cases.
Another idea is to try to relax a given integer program as a sequential knapsack
problem. Given a row
of an integer program, the easiest way to
obtain a relaxation as a sequential knapsack problem is to choose, a priori, a set
of divisible numbers f say. The sequential knapsack problem defined via
the constraint X
is certainly a relaxation of the given integer program.
A more specific relaxation is obtained by generalizing the concept of (1; k)-
configurations. Consider the 0=1 knapsack problem defined by the constraint
with N :=
assume that f(S) - F , f(S)
define a partition S 1 of the set N
of items as
Based on this partition, we define an inequality with the divisibility property that
is valid for the given 0=1 knapsack problem. We set b 1 := 1 and, for
we define
Note that t j - We define finally
If such a t -+1 exists (i.e. if f s ! f then the inequality
is valid for the 0=1 knapsack problem. Before verifying this statement, let us
illustrate the above construction on an example.
Example 5.1. Consider the knapsack problem in 0=1 variables defined via the
constraint
3. We choose
This meets the requirements that the indices
must satisfy, because f 4 - f 3 +f 2 and f 8 - f 7 +f 6 . In this example, the inequality
(?) is of the form
and it is valid for the given knapsack polytope.
Let us now show that the inequality (?) is always valid under the above assump-
tions. It is valid if and only if every subset T ' S with f(T
equivalently if and only if the problem
s
has an optimal value f
Setting Y j :=
first show that there always exists an
optimal solution to this problem with Y First, observe
that Y - t -+1 is infeasible for this problem because b
as by construction
solution obtained by decreasing Y - \Gamma1 by t - and increasing Y - by 1 is at least as
good as the initial solution in terms of objective value
equivalently in
terms of the knapsack constraint
optimal solution
with can be transformed into an optimal solution with
Proceeding in this way for all , we can produce an optimal solution
with
The objective value of such a solution satisfies
because implies that there exists z 2 S j with y
such that f
Summing these inequalities for
Hence r and the inequality is valid.
By construction, b j is a multiple of b j \Gamma1 for all j - 2. It follows that (?) has the
divisibility property and we can apply all of our information for the sequential
knapsack polytope induced by inequality (?). In case that if we impose
a "regularity condition" such as "every subset T in S with b(T
then the corresponding inequality defines a facet of
the 0=1 knapsack polytope [8].
For - 2 one can also derive sufficient conditions under which inequality (?)
defines a facet of the corresponding polytope. Yet, such conditions are quite
technical and we refrain within this paper from explaining further details.
If one finds such generalized (1; k)-configurations or some subset of the items having
the divisibility property with respect to some row of a given integer program
all the knowledge about the sequential knapsack polytope can be
used. Together with lifting this yields a powerful tool that might help solving
integer programs.
Acknowledgement
. This research was partially supported by Science Program
SC1-CT91-620 of the EEC and contract ERB CHBGCT 920167 of the EEC
Human Capital and Mobility Program.
--R
"The sequential knapsack pro- blem"
"Some polyhedra related to combinatorial problems"
"Cutting planes and the sequential knapsack problem"
"Solving sequential knapsack problems "
"Graph theory and integer programming "
"The cutting stock problem and integer rounding"
"A Note on 0-1 Programming"
"(1,k)-Configurations and Facets for Packing Pro- blems"
"Integer knapsack and flow covers with divisible coefficients: Polyhedra, optimization and separation"
"On the 0/1 knapsack polytope"
"Hilbert bases and the facets of special knapsack pro- blems"
"Faces of Linear Inequalities in 0-1 Variables"
--TR | integer programming;linear programming formulation;knapsack problem;knapsack polytope;separation |
589172 | Modified Cholesky Factorizations in Interior-Point Algorithms for Linear Programming. | We investigate a modified Cholesky algorithm typical of those used in most interior-point codes for linear programming. Cholesky-based interior-point codes are popular for three reasons: their implementation requires only minimal changes to standard sparse Cholesky algorithms (allowing us to take full advantage of software written by specialists in that area); they tend to be more efficient than competing approaches that use alternative factorizations; and they perform robustly on most practical problems, yielding good interior-point steps even when the coefficient matrix of the main linear system to be solved for the step components is ill conditioned. We investigate this surprisingly robust performance by using analytical tools from matrix perturbation theory and error analysis, illustrating our results with computational experiments. Finally, we point out the potential limitations of this approach. | Introduction
. Most interior-point codes for linear programming share a common
feature: their major computational operation-solution of a large linear system
of equations-is performed by a direct sparse Cholesky algorithm. In this algorithm,
row and column orderings are determined a priori by well-known heuristics (minimum
degree and enhancements, minimum local fill, nested dissection) that are based solely
on the sparsity pattern and not on the numerical values of the nonzero elements. The
ordering phase is followed by a symbolic factorization phase, in which the nonzero
structure of the Cholesky factor is determined and storage is allocated. Finally, a
numerical factorization phase fills in the numerical values of the lower triangular
In interior-point codes, the first two phases usually are performed
just once, during either the first interior-point iteration or computation of a starting
point.
In the interior-point context, the unadorned Cholesky algorithm can run into difficulties
because of extreme ill conditioning. Some of the diagonal pivots encountered
during the numerical factorization phase can be zero or negative, causing the standard
Cholesky procedure to break down. Instead of crashing, most codes apply a
"patch" to the algorithm to handle such pivots. The offending pivot element is sometimes
replaced by a huge number, as in LIPSOL [17] or PCx [1]. In other codes such
as IPMOS [16], the pivot is replaced by a moderate number, but the corresponding
right-hand side element is set to zero, as are the off-diagonal elements in the corresponding
column of the Cholesky factor. The first practical interior-point code, OB1
[6], explicitly zeroes the components of the solution vector that correspond to small
pivots. All these strategies are essentially equivalent to the algorithm we describe
in this paper. To date, there has been little investigation of them from a numerical
analysis viewpoint.
The "patches" described above have the advantage that they can be implemented
by changing just a few lines in general sparse Cholesky codes. It is therefore possible
to take advantage of the long-term development effort that has gone into designing
such codes and their underlying algorithms. The recent codes LIPSOL [17] and PCx
Mathematics and Computer Science Division, Argonne National Laboratory, 9700 South Cass
Avenue, Argonne, IL 60439. This work was supported by the Mathematical, Information, and Computational
Sciences Division subprogram of the Office of Computational and Technology Research,
U.S. Department of Energy, under Contract W-31-109-Eng-38.
[1] make explicit use of the freely available sparse Cholesky code of Ng and Peyton
[8]. Other codes either modify the well-known SPARSPAK routines of George and
Liu [3] or include customized linear algebra routines that implement well established
algorithmic ideas. (At least one author has experimented with modifications to the
standard heuristics: M'esz'aros [7] describes an inexact version of the minimum local
fill ordering.)
One possible remedy for small pivots is diagonal pivoting. At each iteration, a
"large" diagonal element is selected from the unreduced portion of the matrix and
moved to the pivot position by symmetric row and column pivoting. The algorithm
is terminated when none of the remaining diagonal elements is sufficiently large, and
an approximate solution is computed with the partial factors. (See Higham [4, Chapter
10] for details and error analysis.) This strategy is not particularly appealing in
the context of interior-point linear programming codes because of the loss of efficiency
due to shifting of data during the numerical factorization. Moreover, there is little
incentive to test this strategy because the simple patches described above perform so
well in practice.
In this article, we use standard results from numerical analysis to explain the
good performance of these patching strategies on the vast majority of problems. We
also gain some insight into their limitations and into how and why they fail.
Our error analysis for the modified Cholesky algorithm is rigorous, with explicitly
stated assumptions and precise bounds (see Sections 3 and 4). We revert, however, to
a more informal style when applying these results to the interior-point context (Section
5). The reason is pure pragmatism. A fully rigorous analysis would be impossibly
notationally speaking, and unduly pessimistic. The informal analysis yields
adequate insight into the typical performance of the algorithm, as our computational
results in Section 6 demonstrate.
A number of other papers on linear algebra operations in barrier and interior-point
methods have appeared in recent years. Wright [12] has considered the Newton-
logarithmic barrier method for general constrained optimization, in which the linear
system to be solved for the Newton step is positive semidefinite and ill conditioned
during later iterations. She uses a Cholesky factorization with diagonal pivoting to
identify the subspace spanned by the active constraint Jacobian. From this infor-
mation, an accurate solution of the Newton equations can be obtained, in which the
components of the step in both the range space of the active constraint Jacobian and
the null space of its transpose are well resolved. Our analysis has a similar flavor
to Wright's, but the application is somewhat different. The unknowns in our linear
system are the unconstrained dual variables rather than the primals and, since this
problem is linear, we have little interest in resolving the component of the step in
the near-null space of the coefficient matrix. We focus too on Cholesky algorithms
that perform no pivoting during the numerical factorization, reflecting computational
practice in the current generation of interior-point linear programming codes.
In an earlier paper [14], we considered the stability of algorithms for the symmetric
indefinite form of the step equations at each iteration of a interior-point method for
linear programming. We showed that, despite the ill-conditioning of the coefficient
matrix, the steps obtained by this approach are good search directions for the interior-point
method. Forsgren, Gill, and Shinnerl [2] perform a similar analysis in the context
of logarithmic barrier methods.
The remainder of this paper is organized as follows. In Section 2, we introduce
primal-dual interior-point methods and derive the linear equations to be solved at each
iteration of these methods. Section 3 introduces Algorithm modchol, the modified
procedure, and examines the accuracy of the solution obtained with this
factorization, under certain assumptions on the eigenvalues of the factored matrix.
In Section 4, we account for the effect of finite-precision floating-point arithmetic on
solution accuracy. We return to the interior-point application in Section 5, showing
that Algorithm modchol yields good steps for these methods until the duality gap
becomes very small, even if the linear program is primal or dual degenerate. The
analytical results are verified by computational experiments with an interior-point
code using Algorithm modchol, which are reported in Section 6.
Notation. We summarize here the notation used in the remainder of the paper.
The ith singular value of a matrix A is denoted by oe i (A). We use oe i alone to
denote the ith singular value of the exact Cholesky factor L in Section 3.
For any matrix M and index steps I and J , M IJ denotes the submatrix formed
by the elements . The ith column of M is denoted by M \Deltai ,
and the column submatrix consisting of columns j 2 J is denoted by M \DeltaJ .
Unit roundoff error is denoted by u. Higham [4, Chapter 1] defines u implicitly
by the statement that when ff and i are any two floating-point numbers, op denotes
+, \Gamma, \Theta, and =, and fl(\Delta) denotes the floating-point representation of a real number,
we have
For any positive integer m with mu ! 1, we define
(1)
(see Higham [4, Lemma 2.1]).
The notation k \Delta k denotes the Euclidean vector norm k \Delta k 2 and also its induced
matrix norm, unless otherwise noted. For any matrix A, the matrix consisting of the
absolute values of each element is denoted by jAj. We use 1 to denote the vector
Finally, we mention the parameter ffl that defines the pivot threshold in the modified
algorithm. A second quantity -ffl, which is related to ffl by
appears frequently in the analysis because the incorporation of the scaling term 2m 2
saves notational clutter.
2. Primal-Dual Algorithms for Linear Programming. We consider the
linear programming problem in standard form:
subject to
(2)
. The dual of (2) is
subject to A T -
m . We assume throughout the paper that A has full row
rank, so that m - n. The Karush-Kuhn-Tucker (KKT) conditions, which identify a
vector triple (x; -; s) as a primal-dual solution for (2), (3), can be stated as follows:
(4a)
(4b)
(4c)
(4d)
We assume throughout the paper that a primal-dual solution exists. We make no
assumptions about uniqueness or nondegeneracy; our analysis in Section 5 continues
to hold when the problem (2) is primal or dual degenerate. It is well known that
the index set f1; ng can be partitioned into two sets B and N such that for all
primal-dual solutions
x
Primal-dual interior-point algorithms generate a sequence of iterates (x; -; s) that
satisfy the strict inequality (x; s) ? 0. They find search directions by applying a
modification of Newton's method to the system of nonlinear equations formed by the
first three KKT conditions (4a),(4b),(4c), namely,
In general, the search direction (\Deltax; \Delta-; \Deltas) is obtained from the following linear
\Delta-
\Deltas5 =4 \Gammar c
where the coefficient matrix is the Jacobian of (6) and the right-hand side components
r b and r c are defined by
In a pure Newton (affine-scaling) method, the remaining right-hand side component
r xs is defined by
and, in this case, we denote the solution of (7) by (\Deltax aff ; \Delta- aff ; \Deltas aff ). In a path-following
method, we have
where - is the duality gap defined by
is a centering parameter. In the "Mehrotra predictor-corrector" al-
gorithm, which is used as the basis of many practical codes, the search direction is
calculated by setting
where \DeltaX aff and \DeltaS aff are the diagonal matrices formed from the affine-scaling step
components \Deltax aff and \Deltas aff . Hence, Mehrotra's method requires the solution of two
linear systems at each iteration-the affine scaling system (7), (8), (9), and the search
direction system (7), (8), (12). A heuristic based on the effectiveness of the affine
scaling direction is used to determine the value of i in (12).
Once a search direction has been determined, the primal-dual algorithm takes a
step of the form
where ff is chosen to maintain strict positivity of the x and s components; that is,
In most codes, ff is chosen to be some fraction of the step-to-boundary ff max defined
as
A typical strategy is to set
as the interior-point method approaches the solution
set.
By applying block elimination to (7) and using the notation
we obtain the following equivalent system:
(16a)
(16c)
In many codes, the solution is obtained from just this formulation. A sparse Cholesky
factorization, modified to handle small pivots, is applied to the symmetric positive
definite coefficient matrix AD 2 A T in (16a) and the solution \Delta- is obtained by triangular
substitution with the computed factor. The remaining direction components
are recovered from (16b) and (16c). This technique yields steps (\Deltax; \Delta-; \Deltas) that
are useful search directions for the interior-point algorithm, even when the matrix
happens during later iterations. This observation
is somewhat surprising, since a naive application of error analysis results would
suggest that the combination of ill-conditioning and roundoff would corrupt the direction
hopelessly. The results of Sections 3, 4, and 5 provide an explanation for this
phenomenon.
The following observation is crucial to our analysis: In computing \Delta- from (16a),
we are not interested so much in the error in \Delta- itself as in the effect of this error
on the remaining step components \Deltas and \Deltax that are recovered from (16b) and
(16c), respectively. If the relative errors in these components are large, the positivity
requirement may cause the step length ff to be significantly shortened,
thereby curtailing the algorithm's progress. We return to this issue in Section 5, after
describing and analyzing the modified Cholesky algorithm in Sections 3 and 4.
3. A Modified Cholesky Algorithm. In this section, we describe and analyze
Algorithm modchol, a modified Cholesky algorithm designed to handle ill-conditioned
matrices for which small or negative pivots may arise during the factorization
Algorithm modchol accepts an m \Theta m symmetric positive definite matrix M as
input, together with a small positive user-defined parameter ffl, which defines a threshold
of acceptability for the pivot elements. If a candidate pivot element is smaller than
this threshold, the algorithm simply skips a step of factorization. Algorithm modchol
outputs an approximate lower triangular factor ~
L and an index set J ae
containing the indices of the skipped pivots. In the following specification, we use
M (i) to denote the unfactored part of M that remains after i steps of the algorithm.
Algorithm modchol
Given ffl with
if M (i\Gamma1)
(* skip this elimination step *)
im
.
else
(* perform the usual Cholesky elimination step *)
~
~
L ki .
The ith column of ~
L is zero for each i 2 J ; that is, ~
If we denote
and denote the complement of J in f1;
J , it follows from (17) that
That is, the row or column index of each nonzero element in E must lie in J . It follows
from the algorithm that ~
L is the exact Cholesky factor of the perturbed matrix
which we denote for convenience by ~
M . That is, we have
~
By partitioning this equation into its J and -
J components and using ~
(19), we obtain
~
(21a)
~
The exact Cholesky factor L (whose existence is guaranteed by the assumed positive
definiteness of M ) satisfies
Given the linear system
where M is the matrix factored by modchol, the exact solution obviously satisfies
The approximate solution ~
z is chosen so that the partial vector ~ z -
J solves the reduced
system M -
z -
J , while the complementary subvector z J is set to zero. From
(21a), we see that ~
z -
J can be calculated by performing a pair of triangular substitu-
tions; that is,
~
z -
~
z
Note that z = ~
z when on the other hand, the difference between
~
z and z can be large in a relative sense. We have
z -
z -
and there is no reason to expect z J to be small with respect to the full vector z.
We can show, however, that the difference between L T z and L T ~ z is relatively small
under certain assumptions; this result is the culmination of the analysis of this section
(Theorem 3.6). As we see in Section 5, this difference determines the usefulness of
the computed solution of (16) as a search direction for the interior-point algorithm.
To simplify the analysis, we assume implicitly throughout the paper that
A trivial scaling, which affects neither the algorithm nor its analysis, can always be
applied to the symmetric positive definite matrix M to yield (26).
We start with a sequence of three results that lead to a bound on the difference
between ~
z. These results require few assumptions on the matrix M and
are relatively simple to prove.
Lemma 3.1. The submatrix formed by the last columns of M (i)
is symmetric positive definite, for all 1. Moreover, the diagonal
elements of all these submatrices are bounded by 1.
Proof. This observation follows by a simple inductive argument. By assumption,
the starting matrix M positive definite. Suppose that the desired property
holds for M (i\Gamma1) . If i 2 J , then the lower right (m \Gamma i) \Theta (m \Gamma i) submatrix of
M (i) is identical to the lower right (m \Gamma i) \Theta (m \Gamma i) submatrix of M (i\Gamma1) , which
is positive definite by assumption. Otherwise, if
is obtained by
applying one step of Cholesky reduction to M (i\Gamma1) . It is known that the remaining
submatrix resulting from this operation is positive definite; hence, the lower right
in question is positive definite, and the desired property
holds.
The second claim follows immediately from the fact that M ii -
and the fact that the diagonal elements cannot increase during Algorithm
modchol.
Lemma 3.2. For each i 2 J , we have
Therefore,
Proof. From Lemma 3.1, we have (M (i\Gamma1)
l;l for each
ffl. Since the diagonals of each submatrix
M (i\Gamma1) are bounded by 1, we have M (i\Gamma1)
i;l
l;l
Hence, we have
thereby proving the first claim. By (18), we have
thereby proving (27).
In the case in which all the small pivots appear in the bottom right corner of the
matrix (that is, index p), the estimate (27) can be
improved to
This stronger estimate applies in most instances of the interior-point application of
Section 5.
We are now able to derive an estimate of the difference between ~
Theorem 3.3. For the exact solution z and approximate solution ~
z defined in
(24) and (25), respectively, we have that
Proof. From (24) together with (21), we have
JJ z J
~
~
JJ
z J
while from (25), we have
~
z -
J ~ z J
J ~ z:
By combining these two relations, we obtain
~
Since ~
the result follows immediately.
The remaining analysis of this section requires some additional assumptions on
the distribution of the singular values of M and on the parameter ffl. Accordingly,
we introduce a little more notation. The eigenvalues of M are denoted by oe 2
We define the diagonal matrix \Sigma by
It follows that there exists an orthogonal matrix Q such that
Because the largest diagonal in M is 1, we have by elementary analysis that
In the subsequent analysis, we assume that there is an integer p with 1 -
such that
ffl ffl is small relative to oe 2
there is a significant gap in the spectrum of M between oe 2
p and
p+1 .
(We will be more specific about these two assumptions presently.) By partitioning
the spectrum at the gap, we obtain
From (33), Q can be partitioned accordingly to obtain
are the singular values of L. In
fact, we must have
orthogonal matrix U , where \Sigma and Q are defined as above.
We use ~
to denote the eigenvalues of the perturbed matrix ~
M .
It follows immediately from (20) that the singular values of ~
are ~ oe i ,
The rank of ~
J j, because ~
J is lower triangular with nonzero diagonals while
~
Therefore, we have
~
As in (36), there are orthogonal m \Theta m matrices ~
U and ~
Q such that
~
U ~
where
~
It is an immediate consequence of an eigenvalue perturbation result of Stewart and
Sun [10, Corollary IV.4.13] and Lemma 3.2 that
The main assumption of this section is that j -
correctly identifies the numerical rank of the matrix M . One might expect that we
should not have to assume this equality at all-that it should follow from the spectrum
gap and from a judicious choice of ffl. Practical experience supports this expectation;
the algorithm has little trouble determining the numerical rank on the vast majority
of problems. In fact, part of the result-the bound j -
p-follows from a minimal
assumption on ffl.
Lemma 3.4. If -
Proof. If
we have from (37) and (39) that
contradicting our assumption that -ffl 1=2 ! oe 2
.
However, the conditions on ffl, oe p , and oe p+1 needed to prove the other half of
the result-j -
rigorous to be useful. This is a consequence of the
fact that poorly conditioned triangular matrices need not have particularly small
diagonal elements (see Lawson and Hanson [5, p. 31] for the classic example of this
phenomenon).
Our next result concerns perturbation of the subspace spanned by Q 1 , which is
the invariant subspace of "large" eigenvalues of M .
Lemma 3.5. Suppose that j -
and that the values oe p and oe p+1 from
(31) and ffl from Lemma 3.2 satisfy the conditions
(40a)
(40b)
Then there is a p \Theta p symmetric positive definite matrix ~
and an orthonormal m \Theta p
matrix ~
~
~
~
(The constants used in (40a) and in similar expressions should not be taken too
seriously. We assign them specific values only to avoid an excess of notation.)
Proof. The result is a straightforward consequence of Theorem V.2.8 of Stewart
and Sun [10, p. 238]. Since ~
use (33) and partition as in (35) to obtain
We now make the following identifications with the quantities in the cited result:
~
where sep(\Delta; \Delta) is the minimum distance between the spectra of its two arguments.
From the given result, there is a matrix P of dimension (m \Gamma p) \Theta p such that the
matrix ~
defined by
~
is an invariant subspace for ~
~
Moreover, the representation of ~
M with respect to ~
~
The bound (42) follows from (44), (45), and kQ 2 It follows immediately from
the first equality in (46) that ~
is symmetric, and we have
verifying the inequality (43). This inequality implies that the smallest singular value
of ~
is no smaller than oe 2
is symmetric positive definite.
The cited result states further that the matrix ~
is orthogonal to
~
defines an invariant subspace for ~
M . In fact, we have
for some (m \Gamma p) \Theta (m \Gamma p) symmetric matrix -
. Since ~
and ~
both have rank b,
we must have -
0, so we have
~
~
~
Hence, (41) is also satisfied, and the proof is complete.
Combining (40b) with (39), we obtain
Another quantity that enters into our error bounds is the norm of ~
J . We denote
J jth singular value of ~
J . (The lower bound of 1 in
simplifies our analysis.) Note from (21a) that
1 , we have from (34) and (49) that
Under the assumption j -
the nonzero part of ~
L-the submatrix ~
J -has
full rank p and singular values ~ oe
oe p . Since ~
J differs from ~
J in the presence
of the additional rows ~
J , we have
and therefore
The additional rows ~
J can have nontrivial magnitude relative to ~
J , so ~ oe p may
be significantly larger than - \Gamma1 . However, ~ oe p cannot be too large, since from (39),
(40b), and (34), we have that
~
For the purposes of our analysis, we make the assumption that - ~
p is moderate in
size. Specifically, we assume that
Because of (39) and (40b), we have ~
implies that
and, in addition,
3:
We can now prove the main result of this section.
Theorem 3.6. Suppose that j -
m, that the conditions (40) hold, and
that the estimate (51) is satisfied. We then have
Proof. From (36), we have
since U is orthogonal. Now from the partition (35), and using the fact that kQ 2
(unless of course \Sigma 2 and Q 2 are vacuous), we obtain
~
~
The first term in this expression is easiest to bound. From (35), we have k\Sigma \Gamma1
. Applying the relations (41), (20), (38), (47), (29), (27), and (48), respectively,
we obtain
~
We therefore have
~
The second and third terms in (53) require a bound on k~z \Gamma zk. From (30) and the
fact that ~ z
~
and therefore
kz J k:
From (47), we have k ~
while from (27), we have
Substituting these estimates into (56) and using (52), we obtain
Finally, using ~
z together with - 1, (34), and (57), we obtain
Turning specifically to the second term in (53), we have from (34), Lemma 3.5,
(47), and (40) that
~
oe 22-ffl 1=2
By combining this bound with (58) and k\Sigma \Gamma1
, we obtain
~
For the third term in (53), we have from k\Sigma 2
The result of the theorem is obtained by substituting (54), (59), and (60) into
(53).
Note that if
m), we have ~
so the conclusion of
Theorem 3.6 holds trivially in this case as well is we define oe
4. The Effect of Finite Precision Computations. In the analysis of the
preceding section, we assumed for simplicity that all arithmetic was exact. In this
section, we take account of the roundoff errors that are introduced when the approximate
solution ~ z is calculated in a finite-precision environment.
Our analysis above focused on the approximate solution ~
z obtained from (25),
where the subvector ~
z J satisfies the following system:
z -
while the subvector ~ z J is fixed at zero. In this section, we use - z to denote the finite
precision analog of ~
z. We examine errors in -
z due to
ffl roundoff error in Algorithm modchol,
ffl error arising during the triangular substitutions in (61), and
ffl evaluation error in the right-hand side r.
As we see in Section 5, evaluation error in the right-hand side is a significant feature
of the application to interior-point codes. We denote this error by e, so that the
right-hand side r -
J in the system (61) is replaced by r -
J .
Fortunately, our results follow in a straightforward way from existing results for
the Cholesky factorization, since a close inspection of Algorithm modchol shows that
it simply performs a standard Cholesky factorization on the submatrix M -
J .
Before stating the main results, we introduce two more assumptions. The first
concerns the relative sizes of - and u, specifically,
where fl m+1 is defined as in Section 1. Since - ? 1 and m - 1, it follows immediately
that
The second assumption is that finite precision does not affect cutoff decisions in Algorithm
modchol. That is, the presence of roundoff error in each submatrix M (i\Gamma1)
does not affect whether the threshold criterion M (i\Gamma1)
ii - fi ffl passes or fails for each
i. This assumption concerns the relative sizes of u and ffl, and it requires some ex-
planation. We cannot expect to take care of the "borderline cases" in which some
candidate pivots fall just to one side or the other of the threshold. Rather, we want
the cases in which there is a clear distinction between small and large pivots in exact
arithmetic to retain this distinction in finite precision arithmetic, and we want the
threshold fi ffl to fall comfortably inside the "gap" in both settings. In finite precision,
the size of rounding error introduced into M (i\Gamma1)
ii by earlier steps of Algorithm mod-
chol is comparable to fiu. (Each time M ii is updated by the algorithm, a positive
number no larger than itself is subtracted from it. Since jM ii j - fi, the floating-point
error introduced here is bounded by fiu.) We want these errors to be smaller than
the threshold fi ffl, so that pivots that are tiny in exact arithmetic do not exceed the
threshold in finite precision. Hence, we can state this assumption roughly as follows:
The following lemma accounts for the effects of finite precision on the approximate
solution ~
z obtained from Algorithm modchol and (25).
Lemma 4.1. Suppose that Algorithm modchol and the triangular substitutions
in (61) are performed in finite-precision arithmetic with perturbed right-hand side
J to yield an approximate solution - z. Suppose, too, that (62) holds and that
roundoff error does not affect the composition of J . We then have
where z is the exact solution from (23).
Proof. Algorithm modchol operates as a standard Cholesky factorization on the
J , so we can apply a standard perturbation theorem to bound the error
in the subvector - z -
J . From Higham [4, Theorem 10.4], we find that -
z -
J satisfies
where
Comparing (66) with (61), we find that
z -
Manipulating in the usual way, we obtain
z -
It follows immediately from (67) that
Combining (50), (62), and (63), we obtain
so that the denominator in (68) is bounded below by :5. Hence, by substitution into
(68), using (34), (49), (69), and (63), we have that
Finally, we bound k~z -
J k in terms of kzk. From (34) and (57), we have
By combining this bound with (70), we obtain the result.
The major results of Sections 3 and 4 can be summarized in the following theorem.
Theorem 4.2. Suppose that Algorithm modchol and the triangular substitutions
in (61) are performed in finite-precision arithmetic with perturbed right-hand side
J to yield an approximate solution - z. Suppose, too, that (62) holds and that
roundoff errors do not affect the composition of J . Finally, suppose that either
m, the conditions (40) hold, and the estimate (51) is satisfied.
We then have
ae
oe
Proof. When the result is immediate from Lemma 4.1 and ~
z. For the
remaining case, we obtain (71) by combining the results of Theorem 3.6 and Lemma
4.1. We need note only that kz J k - kzk and that, from (34), we have
zk:
5. Application to the Interior-Point Algorithm. In this section, we return
to the motivating application: primal-dual interior-point software for linear programming
and, in particular, the linear system (16) that is solved at each iteration. We
apply the main result-Theorem 4.2-and examine the effect of the parameter ffl and
unit roundoff u on the quality of the computed search direction ( c
\Deltax; c
\Delta-; c
\Deltas). Our
focus is on the later iterations of the interior-point algorithm, during which - is small
and the ill-conditioning of AD 2 A T can become acute. Our results show how and why
errors arise in ( c
\Deltax; c
\Delta-; c
\Deltas) and what effect these errors have on the step length, the
convergence of the algorithm, and the accuracy that can be attained by this algorithm.
They also suggest an appropriate size for the parameter ffl.
In this section, we revert to an informal style of analysis, using order notation to
hide constants of moderate size. Thus if j and i are two positive numbers, we write
if the ratio j=i is not too large. Similarly, we
O(j). Conventionally, order notation is used only when j and i are quantities
that approach zero in the limit of the algorithm in question. Here, however, we use
it in connection with the unit roundoff u, which is small but fixed. This slight abuse
of notation results in a much clearer insight into the behavior of Algorithm modchol
in the interior-point context.
In the next subsection, we look closely at the affine-scaling step, for which r xs
is defined by (9). This step is important because it closely approximates the steps
taken by most rapidly converging algorithms during their final iterations. Subsection
5.2 shows that the steps calculated during the final stages of Mehrotra's predictor
corrector algorithm (and therefore by most interior-point codes) have essentially the
same properties as affine-scaling steps.
5.1. Affine-Scaling Steps. We start by estimating the sizes of the various constituents
of the equations (16)-the residuals r b and r c , the B and N components of
x, s, and the diagonal matrix D. In standard infeasible-interior-point algorithms (see,
for example, Wright [15, Chapter 6]), we have
These estimates are also observed to hold in practice on the majority of problems for
values of - greater than u 1=2 . An immediate consequence of these estimates and the
definition (15) is that
We assume the coefficient matrix A to be well conditioned; that is, oe 1 (A) and
are both
\Omega\Gammah/1 We assume further that the submatrix A \DeltaB of columns A \Deltai ,
well conditioned. It follows from this assumption together with the estimate
(73) that the matrix A \DeltaB D 2
\DeltaB has full rank min(jBj; m). In fact, since A \DeltaB is well
conditioned, all nonzero singular values of A \DeltaB D 2
\DeltaB
it follows from (15) and (73) that A \DeltaN D 2
so we conclude that
(74a)
Since the largest diagonal element of AD 2 A T is
scaled coefficient
matrix for (16a) is
For consistency with Section 3, the singular values of the matrix in (75) are denoted
by oe 2
. From this definition together with (74) and (75), we deduce that
(76a)
Recalling our notation p of Section 3, we have in this case that
The exact Cholesky factor L (see Sections 3 and
Suppose now that Algorithm modchol is used to compute the solution of (16a),
where the right-hand-side component r xs is set to its affine-scaling value XS1. This
process result in a computed solution c
\Delta- aff
for (16a). The remaining step components
c
\Deltas aff
and c
\Deltax aff
are obtained by substitution into (16b) and (16c), respectively, again
in finite-precision arithmetic. Our main tool for analyzing the errors in the computed
step is Theorem 4.2.
Consider the exact affine scaling step (\Deltax aff ; \Delta- aff ; \Deltas aff ). Standard results for
methods (see, for example, [15, Theorem 7.5]), together with
the conditions (72), imply that
(This estimate holds only when - falls below a data-dependent threshold ffl(A; b; c)
defined by Wright [15, Chapter 3].) From (16b) and (72), we have
so it follows from our assumptions about the well conditioning of A that
\Delta-
We can be more specific about the sizes of the critical components \Deltax aff
and \Deltas aff
we multiply the third block row in (7) by (XS) \Gamma1 and use the
definition (9), we obtain
\Deltax aff
\Deltas aff
Therefore, from (72) and (78), we have for i 2 N that
\Deltax aff
and therefore, using (72) again, we have
\Deltax aff
In a similar way, we obtain
\Deltas aff
From the estimates (78), (80), and (81), we can show that a near-unit step can
be taken along the direction (\Deltax aff ; \Delta- aff ; \Deltas aff ) without violating positivity of the
x and s components. Substituting (\Deltax; \Delta-;
have
To verify this estimate, suppose that s i
(81), we have
so it follows from (72) that
For the corresponding component x i , we have from (72) and (78) that x i
and \Deltax aff
O(-). Hence, for all - sufficiently small and all ff 2 [0; 1], we have
logic can be applied to the remaining indices i 2 N , thereby
completing our verification of (82).
Returning to the computed affine-scaling step ( c
\Deltax aff
\Delta- aff
\Deltas aff
), we now apply
Theorem 4.2 after checking that its assumptions of are satisfied for small enough
- and reasonable values of u and ffl. For double-precision computations, we have
. Hence, since A is well conditioned, we can expect the condition (62) to
hold in all nonpathological circumstances. Because of (76), our assumption (40a) on
the singular value distribution clearly holds for all sufficiently small -. The condition
(40b) is satisfied for any reasonable choice of ffl. The assumption that Algorithm
modchol correctly identifies the numerical rank (that is,
is, as we discussed
in Section 3, difficult to guarantee, but it was observed to hold on all problems that we
tested. The assumption that rounding errors do not interfere with the makeup of the
small pivot index set J is likewise impossible to verify rigorously; but, as discussed
in Section 4, it can reasonably be expected to hold when ffl - u (64).
A good choice for ffl-one that satisfies the assumptions just mentioned while
keeping the bound (71) as small as possible-is therefore
For generality, we continue to use ffl and -
ffl in the analysis that follows, substituting
the specific value (83) only at the end.
Having verified that we can reasonably expect Theorem 4.2 to hold for the system
(16a), we now estimate the quantities on the right-hand side of (71). From (76a), we
have oe 1 =oe O(1), while from (76b), we have oe O(-). The general estimate
while the definition of fl m+1 gives the estimate fl
We need to account, too, for the errors incurred in evaluating the right-hand side
of (16a). The floating-point error in forming r xs = XS1 is only O(-u) in magnitude,
since just a single floating-point multiplication is needed to calculate each component
of this vector, and each such element is O(-) (see (72)). The residuals r b and
r c have magnitude O(-) in exact arithmetic (see (72)), but they are calculated as
differences of O(1) quantities and so contain evaluation error of absolute magnitude
O(u). Specifically, componentwise errors in the computed version of r c are bounded
by
u, and similarly for r b . Because of the estimate (73), the errors
in r c are magnified to (- \Gamma1 u) when we multiply by AD 2 in (16a). In fact, this term
is the dominant one in the total right-hand-side evaluation error. The errors that
occur when we perform floating-point addition of the terms r b , AD 2 r c , and AS \Gamma1 r xs
are less significant; they lead to additional terms of sizes O(u) and O(- \Gamma1 u 2 ). In
summary, the total right-hand-side evaluation error is O(- \Gamma1 u). Hence, after scaling
by the factor ae defined in (75), we have
where e is the error vector of Section 4.
Substituting the estimates (76), (79), and (84) into (71), we have
\Delta- aff
If
(a reasonable estimate when the Cholesky factorization correctly identifies the numerical
rank and A \DeltaB is well conditioned), the error bound above simplifies to
\Delta- aff
From (77) we have that
ae
for some orthogonal matrix Q. Since orthogonal transformations do not affect the
Euclidean norm of a vector, we can substitute ae 1=2 DA T for L T in (86) and use (75)
to write
\Delta-
aff
\Delta-
aff
Note too that from (58), (65), (79), and (84), we have
\Delta- aff
\Delta- aff
\Delta- aff
\Delta- aff
where ~
\Delta- aff
is the approximate solution that would be obtained by Algorithm mod-
chol if it was used to solve (16a) in exact arithmetic.
Next, we examine the effect of the error in c
\Delta- aff
and the evaluation error in the
right-hand side of (16b) on the calculated step c
\Deltas aff
. From (79) and (88), we have
that
\Delta-
aff
\Delta-
aff
Hence, taking into account the O(u) evaluation error in the term r c , we have immediately
from (16b) that
\Deltas
\Deltas aff
\Delta- aff
Clearly, for the "large" components of s-namely, the i 2 N components-errors
of this magnitude do not affect the step length ff max to the boundary defined in (14).
However, for the critical components i 2 B, the estimate (90) is not good enough to
guarantee that ff max is close to 1. (Repeating the argument that follows (82), we find
only that Fortunately, a refined estimate of the error in the B
components is available. As in (90), we have
\Deltas
\Deltas aff
\Delta- aff
where from (87) we have
\Delta-
aff
From (73), we have D B, so from (91) we obtain
c
\Deltas aff
As in the discussion following (82), we find that s i
\Deltas aff
possible only if
This estimate suggests that near-unit steps can be taken, at least in the c
\Deltas aff
com-
ponents, provided that - is significantly larger that u. When all bets are
off!
Finally, we estimate the errors in the computed version of \Deltax aff (obtained from
(16c)) and estimate their effect on the ff max . Again, we consider the components
separately.
For B, the O(-u) evaluation error in (r xs ) i is magnified by the term s \Gamma1
replacement of \Deltas aff by c
\Deltas aff
yields an additional error of size
O(-ffl which is also magnified by
arithmetic errors are less significant. In summary, we find that
c
\Deltax aff
By the usual reasoning, we find that x i
\Deltax aff
satisfying (94).
For the O(-u) evaluation error in (r xs ) i is not magnified appreciably
by s
, while from (90), the O(- + u) error in \Deltas aff is actually diminished after
multiplication by s
We find that
c
\Deltax aff
Hence, we can have x
\Deltax aff
From (94) and (97), we conclude that the value of ff max defined by (14), with the
calculated direction ( c
\Deltax aff
\Delta- aff
\Deltas aff
replacing the exact search direction, satisfies
the estimate
Note from (89), (90), and (96) that, in an absolute sense, the errors in c
\Delta- aff
c
\Deltas aff
, and c
\Deltax aff
are small. By contrast, the O(- \Gamma1 u) term in (95) implies
that the errors in c
\Deltax
aff
may become large as - # 0. These large errors may
in turn cause the residuals r b to grow as - # 0. These expectations are confirmed by
the computational experiments of Section 6.
The estimate (98) and the parameter choice suggest strongly that the
algorithm should be terminated when
-
When - reaches this threshold, all three terms in the estimate (98) are in balance.
Below this threshold, the O(- \Gamma1 u) term in c
\Deltax aff
may cause r b to grow, making
further reduction of - counterproductive. The convergence tolerances used by most
interior-point codes-arrived at by practical experience rather than any theoretical
considerations-are similar to (99). The code PCx is typical. It declares optimality
if the following three conditions are satisfied:
where the default value of tol is 10 \Gamma8 . (Note that 10 \Gamma8 - u 1=2 in double precision
arithmetic on most machines.)
5.2. Mehrotra Predictor-Corrector Steps. Having analyzed the affine-scaling
search direction and its calculated approximation, we turn our attention briefly to the
search direction used by Mehrotra's predictor-corrector algorithm. As mentioned in
Section 2, these steps are obtained by setting r xs as in (12), for some heuristic choice
of the centering parameter i. We can write the search direction as
where (\Deltax cc ; \Delta- cc ; \Deltas cc ) is the "corrector-centering" step component that satisfies
the following linear system:4 0 A T I
\Delta- cc
\Deltas cc5 =4
Block elimination on this system yields the following special case of (16a):
Since we assume full rank of A, and since the diagonal elements of D are all strictly
positive, the coefficient matrix is invertible, and we have
A result of Stewart [9] and Todd [11] states that the norm k(AD 2 A T )
bounded independently of D over the set of all positive definite diagonal matrices D
(and therefore independently of x and s with (x; s) ? 0). Therefore, we have
From (72), we have kX while from (78), it follows that k\DeltaX aff \DeltaS aff
O(- 2 ). Hence, we have
A typical heuristic for choosing the centering parameter i is to set
where - aff is the value of - that results from a full step-to-boundary ff max along the
affine-scaling direction. If the search direction is exact, we have -
this heuristic yields Use of the calculated direction ( c
\Deltax aff
\Delta- aff
\Deltas aff
together with the estimate (98) leads us to expect - case too,
provided that - u 1=2 . Hence, we have from (101) that k\Delta- cc
from (100) and (79), we have
where \Delta- is the - component of the Mehrotra search direction.
We also can apply the Stewart-Todd result to formulae for \Deltax cc and \Deltas cc to show
that k(\Deltax cc ; \Deltas cc Therefore, we have
corresponding to (78).
Because of the estimates (102) and (103), the analysis of the preceding subsection
can be applied without modification to the calculated version of the search direction
(100). In particular, if we redefine the step-to-boundary ff max in terms of this calculated
\Deltax; c
\Delta-; c
\Deltas), we find that the estimate (98) still applies. We conclude
that near-unit steps can still be taken along this direction provided that - u 1=2 .
6. Implementation and Computational Results. Most interior-point codes
use modified Cholesky algorithms with essentially the same properties as Algorithm
modchol. They differ slightly, however, in the implementation. The IPMOS code of
Xu, Hung, and Ye [16] replaces small pivot elements by 1 and fills out the corresponding
column of the Cholesky factor with zeros and also inserts a zero in the right-hand
side. The criterion for identifying a small pivot is not explained in the reference [16],
but otherwise this strategy is equivalent to Algorithm modchol. Zhang's LIPSOL
code [17] and the PCx code of Czyzyk, Mehrotra, and Wright [1] replace small pivots
by a huge number-10 128 -but otherwise leave the Cholesky algorithm unchanged.
The net effect is, however, almost equivalent to Algorithm modchol and the triangular
substitution procedure (25). The advantage of this approach is that it involves
minimal changes to a standard sparse Cholesky code. We need only add a loop to
calculate the largest diagonal element fi, and a small pivot check immediately before
the point at which the computation L
M ii is performed.
To test that the analysis of this paper was reflected in practical computations,
we coded a primal-dual algorithm that used Algorithm modchol in conjunction with
the formulation (16). The code was used to solve some small random linear programs
in which the amount of degeneracy-the composition of index sets B and N -was
carefully controlled. At each iterate, we monitored various quantities and compared
them against the estimates of Section 5.
The linear programming test problems were posed in standard form (2) with
12. The matrix A is fully dense, with elements
are random variables drawn from a uniform distribution on the
interval [0; 1]. (Of course, the values of - 1 and - 2 are different for each element of
the matrix.) We can reasonably expect this matrix A to satisfy the well-conditioning
assumptions of Section 5. The user specifies the number of indices to appear in B,
and we set
A primal solution x is constructed with
x
where - is randomly drawn from the uniform distribution on [0; 1]. We choose the
dual solution - to be the vector (1; fix an optimal dual slack vector
s to be
s
where - is random as above. Finally, we set
The code was an implementation of the infeasible-interior-point algorithm described
by Wright [13]. The details of this algorithm are unimportant; we need note
only that its iterates satisfy the estimates (72) in exact arithmetic and that the algorithm
takes steps along the affine scaling direction during its later iterations. At each
iteration of the algorithm, we calculated the affine scaling direction (whether or not it
was actually used as a search direction) and printed the norms k c
\Deltax aff
k1 , k c
\Delta- aff
k1 ,
and k c
\Deltas aff
k1 alongside the duality measure - and residual norm k(r b ; r c )k 1 for the
current point. We also kept track of the number of small pivots encountered during
the factorization, that is, the number of elements in J . The parameter ffl was set to
\Gamma12 , which is about 100u on the SPARCstation 5 that was used for the experiments.
The results were not particularly sensitive to this parameter.
Results are shown in Tables 1-4. For each iteration of the algorithm, these tables
list the number of small pivots jJ j, the base-10 logarithms of -, k(r
the affine-scaling step norms mentioned above. The step-to-boundary ff max along the
calculated affine-scaling direction is also tabulated. A horizontal line in each table
indicates the iterate at which termination occurs according to the criterion (99).
In
Table
1 we chose making the linear program nondegenerate
and the primal-dual solution unique. It is clear that c
\Delta- aff
and c
\Deltas aff
satisfy the
estimates (88) and (90), respectively, even when the algorithm is continues past the
point of normal termination. The component c
\Deltax aff
, on the other hand, clearly shows
the influence of the O(- \Gamma1 u) error term in (95) when - becomes comparable to or
smaller than u. Note, too, that the error in c
\Deltax aff
is transmitted to the residual r b
on succeeding iterations but that this effect does not become destructive until - is
much smaller than its normal termination threshold. The values of ff max are also
consistent with the estimate (98). This step length approaches 1 until the normal
point of termination is reached, after which the errors in c
\Deltax aff
and r b make further
progress impossible.
Table
2 shows the interesting case in which we choose 4, so that the co-efficient
matrix in (16a) has four singular values of
and two of
-). The second column shows that Algorithm modchol correctly identifies
the numerical rank during the last few iterations and that the interior-point
algorithm continues to generate useful steps and to make good progress even after
encounters small pivots. Apart from this feature, the behavior is the same
as in
Table
1, with errors in c
\Deltax aff
causing the interior-point algorithm to behave
poorly when it is permitted to run past its normal point of termination. We noted
that for all iterations, the "small" pivots were at the bottom right corner of the
so that (28) rather than the general estimate (27) applies to the
perturbation matrix E. In this case, we can replace -ffl 1=2 by -ffl in estimates of Section
5 such as (93), (95), and (98).
Table
3 illustrates another case in which 4, with the added complication
that A is rank deficient. (We forced rank deficiency by setting A
so that the first and second rows each contain a single nonzero
in their last column.) The (2; 2) pivot is skipped at every invocation of Algorithm
modchol. As - becomes small, the final pivot is skipped as well, and the numerical
rank is correctly determined. Since the small pivots are not localized in the bottom
right corner, the special bound (28) does not apply, so we cannot strengthen the
bounds on the step components as in the previous paragraph. The computational
behavior is qualitatively the same as in Tables 1 and 2.
Table
4 illustrates a problem for which 8. Here, the coefficient matrices
retain full numerical rank at all iterates, and the behavior is similar to that reported
in
Table
1. One point of difference is that the errors in c
\Deltax aff
, which start to increase
after iteration 19, do not have an immediate effect on the residual r b . The reason is
simply that this particular interior-point algorithm chose to take a path-following step
at iterations 21 and 22 rather than the affine scaling step, and the \Deltax components were
calculated accurately in the path following step. An affine-scaling step is, however,
taken at iteration 28, and the effect of the error in c
\Deltax aff
on the residual r b at the
following iterate is obvious.
--R
Technical Report OTC 96/01
Stability of symmetric ill-conditioned systems arising in interior methods for constrained optimization
Computer Solution of Large Sparse Positive Definite Systems
Accuracy and Stability of Numerical Algorithms
Solving Least Squares Problems
Computational experience with a primal-dual interior point method for linear programming
Block sparse Cholesky algorithms on advanced uniprocessor com- puters
On scaled projections and pseudoinverses
Matrix Perturbation Theory
A Dantzig-Wolfe-like variant of Karmarkar's interior-point linear programming algorithm
Some properties of the Hessian of the logarithmic barrier function
A simplified homogeneous and self-dual linear programming algorithm and its implementation
Solving large-scale linear programs by interior-point methods under the MATLAB enviroment
--TR
--CTR
Francis R. Bach , Michael I. Jordan, Kernel independent component analysis, The Journal of Machine Learning Research, 3, p.1-48, 3/1/2003 | error analysis;interior-point algorithms and software;cholesky factorization;matrix perturbations |
589182 | Two-Step Algorithms for Nonlinear Optimization with Structured Applications. | In this paper we propose extensions to trust-region algorithms in which the classical step is augmented with a second step that we insist yields a decrease in the value of the objective function. The classical convergence theory for trust-region algorithms is adapted to this class of two-step algorithms. The algorithms can be applied to any problem with whose contribution to the objective function is a known functional form. In the nonlinear programming package LANCELOT, they have been applied to update slack variables and variables introduced to solve minimax problems, leading to enhanced optimization efficiency. Extensive numerical results are presented to show the effectiveness of these techniques. | Introduction
In nonlinear optimization problems with expensive function and gradient evaluations, it is desirable
to extract as much improvement as possible at each iteration of an algorithm. When the objective
function contains a subset of variables that occurs in a predictable functional form, a second,
computationally relatively inexpensive, update can be applied to these variables following a classical
optimization step. The additional step provides a further reduction in the objective function
and can lead to superior optimization e-ciency. The two-step algorithms have been successfully
applied to the updating of slack variables and to a particular formulation of minimax problems, as
is indicated by numerical results on a variety of problems. In these instances a subset of variables
(slack variables and variables introduced to solve minimax problems) appears in a xed, known
algebraic form in the objective function. However, since it can be applied to any problem where
a subset of the variables can be optimized relatively cheaply compared with the cost of evaluating
the entire function (for example if some terms require simulation and other independent terms are
Department of Mathematical Sciences, IBM T. J. Watson Research Center, Route 134 and Taconic, Room 33-206,
Yorktown Heights NY 10598.
y Departamento de Matematica, Universidade de Coimbra, 3000 Coimbra, Portugal. This work started when this
author was visiting the IBM T. J. Watson Research Center at Yorktown Heights and was supported in part by Centro
de Matematica da Universidade de Coimbra, Instituto de Telecomunicac~oes, FCT, Praxis XXI 2/2.1/MAT/346/94,
and IBM Portugal.
z Computer Architecture and Design Automation, IBM T. J. Watson Research Center, Route 134 and Taconic,
Room 33-156, Yorktown Heights NY 10598.
available analytically), their applicability is really rather broad. We propose modications to existing
nonlinear optimization algorithms. An alternative approach, when feasible, is to reformulate
the original problem by eliminating a subset of variables and then to apply the algorithms in the
remaining variables (see, for example, Golub and Pereyra [17]).
This paper deals with two-step algorithms where the second step is required to yield a decrease
in the value of the objective function. The analysis given here covers the global convergence of
two-step trust-region algorithms and it is presented for the unconstrained minimization problem:
continuously dierentiable function. For both trust
regions and line searches, one can consider two versions of the two-step algorithms, one called
greedy and the other called conservative. The greedy version exploits as much as possible the
decrease obtained by the second step, whereas the conservative approach calculates the second step
only after the rst step has been conrmed to satisfy the traditional criteria required for global
convergence. We point out that the conservative two-step line-search algorithm is not new and can
be found in the books by Bertsekas [1], Section 1.3.1, and Luenberger [19], Section 7.10, where the
second step is called a spacer step. A description of the greedy and conservative two-step line-search
algorithms can be found in [11].
In trust regions, if the second step is guaranteed to decrease the value of the objective function,
global convergence of the type lim inf k !+1 krf(y k immediately attained. Further, in
the cases where the rst step would be rejected, the sum of the rst and second steps has a better
chance of being accepted (see Remark 3.1). To obtain lim k !+1 krf(y k either the norm of
the second step has to be controlled by the trust region (see condition (13)) or the decrease on the
objective function attained by the second step has to be of the order of magnitude of the norm of
this step (see condition (12)).
The update of the slack variables referred to above motivated the study of the local rate of
convergence of a two-step Newton's method. We show that a second Newton step in some of the
variables retains the q-quadratic rate of convergence of the traditional Newton's method.
This paper is structured as follows. In Section 2 we introduce the two-step trust-region al-
gorithms, and in Section 3 we analyze their global convergence properties. The local rate of the
two-step Newton's method is studied in Section 4. The application of the two-step ideas to update
slack variables and variables introduced for the solution of minimax problems is described in Section
5. Section 6 presents the numerical results obtained with LANCELOT using these updates for analytic
problems and dynamic-simulation-based and analytic static-timing-based circuit optimization
problems. Finally, some conclusions are drawn in Section 7.
Two-step trust-region algorithms
We rst consider the trust-region framework presented in the paper by More [20] for unconstrained
minimization. The (classical) trust-region algorithm builds a quadratic model of the form
at the current point y k , where H k is an approximation to r 2 f(y k ) (note that m k (y k
Then a step s k is computed by approximately solving the trust-region subproblem
subject to ksk k ;
where k is called the trust-region radius and kk is an arbitrary norm. The new point y
is tested for acceptance. If the actual reduction f(y k larger than a given fraction of
the predicted reduction m k (y k then the step s k and the new point y k+1 are accepted.
In this situation, the quadratic model m k (y is considered to be a good approximation to the
function f(y) in the region . The trust radius may be increased. Otherwise, the
is considered not to be a good approximation to the function f(y) in
the region . In this case, the new point y k+1 is rejected, and a new trust-region
subproblem of the form (2) is solved for a smaller value of the trust radius. This simple trust-region
algorithm is described below.
Algorithm 2.1 (Trust-region algorithm)
1. Given y 0 , the value f(y 0 ), the gradient rf(y 0 ) and an approximation H 0 to the Hessian of f
at y 0 , and the initial trust-region radius 0 . Set
and in (0; 1).
2. Compute a step s k based on the trust-region problem (2).
3. Compute
4. In the case where
set
compute H k+1 , and select k+1 satisfying k+1 k .
Otherwise, set
5. Increment k by one and go to Step 2.
The mechanism used to update the trust radius that is described in Algorithm 2.1 is simple and
su-ces to prove convergence results. In practice, with the goal of improving optimization e-ciency,
one uses updating schemes that are more complex involving several subcases according to the value
of k .
We propose in this paper a modication of this trust-region algorithm. We are motivated by a
situation where it is desirable to update slack variables and variables introduced to solve minimax
problems, at every iteration of the trust-region algorithm [7] implemented in LANCELOT [9]. See
Section 5 for more details on practical applications.
The two-step trust-region algorithm is quite easy to describe. Suppose that after computing a
step s k based on the trust-region subproblem (2) we know some properties of the function f(y) that
enables us to compute a new step ^ s k for which we can guarantee that f(y k
In this situation we would certainly like to have y and to test whether this new
point should be accepted or not. This modication requires a careful redenition of the actual and
predicted reductions given for Algorithm 2.1. The new actual and predicted reductions that we
propose are:
ared(y
pred(y
The new predicted reduction is the predicted reduction obtained by the rst step plus the (actual)
reduction obtained by the second step. The choice pred(y k ; s
not appropriate since the second step ^ s k is not computed using the model m k (y k
The two-step trust-region algorithm is given below.
Algorithm 2.2 (Two-step trust-region algorithm { Greedy)
1. Same as in Algorithm 2.1.
2. Compute a step
s k based on the trust-region problem (2).
3. If possible, nd another step ^ s k such that
4. Compute
pred(y
5. In the case where
set
compute H k+1 , and select k+1 satisfying k+1 k .
Otherwise, set
6. Increment k by one and go to Step 2.
The two-step trust-region Algorithm 2.2 evaluates the new point y acceptance
after both steps s k and ^ s k have been computed. We call this version \greedy" because it tries
to take as much advantage as possible of the decrease obtained by the second step ^ s k . Note that
although the function f is evaluated twice in Algorithm 2.2, the reevaluation is often computationally
inexpensive. The context in which we are particularly interested involves relatively expensive
evaluations at y k and evaluations at y k
involving only a subset of the variables that
are cheap to compute (see Section 5).
We could also consider a two-step trust-region algorithm where rst an acceptable step s k is
determined, and only afterwards a second step ^ s k is computed. This algorithm is outlined below.
Algorithm 2.3 (Two-step trust-region algorithm { Conservative)
1. Same as in Algorithm 2.1.
2. Repeat
(a) Compute a step
s k based on the trust-region problem (2).
(b) Compute
(c) If k > , then set
compute k+1 satisfying k+1 k , and set accepted = true.
If k , set
k and accepted = false.
Until accepted.
3. If possible, nd another step ^ s k such that
4. Set y
s k .
5. Update H k . Increment k by one and go to Step 2.
The same comments about the function evaluations apply to Algorithm 2.3 after the computation
of a successful step s k . However, in the case of Algorithm 2.3, the function f has to be
evaluated twice only in iterations corresponding to successful rst steps s k .
3 Global convergence of the two-step trust-region algorithms
We analyze rst the two-step trust-region Algorithm 2.2, i.e., the greedy version. The analysis for
the conservative Algorithm 2.3 is similar.
In this section we make the assumption that fH k g is a bounded sequence. So, there exists a
> 0 for which
We require the step
s k to satisfy a fraction of Cauchy decrease on the trust-region problem (2). In
other words we ask s k to satisfy
for 2 (0; 1]. The step c k is called the Cauchy step, and it is dened as the solution of the scalar
problem in the unknown
subject to ksk k ;
There is a variety of algorithms that compute steps satisfying this condition (see [3], [22], [23], [25],
and [26]).
Proposition 3.1 If
a fraction of Cauchy decrease then:
krf(y k )k
where and are as in (6) and (5) respectively.
Proof: See Powell [24], Theorem 4, or More [20], Lemma 4.8. 2
If we use this proposition and the fact that f(y k
pred(y
krf(y k )k
krf(y k )k
This inequality is crucial to prove global convergence of the two-step algorithm. In particular, if
the iteration k is successful, then
ared(y
We are ready to prove the rst convergence result.
Theorem 3.1 Consider a sequence fy k g generated by Algorithm 2.2 where s k satises (6). If f is
continuously dierentiable and bounded below on
and fH k g is a bounded sequence, then
lim inf
krf(y
So, if the sequence fy k g is bounded, there exists at least one limit point y for which rf(y
Proof: The proof is similar to the proof given in [20], Theorem 4.10.
Assume by contradiction that fkrf(y k )kg is bounded away from zero, i.e., that there exists an
> 0 such that krf(y k )k for all k. As in [20], Theorem 4.10, we make direct use of (9) and of
the rules that update the trust radius, to obtain:
and so lim k !+1
The next step is to show that lim k
Note that from the denitions (3) and
(4), we have
ared(y
which in turn, by using a Taylor series expansion and ks k k k , implies
This inequality and (8) show that
converges to zero. The rest of the proof follows a
classical argument in trust regions: if ^
k converges to one, the rules that update the trust radius
show that k cannot converge to zero. So, a contradiction is attained and the proof is completed. 2
The result of Theorem 3.1 does not require the step ^ s k to be O( k ) which may seem surprising.
This result shows the appropriateness of the denitions given in (3) and (4) for the actual and
predicted reductions. These denitions allow us to obtain the conditions (9) and (11) that are
crucial to establish (10).
Remark 3.1 It is also important to note that the denitions (3) and (4) can improve the acceptability
of a step. In fact, we have
before. We now note that ^
k and the function ^
strictly increasing if k < 1. In other words, in cases where a
standard trust-region algorithm rejects a step the modied criterion is always better than the usual
one. Further, it can be noted that ^
which indicates that all successful iterations of
the the standard algorithm will also be successful in the modied two-step algorithm. In particular,
The next step in the analysis is to prove that, with additional conditions on the second step,
Theorem 3.2 Consider a sequence fy k g generated by Algorithm 2.2 where
that f is continuously dierentiable and bounded below on L(y 0 ) and that fH k g is a bounded
sequence. If rf is uniformly continuous on L(y 0 ) and if either
or
are positive constants independent of k, then
lim
krf(y
So, if the sequence fy k g is bounded, every limit point y satises rf(y
Proof: The proof is similar to the proof given in [20], Theorem 4.14. See also Thomas [27].
We show the result by contradiction. Assume therefore that there exists an 1 2 (0; 1) and
a subsequence indexed by fm i g of successful iterates such that, for all m i in this subsequence,
Theorem 3.1 guarantees the existence of another subsequence indexed by fl i g
such that krf(y k )k 2 , for m i k < l i and krf(y l i )k < 2 (where fm i g is without loss
of generality the subsequence previously mentioned). Here 2 is any real number chosen to be in
converges to zero, for k su-ciently large corresponding to successful
iterations
holds if (12) is satised, and
holds otherwise with
.
We consider the cases (12) and (13) separately. In both cases we make use of:
In the sums
we consider only indices corresponding to successful iterations.
If (12) holds then we use (15) to obtain
[ks
If (13) holds then we appeal to (16) and write
[ks
In either case we obtain
and since the right hand side of this inequality goes to zero, so does the left hand side
Since the gradient of f is uniformly continuous, we have for i su-ciently large that
can be any number in (0; 1 ) this inequality contradicts the supposition. 2
In the theorem above we required the norm of the step ^
s k to either be O( k ) or O (f(y k
)). The former condition can be enforced in Step 2 of the Algorithm 2.2, although
this might not be benecial and could lead to an inferior decrease.
We can obtain global convergence to a point that also satises the necessary second-order
conditions for optimality. For this purpose, we require the step s k to satisfy a fraction of optimal
decrease for the trust-region problem (2). In other words we ask s k to satisfy
where 2 (0; 1], and s
k is an optimal solution of (2). (This condition can be weakened in several
ways [20].) A step s k satisfying a fraction of optimal decrease can be computed by using the
algorithms proposed in [22] and [25] in the case where the trust-region norm is Euclidean. The
global convergence result is the following.
Theorem 3.3 Consider a sequence fy k g generated by Algorithm 2.2 where H
satises (17). If L(y 0 ) is compact and f is twice continuously dierentiable on L(y 0 ), then there
exists at least one limit point y for which rf(y positive semi-denite.
Proof: The proof is basically the same as the proof of Theorem 4.7 in [22]. 2
To obtain stronger global convergence results to second-order points, for instance the results in
Theorems 4.11 and 4.13 in [22] (see also [21], Theorem 4.17, c and d), other conditions are required
like k^s k k being of O( k ).
The next results show that the second step can preserve the nice local properties of the behavior
of the trust radius that are typical in trust-region algorithms.
Theorem 3.4 Let fy k g be a sequence generated by Algorithm 2.2 where
In addition, assume that the step ^
s k satises either condition (12) or condition (13). If
f is twice continuously dierentiable and bounded below on L(y 0 ) and fy k g has a limit point y
such that H positive denite, then fy k g converges to y , all iterations are eventually
successful, and f k g is bounded away from zero.
Proof: From Theorem 3.2 we can guarantee that lim k !+1 krf(y k the proof is
basically the same as the proof of Theorem 4.19 in [20]. 2
An alternative to this result where we do not impose conditions (12) or (13) on the second step
is given below. However we need to assume that fy k g converges to y .
Theorem 3.5 Let fy k g be a sequence generated by Algorithm 2.2 where
continuously dierentiable on L(y 0 ) and fy k g converges to a point y such
that H positive denite, then all iterations are eventually successful and f k g is
bounded away from zero.
Proof: The rst step
s k yields a decrease in the quadratic model:
Thus, the assumptions made on H k and H guarantee
ks
for su-ciently large k, which in turn, by using (8), implies
pred(y ks k
(The constants c 3 and c 4 are independent of k.)
A Taylor series expansion for the expression (11) gives
The fact that fy k g converges and the result lim inf k !+1 krf(y k Theorem 3.1, together
imply lim k !+1 krf(y k Thus, from (18) we get lim k !+1 ks k
The proof is terminated with a typical argument in trust regions. From (19), (20) and lim k !+1
ks we obtain the limit
lim
pred(y
which shows, by appealing to the rules that update the trust radius, that all iterations are eventually
successful and the trust radius is uniformly bounded away from zero. 2
The global convergence analysis for Algorithm 2.3 is identical to the analysis given above for
Algorithm 2.2. We point out that Algorithm 2.3 is well dened since at a nonstationary point it is
always possible to nd an acceptable rst step. Also, for every k,
krf(y k )k
krf(y k )k
Thus, the results given in Theorems 3.1-3.5 hold for Algorithm 2.3. The lim inf-type result (10) is
obtained under the classical assumptions for trust-region algorithms for unconstrained optimization.
To obtain the lim-type result (14) one of the two conditions (12) and (13) is required.
In the case of the applications considered in Section 5, the decrease obtained by the second step
s k is always guaranteed to satisfy
Moreover, the objective function strictly decreases along the segment between the points y k
and
s k . In this case we can modify Step 3 of Algorithms 2.2 and 2.3 in such a way that
we meet the requirements of Theorem 3.2. This modication is given below. It is easy to verify
that
either (12) or (13).
Algorithm 3.1 (Step 3 for Algorithms 2.2 and 2.3 { Quadratic decrease case)
3. Compute a step ^
s k such that
g so that k^s k k c 2 k and ^
s k is not enlarged.
(Otherwise (12) holds with c
The positive parameters and c 2 should be set a priori in Step 1 of Algorithms 2.2 and 2.3.
Of course, we would like to prove the result of Theorem 3.2 for the case where the condition
(12) is replaced by the condition (21). However, such a result is unlikely to be true.
4 Local rate of convergence of a two-step Newton's method
In the next section we are interested in two-step algorithms where the second step is calculated
as a Newton-type step in some of the variables. In this section we investigate the local rate of
convergence for an algorithm where each step is composed of two Newton steps, the second being
computed only for a subset of the variables. For this purpose let
x
Suppose the rst step s k is a full Newton step, i.e., s
At the intermediate point y k , a Newton step is applied in the variables u with
x k xed. This
two-step Newton's method is described below.
Algorithm 4.1 (Two-step Newton's method)
1. Choose y 0 .
2. For do
2.1 Compute
s k .
2.2 Compute
and set s
s k .
2.3 Set y
The proof of the local convergence rate of the two-step Newton's method requires a few modications
from the standard proof of Newton's method [12], Theorem 5.2.1. Recall that that proof
of Newton's method is by induction.
Corollary 4.1 Let f be twice continuously dierentiable in an open set D where the second partial
derivatives are Lipschitz continuous. If fy k g is a sequence generated by Algorithm 4.1 converging
to a point y 2 D for which rf(y positive denite, then fy k g converges with
a q-quadratic rate.
Proof: If y k is su-ciently close to y , the perturbation result [12], Theorem 3.1.4, can be used to
prove the nonsingularity of the Hessian matrix r 2 f(y k ). Furthermore,
Now we show that r 2
First we point out that r 2
uu f(y) is Lipschitz
continuous on D and r 2
uu f(y ) is positive denite. Thus, inequality (22) and the perturbation
lemma cited above, together imply the nonsingularity of r 2
Hence the method is locally
well-dened, and the second step yields
since r u f(y) is Lipschitz continuous near y . Now we use inequalities (22) and (23), and write
This last inequality establishes the q-quadratic rate of convergence. 2
Applications
We begin by considering updating the slack variables in LANCELOT. Suppose the problem we are
trying to solve has the form
minimize f(x)
subject to c i (x)
are positive integers. The technique implemented in the LANCELOT
package [9] is the augmented Lagrangian algorithm proposed by Conn, Gould, and Toint in [8]. For
the application of the augmented Lagrangian algorithm this problem is reformulated as:
minimize f(x)
subject to c i (x)
by adding the slack variables u i , m. This algorithm considers the following augmented
Lagrangian merit function:
where:
i is an estimate for the Lagrange multiplier associated with the i-th constraint,
is a (positive) penalty parameter,
s ii is a (positive) scaling factor that is associated with the i-th constraint, and
solves a sequence of minimization problems with simple bounds of the
following
subject to u
for xed values of , s ii , and i , m. The two-step trust-region framework and analysis
described in this paper for unconstrained minimization problems can be extended in an entirely
straightforward way to a number of algorithms for minimization problems with simple bounds, in
particular to the algorithms [7] used by LANCELOT to solve problem (25).
If x is xed, the function (x; in the slack variables u. Let us denote this
quadratic by q(u; x):
where d(x) and e(x) depend on x but F is constant. (The dependency on i , s ii , and is not
important since these are constants xed before the minimization process is started.)
The key idea is to update these slack variables at every iteration k of the trust-region algorithm
[7] that is used in LANCELOT to solve problem (25). The trust-region algorithm computes, at the
current point y k , a rst step s k . Now, at the new point y
s k we compute the step ^ s k by updating
the slack variables u. So, we have
where
(Here f represents the objective function of Sections 1-4.) Note that the second step ^ s k is exclusively
in the components associated with slack variables. This step is computed as u k+1
is the optimal solution of
subject to u
Due to the simple form of this quadratic, the solution is explicit:
s ii
It is important to remark that these updates require no further function or gradient evaluations.
They have also been considered in the codes NPSOL and SNOPT [15], [16] to update slack variables
after the application of a line search to the augmented Lagrangian merit function and prior to the
solution of the next quadratic programming problem. Other ways of dealing with slack variables
have been studied in the literature (see Gould [18] and the references therein).
For the study of the impact of the slack variable update on the global convergence of the trust-region
algorithm, the step in these variables is required only to decrease the quadratic q(u;
u k to u k +u k . In such a case, we can always guarantee that the decrease in the objective function
is larger than k^s k k 2 , that is that (21) holds. This result is shown in the following proposition. We
drop
x k from q( ; x k ) to simplify the notation.
Proposition 5.1 There exists a positive constant c 5 such that, whenever q(u k +u k ) < q(u k ), we
have
Proof: First we write down a few properties of the quadratic q(u). Simple algebraic manipulations
lead to:
Also, since q(u) is convex:
Let c be a positive constant such that c < min
is the smallest eigenvalue of
F . Now we consider two cases.
1.
cku k k 2 . In this case we use (29), to obtain
2.
. In this case we appeal to (28) and
to get
min
The proof is completed by setting c
cg. 2
Another example of the application of two-step algorithms arises in one approach to the solution
of minimax problems. Consider the following
where each f i is a real-valued function dened in IR n . One way of solving this minimax problem is
to reformulate it as a nonlinear programming problem by adding an articial variable z. See [18]
for more details. This leads to
minimize z
subject to z f i (x)
where the slack variables have also been introduced. If LANCELOT is used to solve this nonlinear
programming problem, then the augmented Lagrangian algorithm requires the solution of a
sequence of problems with simple bounds of the type:
subject to u
where
In this situation the function (x; z; in the variables u and z for xed values
of x. (Again, , S, and are constants and not variables for problem (32).) The application of the
two-step trust-region algorithm follows in a similar way. The Hessian of the quadratic is positive
semi-denite with the following form
where the last row and the last column correspond to the variable z. The solution of the quadratic
program
minimize q(z; u;
subject to u
is given by
s ii
where z k+1 is the solution of the equation
s ii
s ii
with right hand side
s ii
The equation (35) is solved easily with O(m)
oating point operations and comparisons, showing
that the solution of the quadratic program (33) is a relatively inexpensive calculation.
There are several nonlinear optimization problems in which some subset of the problem variables
occur linearly, for example, arrival times in static-timing-based circuit optimization problems [6].
Such problems can also benet from two-step updating.
6 Numerical tests
6.1 Analytic problems
We modied LANCELOT (Release A) [9] to include the slack variable update (27) and the slack
and minimax variable updates (34)-(36). These updates were incorporated in LANCELOT using
a greedy two-step modication of the trust-region algorithm [7] for minimization problems with
simple bounds that is implemented in the subroutine SBMIN. (The greedy two-step trust-region
algorithm for unconstrained minimization problems is Algorithm 2.2.) We tested the following
versions of LANCELOT:
1. LANCELOT (Release A) with the default parameter conguration SPEC.SPC le, except that
we increased the maximum number of iterations to 4000.
2. Version 1 with the slack and minimax variable updates (27) and (34)-(36) incorporated in
SBMIN using a greedy two-step trust-region algorithm.
3. The same as Version 2 but with no update of the variable z for minimax problems, i.e., z
xed in (34)-(36).
We compared the numerical performance of these three versions on a set of problems 1 from
the CUTE collection [2]. This set of problems is listed in Table 1, and in the case of minimax
formulations in Table 2, where we mention the number of variables (including slacks and, where
applicable, the minimax variable z), the number of slack variables, and the number of equality and
inequality constraints (excluding simple bounds on the variables). Note that the minimax problems
were reformulated as nonlinear programming problems by the introduction of an additional minimax
variable z as shown above (31).
The computational results are presented in Tables 3, 4, and 5. All tests were conducted on an
IBM Risc/System 6000 model 390 workstation. In Table 3 we compare the results of Versions 1
and 2 for problems that are not minimax problems. In Table 4 we present the results of Versions
1 and 2 for minimax problems. In Table 5 we include the results of Versions 1 and 3 for minimax
problems. In Tables 4 and 5 we include the majority of the minimax problems but not all (see
Section 6.3 for numerical results on the remaining problems). In these tables we report the value of
the
ag INFORM, the number of iterations, the total CPU time, and the determined values (a single
value if they are both the same) of the objective function. The values of INFORM have the following
meaning:
meaning that the norm of the projected gradient of the augmented
Lagrangian function has become smaller than 10 5 .
cases where the maximum number of iterations (4000) has been reached.
cases where the norm of the step has become too small.
Our conclusion based on these sets of problems is that the version with the slack and minimax
variable updates exhibits superior numerical behavior. In fact, this version required an average of
15% fewer iterations than the version without these updates (the problems HS109, HAIFAM, and
POLAK6 were excluded from this calculation, mainly because the comparison was extraordinarily
favorable in the case of the rst two and worse in the last). Comparing Tables 4 and 5, updating the
variable z in addition to two-step updates on just the slacks is seen to yield a signicant
benet. However, there are some minimax problems where the two-step algorithm performs poorly
and this situation is analyzed in detail in Section 6.3.
Although CUTE contains more than 56 problems with general constraints the majority of these are equality
constrained problems. We excluded all problems that took more than 4000 iterations with both Versions 1 and 2.
We included the rest, with the exception of some problems that are too easy, making a total of 56 problems of which
are minimax problems and 26 are non-minimax problems.
Problem Name Variables Slacks Constraints
CORE1
CORE2 157 26 134
CORKSCRW
HADAMARD 769 512 648
HS85 26 21 21
Table
1: Non-minimax problems from the CUTE collection that were used.
6.2 Circuit optimization problems
We have built extensive experience with circuit optimization problems, where { due to expensive
function evaluations, modest numerical noise levels, and practical stopping criteria { the implementation
is designed to terminate before many \asymptotic" iterations are taken. The algorithms
described in this paper have been used in a dynamic-simulation-based circuit optimization tool
called JiyTune (see [4], [5], and [10]). JiyTune optimizes transistor and wire sizes of digital integrated
circuits to meet delay, power, and area goals. It is based on fast circuit simulation and
time-domain sensitivity computation in SPECS (see [13] and [28]). To optimize multiple path delays
through a high-performance circuit, the tuning is often formulated as a minimax problem or a
minimization problem with nonlinear inequality constraints.
We remark that many of the analytic problems (especially the minimax problems) are rather
small and involve inexpensive function evaluations. Moreover, it is clear that two-step updating is
unlikely to be helpful asymptotically in these situations. Consequently we also report numerical
results with circuit optimization problems which are indicative of problems with expensive function
evaluations, where termination (because of inherent noise and practical considerations) is encouraged
to be before any signicant asymptotic behavior. The numerical results are presented in Table
6. As in Version 1, the second step consisted of the slack and minimax variable updates (27) and
(34)-(36). However the gradient and constraint tolerances used were 10 respectively,
Problem Name Variables Slacks Constraints
COSHFUN 81 20 20
GOFFIN 101 50 50
HAIFAL 9301 8958 8958
HALDMADS 48 42 42
MINMAXBD
Table
2: Minimax problems from the CUTE collection that were used.
with some safeguards related to an expected level of numerical noise. We can clearly observe from
Table
6 that the two-step algorithm leads to better nal objective function values. In practical
applications where a simple function evaluation takes more than ten minutes of CPU time the
eectiveness of such a simple addition is indeed signicant. (There are situations where the greedy
two-step trust-region algorithm is able to take advantage of the decrease given by the slack and
variable updates and, by doing so, this algorithm can accept steps that otherwise would
have been rejected, see Remark 3.1.)
We also applied the algorithms of this paper to analytic static-timing-based circuit optimization
problems (see Table 7), where it is clear that the advantage of the two-step approach is increasingly
apparent for larger problems.
Problem Name Inform Iterations Total CPU Obj. Function
CORE1 0/0 953/983 7.41/17 91.1
CORE2 0/0 1048/1086 25.6/25.7 72.9
CORKSCRW 0/0 41/42 0.55/0.54 1.16
HADAMARD 0/0 1709/548 2290/276 1.14/1
TFI3 0/0 23/34 0.38/0.38 4.3
Table
3: Comparison between Versions 1 and 2 for non-minimax problems (LANCELOT with-
out/with two-step updating).
6.3 Further experiments with minimax problems
In this section we consider those minimax problems in our test set for which the two-step algorithm
not only does not improve numerically the results obtained in the one-step case, but also makes
them considerably worse (see the rst part of Table 8). We analyze the reasons for the failure of the
two-step updating on some minimax problems and discuss a few ways to enforce better numerical
behavior.
We consider the general minimax problem (30). Our aim is to show that for some types of
problems the second step has a tendency to make the Hessian of ill-conditioned. Let us
assume that (as happens by default for the rst LANCELOT
major iteration). Under these circumstances, we have:
By using the notation g i (x; z; we have the following expressions for the elements
TWO-STEP ALGORITHMS FOR NONLINEAR OPTIMIZATION 20
Problem Name Inform Iterations Total CPU Obj. Function
CONGIGMZ 0/0 32/19 0.04/0.05 28
COSHFUN 0/0 127/69 1.31/1.06 -0.773
GOFFIN 0/0 14/4 1.03/0.67 0
Table
4: Comparison between Versions 1 and 2 for minimax problems (LANCELOT without/with
two-step updating).
of the gradient of :
r z
r
Similarly the elements of the Hessian matrix of are given by:
for If the magnitudes of the products r 2
are small compared to those of the products r x then the Hessian of is given
Problem Name Inform Iterations Total CPU Obj. Function
CONGIGMZ 0/0 32/25 0.04/0.1 28
COSHFUN 0/0 127/92 1.31/1.08 -0.773
GOFFIN 0/0 14/8 1.03/0.66 0
Table
5: Comparison of Versions 1 and 3 for minimax problems (LANCELOT without/with two-step
updating only on slacks).
approximately
a i1 a
a i1 a in
a i1 a
i a in a
i a in a in
i a in a
i a in
a
and the indices i in the sums go from 1 to m. This matrix is clearly
singular. In fact, the n 1-st row is the negative sum of the last m rows. Moreover, any of the
rst n rows is a linear combination of the last m rows. As result of these observations, the Hessian
(and the projected Hessian) of is ill-conditioned if
(37)
happens for \many" indices j and k. This is the key point in this analysis: the second step has
a tendency to produce iterates that worsen property (37) because it produces a decrease on the
Problem Name Variables Ineq. Iterations Total CPU Obj. Function
IOmuxpower 102 42 21/29 7230/9220 -15100/-16000
coulman cold 33 17 22/22 69.5/68.3 271/262
clkgen 22 5 25/5 35/10.8 1.98/1.82
coulman hot 33 17 16/32 46.2/100 283/253
coulman delay 33 17 26/24 72.6/73.5 116/111
Minimax:
bultmann latch
stall1
coulman cold minmax 34 17 61/80 184/229 69.4/66.9
coulman hot minmax 34 17 66/44 197/134 74.4/75.1
eischer
northrop xor
coulman delay minmax 34 17 100/100 290/306 67.4/70.5
Table
updating for dynamic-simulation-based circuit optimization
problems. Ineq. stands for the number of inequality constraints.
values of g i (x; z; u) for some indices i. The Hessian of might very well be ill-conditioned if no
second steps are applied, but there is no doubt (and the numerical results are a evidence of this
claim) that the second step for some problems worsens the situation by making the Hessian of
more ill-conditioned.
In the presence of nonzero Lagrange multipliers i , m, the formulae for the gradient
and Hessian of are the same with g i (x; z; u) substituted by g i (x; z; u)+ i and similar conclusions
could be drawn.
The second step may produce very bad results on some minimax problems because it points
towards the set f(x; z; (where the Hessian of the augmented Lagrangian
is ill-conditioned) and this eect in
uences negatively the calculation of the rst step at
the next iteration. Given this undesirable feature of the Hessian of at points close to this set,
one possible improvement to the two-step algorithm is to make sure that the calculation of the rst
step is accurate (in the LANCELOT context this could be achieved by choosing a smaller tolerance
for the stopping criterion of the conjugate-gradient technique). Another possible improvement is to
reduce the ill-conditioning of the Hessian of (for instance by increasing the value of the penalty
parameter as can be seen in examples with a few variables). Indeed, these modications improve
the bad numerical results presented before: in the second part of Table 8 we compare the results
obtained by the following modications of Versions 1 and 2:
4. Version 1 with an initial value for the penalty parameter of 100 (the default value is 0:1).
5. Version 2 with an initial value for the penalty parameter of 100 and a tolerance of 10 12 in
Problem Name Variables Ineq. Iterations Total CPU Obj. Function
Symmetric 9
Table
7: LANCELOT without/with two-step updating for analytic (minimax) static-timing-based
circuit optimization problems. Ineq. stands for the number of inequality constraints.
the stopping criterion for conjugate gradients.
The study of strategies that can make two-step updating more eective for minimax problems in
general is the subject for future research.
7 Concluding remarks
In this paper we presented and analyzed a framework under which classical algorithms for nonlinear
optimization can be modied to allow second computationally e-cient steps that are not generated
in the conventional way but that are guaranteed to yield decrease in the objective function. We
gave as examples of the two-step algorithms the update of slack variables in LANCELOT, and the
update of variables introduced to solve minimax problems. However, we emphasize that the two-step
algorithms can be very generally applied, for example, whenever the functions dening the
problem are in a known functional form in some of the variables.
We considered trust-region algorithms for which we proposed a greedy and a conservative two-step
algorithm. We analyzed the convergence properties of the trust-region two-step algorithms
(see [11] for line-search two-step algorithms), deriving the conditions under which they attain
global convergence. We also showed that a two-step Newton's method (for which the second step
is computed only for a subset of the variables) has a q-quadratic rate of convergence.
The greedy two-step algorithms are designed to exploit as much as possible the decrease attained
by the second step. The trust-region framework allowed to us to design a greedy two-step trust-region
algorithm that is particularly well tailored to achieve this purpose.
Finally, we included numerical evidence that this technique is eective, particularly for problems
with expensive function evaluations. The two-step algorithms have already found practical
applications in optimization of high-performance custom microprocessor integrated circuits.
Problem Name Inform Iterations Total CPU Obj. Function
MINMAXBD 0/0 267/952 1.34/3.59 116
POLAK3 0/0 71/125 0.4/0.8 5.93
MINMAXBD 0/0 47/43 0.25/0.22 116
Table
8: In the rst part, comparison of Versions 1 and 2 for minimax problems (LANCELOT
without/with two-step updating). In the second part, comparison of Versions 4 and 5 for minimax
problems (LANCELOT without/with two-step updating).
Acknowledgments
We are grateful to N. I. M. Gould (Rutherford Appleton Laboratory) for his comments and suggestions
on an earlier version of this paper that led to many improvements. We are also grateful
to K. Scheinberg (IBM T. J. Watson Research Center) for helping with the numerical results and
explanation in Section 6.3. We would like to thank I. M. Elfadel (IBM T. J. Watson Research
Center) for providing the analytic static-timing-based optimization circuit problems. Finally, we
are grateful to the referees for their useful comments and suggestions.
--R
Computer Science and Applied Mathematics
CUTE: Constrained and Unconstrained Testing Environment
Approximate solution of the trust-region problem by minimization over two-dimensional subspaces
Optimization of custom MOS circuits by transistor sizing
Global convergence of a class of trust-region algorithms for optimization problems with simple bounds
Circuit optimization via adjoint Lagrangians
Numerical Methods for Unconstrained Optimization and Nonlinear Equations
Sensitivity computation in piecewise approximate circuit simulation
Practical Methods of Optimization
SNOPT: An SQP algorithm for large-scale constrained optimization
User's guide for NPSOL 5.0: A Fortran package for nonlinear programming
On solving three classes of nonlinear programming problems via simple di
Linear and Nonlinear Programming
A new algorithm for unconstrained optimization
Minimization of a large-scale quadratic function subject to a spherical con- straint
The conjugate gradient method and trust regions in large scale optimization
Sequential Estimation Techniques for Quasi-Newton Algorithms
Piecewise approximate circuit simulation
--TR
--CTR
Tong Zhang, On the Dual Formulation of Regularized Linear Systems with Convex Risks, Machine Learning, v.46 n.1-3, p.91-129, 2002
Andreas Wchter , Chandu Visweswariah , Andrew R. Conn, Large-scale nonlinear optimization in circuit tuning, Future Generation Computer Systems, v.21 n.8, p.1251-1262, October 2005
Xiaoliang Bai , Chandu Visweswariah , Philip N. Strenski, Uncertainty-aware circuit optimization, Proceedings of the 39th conference on Design automation, June 10-14, 2002, New Orleans, Louisiana, USA
Nicholas I. M. Gould , Dominique Orban , Philippe L. Toint, CUTEr and SifDec: A constrained and unconstrained testing environment, revisited, ACM Transactions on Mathematical Software (TOMS), v.29 n.4, p.373-394, December
A. R. Conn , I. M. Elfadel , W. W. Molzen, Jr. , P. R. O'Brien , P. N. Strenski , C. Visweswariah , C. B. Whan, Gradient-based optimization of custom circuits using a static-timing formulation, Proceedings of the 36th ACM/IEEE conference on Design automation, p.452-459, June 21-25, 1999, New Orleans, Louisiana, United States
Andrew R. Conn , Ruud A. Haring , Chandu Visweswariah, Noise considerations in circuit optimization, Proceedings of the 1998 IEEE/ACM international conference on Computer-aided design, p.220-227, November 08-12, 1998, San Jose, California, United States
Andrew R. Conn , Chandu Visweswariah, Overview of continuous optimization advances and applications to circuit tuning, Proceedings of the 2001 international symposium on Physical design, p.74-81, April 01-04, 2001, Sonoma, California, United States | spacer steps;expensive function evaluations;LANCELOT;circuit optimization;trust regions;two-step algorithms;minimax problems;line searches;slack variables |
589186 | A Reflective Newton Method for Minimizing a Quadratic Function Subject to Bounds on some of the Variables. | We propose a new algorithm, a reflective Newton method, for the minimization of a quadratic function of many variables subject to upper and lower bounds on some of the variables. The method applies to a general (indefinite) quadratic function for which a local minimizer subject to bounds is required and is particularly suitable for the large-scale problem. Our new method exhibits strong convergence properties and global and second-order convergence and appears to have significant practical potential. Strictly feasible points are generated. We provide experimental results on moderately large and sparse problems based on both sparse Cholesky and preconditioned conjugate gradient linear solvers. | Introduction
. In this paper we propose a new algorithm for solving the box-
constrained quadratic programming problem
The matrix H is symmetric and, in general, indefinite; l
We denote the feasible region and the strict
ug. When H is indefinite we are interested in locating
a local minimizer.
Problem (1.1) arises as a subproblem when minimizing general nonlinear functions
subject to bounds and as a problem in its own right. The box-constrained quadratic
programming problem represents an important class of optimization problems and has
been the subject of considerable recent work (e.g., [1, 5, 12, 13, 16, 19, 21, 24, 26, 27, 32]).
A special subclass deserves mention: the box-constrained least-squares problem,
where A is a rectangular m-by-n matrix with m ? n. Our proposed algorithm can
of course be applied to this special case if we form b. The
determination of a version of our algorithm which does not involve the formation of the
matrix H is an open question.
We propose a new approach, a reflective Newton algorithm. The algorithm generates
a sequence of strictly feasible iterates, fx k g, which converges under standard
assumptions to a local solution of (1.1), x , at a quadratic convergence rate. Coleman
and Li [10] establish theoretical properties of the reflective Newton approach applied to
the general nonlinear box-constrained problem - as we indicate in Section 5 these results
apply directly to the reflective Newton procedure proposed here for the quadratic
minimization problem (1.1). In this paper we discuss the nature of the reflective transformation
(Section 2); we discuss the reflective Newton approach as applied to problem
with emphasis on a specialized line search to exploit the special structure of this
problem (Section 3); numerical experiments involving an implementation of the reflective
Newton method applied to box-constrained quadratic minimization (1.1) are
discussed (Section 4).
The sequence fx k g generated by the algorithm is strictly feasible: therefore, the
algorithm can be regarded as an "interior-point" algorithm. However, this is a misleading
classification. The algorithm differs markedly from methods commonly referred
to as "interior-point" algorithms. For example, the proposed algorithm does not use a
barrier function to ensure feasibility. The algorithm generates descent directions for q
and then follows a piecewise linear path, reflecting off constraints as they are encoun-
tered. Most interior point methods, on the other hand, generate descent directions (for
some function) and then restrict the step, along this straight-line direction, to ensure
feasibility.
The algorithm most similar to our current proposal is probably the recent method
due to Coleman and Hulbert [6]. (There is also a strong connection to previous work
by Coleman and Li [7, 8, 9, 20] on various norm minimization problems.) Both are
driven by the nonlinear system of equations representing first-order optimality condi-
tions. Both methods require piecewise quadratic minimization. The methods differ
in that our new algorithm is more general: positive definiteness of H is not required
and it is not necessary to have finite upper and lower bounds on all the variables -
the Coleman/Hulbert method requires both these restrictive properties. Finally, the
Coleman/Hulbert method is an exterior-point method, requiring strict decrease in a
piecewise quadratic "dual" function, whereas the new method generates feasible iter-
ates, requiring strict decrease in the original quadratic function q.
There are four key observations that underpin our new approach.
First, it is possible to change the constrained problem (1.1) to an unconstrained
problem without using a penalty parameter. We can replace (1.1) with an unconstrained
problem,
min
q(y)
where - q(y) is a continuous piecewise quadratic function of y, and y 2 R n is unrestricted.
Details of this transformation are given in the next section including a result, Theorem
1, proving the equivalence of (1.3) with (1.1). One view of our algorithm is that it is
designed to find a local minimizer of -
q(y). Variables x and y are related by a piecewise
linear transformation, a reflective transformation,
a feasible sequence fx k g. Moreover, evaluation of -
q(y) corresponds to evaluation of
q(R(y)).
Alternatively, one can view our approach entirely in the original variables x. Then,
instead of describing the method as a descent algorithm for the transformed problem
q(y), our method can be described as a method that generates feasible iterates by
following a piecewise linear path induced by the reflective mapping R. We discuss this
below.
The second key observation is that the first-order optimality conditions for (1.1),
or equivalently (1.3), can be expressed as a single system of nonlinear equations,
and a Newton step for this system is a descent direction for -
q in a neighbourhood of
a local solution y . Moreover, in a neighbourhood of a local solution to (1.3) a full
Newton step for (1.4), i.e., a unit step size in the Newton direction, yields decrease in
q(y). This is a very important point because it suggests that a Newton process for (1.4)
is compatible with (1.3), at least in the neighbourhood of a solution. It suggests that
ultimate second-order convergence can be achieved while decreasing -
q(y).
The third observation leads to globalization of the Newton process. It turns out that
the Newton equation for (1.4), the nonlinear system representing first-order optimality
conditions, can be written in the form:
where M is a symmetric matrix. 3 Moreover, it turns out that M is positive definite in
a neighbourhood of a minimizer of -
and M can be interpreted, loosely, as a second
derivative matrix for -
q(y) . This suggests the use of an ellipsoidal constraint to ensure
a descent direction when far from the solution. Specifically, solve
\Deltag
where D is a positive diagonal scaling matrix and \Delta is positive. As we discuss in [10],
a good choice for matrix D(x) is
i.e., D is a diagonal matrix with the i th diagonal component equal to jv i (x)j 1
2 . Vector
defined in Figure 1 where
Diagonal matrix D plays an important role in this paper - henceforth we reserve the
notation D, without superscript, to refer to definition (1.7). Of course D k refers to (1.7)
with all quantities defined at the current point x k .
(iv) If g i - 0 and l
Fig. 1. Definition of v(x)
Solving (1.6) involves solving a symmetric positive definite system,
for a suitable -, and then s / D-s y . Thus it is easy to see that (1.6) leads to a descent
direction for - q at the current point. It may be felt that solving (1.6) is an expensive way
to determine a descent direction in the large-scale setting. With this is mind a restricted
version of (1.6) is used in our algorithm. In particular, similar to [2] we usually restrict
s to be in a low-dimensional subspace S. So (1.6) is replaced with
where S is a low-dimensional subspace of ! n . Provided the ellipsoidal constraint
inactive near the solution, and the low-dimensional subspace S
3 The function -
q is a piecewise quadratic function of y: therefore, r y -
q does not always exist. How-
ever, the proposed algorithm only generates points where r y -
q is defined.
ultimately includes the Newton direction, the solution to (1.9) will eventually be the
Newton step (1.5).
The fourth major ingredient of our approach is the line search. Once a descent
direction s k is determined, a one-dimensional line search is performed to approximately
locate a minimizer of -
q k has structure: -
q k is a one-dimensional
piecewise quadratic function and so an efficient specialized line search procedure can be
used. (Alternative view: A one-dimensional piecewise linear search is performed along
a "reflective path", p k (ff), defined by the reflective transformation R and beginning at
We conclude the introduction with a short review of optimality conditions for problem
(1.1).
The first-order optimality conditions can be written: If a feasible point x is
a local minimizer of (1.1) then
Let F ree denote the set of indices corresponding to "free" variables at point x
F ree
Second-order necessary conditions can be written 4 : If a feasible point x is a local
minimizer of (1.1) then D 2
ree is the submatrix of H
corresponding to the index set F ree
These conditions are necessary but not sufficient. To state practical sufficiency
conditions we first need a definition of degeneracy.
Definition 1. A point x nondegenerate if, for each index i:
With this definition we can state second-order sufficiency conditions: If a
nondegenerate feasible point x satisfies D 2
ree ? 0, then x is a local
minimizer of (1.1).
2. The Reflective Transformation. One interpretation of our approach to solving
the box-constrained quadratic programming problem (1.1) involves a transformation
to an unconstrained piecewise quadratic minimization problem (1.3). The purpose of
this section is to introduce this transformation. Since some of the ideas involved are
more generally applicable, we begin our discussion at a more abstract level and gradually
work our way back to the box-constrained quadratic programming situation.
4 Notation: If a matrix A is a symmetric matrix then we write A ? 0 to mean A is positive definite;
means A is positive semi-definite.
Consider the problem
where f is a continuous and C is a closed connected region of ! n .
We consider when the constrained problem (2.1) can be replaced with an unconstrained
minimization problem of the form,
min
where R is a continuous onto mapping,
What further restrictions on the mapping R make this an acceptable transfor-
mation? To see that continuity is not enough consider the following one-dimensional
example. Let Obviously there is only one local solution
(the global solution), x = 1. However, let R(y) be any continuous function, mapping
onto [0; 1], with a strict local maximizer at -
y with -
1). It is easy to
see that -
y is a local minimizer of f(R(y)) but - x is clearly not a local solution to the
original problem.
The following property plays the key role in answering this question.
R is an open mapping if for each ffl ? 0 and pair
See Munkres [25], for example, for a discussion of open mappings. We can now
answer our question.
Theorem 1. C be a continuous onto mapping. Further, assume R
is an open mapping. Then,
(i) If y is a local minimizer of (2.2) then -
local minimizer of (2.1).
(ii) If x is a local minimizer of (2.1) then for each -
- y is a local minimizer of (2.2). Moreover, there exists at least one -
y such that
(iii) Problem (2.1) is unbounded below if and only if problem (2.2) is unbounded
below.
Proof. (i) Assume y is a local minimizer of (2.2). Let -
local minimizer, there exists ffl ? 0 such that
But by (2.3) there exists
Hence for each x 2 N
Therefore, - x is a local minimizer of (2.1).
(ii) Assume x is a local minimizer of (2.1). But R is an onto mapping and therefore
there exists - y is a local minimizer, there exists ffl ? 0
such that
By continuity there exists
Therefore, for all y
Hence, - y is a local minimizer of (2.2).
(iii) Suppose fy k g is a sequence such that
lim
Alternatively, assume fx k g 2 C and
lim
But R is an onto mapping; therefore, for each x k there exists y k such that R(y k
and
lim
and therefore (iii) is established.
To illustrate, consider the problem
0g. A definition of R that clearly satisfies the open mapping property
is multiplication. Note that R is
differentiable and the Jacobian of R, J R (y), is nonsingular if and only if R(y) 2 int(C),
where int(C) is the interior of C. Specifically, r y
x
where D g is the diagonal matrix diag(r x f) and D y is the diagonal matrix diag(y). (Note
that r 2
y R is a tensor term and r x fr 2
y R is a matrix - diagonal in this case.) Therefore,
this definition of R leads to an unconstrained twice-differentiable minimization problem
and standard techniques can be used to solve (2.2). Unfortunately, our numerical
experience with this approach has been mixed: In particular, as problems become large
and ill-conditioning (and near-degeneracy) increases, the number of iterations required
by standard minimization algorithms, to achieve good accuracy, becomes quite large.
We feel this is due in part to the fact that this transformation causes an increase in
the complexity of the function to be minimized: e.g., a quadratic function becomes a
quartic. Our objection to this approach is largely numerical - ill-conditioning in the
original problem is accentuated when the problem is transformed to a more complex
form. Note also that the transformed problem may have many more local minimizers
- by Theorem 1 this, in itself, is not a problem. However, along with this increase in
the number of local minimizers comes an increase in negative curvature and this may
cause some optimization algorithms some difficulty. In any event, our experience with
this simple differentiable transformation has not been satisfactory: the subject of this
paper is an alternative definition of R.
For problem (2.5) consider is a vector, jvj denotes the vector
whose components are the absolute values of the vector v. It is clear that the open
mapping property holds. Note that R is not everywhere differentiable. In particular,
R is differentiable at point y if and only if R(y) 2 int(C), i.e., y i 6= 0. In this case
obviously nonsingular. Using this transformation,
f(R(y)) has a piecewise differentiable nature as a function of y. For example, if f(x) is
a quadratic function then f(R(y)) is a piecewise quadratic function of y.
We now extend the absolute value approach, to handle the more general
situation,
where for each index i either u i is finite or u Similarly, for each index i either l i
is finite or l
For simplicity we assume that the finite values of u are all equal to unity and the finite
values of l are all equal to zero (a simple translation and scaling can achieve this form 5 ).
The transformation we propose, x = R(y), is a diagonal transformation, i.e., for
5 A definition of the reflective transformation applied directly to the general problem is given in [10]
y
Fig. 2. The 1-Dimensional Reflective Transformation (Finite Upper and Lower Bound)
each index i, x i depends only on y i . This transformation, induces a piecewise
linear "reflective" path in x.
For example, if u Figure 2.
If l is the absolute value function; if l
then R i is illustrated in Figure 3.
Finally, if u . The four cases are described
more formally in Figure 4.
It is easy to verify that R satisfies the requirements of Theorem 1; therefore, use
of R does not introduce extraneous local minimizers nor does it restrict the set of local
minimizers.
Using the reflective transformation, problem (1.1) can be replaced with the unconstrained
piecewise quadratic minimization (1.3). In principle, problem (1.3) can be
solved by a descent direction algorithm, e.g., Algorithm 1 in Figure 5.
An advantage of using this piecewise linear transformation R is the linear aspect
of the transformation: when y is a differentiable point the local complexity of
is the same as the local complexity of f(x). When
q is a piecewise quadratic function. The apparent disadvantage is the
piecewise nature of -
f(y). This lack of differentiability means that conventional nonlinear
minimization methods cannot be used.
In particular, in order to guarantee convergence, restrictions on the nature of the
descent direction s y must be imposed. To see this suppose that y k is very close to a
y is a descent direction for - q at y k . If s y is nearly
perpendicular to this hyperplane then the usual descent condition, r y -
only result in a very small step since r y -
q is not continuous at x In [10]
we describe two properties, "constraint-compatibility" and "consistency", which help
guarantee sufficiently long steps and, consequently, global convergence. We discuss this
y
OE
x
Fig. 3. The 1-Dimensional Reflective Transformation with Infinite Upper Bound
briefly in Section 3.
The straight-line direction s y
k corresponds to a piecewise linear path in x. This
piecewise linear path can be described as follows. For simplicity, and without loss of
generality, assume y
k . Define the vector 6
Component i of vector BR k records the positive distance form x k to the breakpoint
corresponding to variable x k i in the direction s x
k . The piecewise linear (reflective) path
is defined by Algorithm 2 in Figure 6. Since only a single outer iteration is considered,
we do not include the subscript k with the variables in our description of Algorithm 2
- dependence on k is assumed.
Given the current point x k and a descent direction s x
denote the piecewise
linear path defined by Algorithm 2: For fi
Note that it is now possible to describe Algorithm 1 entirely in x-space without
explicitly introducing either the function -
q or the variables y. We do this in Algorithm
Figure
8.
The difference between Algorithm 1 and Algorithm 3 is notational. The view
presented by Algorithm 3 has the advantage that it is in the original space - visualization
6 For the purpose of computing BR we assume the following rules regarding arithmetic with infinities.
If a is a finite scalar then a
Case 1: (l
To evaluate x
then we can differentiate R to obtain the i th diagonal element of
the diagonal Jacobian matrix
else J R
Case 2: (l
To evaluate x
then we can differentiate R to obtain the i th diagonal element of the
Jacobian matrix
Case 3: (l
To evaluate x
then we can differentiate R to obtain the i th diagonal element of the
Jacobian matrix
If y
Case 4: (l
In this case there are no constraints on x i and so x
Fig. 4. The Reflective Transformation R
Algorithm 1
Choose
For
1. Determine a descent direction s y
for -
q(y) at y k
2. Perform an approximate line minimization of -
k ), with respect to
ff, to determine an acceptable stepsize ff k (such that ff k does not correspond
to a breakpoint)
3. y
Fig. 5. Descent dir'n algorithm for -
f (y)
Algorithm 2 [Let fi
[i u is a finite upper bound on the number of segments of the path to be determined]
For
1. Let fi i be the distance to the nearest breakpoint along
2.
3. Reflect to get new dir'n and update BR:
(a)
(b) For each j such that (b i
Fig. 6. Determine the linear reflective path p
of the reflective process is natural. The advantage of the first view, Algorithm 1, is that
the algorithm is a straight line descent direction algorithm, a familiar structure. It is
probably useful for the reader to keep both views in mind. In this paper we will primarily
work in the x-space and Algorithm 3. For simplicity we now drop the superscripts y
and x (e.g., s x becomes s).
It is well known that a descent direction algorithm demands sufficient decrease at
every step in order to achieve reasonable convergence properties. We use conditions
suggested by Goldfarb [18] for use in the unconstrained setting: Given
and a descent direction s k , ff k satisfies our approximate line search conditions if
and
Fig. 7. A Reflective Path
Algorithm 3
Choose
For
1. Determine an initial descent dir'n s x
k for q at x k 2 int(F ). Determine the
piecewise linear reflective path p k (ff) via Algorithm 2.
2. Perform an approximate piecewise line minimization of q(x k +p k (ff)), with
respect to ff, to determine an acceptable stepsize ff k (such that ff k does
not correspond to a breakpoint).
3. x
Fig. 8. A reflective path algorithm
is the
piecewise linear path defined by (2.8). Condition (2.9) can be interpreted as restricting
the step length from being too large relative to the decrease in f ; condition (2.10) can
be interpreted as restricting the step length from being relatively too small.
A basic reflective path algorithm for problem (1.1) can now be stated, Algorithm
4. To allow for flexibility, especially with regard to the Newton step, we do not always
require that both (2.9) and (2.10) be satisfied. Instead, we demand that either both
these conditions are satisfied or (2.9) is satisfied and ff k is bounded away from zero,
e.g., latter conditions are used to allow for the liberal use of Newton
steps and do not weaken the global convergence results.
Note that Algorithm 4 generates strictly feasible points; i.e., since x 1 2 int(F ), it
follows that x k 2 int(F ).
Algorithm 4 [ ae is a positive scalar.]
Choose
For
1. Determine an initial descent dir'n s k for q at x k . Note that the piecewise
linear path p k is defined by x
2. Perform an approximate piecewise line minimization of q(x k +p k (ff)), with
respect to ff, to determine ff k such that:
(a) ff k does not correspond to a breakpoint
(b) condition (2.9) is satisfied
(c) Either
i. ff k satisfies condition (2.10), or
ii.
3. x
Fig. 9. A reflective path algorithm satisfying line search conditions
3. Algorithm Specifics. A framework for our reflective Newton approach was
presented in the previous section, Algorithm 4. In this section we specify more precisely
how the search directions will be generated as well as the mechanics of the line search,
specialized to the quadratic problem (1.1).
The convergence analysis given in [10] uses two important properties of the sequence
of search directions, "constraint-compatibility" and "consistency". "Constraint-
compatability" is needed to guarantee that a sufficiently long step is taken before the
first constraint is encountered. The usual descent condition, that g T
sufficiently
negative, is not enough in the context of a reflective algorithm because this condition
takes no account of the proximity of the constraints. "Consistency" is a more standard
notion capturing the idea that first-order descent, represented by the term g T
sisitent with first-order convergence. Following (1.7), define D 2
Definition 2. A sequence of vectors fw k g is constraint-compatible if the sequence
fD \Gamma2
is bounded.
Definition 3. A sequence of vectors fw k g satisfies the consistency condition
Central to our approach, both in terms of achieving quadratic convergence and the
satisfaction of constraint-compatibility and consistency, is the frequent use of a reduced
trust region problem to determine s
where S k is a subspace of R n , D k is a positive diagonal scaling matrix, and
The matrix M k is defined:
where J v is the Jacobian 7 of v, where v is defined in Figure 9. Matrix D g
v is a diagonal
matrix with component i defined D
is an "extended
gradient", extended to deal with possible degeneracy. In particular,
where - g is a small positive constant. Clearly if x is a nondegenerate point and - g is
sufficiently small then (which implies that x is
degenerate) then
The diagonal matrix D(x), used in (3.1), is defined by (1.7), i.e. 8 ,
This choice yields a well-defined trust region problem (3.1). To see this note that using
(3.4), (3.1) becomes
where
s;
and
is a diagonal matrix, D
M k is positive definite in a
neighbourhood of a nondegenerate point satisfying the second-order sufficiency condi-
tions. Moreover, unlike fM k g, f -
k g is bounded. Matrix -
M k is a featured performer in
our reflective Newton algorithm. A Newton step is defined when -
M k is positive definite:
A final remark on the choice of scaling matrix (3.4). If we assume that D has the
1is the only reasonable choice. To see this suppose
consider that
7 Matrix J v is a diagonal matrix with each diagonal component equal to zero or unity. For example,
if all the components of u and v are finite then J has a finite lower bound and an
infinite upper bound (or vice-versa) then strictly speaking v i is not differentiable at a point
define J v
at such a point. Note that v i is discontinuous at such a point but v i \Delta g i is continuous.
8 Notation: If z is a vector then jzj 1
2 denotes a vector with the i th component equal to jz
and the calculation of DMD involves division by jv(x)j 1\Gamma2p
which includes components which go to zero as x ! x . On the other hand, if approaches singularity as x ! x (consider v
We will specify subspace S k below; it is important to realize that the cardinality of
our implementation. Therefore, the cost of solving (3.1)
is negligible. Given S k , the subspace trust region problem (3.5) can be approached
in the following way. Let S k be defined by the t k independent columns of an n-by-t k
be an
orthonormalization of the columns of D \Gamma1
for some vector s Y k . Therefore problem (3.5) becomes
ks Y k
and set s
. The solution to (3.7) is of negligible cost once the matrices are
small (see Appendix).
Algorithm 5 in Figure 10 presents a second-order reflective path algorithm.
Algorithm 5
Choose
For
1.
2. Determine initial descent dir'n s k for q at
M k is positive definite
and k-s N
k . If -
k is not positive definite choose
choose subspace S k , and solve (3.1) to get s k .
3. Determine ff
k and x k
otherwise, perform an approximate piecewise line minimization of q(x k
with respect to ff, to determine ff k such that
(a) ff k is not a breakpoint
(b) ff k satisfies (2.9) and (2.10).
4. x
Fig. 10. A second-order reflective path algorithm
Note: If ff accepted by the line search but corresponds to a breakpoint, then
modify
where ~
ff k is not a breakpoint, ~
9 If A is a matrix then ! A ? denotes the space spanned by the columns of A.
It remains to be more precise about the determination of s k and S k and to fully
specify the line search. We begin with s k and S k . Algorithm 6 in Figure 11 describes
our procedure.
Algorithm 6 [Let positive constants.]
Case 0: -
M k is positive definite and k-s N
Case 1: -
M k is positive definite and k-s N
if kr(-s N
solve (3.1) to get s k .
else
set s
Case 2: -
k is not positive definite. Compute
w k is a unit vector
such that fw k g is constraint-compatible and
z
solve (3.1) to get s k .
else
solve (3.1) to get s k .
Fig. 11. Determination of the descent direction s k
Remark on Case 2: We determine an appropriate negative curvature direction in
the following way. If a (sparse) Cholesky factorization of -
k does not complete then
k is not positive definite and a unit direction of non-positive curvature, -
readily
available and easily computable (e.g., [17]). Algorithm 6 can make use of -
sufficient negative curvature is displayed by -
and fw
is constraint-compatible. A constraint-compatibility test is implemented
by introducing a large constant, - cp , and requiring,
If either condition (3.9) or condition (3.10) is not satisfied then -
must be rejected. In
this case we can turn to a Lanczos process [10] to get a unit vector -
w k such that both
and (3.10) are satisfied. It is interesting to note that in our extensive numerical
experimentation, with results reported in Section 4, conditions (3.9) and (3.10) were
always satisfied by the (partial) Cholesky factorization method - the backup Lanczos
procedure was never required.
The Line Search. We have designed a specialized approximate line search procedure
to efficiently exploit the structure of this problem and to guarantee the line search
conditions in Algorithm 5. Before describing the approximate procedure, we develop
an exact line search procedure - this is possible because the problem is to find a local
minimizer of a quadratic function along a piecewise linear path. In the end we do not
use the exact line search per se but rather we use a truncated version of it, subroutine
"improve", within an overall approximate strategy. But we begin with the exact search.
The Exact Line Search. We are initially concerned with the exact determination
of ff
k where ff
k is a local minimizer of q(x k (ff)). Note: It is convenient to describe
the exact line search in terms of the y-variables, i.e., y
R(y view we have a straight-line minimization of a
piecewise quadratic function -
Henceforth in this section we omit the major iteration subscript.
The function -
is a continuous piecewise quadratic function. The ray
can be divided into intervals from left to right, I
q(ff) is smooth on each interval. Denote the restriction
of -
q(ff) to the j th interval by q j (ff) - note that q j (ff) is a quadratic function of a single
variable.
Our exact line search algorithm visits the intervals I 1 ; I 2 ; :::; in a left-to-right fashion
in an attempt to locate the first local minimizer of -
Assume we have not located
a local minimizer on intervals I 1 ; I 2 ; :::; I j \Gamma1 and assume that (q j There are
two possibilities: either q j (ff) has a minimizer strictly within the interval I j or it does
not. If it does, i.e., ff j
. However, if
does not admit a minimizer within I j then we must consider the possibility that fi j
is a minimizer of -
(ff). This is now the case if (q j+1
is not in int(I j )
and (q j+1 process is repeated on interval I j+1 .
Algorithm 7 in Figure 12 presents a compact description of the exact line search
algorithm we have sketched above.
Step (1.2) in Algorithm 7 follows from the observation that if (ff k
then s is a direction of infinite descent for -
q, beginning at y and therefore, by Theorem
descent for (1.1).
Algorithm 7 [Exact line search along direction s beginning at point y]
(0) Determine the array of breakpoints BR according to (2.7). fi 0 / 0.
(1) For
, the minimizer of q k , if it exists; otherwise, set ff k
1.
(problem (1.1) is unbounded).
, exit.
exit.
is the index such that according to
Algorithm 2.
Fig. 12. The Exact Line Search Algorithm
Our final concern, with regard to the exact line search, is an efficient implementation
of step (1.1). In theory this computation is straightforward. Assume q k
a k
. If a k
unbounded below; if a k
then q k (ff) is unbounded below (unless a k
which case q k is constant). However,
the challenge is to determine a k
1 and a k
efficiently, for (Note that a k
0 is not
The key to efficiency here is the observation that the reflective transformation R
is linear on each interval I is constant within each interval. For
each interval I k , define oe k to be the vector of diagonal elements of J R evaluated at any
point in the interval (y it follows that
for It follows that
Therefore, in the terminology used above, a k
a k
A straightforward implementation for determining a k
breakpoint (a k
0 is not needed). However, there is considerable structure that can be
exploited; in particular, oe k+1 , a vector with each component equal to \Sigma1, differs from
oe k in exactly one component. We can exploit this to reduce the work in the line search
to O(n) per breakpoint.
Suppose we have determined that I k does not contain ff k
and we have at hand the
following quantities:
a k
a k
and x k , the value of x at the k th breakpoint. If j is the index such that
then
The vector w k can be updated as follows:
where H j is column j of H. Coefficient a k+1
2 is simply computed:
a k+1
Coefficient a k+1
1 can be efficiently computed by considering the following equalities:
a k+1
Finally, x k+1 is computed:
In summary, the coefficients a k+1
2 and the intermediate quantities x
can be computed, given a k
using (3.16),(3.17),(3.18) and (3.19). This amounts
to approximately 4n work. Of course the initial quantities, w
a 0must be computed
from scratch requiring O(n 2 ) work. Therefore, if k br denotes the number of breakpoints
crossed, the total cost of the exact line search is:
initialization of w
(ii) O(n) for determination of BR,
(iii) O(k br n) for steps (1.0) - (1.5).
The Approximate Line Search. The exact line search described above is not
practical, for two reasons. First, an exact minimizer along a line may correspond exactly
to a breakpoint, i.e., a boundary point, and the algorithm requires strictly feasible
points. This is actually not a serious problem since a small perturbation would yield
strict feasibility and the reflective Newton method is not very sensitive to boundary
proximity.
A more serious objection to the exact line search is economy: despite the efficient
implementation described in the previous section, the relative cost can be high if the
number of breakpoints crossed, k br , is large. Certainly if there are a large number of
tight variables at the solution, say something close to n, then the total cost of the exact
line search algorithm ultimately becomes O(n 2 ) per line search. This is unsatisfactory
and unnecessary since an economical approximate line search can be just as effective.
In this section we describe an approximate line search, henceforth refer to as subroutine
improve, which uses the exact line search, described above, in a limited fashion-
beginning at an approximate minimizer, subject to a bound, k u , on the number of
breakpoints permitted to cross. In particular, improve is used in a cleanup role: after
determining an initial approximate minimizer by a bisection strategy, improve is called
upon to apply the exact line search strategy. Thereby the approximate minimizer is
further improved at cost O(k u n), where k u is typically chosen to be small, e.g., k
(In improve we also impose an approximate upper bound ff max on the size of ff. That
is, the size of the improvement is bounded by ff
Subroutine improve has the following calling sequence:
A more precise description of the approximate line search algorithm is given in
Figure
13, Algorithm 8. The basic idea is as follows. First, if the direction s k is a
Newton direction s N
k then a unit step is attempted. If (2.9) is satisfied then the full
Newton step is accepted subject to further improvement by subroutine improve and
possible (slight) adjustment to avoid a breakpoint. Second, if s k does not corrspond to
a Newton direction or if it does but a unit step does not satisfy (2.9), then a bisection
procedure is executed on the interval (0; ff u ) where ff u - 1 is an upper bound on the
step size. A point is located satisfying both (2.9) and (2.10) and then possibly further
improved with subroutine improve.
Algorithm 8 [Approximate line search along direction s beginning at point y]
k and a unit step along s k satisfies (2.9)
ff k corresponds to a breakpoint, set ff
is not a breakpoint
else f s k 6= s N
k or a unit step along s N
k does not satisfy (2.9)g
ffl use bisection to find -
and (2.10) such that -
ff k is not
a breakpoint
set ff
ff k corresponds to a breakpoint, set ff
is not a breakpoint
Fig. 13. Approximate Line Search Algorithm
4. Numerical Experiments. We have implemented our algorithm in a version
of Matlab which allows for sparse matrix data structures [15], now Matlab 4:0. In this
section we present some preliminary numerical results.
With the exception of the results reported on Table 12, all experiments were performed
on Sun Sparc workstations in the Matlab environment [22]. Experiments reported
in Table 12 were performed in a heterogeneous environment involving an Intel
IPSC/860 32-node multiprocessor as the "backend", and a Sun Sparc workstation as the
"frontend". Matrix factorizations and solves were performed on the "backend", in C,
while the main Matlab program executed on the "frontend". Communication between
"frontend" and "frontend" over ethernet was implemented through the use of Matlab
"MEX" files. We used this environment to facilitate the solving of very large problems.
(Details on this heterogeneous environment are given in [3].)
Starting and Stopping: In all the experiments reported in this paper the starting value
of x, i.e., x 1 , is as follows. For component j where both upper and lower bounds are
finite, choose the midpoint, j. If both upper and lower bound corresponding
to component j are infinite in size, choose choose
(Note: The
reflective Newton approach is not particularly sensitive to starting value. For example,
we repeated many of the experiments reported here using a random (strictly) feasible
starting point - very little difference in behaviour was detected.)
Choosing a robust stopping rule in optimization is not easy. Our primary stopping
rule is based on the relative difference in function value. This is reasonable partly
because strict feasibility is always maintained, and partly because often the real objective
in practical optimization is to achieve a point of relatively low function value.
Specifically, are primary stopping rule is:
We choose in Matlab on
a Sun Sparc workstation, . We do have secondary stopping criteria
as well - designed to determine when progress is deemed too slow. This secondary rule
tends to kick in when solving degenerate or ill-conditioned problems and a very flat
region around the solution has been entered.
Parameter settings: There are a number of parameters in the algorithm: most are
either in the very large or very small category. Here are the settings we used in our
experiments:
Used in the determination of scaling matrix D, see (3.3): -
Used in the line search, see (2.9): oe l = :1
Used in the line search, see (2.10): oe
A bound on the number of breakpoints crossed in subroutine improve:
ffl ae: A lower bound on the stepsize, see Algorithm 8:
If the line search produces a unit step which turns out to be a breakpoint,
this point is perturbed by an amount bounded by - ff kD k g k k, see (3.8): -
Used to test for constraint-compatibility, see (3.10): -
Used in the negative curvature test, see (3.9): ffl
Used in Algorithm
Used in Algorithm 8:
An upper bound on the bisection process used in Algorithm 8: ff
4.1. Positive Definite Problems. We have generated a number of quadratic
test problems with certain properties. In the first set of results we concentrate on
the case where H is symmetric positive definite. In the results reported below we use
sparse matrices H with sparsity patterns representing 3-dimensional grid using a 7-
point difference scheme. The Mor'e/Toraldo [24] QP-generator was adapted to generate
problems with a given sparsity pattern (see also [6]). We will not review the generator
characteristics here: our generator is a straightforward adaptation of the Mor'e/Toraldo
scheme to the sparse setting. We use several sparse Matlab functions (e.g., "sprandsym",
"sprand").
In
Tables
1-3, the dimension of the test problems is in each case. The
parameter "pctbnd" indicates the percentage of variables tight at the solution - approximately
evenly divided between upper and lower bounds. Parameter "deg" reflects
the degree to which the solution is (nearly) degenerate - the larger the value of "deg",
the greater the amount of (near) degeneracy. Technical details of "deg" are discussed in
[6]. Parameter "cond" reflects the conditioning of the matrix H: the condition number
of H is approximately 10 cond .
The upper and lower bound vectors, u and l, were generated as follows. Approximately
75% of the components of l were chosen to be finite and assigned the value of zero
- the index assignment was made in a random fashion. Similarly, approximately 75%
of the components of u were chosen to be finite and assigned the value of unity. Again,
the index assignment was made in a random fashion, independent of the assignment of
l.
Each row of Tables 1-3 reflects the results of 10 independent runs with the same
parameter settings. The third column, labelled "max", indicates the maximum number
of iterations required, over the set of 10 independent runs, to achieve the stopping
criteria; the fourth column, labelled "avg" records the average number of iterations
required to reach the stopping criteria over the 10 problems; the last column, labelled
"acc", records the number of digits of accuracy achieved in the function value (the true
solution is known).
Table
Positive
deg cond max avg acc
9 6 15 15 15
9 9
Observations on Tables 1-3: First, we observe the remarkable consistency of our reflective
Newton method on these problems. In terms of iterations required to achieve the
stopping criteria and accuracy attained in the function value, there is apparently very
little sensitivity to degeneracy, conditioning, or number of variables tight at the solution.
Of course we do not claim that accuracy in x is independent of condition/degeneracy
it surely is not. However, it is usually acceptable in optimization to attain a point
with nearly optimal function value and we have been quite successful in that (on this
test collection).
Second, the absolute number of iterations required to obtain a very accurate solution
(in terms of the function value q) is modest in every case, i.e., less than 20.
Positive
deg cond max avg acc
9 6 17 17 15
6 9 17 17 15
9 9 17 16.3 15
Table
Positive
deg cond max avg acc
9 3 17 16.3 15
9 6
Positive Definite Problems: Timing Breakdown
1000
This is very encouraging considering the dimension of the problems (n = 1000) and the
spectrum of problem characteristics being considered.
It is important to know where the algorithm spends its time. To this end we generated
larger problems, with the same structure, and we have broken down the timing
information. In Table 4 we consider a representative positive definite problem with "av-
erage characteristics", i.e., vary the problem
dimension n. (The sparsity structure remains the same.) The second column, labelled
"it", records the number of iterations required to achieve the stopping criteria; "totM"
records the total number of flops used by the (partial) Cholesky factorization ("m"
represents a million); "totls" records the number of flops used in the approximate line
search algorithm. Over 95% of the total flop count on these problems is represented by
the sum of the "totM" and "totls" columns - the remaining work in the algorithm, such
as the 2-dimensional trust region solution, is negligible in comparison (see Appendix
for more detail on the solution of trust region problems).
Observations on Table 4: First, there is no significant growth in number of iterations
as the problem dimension n increases. High accuracy is maintained for larger values
of n as well. As n increases the sparse matrix factorization work, "totM", increases
relative to the lines search cost, "totls". Therefore, speedup of the (partial) sparse
factorization aspect of the algorithm (e.g., use of parallelism, exploitation of
specific particular structure) will have significant impact on the overall computing time.
Conversely, improving the approximate line search (in terms of cost) is not a crucial
computing issue, at this point, for large-scale problems.
In addition to these randomly generated, but structured, positive definite problems,
we have experimented with three specific test cases. Two of these problems are from
the literature (e.g., [12, 24]) and the third example is new. In Tables 5 and 6 we report
on the "obstacle" problem - in the first case there are lower bounds only, in the second
case there are lower and upper bounds. In defining the specific example used we have
chosen the same parameter settings and specific functions used in [24]. Table 7 reports
on the elastic-plastic torsion problem. Again we used the same parameters as reported
in [24] to define the problem.
In
Table
8 we report on a linear spline approximation problem. This type of problem
arises, for example, in a particle method approach to turbulent combustion simulation
[28]. The problem results in a large sparse least-squares problem subject to
nonnegativity constraints on the variables. To set up a sample problem we assume
an m-by-m-by-m 3-dimensional grid. Within each cell are a set of particles randomly
located (we use approximately 10 particles per cell in our experiments). Each particle
p has a known function value, OE(p). Associate with each grid intersection point a linear
basis function and determine the best set of coefficients, x, for the basis functions, in
the least-squares sense, subject to nonnegativity constraints on x. The function OE we
used in our experiments is defined: given a point in 3-space,
Table
Obstacle Problem: Lower Bounds Only
its norm
50 2500 14 13
100 10,000 15 12
Table
Obstacle Problem: Lower and Upper Bounds
its norm
50 2500 14 12
100
Table
Elastic-plastic Torsion Problem
its norm
100 10,000 11 12
Observations on Tables 5-8. The most noteworthy observation is the apparent
insensitivity of our method to problem size for each of these problems. The number of
Linear Spline Approximation
its norm
22 10,648 17 11
iterations does not grow, for a given problem class, as the dimension of the problem
increases. For example, for the linear spline problem, 16 iterations are required when
iterations are required when Moreover, the number of
iterations is always modest, on this test set, i.e., less than 20. High accuracy is achieved
in all cases.
4.2. Indefinite Problems. We have adapted the Mor'e/Toraldo QP generation
scheme, in combination with sparse matrix functions in Matlab 4.0, to generate large
sparse indefinite matrices with a given sparsity pattern and given approximate set of
approximate eigenvalues. In the indefinite case we chose finite upper and lower bound
vectors, 1. (This is to avoid the generation of unbounded problems.)
In each of the problems in Tables 9-12 roughly 10% of the eigenvalues of H are neg-
ative. The column labels are the same as before however here "acc" does not represent
the number of accurate digits compared with the true solution since the true solution
is unknown due to indefiniteness of H. Instead, "acc" records the number of matching
digits in the objective function q in the last 2 iterations. (In each case the optimality
conditions were verified to hold at the final point.)
Table
Indefinite problems.
deg cond max avg acc
9 3 23 19.3 15
9 6 26 21.7 15
9 9
Observations on Tables 9-11: Iteration counts indicate that our method is not quite
as consistent or efficient on indefinite problems compared to the performance on positive
definite problems. Still, the overall efficiency seems very good - the average number of
Indefinite Problems.
deg cond max avg acc
9 3
9 6 19 17.7 15
3 9 14 11.3 15
9 9 25 18.3 15
Table
Indefinite problems.
deg cond max avg acc
9 3
9 6
6 9 15 13.
iterations required for any problem category is always less than 23.
In
Table
12 we indicate where the algorithm spends its time on indefinite problems
by considering a representative example and increasing the dimension.
Table
Indefinite Problems: Timing Breakdown
Remark on Table 12: We see no apparent growth in required iterations as n increases.
Clearly the "totM" column dominates the "totls" column as n increases. Recall that
"totM" represents the matrix factorization flop count while "totls" represents the number
of total flops required by the line search procedure. Therefore, to obtain further
improvements in efficiency for this type of approach it is best to focus on the matrix
factorization aspect of the overall procedure.
5. Theory and Conclusions. The numerical results obtained to date strongly
support the notion that a reflective Newton method represents an efficient way to accurately
locate local minimizers of large-scale (indefinite) quadratic functions subject to
bounds on some of the variables. The theory is supporting also: our reflective Newton
method is globally and quadratically convergent. Coleman and Li [10] present important
theoretical properties of reflective Newton methods for general nonlinear functions,
subject to bounds on some of the variables. The method in this paper is a specialization
of the general method to the quadratic case. Therefore, the general theory applies.
We make a compactness assumption before formally stating the main result.
Compactness Assumption: Given initial point x 1 2 F , it is assumed that the level
set
Theorem 2. Let fx k g be generated by Algorithm 5 with fs k g generated by Algorithm
6 and with fff k g determined by the approximate line search algorithm (Algorithm
7). If -
ffl Every limit point of fx k g is a first-order point.
ae M is the maximum spectral radius of -
M (x) on
)g. Since ae( -
is continuous on L, a compact set, the upper bound ae M exists. Recall that - 3
is a constant used in
Algorithm 6.
ffl Every nondegenerate limit point satisfies the second-order necessary conditions.
ffl If a nondegenerate limit point x satisfies second-order sufficiency conditions
is sufficiently small, fx k g is convergent to x ; the convergence
rate is quadratic, i.e.,
Proof. This algorithm is in the class of algorithm described in [10] and all the
assumptions of Theorem 20 in [10] are satisfied. The result follows from Theorem 20 in
[10].
In conclusion, strong theoretical and computational results indicate that a reflective
Newton method is an efficient and reliable way to solve problem (1.1) to high accuracy.
The computational results reported in this paper support this claim.
6.
Acknowledgement
. We thank our colleagues Jianguo Liu and Danny Ralph
for many helpful discussions on this work. Danny Ralph drew our attention to the
topology reference [25].
7.
Appendix
: The trust region problem. The trust region problem is
where A is a real symmetric matrix and k \Delta k denotes the 2-norm. The purpose of this
section is to review the nature of problem (7.1) and discuss a possible solution suitable
for low-dimensional problems. (In the context of our reflective Newton method for
problem (1.1), A is matrix -
M(x), a symmetric matix of order 2. The computational cost
of the procedure we describe to solve (7.1) is negligible compared to the other required
computations in the reflective Newton algorithm we propose.) For larger problems a
more approximate procedure is usually preferred, e.g., [14, 23, 30, 31]. Much of the
material in this section can be found elsewhere, e.g., [2, 4, 11, 14, 23, 29, 30, 31].
Diagonalization. We begin with an extremely useful characterization of the global
solution to (7.1).
Theorem 3. Vector s solves (7.1) if and only if there exists a scalar - 0 such
that
(a)
positive semidefinite;
(c) ksk - \Delta;
Proof. A proof is given, for example, in [30].
The usefulness of this result is best revealed after diagonalization. Suppose
where the columns of V are the orthonormal eigenvectors of A and
Obviously then so (a) is equivalent to
s:
By (b), - \Gamma- 1 , and so all vectors s satisfying (a, b) are of the form
The vector fi is arbitrary with respect to (a,b) but can sometimes help with respect
to satisfying (c,d). A basis for an algorithmic approach to this problem is to assume
the form given in (7.4) and strive to satisfy (c,d) by choosing -, and in some cases
fi, appropriately. (fi plays a role only if - is the value of - at the
solution.)
The situation where (c,d) can be satisfied with easily
dispensed with (first half of Case 1 below). Therefore the primary task, assuming form
(7.4), is to determine -, and sometimes fi, to satisfy
We divide our approach into three possibilities. g.
Case 1: In this case either the Newton step is within the sphere ksk 2 - \Delta or it
is not. If kA then the optimal solution s
Case 2:
Figure
Obviously ks(-)k !1 as - ! \Gamma-
1 , and
1. Moreover, ks(-)k is convex; therefore, ks(-)k intersects \Delta in
exactly one place for - ? \Gamma- 1 .
Case 3:
ks 1 (\Gamma- 1 )k
is finite. Consider figures 3,4. There are now two possibilities. If ks 1 (\Gamma- 1
then there is a solution to (7.5) to the right of \Gamma- 1 ,
chosen to ensure (7.5), and - . Note that if jIj ? 1
then the null space component of s may not be unique.
The Reciprocal Secular Equation. In theory we can build an algorithm on the
remarks given above. However it is better, numerically, to replace condition (7.5) with
\Gammaks(-)k
0:
Equation (7.6) is more linear in shape than equation (7.5); therefore, equation (7.6) is
more amenable to solution via Newton's method.
Considering the definition of rsec(-),
-n+-
it is easy to verify the following:
1. rsec(-) is convex on (\Gamma- 1 ; 1] and lim -!1
2. lim -!\Gamma- +rsec(-) is finite.
3. If ff i 6= 0, for some i 2 I, then rsec(\Gamma-
. Obviously
a single zero of rsec exits to the right of \Gamma- 1 in this case.
4. If 8i 2 I; ff
Algorithmic and Numerical Concerns. We assume that a solution to (7.1) is
sought and we are willing and able to compute full eigenvalue information,
(If this is not the case, perhaps due to the cost, then it is possible to approximately
solve (7.6) using an iterative scheme involving the Cholesky factorization of A
[14,
The method using appears to be straightforward.
Case 1: then the Newton step, \GammaA \Gamma1 g, is the solution with
then we can determine the zero of rsec(- ? 0.
Case 2:
and rsec admits
a single solution to the right of \Gamma- 1 .
Case 3:
If ks 1 (\Gamma- 1 )k - \Delta then there is a zero of rsec(-) to the right of \Gamma- 1 with
Otherwise, a solution to (7.1) is given by (7.4),
ks
Unfortunately, the situation is not quite so clean from a numerical point of view:
there is fuzziness between the second and third cases. In particular, if ff 1 is small the
equation very ill-conditioned for - near \Gamma- 1 and it can be quite difficult
(nigh impossible) to compute - such that rsec(-) is small. This extreme ill-conditioning
is due to the following "disagreement".
Assume I = f1g, for simplicity, and note that, if ff
On the other hand, if
which is not, in
general, the limiting value of (7.8). Therefore nearby problems (ff
can yield very different solutions to (7.6), and this is the cause of the ill-conditioning
of (7.6). Our solution to this ill-conditioning problem (trust.m) is to first attempt to
find a solution to However, if jrsec( -)j is not small, where -
- is
the computed "zero" returned by the zero-finder, then we set -
solution to (7.1) via (7.4).
This strategy works because the solution to (7.6) with ff i small for i 2 I is close to
a solution of (7.1) with the corresponding ff i at zero. To see this, initially assume that
Now first consider the case where ff the solution to (7.1) is:
where
otherwise, the result
is obvious). Define s
the solution to (7.1) can be written
Next consider ff small number. We can write the solution to (7.1) as
But as
which implies
Therefore, the solution to (7.1) with ff is near to the solution to (7.1) with ff
In general, if jIj ? 1 a solution to (7.1), with some components ff i near to zero, is
near to a solution of (7.1) with those components set to zero. In this case, where several
coefficients ff i equal to zero, i 2 I, problem (7.1) does not enjoy a unique solution [see
(7.4)] but the range space component is unique.
--R
Approximate solution of the trust region problem by minimization over two-dimensional subspaces
Advanced Computing Research Institute
Computing a trust region step for a penalty function
A direct active set algorithm for large sparse quadratic programs with simple bounds
A quadratically-convergent algorithm for the linear programming problem with lower and upper bounds
Global convergence of a class of trust region algorithms for optimization with simple bounds
On the minimization of quadratic functions subject to box constraints
Minimization of a quadratic function of many variables subject only to lower and upper bounds
Computing optimal locally constrained steps
Sparse matrices in matlab: Design and implemen- tation
Minimization subject to bounds on the variables
Curvilinear path steplength algorithms for minimization algorithms which use directions of negative curvature
A globally convergent method for l p problems
Solving the minimal least squares problem subject to bounds on the variables
A generalized conjugate gradient algorithm for solving a class of quadratic programming problems
Application of the velocity-dissipation PDF model to inhomogeneous turbulent flows
A family of trust-region-based algorithms for unconstrained minimization with strong global convergence properties
Trust region methods for unconstrained optimization
The conjugate gradient methods and trust regions in large scale optimization
A class of methods for solving large convex quadratic programs subject to box constraints
--TR
--CTR
Kamin Whitehouse , David Culler, Calibration as parameter estimation in sensor networks, Proceedings of the 1st ACM international workshop on Wireless sensor networks and applications, September 28-28, 2002, Atlanta, Georgia, USA
Kamin Whitehouse , David Culler, Macro-calibration in sensor/actuator networks, Mobile Networks and Applications, v.8 n.4, p.463-472, August
D. C. Jamrog , R. A. Tapia , Y. Zhang, Comparison of two sets of first-order conditions as bases of interior-point Newton methods for optimization with simple bounds, Journal of Optimization Theory and Applications, v.113 n.1, p.21-40, April 2002
Keiji Yanai , Nikhil V. Shirahatti , Prasad Gabbur , Kobus Barnard, Evaluation strategies for image understanding and retrieval, Proceedings of the 7th ACM SIGMM international workshop on Multimedia information retrieval, November 10-11, 2005, Hilton, Singapore
V. G. Domrachev , O. M. Poleshuk, A Regression Model for Fuzzy Initial Data, Automation and Remote Control, v.64 n.11, p.1715-1723, November | interior-point method;interior Newton method;quadratic programming |
589192 | Convergence Properties of Minimization Algorithms for Convex Constraints Using a Structured Trust Region. | In this paper, we present a class of trust region algorithms for minimization problems within convex feasible regions in which the structure of the problem is explicitly used in the definition of the trust region. This development is intended to reflect the possibility that some parts of the problem may be more accurately modelled than others, a common occurrence in large-scale nonlinear applications. After describing the structured trust region mechanism, we prove global convergence for all algorithms in our class. | Introduction
Trust region algorithms have enjoyed a long and successful history as tools for the solution of non-
linear, nonconvex, optimization problems. They have been studied and applied to unconstrained
problems (see [7], [17], [25], [28], [29], [30], [31], [34], [35], [38]) and to problems involving various
classes of constraints, including simple bounds ([6], [10], [11], [27], [32]), convex constraints ([2],
[3], [14], [41]), and nonconvex ones ([5], [8], [16], [36], [44]). This long lasting interest is probably
justified by the attractive combination of a solid convergence theory, a noted algorithmic
robustness, the existence of numerically efficient implementations and an intuitively appealing
motivation. The main idea behind trust region algorithms is that, if a nonlinear function (ob-
jective and/or constraints) is expensive to compute or difficult to handle explicitly, it should be
replaced by a suitable model. This model is deemed to be trustworthy within a certain trust
region around the current point. The trust region is defined by its shape and its radius. The
minimization involving the difficult nonlinear function(s) is then replaced by a sequence of minimizations
of the simpler model(s) within appropriate trust regions. The trust region radii are
adjusted to reflect the agreement between the model and true functions as the process proceeds.
It is remarkable that, up to now, all algorithms that we are aware of use a single trust
region radius to measure the degree of trustworthiness of the models employed, even if several
This research was supported in part by the Advanced Research Projects Agency of the Department of Defense
and was monitored by the Air Force Office of Scientific Research under Contract No F49620-91-C-0079. The United
States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding
any copyright notation hereon.
This work was also supported by the Belgian national Fund for Scientific Research.
different functions are involved. This choice is somewhat surprising if one admits that some of
the modelled functions could be substantially "better behaved" than others in the same problem,
as this implies that the region in which their models can be trusted might also be substantially
larger. In this context, the unstructured trust region choice might be viewed as a conservative
strategy ensuring that all models may be trusted in what amounts to a "safe minimal" region.
While this strategy might be reasonable for small problems, where each involved function depends
on all the problem's variables, it is clearly questionable for large-scale applications, where each
of the problem's function typically depends only on a small number of variables. For instance,
one might consider the minimization of an unconstrained objective function consisting of the sum
of many quadratic and a few highly nonlinear terms, the latter involving a small subset of the
variables. If a classical unstructured trust region algorithm, with a quadratic model, is used, the
quadratic terms are perfectly modelled, but the steps that one can make at each iteration are
(unnecessarily) limited by the highly nonlinear behaviour of a small subset of the variables.
It is the purpose of this paper to present and analyze a class of algorithms that use the
problem's structure in the definition of the trust region, allowing large steps in directions in
which the model has proved to be adequate while restricting the movement in directions where the
model seems unreliable. To be more precise, we will consider the problem of minimizing a partially
separable objective function subject to convex constraints; we will then use the decomposition of
the objective function into element functions as the basis for our structured trust region definition.
The choice of the partially separable structure, a concept introduced in [21], is motivated by the
very general geometric nature of this structure and by the increasing recognition of its practical
use (see [4], [9], [12], [13], [18], [19], [20], [22], [26], [39], [42], [43], amongst others). More
significantly, partial separability provides a decomposition of the considered nonlinear function
into a linear combination of smaller element functions, each of which may then be modelled
separately (see [40]). It is then quite natural to assign one trust region radius per element
function and to decide on its increase or decrease separately. Because different element functions
typically involve different sets of variables, each element trust region only restricts the components
of the step corresponding to its elemental variables.
An obvious approach is to use the norm-scaling matrices allowed in the theory for unstructured
trust region methods ([10], for instance) to account for differences in model adequacy among
elements when constructing the trust region. This would be satisfactory if the existing theory did
not require that the scaling matrices be of uniformly bounded condition number. Unfortunately,
it is easy to conceive of instances where this is a severe handicap. For example, it would prevent
the trust region radius of a well-modelled (perhaps linear or quadratic) element from increasing to
infinity while at the same time ensuring that that of a badly behaved nonlinear element function
remains of modest size. Moreover, this strategy may well cause numerical difficulties when
attempting to solve the trust region problem. In fact, as we will shortly see, additional algorithmic
safeguards are important when simultaneously handling trust regions of vastly different sizes.
Thus, we do not consider such an approach further in this paper.
Section 2 of the paper presents the problem in more detail and the new class of algorithms
using the principle of structured trust regions. Global convergence for all algorithms in the class
is proved in Section 3. We briefly discuss the identification of active constraints in Section 4. We
examine in Section 5 some extensions of the results of the previous sections. We finally give some
comments and perspectives in Section 6.
Structured trust region for partially separable problems
2.1 A structured model of the objective and the corresponding structured
trust region
2.1.1 The problem
The problem we consider is that of minimizing a smooth objective function subject to convex
constraints. That is, we wish to solve the problem
minimize
where X is a closed convex subset of R n . We denote the Euclidean inner product on R n by h\Delta; \Deltai,
and the associated ' 2 -norm by k \Delta k. Given Y a closed convex subset of R n , we define the operator
Y (\Delta) to be the orthogonal projection onto Y . We now list our additional assumptions on (2.1).
AS.1 X has a non-empty interior.
AS.2 f is bounded below on X .
AS.3 f is partially separable, which means that
and that, for each i 2 there exists a subspace N i 6= f0g such that, for all w 2 N i
and all x 2 X ,
AS.4 For each continuously differentiable in an open set containing X and
its gradient is uniformly bounded on X .
Note that we admit the case where X is unbounded or even identical to R n itself, in which
case we obtain an unconstrained problem. In relation to the partial separability of the objective
function, we also consider the range subspace (see [23]) associated with each element function f i ,
which is defined as
We are mostly interested in the case where the dimension of each R i is small compared to n.
A commonly occurring case is when each element function f i only depends on a small subset
of the problem's variables: R i is then the subspace spanned by the vectors of the canonical
basis corresponding to the variables that occur in f i (the elemental variables). The range of the
projection operator PR i (\Delta) is therefore of low dimensionality. The reader is referred to [12] for a
more detailed introduction to partially separable functions.
We note that f is invariant for any translation in the subspace (
We may therefore
restrict our attention to the case where
without loss of generality.
2.1.2 The element models
The algorithm we have in mind is iterative and generates feasible iterates (in the sense that all
iterates belong to X). At iteration k, we will associate a model m i;k with each element function
f i . This model, defined on R i in a neighbourhood of the projection of the k-th iterate x k on this
subspace, is meant to approximate f i for all x in the element trust region
where is the i-th trust region radius at iteration k and the norm k \Delta k is chosen to be the
usual Euclidean norm in order to simplify the exposition. In what follows, we will slightly abuse
notation by writing m i;k (x) for an x 2 R n , instead of the more complete m i;k (PR i
(x)). We will
furthermore assume that each model m i;k (i differentiable and has
Lipschitz continuous first derivatives on an open set containing B i;k , and that
Moreover, we assume that g i;k
in the sense that,
for all
where e i;k
is a constant and where \Delta min;k is defined by
i2f1;:::;pg
Condition (2.8) is quite weak, as it merely requires that the first order information be reasonably
accurate whenever some trust region radius is small (i. e. the corresponding model fits
badly). Indeed, one expects the coherency of this first order behaviour to be of crucial importance
in such cases. Further arguments supporting a choice similar to (2.8) for problems with convex
constraints are presented in [14].
Amongst the most commonly used element models, linear or quadratic approximations are
pre-eminent. One can, for instance, consider the quadratic model given by the first three terms
of the element function Taylor series around the current iterate. Another popular choice is a
quadratic model where the second derivative matrix is recurred using quasi-Newton formulae.
2.1.3 The overall model and trust region
With all the element models at hand, we are now in position to define the overall model at
iteration k, denoted m k , whose purpose is to approximate the overall objective function f in a
neighbourhood of the current iterate x k . From (2.2), it is natural to use the overall model
for all x in the overall trust region defined by
i2f1;:::;pg
Indeed B k is the intersection of all element trust regions, that is the region in which all element
models may be trusted, irrespective of the additional limitation possibly imposed by the feasible
set X .
Of course, the actual shape of the trust region B k is determined by the choice of the Euclidean
norm: it corresponds to the intersection of cylinders whose axis are aligned with the subspaces N i
and whose radii reflect the quality of the element models: large in subspaces where the element
models predict the element function correctly and smaller in subspaces where the prediction is
poorer. In practice, one might wish to choose other norms, such as the ' 1 -norm. In this case,
and assuming that the subspaces R i are spanned by subsets of the canonical basis vectors, the
shape of the trust region is that of a box, the length of whose sides again reflects the quality of the
element models. The extension of the theory to more general norms is considered in Section 5.4.
2.1.4 Curvature
We now follow [14] and [41] and define the generalized Rayleigh quotient of f at x along s 6= 0 by
Obviously, this definition is valid only if s is such that x belongs to the domain of definition
of f . Note that, by convention,
If we assume that f is twice continuously differentiable, the mean-value theorem (see [24]) implies
that
Z 1Z 1t
dv dt: (2:14)
Furthermore, if f is quadratic, then one easily verifies that !(f; x; s) is independent of x and is
equal to the Rayleigh quotient of the matrix r 2 f in the direction s. We note that, because of
is bounded by some constant L i - 0 (see [24]). Hence we obtain that
i2f1;:::;pg
for all pg. The quantity that we need in our algorithm statement
and analysis is a monotonically increasing upper bound on the magnitude of the generalized
Rayleigh quotient !(m i;k defined by
q2f0;:::;kg
i2f1;:::;pg
where s i;k
the actual trial step computed by the algorithm, as defined below.
The quantity !(m i;k measures the curvature of the model m i;k in the direction of the trial
step s k . If quadratic models m i;k are considered, an upper bound on fi k is given by the largest
singular value of all Hessian matrices, plus one. We will assume that our choice of models is such
that this curvature does not increase too fast, which could lead to premature convergence of the
algorithm to a non-critical point (see [41]). More precisely, we make the following assumption,
as in [14], [10], [35] and [41].
This condition is weaker than the common assumption that the model's second derivative
matrices are uniformly bounded [32], which holds, for instance, for the classical Newton's method,
where quadratic models using analytical second derivatives are used on a compact domain. It is
also weaker than the condition
for some constant c 0 ? 0, which holds in the case where quadratic element models are used and
updated using either the BFGS or the safeguarded Symmetric Rank One quasi-Newton formulae.
2.1.5 Criticality
Before we can describe our algorithm in detail, we also need a criticality criterion for our problem.
A critical point of our problem is a feasible point x where the negative gradient of the objective
function \Gammarf (x) belongs to the normal cone of X at x 2 X , which is defined by
fy
The associated tangent cone of X at x 2 X is the polar of N (x), that is
Thus every measure of criticality has to depend on the (differentiable) objective f and on the
geometry of the feasible set at the current point. We will use the symbol ff(x; f; X) to denote
such a criticality measure.
AS.6 The criticality measure ff(x; h; X) is non-negative for all x 2 X and all functions h differentiable
in an open neighbourhood of x. Moreover ff(x; h; only if x is critical
for the problem
minimize x2X h(x): (2:21)
But, within the algorithm, only approximate gradient vectors might be available, namely the
vectors g k and g i;k , the gradients of the models. It is therefore natural to use
the criticality measure for the problem
as an "approximate" criticality measure for (2.1). Note that ff k ? 0 implies that g k 6= 0.
In unconstrained optimization, one typically chooses
the obvious criticality measure (see [31] or [34]). When bound constraints are present, the choice
is made in [10]. For the infinite dimensional case, the definition
is used in [41]. For the case where convex constraints are considered,
is chosen in [32], where t C
is the line coordinate of the so-called "generalized Cauchy point"
to be discussed below. In a similar context,
is used in [14].
2.2 Ensuring sufficient model decrease
2.2.1 An overview of the classical sufficient decrease condition
A key to trust region algorithms is to choose a step s k at iteration k that is guaranteed to provide
a sufficient decrease on the overall objective function model m k . In other words, a step such that
is sufficiently positive, given the value of a suitable criticality measure ff k satisfying AS.6. This
concept of "sufficient decrease" is usually made more formal by introducing the notion of the
(generalized) Cauchy point. This remarkable point, denoted x C
k , is typically computed by trust
region algorithms as a point on (or close to) the projected gradient path PX
that is also within the trust region and sufficiently reduces the overall model in the sense that
is a constant and ff k a criticality measure satisfying AS.6. However, such a point
may not exist when the trust region radius \Delta k is small compared with ff 2
. In this case, the
generalized Cauchy point is chosen as (or close to) the intersection of the projected gradient path
with the boundary of the trust region, yielding an inequality of the form
A point on the projected gradient path satisfying (2.30) may also fail to exist because the projected
gradient path itself ends on the boundary of X , well inside the trust region. In that case, this
end point (or another feasible point close to it) is typically chosen as generalized Cauchy point,
and it is then typically shown that
One then ensures the "sufficient decrease" by requiring that the chosen step s k produces at least
a fixed fraction of the overall model reduction achieved by the generalized Cauchy point, which
is to say that
ae ff k
oe
Many variants on the above scheme exist in the literature for the unstructured trust region
case. All of these variants ensure that a suitable step is found after a finite number of trials. The
best known is for unconstrained problems when the ' 2 -norm is used to define the trust region
shape. In that case, the projected gradient path is simply given by all negative multiples of
the gradient g k and the Cauchy point is simply the point that minimizes the model m k in the
intersection of the steepest descent direction and the trust region (see, for instance, [34] and [37]).
When other norms are used, for example the ' 1 -norm, one can then choose either to minimize
the model in the intersection of this steepest descent direction and the trust region, as before
(see [10]), or to "bend" the projected gradient path onto the boundary of the trust region and to
choose the generalized Cauchy point as a point which satisfies classical Goldstein-type linesearch
conditions along that path while staying within the trust region (see [33] and [41]). Both these
latter strategies are used in the LANCELOT software [13]. When additional convex constraints
are present, the projected gradient path is additionally "bent" to follow the boundary of the
feasible domain. Thus the philosophy is the same, in that (2.33) is guaranteed in the above cases.
Indeed satisfaction of this condition has been derived for each of the choices (2.24)-(2.28) for ff k
in the papers where they were respectively introduced.
2.2.2 Sufficient decrease for structured model and trust region
We will use a similar approach in our structured model and trust region framework to determine
what is a sufficient decrease of the overall model m k within the region B k , whose shape is chosen
to reflect the structure of the problem. Special care is needed because this region might be very
"asymmetric" in the sense that it may allow very large steps in some directions and only very
short ones in others. As a consequence, we have to adapt the notion of trust region "radius" to
our context and adequately reformulate condition (2.33).
From a practical point of view, one might use a two-stage approach. In this, one first aims
to find a step producing a sufficient model decrease in a smaller, but more symmetric, region.
Following this, one then allows the step to increase within the trust region while maintaining
control over the model decrease.
To be specific, let
be the trust region whose radius is determined by the possibly most nonlinear part of the model.
Applying the results discussed in the previous section after condition (2.33), one may deduce that
it is possible to find, in a finite number of trials, a step s min;k such that x k
and
ae ff k
oe
for some suitably chosen criticality measure ff k satisfying AS.6 and some constant -
However, the restriction that the length of s min;k is bounded by \Delta min;k makes the whole
exercise of shaping B k to reflect the problem's structure entirely irrelevant. One might therefore
be prepared to accept a larger step provided it remains feasible, within the trust region B k , and
produces a further significant model decrease. More specifically, we allow our algorithm to choose
any step s k such that x k which guarantees that
ae ff k
ks k k]; 1
oe
Note that, since (2.36) holds for s this condition can therefore be achieved in
practice after a finite number of trials. Observe also that (2.36) is fundamentally different from
an angle test of the form
as (2.36) does not prevent s k from being orthogonal to the steepest descent direction, so long
as a sufficient model reduction is obtained. This is useful because such a step may occur when
moving away from a saddle point of the objective function. Finally note that, as expected, (2.36)
reduces to (2.33) in the case where only one trust region is considered.
2.3 A class of structured trust region algorithms
We now describe the class of algorithms that we consider for solving (2.1). Besides - 1 used in
used in (2.36), it depends on the constants
and
In addition to the above conditions, we also require a compatibility condition between the j i 's
and the - i 's. Specifically, we request that
Typical values for these constants are -
Algorithm
step 0: initialization.
The starting point x together with the element function values ff i
and the initial trust region radii
step 1: model choice.
For choose the model m i;k of the element function f i in the trust region B i;k
centered at x k (as defined in (2.6)), satisfying (2.7) and (2.8).
step 2: determination of the step.
Choose a step s k such that the sufficient decrease condition (2.36) holds and
step 3: measure overall model fit.
If
then
else
step 4: update the element trust region radii.
Denote the achieved changes in the element functions and their models by
ffif i;k
and
respectively. Then define the set of negligible elements at iteration k as
and the set of meaningful elements as its complement, that is
Then, for each i 2 perform the following.
Case 1:
ffl If
and (2.43) both hold, then choose
ffl If (2.50) holds but (2.43) fails then choose
ffl If (2.50) fails, but
holds, then choose
ffl If (2.53) fails, then choose
Case 2:
ffl If
and (2.43) both hold, then choose
ffl If (2.56) holds but (2.43) fails, then choose
ffl If (2.56) fails, then choose
Increment k by one and return to step 1.
End of Algorithm
As is traditional in trust region algorithms, we will call an iteration successful if the test
(2.43) is satisfied, that is when the achieved objective reduction ffif k is large enough compared
to the reduction predicted by the overall model. If (2.43) fails, the iteration is said to be
unsuccessful. In what follows, we will denote by S the set of all successful iterations.
We now comment on various aspects of the algorithm.
1. The algorithm is constructed in such a way that a successful step is always possible, for
sufficiently small trust region radii, if the current iterate x k is not critical. This result is
formally proved in Corollary 8.
2. The choice of the element models m i;k is left rather open in the above description. It clearly
needs to be made precise for any practical implementation of the algorithm. One common
choice would be to set
where H i;k is a symmetric approximation to r 2 f i nullspace contains the subspace
In particular, Newton's method corresponds to the choice g
which is guaranteed to satisfy this latter condition. Another possible choice is
which may be attractive for the simpler element functions. In
this case, the model's fit to the true function is always good for the i-th element, and the
algorithm guarantees that the \Delta i;k form a non-decreasing sequence.
3. If the model change for an element is negligible, that is small compared to the overall
predicted change, we do not need to restrict its element trust region size unless the true
element change is relatively large compared with the same overall predicted change. We
can therefore afford to ignore negligible items until they stop being relatively negligible,
something which is inevitable when convergence occurs. Hence our distinction between
"negligible" elements (in N k ) and "meaningful" ones (in M k ).
Condition (2.41) can be viewed in this context as a guarantee that a new iterate will be
accepted in (2.43) whenever the model reduction obtained for all meaningful elements is
also acceptable (i.e. (2.53) holds for all irrespective of the contribution of the
negligible ones. This interpretation is clarified in Lemma 2.
4. The apparent intricacy of (2.50) and (2.53) is caused by two complications which arise in
the context of multiple elements. The first is that, although (2.36) ensures that
always positive, we may not assume in general that the same is true for ffim i;k . The second
is that possible cancellation between elements makes it necessary to consider the "accuracy
of model fit" for an element to be relative to the overall model fit. Indeed, requiring small
relative errors for models with very large values may result in large absolute errors. If
these large errors will then cause to be a poor prediction of ffif k and the
iteration might be unsuccessful. This explains why the perhaps more intuitive tests
cannot be used instead of (2.53) (j = 2) and (2.50) (j = 3).
Observe also that conditions (2.50) and (2.53) reduce to the familiar
when
5. Note again the consistency between the trust region radii updates in step 4 and the case
1. In this latter case, the set N k is always empty and (2.50) then implies (2.43),
because of (2.39). Equation (2.52) is thus never invoked.
stopping criterion has been explicitly included in our algorithm description. This is
adequate for the theoretical analysis that we consider in the present paper, where we are
interested in the asymptotic behaviour of the method, but it should be completed for any
practical use. The choice of a particular stopping criterion will depend on the type of
models being used.
7. The mechanism that we specified for updating the trust region radii does not exclude the
additional requirement that the radii be uniformly bounded, if that is judged suitable for
the type of models used. In practice, keeping the radii bounded is essential to prevent
numerical overflow.
8. One possible implementation of Step 2 first computes a feasible step s C
k that minimizes
trust region of radius \Delta min;k . Note that s C
satisfies (2.35) and
by construction. This step may then subsequently be increased by progressing further
along the arc PX long as the overall model m k continues to decrease and
holds. Additional decrease in m k may then be obtained (for instance by applying
conjugate-gradient steps) provided condition (2.36) is maintained.
Before starting our global convergence analysis, we first state, for future reference, some
properties that result from the mechanism of the algorithm.
Assume that AS.3 holds. At each iteration k of the algorithm,
1. M k contains at least one element. Furthermore
2.
for all pg.
Proof. The first result immediately follows from the definition of N k and the inequality
1. One then deduces that N k contains at most
from which the first part of (2.63) may be deduced. The second inequality in this result is
obtained from X
the relation (2.48) and jN k 1. The bound (2.64) results from (2.51), (2.54), (2.55), (2.57)
and (2.59). 2
We also investigate the coherency between the measure of fit for individual elements and that
for the overall model.
Assume AS.3 holds and that, at iteration k of the algorithm, (2.53) holds for all
and that (2.56) holds for all i 2 N k . Then iteration k is successful, i.e. k 2 S.
Proof. Because (2.53) holds for , one has that
for all such i, where we used the inequality jM k j - p and Lemma 1 to deduce the second inequality.
On the other hand, since (2.56) holds for i 2 N k , one obtains for these i that
where we used item 1 of Lemma 1 to bound jN k j. Now,
jffif i;k j: (2:69)
Combining this last inequality with (2.67) and (2.68) gives that
which then yields (2.43) because of (2.41). 2
We observe from this proof that the weaker condition
could be imposed instead of (2.41). However (2.71), and hence the setting of the algorithm's
constants, would then be problem dependent, which one might consider to be undesirable.
Of course, (2.53) holds whenever (2.50) holds because of (2.39). Lemma 2 therefore shows that
(2.43) is coherent with the measure of the fit between the element models and element functions.
3 Global convergence
We now study the convergence properties of the class of algorithms that we introduced in the
preceding section. Our analysis follows the pattern of similar proofs with an unstructured trust
region (see [14] or [41]). The central idea in the proof is that the algorithm will continue to make
progress as long as a critical point is not reached. We first start by bounding the error between
the true element functions and their models. We next derive a lower bound on the size of the
smallest trust region radius at a non-critical point. This lower bound ensures that the trust region
constraint will not prevent further progress towards a critical point. Only with this bound can
we then prove that limit points of the sequence of iterates produced by the algorithm are indeed
critical for the models used. We close the section by deriving some simple consequences of these
results on the criticality of the limit points for the true objective function.
We first start by bounding the error made between the model of any element function and
the element function itself at x k
Lemma 3 Assume that AS.4 holds and consider a sequence fx k g of iterates generated by the
algorithm. Then there exists a positive constant c 1 - 1 such that
for all
Proof. We first observe that, for each i 2 the definition (2.12),
(2.7) and the Cauchy-Schwarz inequality imply that
ks i;k k
But ks i;k k - \Delta i;k because of (2.6), and hence we obtain from (2.8), (2.15) and (2.16) that
Using (2.9), this then yields (3.1) with
where the last inequality results from (2.15). 2
We now derive an upper bound on the change predicted for an element at a non-critical point,
as a function of the size of the step in the corresponding range subspace.
Lemma 4 Assume that AS.1, AS.3 and AS.4 hold. Consider iteration k of the algorithm and
assume that, for some
Then one has that
ks i;k k (3:6)
for some constant c 2 ? 0 independent of i and k.
Proof. We first note that (2.9), (2.16) and (3.5) imply that
Using (2.12) and (2.16), we also obtain that
ks i;k ks i;k
Remembering now (2.8), (2.6), (3.5) and (3.7), we can deduce that
ks ks ks i;k k 2
ks i;k k:
Inequality (3.9) then gives (3.6) with
i2f1;:::;pg
next prove the important fact that, so long as a critical point has not been determined, the
trust region radii stay sufficiently bounded away from zero, therefore allowing further progress
to be made.
Lemma 5 Assume that AS.1-AS.4 hold. Consider a sequence fx k g of iterates generated by the
algorithm and assume that there exists a constant ffl ? 0 such that
for all k. Then there is a constant c 3 ? 0 such that
for all k.
Proof. Assume, without loss of generality, that
In order to derive a contradiction, assume that there exists a k such that
define r to be the smallest iteration number such that (3.14) holds.
(Note that r - 1 because of (3.13) and the inequality
The monotonic nature of the sequence ffi k g and the bound (2.64) then ensure that
where we used (3.14) and the inequality (3.13). We note that the definitions of i and r give that
which in turn implies that \Delta because of the monotonic nature of the sequence
g. Using this inequality with (2.36), (3.11), and (3.15), we obtain that
ae ffl
ks
oe
ae ffl
oe
which ensures, because of (2.64), that
But (3.15) guarantees that fi We may thus apply Lemma 4 and deduce that
ks
where we also used (2.6) and (3.18).
Assume first that i 2 M r\Gamma1 , which guarantees that using (2.48) and
(3.18),
Because of (2.7), (3.1) and (3.20), we therefore obtain that
ffif
But (3.14) and (3.15) together give that
which, with (3.21), implies that
ffif
Consider first the case where ffim may then apply (3.19) and deduce that
Using (3.23), we now deduce that
ffif
and therefore, because of (3.24), that
which implies that (2.50) holds for element i at iteration r \Gamma 1. Now turn to the case where
Because of (3.19), we deduce that
As above, we use (3.23) to obtain that
ffif
and therefore, because of (3.27), that
which again implies that (2.50) holds for element i at iteration r \Gamma 1.
Assume now that i 2 N r\Gamma1 . Then, because of (2.7), (2.48) and (3.1), we have that
multiplying (3.18) by \Delta i;r\Gamma1 , we obtain that
Combining (3.30) and (3.31), we deduce that
Observing now that (3.14) and (3.15) imply that
we obtain from (3.32) that
But this inequality implies that (2.56) holds for element i at iteration r \Gamma 1. Thus either (2.50)
or (2.56) holds for element i at iteration r \Gamma 1 and the mechanism of the algorithm then implies
that But we may deduce from this inequality that
which contradicts the assumption that r is the smallest iteration number such that (3.14) holds.
The inequality (3.14) therefore never holds and we obtain that (3.12) is satisfied for all k. 2
We now turn to one of the main results in this section, which proves a weak form of global
convergence. The technique is inspired by [35].
Theorem 6 Assume that AS.1-AS.6 hold. Consider a sequence fx k g of iterates generated by
the algorithm. Then
lim inf
Proof. Assume, for the purpose of obtaining a contradiction, that there exists an ffl 2 (0; 1)
such that (3.11) holds for all k - 0. Then
ks k k]; 1
where we used successively (2.43), (2.36), (3.11) and Lemma 5. We note that (3.37) and AS.2
then imply that X
Now let r be an integer such that
and define
the number of successful iterations up to iteration 1). Then define
We now wish to show that both sums
and
are finite. Consider the first. If it has only finitely many terms, its convergence is obvious.
Otherwise, we may assume that F 1 has an infinite number of elements, and we then construct
two subsequences. The first consists of the indices of F 1 in ascending order and the second, F 3
say, of the set of indices in S (in ascending order) with each index repeated r times. Hence the
j-th element of F 3 is no greater than the j-th element of F 1 . This gives that
because of the nondecreasing nature of the sequence ffi k g and (3.38). Now turn to the second
sum in (3.42). Lemma 2 and the mechanism of the algorithm imply that, at each unsuccessful
iteration, at least one element trust region radius satisfies (2.55) or (2.59) and none of them is
allowed to increase. Hence p
Y
Y
which immediately implies that
where
We deduce from this inequality that, for k 2 F 2 ,
where we have also used Lemma 5 and the definition of F 2 in (3.41). Using (3.39), this gives that
and the second sum is convergent. Therefore the sumX
is finite, which contradicts AS.5. Hence condition (3.11) is impossible and (3.36) follows. 2
Notice that the relation between ff k , the criticality measure for problem (2.23), and ff(x k ; f; X),
the criticality measure for problem (2.1), has been left rather unspecified up to this point. It
is indeed remarkable that we can prove Theorem 6 assuming so little on ff. In order to derive
convergence properties for the original problem from Theorem 6, we have to be slightly more
specific and request that, if both function and model have the same first order information, then
the criticality measures on the original problem and on the model problem agree.
AS.7 Let h 1 and h 2 be two continuously differentiable functions in the intersection of X with
a neighbourhood of the feasible point x, such that h 1 Then, the difference
tends to zero.
In other words, we require the criticality measure to be continuous (near zero) in the gradient
of its second argument. Again, this is true for the choices (2.24)-(2.25) and (2.28).
With this additional assumption, we are now ready to examine the criticality of the limit
points of the sequence of iterates generated by the algorithm for the original problem (2.1).
Corollary 7 Assume that AS.1-AS.7 hold. Consider a sequence fx k g of iterates generated by
the algorithm and assume that
lim
for all pg. Then this sequence has at least one critical limit point x .
Proof. From AS.7 and (3.49), we obtain that
lim
which, with (3.36), guarantees
lim inf
The desired conclusion then follows by taking a subsequence of fx k g if necessary. 2
Condition (3.49) is important, otherwise the situation might arise that an iterate is critical for
the current overall model (because its gradient is inexact) while not being critical for the original
problem. There are various ways in which (3.49) can be achieved in a practical algorithm, the
simplest being to make the size of e i;k also depend on ff k itself, ensuring that the first goes to
zero if the latter does.
Corollary 8 Assume that AS.1-AS.7 hold. If S, the set of successful iterations generated by the
algorithm is finite, then all iterates x k are equal to some x for k large enough, and x is critical.
Proof. Assume indeed that S is finite. It is then clear from (2.45) that x k is unchanged for
large enough, and therefore that x is the largest index in S. Note now that
Lemma 2 implies that, if k 62 S, then (2.53) or (2.56) must be violated for at least one element.
Hence we obtain that \Delta min;k converges to zero. But (2.8) then implies that e i;k also converges to
zero for all k converges to rf(x k ). Thus AS.7 and Corollary 7 then guarantee
the criticality of x . 2
As in existing theories for the unstructured trust region case, it is possible to replace the limit
inferior in (3.36) by a true limit, therefore ensuring (if the gradients are asymptotically exact)
that all limit points are critical. As in these theories, a slight strengthening of our assumptions
is however necessary.
AS.8 We assume that
lim
This assumption is similar to that used in [14] and [41], where it is motivated in detail. We only
mention here that (3.52) holds for Newton's method on bounded domains, because fi k is bounded
above in that case.
With this additional assumption, we are now able to replace the limit inferior by a true limit.
Theorem 9 Assume that AS.1-AS.8 hold. Consider the sequence fx k g of iterates generated by
the algorithm and assume that there are infinitely many successful iterations. Then
lim
where S is, as above, the set of successful iterations.
Proof. We again proceed by contradiction. Assume therefore that there exists an ffl 1 2 (0; 1)
and a subsequence fq j g of successful iterates such that, for all q j in this subsequence
Theorem 6 guarantees the existence of another subsequence fl j g such that
where we have chosen ffl 2 2 (0; ffl 1 ). We may now restrict our attention to the subsequence of
successful iterations whose indices are in the set
where q j and l j belong, respectively, to the two subsequences defined above. Applying now (2.36)
we obtain from (2.43), (2.16) and ffl
ks k k]; 1
oe
ks k k]
oe
But AS.8, along with (3.57), imply that
lim
ks
and, because of (2.16), that
lim
ks
Therefore, we can deduce from (3.57) and (3.58), that, for j sufficiently large,
ks k k
where the sums with superscript (K) are restricted to the indices in K, and
But AS.2 and the decreasing nature of the sequence ff(x k )g imply that the last right-hand side
of (3.60) converges to zero as j tends to infinity. Hence the continuity of rf and AS.7 give that
sufficiently large. On the other hand, the second part of (3.59) and (2.8) imply that g q j
is
arbitrarily close to rf(x q j
large enough, and AS.7 hence guarantees that
sufficiently large. We note also that, because of (2.8),
But the mechanism of the algorithm guarantees that no \Delta i;k can increase between iterations
is the largest integer in K that is smaller than l j .
This yields that
We now deduce from the second part of (3.59) that the left-hand side of (3.65) tends to zero
when j tends to infinity, and therefore that, for j sufficiently large,
because of AS.7. Combining (3.62), (3.63) and (3.66), we obtain, using (3.55), that
which is impossible because of (3.54). Hence our initial assumption cannot hold and the theorem
is proved. 2
As above, we now consider the case where we impose that the element gradients are asymptotically
exact.
Assume that AS.1-AS.8 hold. Consider the sequence fx k g of iterates generated
by the algorithm and assume furthermore that (3.49) holds for all pg. Then all limit
points of this sequence are critical.
Proof. If the set S is finite, the conclusion immediately follows from Corollary 8. If, on
the other hand, S has an infinite number of elements, (3.49) implies that g k is arbitrarily close
to rf(x k ) and the combination of AS.7 and Theorem 9 ensures the criticality of any limit point
of the sequence of successful iterates. 2
Of course, (3.49) might be impossible to achieve in practice, and one might consider the case
where we can only assert that
lim sup
i2f1;:::;pg
for some small constant - 3 ? 0. This is the case, for instance, if gradients are approximated by
finite differences.
Corollary 11 Assume that AS.1-AS.6 and AS.8 hold. Consider the sequence fx k g of iterates
generated by the algorithm. Assume furthermore that (3.68) holds and that, for some constant
the criticality measure ff satisfies
for all x 2 X and all functions h 1 and h 2 continuously differentiable in a neighbourhood of x such
that h 1 for each limit point x of the sequence,
Proof. As in Corollary 10, the desired conclusion immediately follows from Corollary 8 if
S is finite. Assume therefore that S has infinitely many elements. We then deduce that, for all
Taking the limit for k tending to infinity in S and using Theorem 9 and (3.68) then gives the
desired conclusion. 2
Finally observe that although (3.69) is stronger than AS.7, it is not a very strong condition.
For instance, it is satisfied with L for the choices (2.24), and also for (2.25) and (2.26)
because of the non-expansive character of the projection operator PX (see [41], for example).
The same property also holds for the choice (2.28), as discussed in [14].
Finite identification of the correct active set
When applied to constrained problems, trust region algorithms typically use the notion of projected
gradient or projected gradient path in order to identify a subset of inequality constraints
that are satisfied as equalities. Ultimately, the aim thereby is to identify the constraints satisfied
as equalities at the solution well before the solution is reached. The methods then reduce to an
unconstrained calculation in the manifold defined by the currently "active" constraints. As a
consequence, it is possible to guarantee fast asymptotic rates of convergence when using accurate
models, as is the case when analytical second order information of the objective and constraint
functions is available.
It is possible to show that structured trust regions do not upset the theory developed in the
unstructured case: it can indeed be shown that the constraints active at a particular limit point
of the sequence of iterates are identified after a finite number of iterations, provided the normals
of the active constraints are linearly independent and strict complementarity holds, and provided
the step s k+1 satisfies the inequality
ks ks k k (k) (4:1)
for each k 62 S and for some constant This latter condition is meant to avoid
a situation where the successful iterates converge to a critical point while a subsequence of
unsuccessful iterates converges to another point with a different active set. It does not constitute
a severe restriction in the step selection procedure and is automatically verified if s k is determined
by a succession of steps of increasing norm such that they remain feasible, within the trust region
ensure (2.36). This is the case, for instance, if truncated conjugate gradients are used
for computing the step in the solution of an unconstrained problem (see [37] or [38]).
The theory considers the active constraint identification problem from a quite general point
of view. The main observation is that a number of the existing theories for active constraint
identification are based on the definition of a special criticality measure that satisfies AS.6 while
not satisfying AS.7 (see [2] or [3], for instance). Let us denote this measure at iteration k by -
ff k .
The steps leading to constraint identification are then as follows.
1. The first step is to prove that a sufficient decrease condition of the type (2.33) also holds
with -
ff k instead of ff k .
2. One then proceeds to prove that
lim inf
much in the same way as for (3.36).
3. The measure -
ff k is also constructed to ensure that it is asymptotically bounded away from
zero for all points such that their active set is not identical to that of a (close) critical point.
(This, in particular, prevents AS.7 from holding.)
4. Some contradiction is then deduced from these last two properties.
However, since this development is rather technical and lengthy, we do not include it in the
present paper, but refer the interested reader to [15] for details of the results and additional
assumptions. This reference also contains the theory concerning the convergence of the iterates
to a single limit point, adapted from [14].
Our experience with the solution of practical problems however indicates that the identification
of active constraints is seldom observed in practice before the very last iterations of the
algorithm, which makes the results discussed in this section mainly of theoretical interest.
Extensions
We examine in this section some extensions and variants of the results presented above.
5.1 A hybrid technique
One of the possible drawbacks of the algorithm of Section 2.3 is that steps might be constrained to
be unnecessarily small in directions corresponding to highly nonlinear element functions. Indeed,
the negative effect of inaccurate models for these elements might be compensated by a successful
step in directions corresponding to less nonlinear elements. This compromise between the different
parts of the objective is, of course, inherent to the classical method using an unstructured trust
region.
We might try to obtain the best of both classical and structured approaches by using a hydrid
technique. In this technique, a global trust region radius \Delta k is recurred for the objective function
considered as a single element (using the algorithm analyzed above, which is then equivalent to
the classical one), along with the individual radii \Delta i;k . We then define the individual "hybrid"
radii by
for each i 2
i;k g: (5:2)
We can then apply our algorithm with these new quantities, to the effect that well-modelled
elements have their associated trust regions possibly extended without having to contract those
corresponding to badly-modelled ones, as long as the global agreement is satisfactory.
It is not difficult to verify that the theory presented above still holds for this hybrid mod-
ification. The key points are to observe that the revised definition of our trust region implies
that
ae ff k
oe
which is the classical sufficient decrease condition (2.33), that the inequalities (2.64) are still valid
with \Delta i;k replaced by \Delta h
i;k , and also that an analogous result to Lemma 5 also holds for the global
trust region radius, as is already well-known from the unstructured trust region case (see [14],
for instance).
5.2 An alternative definition of success
An immediate consequence of inequality (2.63) in Lemma 1 is that it would be possible to replace
the condition (2.43) for an iteration to be successful by
without altering the developments presented above. Indeed, (2.63) shows the equivalence between
(2.43) and (5.4). We have chosen to use seems natural to consider the
same collection of elements on both sides of the inequality.
5.3 Weaker sufficient decrease conditions
It is remarkable to note that Lemma 5 and Theorem 6 can be proved in a weaker context. Indeed,
we could require the weaker sufficient decrease condition
ae ff k
oe
instead of (2.36), and still prove Lemma 5 and Theorem 6. However, we have not been able to
prove Theorem 9, nor active constraint identification, with these assumptions, because (5.5) only
involves the length of the step in a possibly small subspace of R n .
5.4 Using uniformly equivalent norms
Another possible generalization of the theory developed above allows the use of different norms
for each element and for each iteration. Let us denote these norms by the . The
element trust region definition (2.6) then becomes
while the gradient approximation condition (2.8) may be written as
where the norm k \Delta k [i;k] is any norm that satisfies
for all x; y 2 R n . In particular, one can choose the dual norm of k \Delta k (i;k) defined by
With iteration k, we may also associate an overall norm k \Delta k (k) defined on the whole of R n , whose
purpose is to reflect the relative weighting of the different elemental norms k \Delta k (i;k) in a global
measure.
If we assume that all the considered norms are uniformly equivalent, that is if there exists a
constant oe - 1 such that, for all x,oe
is any pair of the above defined norms, then the theory developed in all
the preceding sections is still valid without any substantial modification. Again the details of
the proofs in this more general setting are provided in [15]. Note that this extension covers
the possible introduction of iteration dependent scaling in a practical implementation of our
algorithm, which can be highly desirable for some difficult problems.
6 Conclusions
We have shown in this paper that the trust region concept, one of the most powerful tools for
building efficient and robust algorithms for optimization, can be extended in a very natural way
to reflect the structure of the underlying problem. The algorithm proposed above is indeed a
direct generalization of the more usual case where only an unstructured uniform trust region is
considered. Similar global convergence properties can be proved for the new algorithm, including
the case where dynamic scaling is performed on the variables and the situation where the gradients
are only known approximately.
It remains to see if this modification of a trust region algorithm will prove efficient in practice
and justify the slight additional complexity of the method. Note that the results of preliminary
numerical experiments (based on a modification of LANCELOT using the implementation described
after the algorithm) have been encouraging. Tests on unconstrained problems from the
collection [1] have shown that the new method, although very comparable to LANCELOT
in many cases, sometimes produces substantial improvements. However, we anticipate the real
power of the concept to appear when minimizing augmented Lagrangians or other penalty-like
because scaling is much more critical there than in many of the classical unconstrained
test examples. The authors are planning to include the new technique described in this paper
within the next release of LANCELOT.
One of the nice features of the partially separable functions considered in the present theory
is that the objective is a linear combination of its elements. While group partially separability, as
used in [12] or [13], has computational advantages in terms of economy of derivative calculation,
this structure involves a nonlinear relationship between the elements and the overall function.
This seems to make exploiting the link between local and global models much harder. While
we would be interested in deriving structured trust region methods for group partially separable
functions, the methods would undoubtedly be more complicated and less amenable to analysis.
Thus, we are content, in the present paper, to consider the simpler, but nonetheless very general,
partially separable structure.
Finally, there might be other ways to introduce structure in trust region methods than considering
(group) partially separable objective functions. In particular, trust region methods for
nonlinearly constrained problems seems attractive candidates for an alternative approach that
would separate the trust region(s) on the objective from those on the constraints.
Acknowledgments
The authors are indebted to Johara Shahabuddin for twice pointing out an unsuitable definition
of the sufficient decrease condition (2.36) in Section 2.2.2.
--R
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Parallel global optimization: numerical methods
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Projected gradient methods for linearly constrained problems.
On the global convergence of trust region methods using inexact gradient information.
A trust region strategy for nonlinear equality constrained optimization.
Performance of a multifrontal scheme for partially separable optimization.
Global convergence of a class of trust region algorithms for optimization with simple bounds.
Testing a class of methods for solving minimization problems with simple bounds on the variables.
An introduction to the structure of large scale nonlinear optimization problems and the lancelot project.
LANCELOT: a Fortran package for large-scale nonlinear optimization (Release
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Convergence properties of minimization algorithm for convex constraints using a structured trust region (revised).
A global convergence theory for the Dennis-Celis-Tapia trust-region algorithm for constrained optimization
Practical Methods of Optimization: Unconstrained Optimization.
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On the existence of convex decomposition of partially separable functions.
Algorithmic Methods in Optimal Control.
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Partially separable optimization and parallel computing.
Convergence of trust region algorithms for optimization with bounds when strict complementarity does not hold.
A method for the solution of certain problems in least squares.
An algorithm for least-squares estimation of nonlinear parameters
The Levenberg-Marquardt algorithm: implementation and theory
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Trust regions and projected gradients.
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A trust region algorithm for equality constrained optimization.
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--TR
--CTR
Nicholas I. M. Gould , Dominique Orban , Philippe L. Toint, GALAHAD, a library of thread-safe Fortran 90 packages for large-scale nonlinear optimization, ACM Transactions on Mathematical Software (TOMS), v.29 n.4, p.353-372, December | partial separability;trust region methods;large-scale optimization;convex constraints;structured problems |
589230 | Tensor Methods for Large, Sparse Unconstrained Optimization. | Tensor methods for unconstrained optimization were first introduced by Schnabel and Chow [SIAM J. Optim., 1 (1991), pp. 293--315], who described these methods for small- to moderate-sized problems. The major contribution of this paper is the extension of these methods to large, sparse unconstrained optimization problems. This extension requires an entirely new way of solving the tensor model that makes the methods suitable for solving large, sparse optimization problems efficiently. We present test results for sets of problems where the Hessian at the minimizer is nonsingular and where it is singular. These results show that tensor methods are significantly more efficient and more reliable than standard methods based on Newton's method. | Introduction
In this paper we describe tensor methods for solving the unconstrained optimization problem
where D is some open set containing x . We assume that f is at least twice continuously
differentiable, and n is large.
Tensor methods for unconstrained optimization are general purpose methods primarily intended
to improve upon the performance of standard methods, especially on problems where
deficiency. They are also intended to be at least as efficient as standard
methods on problems where r 2 f(x ) is nonsingular.
Tensor methods for unconstrained optimization base each iteration upon the fourth order
model of the objective function f(x)
is the current iterate, rf(x c ) and r 2 f(x c ) are the first and second analytic
derivatives of f at x c , or finite difference approximations to them, and where the tensor terms
at x c , T c 2 ! n\Thetan\Thetan and V c 2 ! n\Thetan\Thetan\Thetan , are symmetric. (We use the notation rf(x c ) \Delta d for
)d to be consistent with the tensor notation T c \Delta d 3 and
Also, for simplicity, we abbreviate terms of the form dd; ddd, and dddd by d 2 ; d 3 , and d 4 ,
respectively.) Before proceeding, we define the tensor notation used above.
n\Thetan\Thetan . Then for
n\Thetan\Thetan\Thetan . Then for
The tensor terms are selected so that the model interpolates a small number of function and
gradient values from previous iterations. This results in T c and V c being low-rank tensors, which
is crucial for the efficiency of the tensor method. The tensor method requires no more function
or derivative evaluations per iteration and hardly more storage or arithmetic operations, than a
standard method based on Newton's method.
Standard methods for solving unconstrained optimization problems are widely described in
the literature; general references on this topic include Dennis and Schnabel [9], Fletcher [11],
and Gill, Murray, and Wright [13]. In this paper, we propose extensions to standard methods
that use analytic or finite difference gradients and Hessians.
The standard method for unconstrained optimization, Newton's method, bases each iteration
upon the quadratic model of f(x)
This method is defined when r 2 f(x c ) is nonsingular, and consists of setting the next iterate x+
to the minimizer of (1.3), i.e.,
A distinguishing feature of Newton's method is that if r 2 f(x c ) is nonsingular at a local
minimizer x , then the sequence of iterates produced by (1.4) converges quadratically to x .
However, Newton's method is generally linearly convergent at best if r 2 f(x ) is singular [14].
Methods based on (1.2) have been shown to be more reliable and more efficient than standard
methods on small to moderate size problems [18]. In the test results obtained for both non-singular
and singular problems, the improvement by the tensor method over Newton's method
is substantial, ranging from 30% to 50% in iterations, and function and derivative evaluations.
The improvement is even more dramatic for singular problems. Furthermore, the tensor method
solves several problems that Newton's method fails to solve.
The tensor algorithms described in [18] are QR-based algorithms involving orthogonal transformations
of the variable space. These algorithms are very effective for minimizing the tensor
model when the Hessian is dense because they are very stable numerically, especially when the
Hessian is singular. However, they are not efficient for sparse problems because they destroy the
sparsity of the Hessian due to the orthogonal transformation of the variable space. To preserve
the sparsity of the Hessian, we have developed an entirely new way of solving the tensor model
that employs a sparse variant of the Cholesky decomposition. This makes our new algorithms
very well suited for sparse problems.
The remainder of this paper is organized as follows. In x2 we briefly review the techniques
used to form the tensor model, that were introduced in Schnabel and Chow [18]. In x3 we describe
efficient algorithms for minimizing the tensor model when the Hessian is sparse. xx4 and
5 discuss the globally convergent modifications for tensor methods for unconstrained optimiza-
tion. These consist of line search backtracking and model trust region techniques. A high level
implementation of the tensor method is given in x6. In x7 we describe comparative testing for
an implementation based on the tensor method versus an implementation based on Newton's
method, and present summary statistics of the test results. Finally, a summary of our work and
a discussion of future research is given in x8.
2. Forming the Tensor Model
In this section, we briefly review the techniques for forming the tensor model for unconstrained
optimization that were introduced in [18].
As was stated in the previous section, the tensor method for unconstrained optimization
bases each iteration upon the fourth order model of the nonlinear function f(x) given by (1.2).
The choices of T c and V c in (1.2) cause the third order term T c \Delta d 3 and the fourth order
to have simple and useful forms. These tensor terms are selected so that the tensor
model interpolates function and gradient information at a set of p not necessarily consecutive
past iterates x
In the remainder of this paper, we restrict our attention to 1. The reasons for this
choice are that the performance of the tensor version that allows p - 1 is similar overall to that
constraining p to be 1, and that the method is simpler and less expensive to implement in this
case. (The derivation of the third and fourth order tensor terms for p - 1 is explained in detail
in [18].)
The interpolation conditions at the past point x are given by
and
where
Schnabel and Chow [18] choose T c and V c to satisfy (2.1) and (2.2). They first show that
the interpolation conditions (2.1) and (2.2) uniquely determine T c \Delta s 3 and V c \Delta s 4 . Multiplying
(2.2) by s yields
Let ff, fi 2 ! be defined by
Then from (2.1) and (2.3) they obtain the following system of two linear equations in the two
unknowns ff and fi:2 ff
are defined by
The system (2.4)-(2.5) is nonsingular, therefore the values of ff and fi are uniquely determined.
Hence, the interpolation conditions uniquely determine T c \Delta s 3 and V c \Delta s 4 . Since these are the
only interpolation conditions, the choice of T c and V c is vastly underdetermined.
Schnabel and Chow [18] choose T c and V c by first selecting the smallest symmetric V c , in the
Frobenius norm, for which
where fi is determined by (2.4)-(2.5). Then they substitute this value of V c into (2.2), obtaining
where
This is a set of n linear equations in n 3 unknowns T c (i; j; k), 1 - Finally, Schnabel
and Chow [18] choose the smallest symmetric T c and V c , in the Frobenius norm, which satisfy
the equations (2.6)-(2.7). That is,
min
Vc2! n\Thetan\Thetan\Thetan
subject to V c \Delta s
and
min
n\Thetan\Thetan
subject to T c \Delta s
The solution to (2.8) is
(s\Omega s\Omega s\Omega s);
where the tensor V
n\Thetan\Thetan\Thetan is called a fourth order rank-one tensor for
which use the
notation\Omega to be consistent
with [18].)
The solution to (2.9) is
s\Omega s\Omega b; (2:10)
where the notation
n\Thetan\Thetan , is called a third order rank-one
tensor for which T (i; j; is the unique vector for which (2.10) satisfies
(2.6), and is given by
determined by the minimum norm problems (2.9) and (2.8) have rank 2 and 1,
respectively. This is the key to form, store, and solve the tensor model efficiently.
The whole process of forming the tensor model requires only O(n 2 ) arithmetic operations. The
storage needed for forming and storing the tensor model is only a total of 6n.
For further information we refer to [18].
3. Solving the Tensor Model when the Hessian is Sparse
In this section we give efficient algorithms for finding a minimizer of the tensor model (1.2),
when the Hessian is sparse.
The substitution of the values of T c and V c into (1.2) results in the tensor model
As we stated in x2, we only consider the case where the tensor model interpolates f(x) and
rf(x) at the previous iterate, i.e., 1. The generalization for p - 1 is fairly straightforward.
This constraint is mainly motivated by our computational results. When we allow p - 1, our
test results showed almost no improvement over the case where 1. The tensor method is
therefore considerably simpler, and cheaper in terms of storage and cost per iteration.
3.1. Case 1: the Hessian is Nonsingular
We show that the minimization of (3.1) can be reduced to the solution of a third order polynomial
in one unknown, plus the solution of three systems of linear equations that all involve the same
coefficient matrix r 2 f(x c ). For conciseness, we use the notation
A necessary condition for d to be a local minimizer of (3.1) is that the derivative of the
tensor model with respect to d must be zero. That is,
which yields
If we first premultiply equation (3.2) by s T on both sides, we obtain a cubic equation (in fi) in
the unknowns
If we then premultiply equation (3.2) by b T on both sides, we obtain another cubic equation (in
fi) in the unknowns fi and ',
Thus, we obtain a system of two cubic equations in the two unknowns fi and ' which can be
solved analytically.
We now show how to compute the solutions of this system of two cubic equations in two
unknowns by computing the solutions of a single cubic equation in the unknown fi. Let
first calculate the
value of ' as a function of fi using equation (3.3), i.e.,
Note that the denominator of equation (3.5) is equal to zero if either
assume that fi 6= 0, otherwise the tensor model would be reduced to the Newton model. Now,
would be quadratic in fi, therefore
Thus, real valued minimizers of the tensor model (3.1) may exist only if 0: It is easy
to check that in order for ' to have a defined cannot be zero.
If fi 6= 0 and w 6= 0, we substitute expression for ' into equation (3.4) and obtain
which is a third order polynomial in the one unknown fi. The roots of equation (3.6) are
computed analytically. We substitute the values of fi into equation (3.5) to calculate the values
of '. Then we simply substitute the values of fi and ' into equation (3.2) to obtain the values
of d. The major cost in this whole process is the calculation of H
After we compute the values of d, we determine which of them are potential minimizers.
Our criterion is to select those values of d which guarantee that there is a descent path from x c
to x c + d for the model M T among the selected steps, we choose the one that is
closest to the current iterate x c in the euclidean norm sens. If the tensor model has no minimizer
we use the standard Newton step as the step direction for the current iteration.
3.2. Case 2: the Hessian is Rank Deficient
If the Hessian matrix is rank deficient we transform the tensor model given in (3.1) by the
following procedure. Let
d, and ffi is the new unknown. Substituting this
expression for d into (3.1) yields the following tensor model which is a function of ffi,
d) 2
d)
d)s
d)
(b T -
If we let -
d, -
d, -
we obtain the modified tensor model,
d)
The advantage of this transformation is that the matrix -
H is likely to be nonsingular if the rank
of (r 2 f(x c )) is at least n \Gamma 1. A necessary and sufficient condition for -
H to be nonsingular is
given in the following lemma. Let g and H denote rf(x c ) and r 2 f(x c ), respectively.
Lemma 3.1. Let H 2 ! n\Thetan , s
css T is nonsingular if and only if M =6 6 6 6 6 4
H cs
cs
is nonsingular:
(Note that the
submatrix was premultiplied by the constant c to symmetrize the
augmented matrix M .)
Proof. We prove that there exists only if there
exist
H cs
cs
Suppose first that (H
Conversely, if there exists (-v; w) satisfying (3.9), then s T -
otherwise, contradicts (3.9). Thus (H singular if and only if M is
singular.
Corollary 3.2. Let H 2 ! n\Thetan , s
css T is nonsingular then
H cs
has full row rank
Proof. follows from Lemma 3.1.
Lemma 3.3. Let H 2 ! n\Thetan ,
css T is nonsingular if and only if
H cs
has full row rank.
Proof. The only if part follows from Corollary 3.2. Now assume
H cs
has full row rank.
Since H has rank
n\Theta(n\Gamma1) have full column rank. Since
H cs
has full row rank,
(v
From
T and H 2 has full column rank, (3.10) is equivalent to
(v
Thus the n \Theta n matrix
is nonsingular. Analogously, the n \Theta n matrix
is
nonsingular. Therefore
is nonsingular. 2
For ffi to be a local minimizer of (3.8) the derivative of the tensor model (3.8) with respect
to ffi must be zero. That is,
which yields
Premultiplying equation (3.12) by s T on both sides results in a cubic equation (in fi) in the two
unknowns
fis T -
fis T -
fis T -
The premultiplication of equation (3.12) by b T on both sides yields another cubic equation (in
fi) in the two unknowns fi and '
Therefore, we obtain a system of two cubic equations in the two unknowns fi and ' which we
can solve analytically.
Since equation (3.13) is linear in ', we can compute ' as a function of fi, and then substitute
its expression into equation (3.14) to obtain an equation in the one unknown fi. Let
g, and
The denominator of equation (3.15) is equal to zero if either -
then equation (3.13) would be quadratic in fi, therefore
Hence, real valued minimizers of the tensor model (3.8) may exist only if (1
It is straightforward to verify from (3.14) that for ' to be defined
reduces to the following cubic equation in fi
Once we calculated the expressions for fi from equation (3.16), we substitute them into the
following equation for ' obtained from equation (3.14)
If neither -
equation (3.14) and obtain
fiw \Gamma2
which is a third order polynomial in the one unknown fi. The roots of equation (3.17) are then
computed analytically. After we determine the values of fi, we substitute them into equation
(3.15) to calculate the corresponding values of '. then, we simply substitute the values of fi
and ' into equation (3.12) to obtain the values of ffi . The dominant cost in this whole process is
the computation of -
Similar to the nonsingular case, a minimizer ffi is selected such that there exists a descent
path from the current point x c to x c that it is closest to x c .
To obtain the tensor step d we set d to -
An appropriate choice of -
d is the step used in
the previous iteration simply because it has the right scale.
To solve linear systems of the form -
n\Thetan sparse
and we use the augmented matrix M defined in Lemma 3.1. That is, we write
H cs
cs
x
The (n in (3.18) is sparse and can be factorized efficiently as long as the
last row and column are not pivoted until the last few iterations. In fact, we can combine the
nonsingular and singular cases by factorizing H , but we shift to a factorization of the augmented
matrix if H is discovered to be singular with rank n \Gamma 1. However, we use a Schur complement
method to obtain the solution of the augmented matrix by updating the solution from the system
b. This choice was motivated by the fact that the Schur complement method was simpler
and more convenient to use than the factorization of the augmented matrix M . We describe
this updating scheme in x6.
If the Schur complement method shows that M is rank deficient (a case that is very rare in
practice), or H has rank less than use the standard Newton step as the step direction
for the current iteration.
4. Line Search Backtracking Techniques
The line search global strategy we used in conjunction with our tensor method for large sparse
unconstrained optimization is similar to the one used for nonlinear equations [4, 6]. This strategy
has shown to be very successful for large sparse systems of nonlinear equations. We also
found that it is superior to the approach used by Schnabel and Chow [18]. The main difference
between the two approaches is that ours always tries the full tensor step first. If this provides
enough decrease in the objective function then we terminate, otherwise we find acceptable next
iterates in both the Newton and tensor directions and select the one with the lower function
value as the next iterate. Schnabel and Chow on the other hand, always find acceptable next
iterates in both the Newton and tensor directions and choose the one with the lower function
value as the next iterate. In practice, our approach almost always requires fewer function evaluations
while retaining the same efficiency in iteration numbers. The global framework for line
search methods for unconstrained minimization is given in Algorithm 4.1.
Algorithm 4.1. Global Framework for Line Search Methods for Unconstrained Minimization.
Let x c be the current iterate
d t the tensor step
d n is the Newton step
and
if (minimizer of the tensor model was found) then
else
Find an acceptable x n
in the Newton direction d n
using Algorithm A6.3.1 page 325 ([9])
Find an acceptable x t
in the tensor direction d t
using Algorithm A6.3.1 page 325 ([9])
if f(x n
else
endif
endif
else
Find an acceptable x n
in the Newton direction d n
using Algorithm A6.3.1 page 325 ([9])
endif
5. Model Trust Region Techniques
The two computational methods that are generally used for approximately solving the trust
region problem based on the standard model,
subject to jj d jj
where ffi c is the current trust region radius, are the locally constrained optimal (or "hook") step,
and the dogleg step. When ffi c is shorter than the Newton step, the locally constrained optimal
step [16] finds a - c such that jj d(- c
takes The dogleg step is a modification of the trust region algorithm introduced
by Powell [17]. However, rather than finding a point on the curve d(- c ) such
that it approximates this curve by a piecewise linear function in the subspace
spanned by the Newton step and the steepest descent direction \Gammarf (x c ), and takes x+ as the
point on this approximation for which jj x+ \Gamma x c e.g. [9] for more details.)
Unfortunately these two methods are hard to extend to the tensor model, which is a fourth
order model. Trust region algorithms based on (5.19) are well defined because it is always
possible to find a unique point x+ on the curve such that jj x+ \Gamma x c . Additionally, the
value of f(x c )+rf(x c ) along the curve d(- c ) is monotonically decreasing from
x c to x n
which makes the process reasonable. These properties do not
extend to the tensor model which is a fourth order model that may not be convex. Furthermore,
the analogous curve to d(- c ) is more expensive to compute. For these reasons, we consider a
different trust region approach for our tensor methods.
The trust region approach that is discussed in this section is a two-dimensional trust region
step over the subspace spanned by the steepest descent direction and the tensor (or standard)
step. The main reasons that lead us to adopt this approach is because it is easy to construct,
closely related to dogleg type algorithms over the same subspace. This step may be close to
optimal trust region step algorithms in practice. Byrd, Schnabel, and Shultz [7] have shown that
for unconstrained optimization using a standard quadratic model, the analogous two-dimensional
minimization approach produces nearly as much decrease in the quadratic model as the optimal
trust region step in almost all cases.
The two-dimensional trust region approach for the tensor model computes an approximate
solution to
subject to jj d jj
by performing a two-dimensional minimization,
subject to jj d jj
where d t and g s are the tensor step and the steepest descent direction, respectively, and ffi c is the
trust region radius. This approach will always produce a step that reduces the quadratic model
by at least as much as a dogleg type algorithm which reduces d to a piecewise linear curve in the
same subspace. At each iteration of the tensor algorithm, the trust region method either solves
(5.20), or minimizes the standard linear model over the two-dimensional subspace spanned by
the standard Newton step and the steepest descent direction. The decision of whether to use
the tensor or standard model is made using the following criterion:
if (no minimizer of the tensor model was found) or (rf(x c
then
selected by trust region algorithm
else
selected by trust region algorithm
endif
Before we define the two-dimensional trust region step for tensor methods, we show how to
convert the problem
subject to jj d jj
to an unconstrained minimization problem.
First, we make g s orthogonal to d t by performing the Householder transformation:
then, we normalize both - g s and d t to obtain:
~
~
s
Since d is in the subspace spanned by the tensor step ~
d t and the steepest descent direction ~
s ,
it can be written as
If we square the l 2 norm of this expression for d and set it to ffi 2 , we obtain the following equation
for fi as a function of ff
Substituting this expression for fi into (5.25) and then the resulting d into (5.21), yields the
global minimization problem in the one variable ff, given by (5.26) bellow. Thus, problems
and (5.21) are equivalent. Let g hg = ~
s .
c
)ff
To transform the problem
subject to jj d jj
to an unconstrained minimization problem, we use the same procedure described above to show
that (5.27) is equivalent to the following global minimization problem in the one variable ff:
c
Algorithm 5.1. Two-Dimensional Trust Region for Tensor Methods
Let d t be the tensor step
d n the standard step
x c the current iterate
x+ the next iterate
steepest descent direction
and ffi c the current trust region radius.
~
are given by (5.23) and (5.24), respectively.
~
d n is obtained in an analogous way to ~
applying transformations (5.22) and (5.23) to it.
1. if tensor model selected then
Solve problem (5.26) using the procedure described in Algorithm 3.4 [6]
else fstandard Newton model selectedg
Solve problem (5.28) using the procedure described in Algorithm 3.4 [6]
endif
2. if tensor model selected then
~
s
where ff is the global minimizer of (5.26)
else fstandard Newton model selectedg
~
s
where ff is the global minimizer of (5.28)
endif
3. f Check new iterate and update trust region radius.g
pred
the global step d is successful
else
decrease trust region
go to step 1
endif
where
pred
pred
standard Newton model selected.
The methods used for adjusting the trust radius during and between steps, are given in Algorithm
page 338 ([9]). The initial trust radius can be supplied by the user, if not, it is set to the
length of the initial Cauchy step.
6. A High Level Algorithm for the Tensor Method
In this section, we present the overall algorithm for the tensor method for large sparse unconstrained
optimization. Algorithm 6.1 is a high level description of an iteration of the tensor
method, that was described in xx3-5. A summary of the test results for this implementation is
presented in x7.
Algorithm 6.1. An Iteration of the Tensor Method for Large Sparse Unconstrained Optimization
Let x c be the current iterate
d t the tensor step
and d n the Newton step.
1. Calculate rf(x c ) and decide whether to stop. If not:
2. Calculate r 2 f(x c ).
3. Calculate the terms T c and V c in the tensor model, so that the tensor model interpolates
f(x) and rf(x) at the past point.
4. Find a potential minimizer d t of the tensor model (3.1).
5. Find an acceptable next iterate x+ using either a line search or a two-dimensional trust
region global strategy.
go to step 1.
In step 1, the gradient is either computed analytically or approximated by the algorithm
A5.6.3 given in Dennis and Schnabel [9]. In step 2, the Hessian matrix is either calculated
analytically or approximated by a graph coloring algorithm described in [8]. Note that it is
crucial to supply an analytic gradient if the finite difference Hessian matrix requires many
gradient evaluations. Otherwise, the methods described in this paper may not be practical, and
inexact type of methods may be preferable. The procedures for calculating T c and V c in step 3
were discussed in x2. Step 4 calculates d t as described in xx 3-4. The Newton step d n is also
computed as a by product of the minimization of the tensor model. The Newton step d n is
the modified Newton step (r 2 f(x c safely positive
definite, and - ? 0 otherwise. To obtain the perturbation - we use a modification of MA27 [10]
advocated by Gill, Murray, Ponceleon, and Saunders in [12]. In this method we first compute the
LDL T of the Hessian matrix using the MA27 package, then change the block diagonal matrix
D to D + E. The modified matrix is block diagonal positive definite. This guarantees that the
E)L T is positive definite as well. Note, that the Hessian matrix is not
modified if it is already positive definite.
The tensor and Newton algorithms terminate if jj rf(x c ) jj
Another implementation issue that deserves some attention is how to find a solution to the
augmented system (3.18), when the Hessian matrix is rank deficient. To do this, we use a Schur
complement method to update the solution x obtained from solving Hx = b. This requires
that H must have full rank. Thus, some modifications are necessary in order for this method
to work. We have modified the factorization phase of MA27 to be able to detect the row and
column indices of the first pivot that is less or equal than some given tolerance tol. Note that if
the rank of the Hessian matrix is less than we skip this whole updating scheme and
perturb the matrix as described in the previous paragraph. We also modified the solve phase
of MA27 such that whenever there is a zero pivot, the corresponding solution component is set
to zero. This way the solution of is the same as the solution of H e
is the matrix H minus the row and column at which singularity occurred. Since y has
components, the remaining one, which is also the component corresponding to the zero pivot, is
set to 0.) Afterwards, we obtain the solution of an augmented system using a Schur complement
method, where the coefficient matrix is the matrix H augmented by two rows and columns, i.e.,
the (n+ 1)-st row and column are the ones at which singularity was detected, and the (n+2)-nd
row and column are cs T and cs, respectively. The Schur complement method is implemented by
first invoking MA39AD [1] to form the Schur complement H in the extended
matrix, where D is the 2 by 2 lower right submatrix, C is the lower left 2 by n submatrix, and
B is the upper right n by 2 submatrix, of the augmented matrix. The Schur complement is then
factorized into its QR factors. Next, MA39BD [1] solves the extended system (3.18) using the
following well-known scheme:
1. Solve
2. Solve y.
3. Solve
4.
7. Test Results
We tested our tensor and Newton algorithms on a variety of nonsingular and singular test
problems. In the following we present and discuss summary statistics of the test results.
All our computations were performed on a SUN using double
precision arithmetic.
First, we tested our program on the set of unconstrained optimization problems from the
[3] and the MINPACK-2 [2] collections. Most of these problems have nonsingular Hessians
at the solution. We also created singular test problems as proposed in [4, 19] by modifying the
nonsingular test problems from the CUTE collection as follows. Let
be the function to minimize, where f is the number of element functions,
and
In many cases, F at the minimizer x , and F 0 (x ) is nonsingular. then according to
[4, 19], we can create singular systems of nonlinear equations from (7.1) by forming
n\Thetak has full column rank with 1 - k - n. Hence, -
k. For unconstrained optimization, we simply need to define the singular function
From (7.3) and -
and
we know that r 2 -
By using (7.2) and (7.3), we created two sets of singular problems, with r 2 -
respectively, by using
and
respectively. The reason for choosing unit vectors as columns for the matrix A is mainly to
preserve the sparsity of the Hessian during the transformation (7.2).
For all our test problems we used a standard line search backtracking strategy. All the test
problems with the exception of rank problems were ran with analytic
gradients and Hessians provided by the CUTE and MINPACK-2 collections. For rank
test problems, we have modified the analytic gradients provided by the CUTE collection
to take into account the modification (7.2). On the other hand, we used the graph coloring
algorithm [8] to evaluate the finite difference approximation of the Hessian matrix.
A summary for the test problems whose Hessians at the solution have ranks n,
presented in Table 1. The descriptions of the test problems and the detailed results are
given in the Appendix. In Table 1 columns "better" and "worse" represent the number of times
the tensor method was better and worse, respectively, than Newton's method by more than one
gradient evaluation. The "tie" column represents the number of times the tensor and standard
methods required within one gradient evaluation of each other. For each set of problems, we
summarize the comparative costs of the tensor and standard methods using average ratios of
three measures: gradient evaluations, function evaluations, and execution times. The average
gradient evaluation ratio (geval) is the total number of gradients evaluations required by the
tensor method, divided by the total number of gradients evaluations required by the standard
method on these problems. The same measure is used for the average function evaluation
(feval) and execution time (time) ratios. These average ratios include only problems that were
successfully solved by both methods. We have excluded all cases where the tensor and standard
methods converged to a different minimizer. However, the statistics for the "better", "worse",
and "tie" columns include the cases where only one of the two methods converges, and exclude
the cases where both methods do not converge. We also excluded problems requiring a number
of gradient evaluations less or equal than 3 by both methods. Finally, columns "t/s" and "s/t"
show the number of problems solved by the tensor method but not by the standard method
and the number of problems solved by the standard method but not by the tensor method,
respectively.
The improvement by the tensor method over the standard method on problems with rank
averaging 48% in function evaluations, 52% in gradient evaluations, and 59% in
execution times. This is due in part to the rate of convergence of the tensor method being faster
than that of Newton's method, which is known to be only linearly convergent with constant3
. On problems with rank the improvement by the tensor method over the standard
method is also substantial, averaging 30% in function evaluations, 37% in gradient evaluations,
and 34% in execution times. In the test results obtained for the nonsingular problems, the tensor
method is 9% worse than the standard method in function evaluations, but 31% and 33% better
in gradient evaluations and in execution times, respectively. The main reason for the tensor
method requiring on the average more function evaluations than the standard method is because
on some problems, the full tensor step does not provide sufficient decrease in the objective
function, and therefore the tensor method has to perform a line search in both the Newton
and tensor directions, which causes the number of function evaluations required by the tensor
method to be inflated. As a result, we intend to investigate other possible global frameworks for
line search methods that could potentially reduce the number of functions evaluations for the
tensor method.
To obtain an experimental indication of the local convergence behavior of the tensor and
Newton methods on problems where rank(r 2 f(x examined the sequence of ratios
produced by the Newton and tensor methods on such problems. These ratios for a typical
problem are given in Table 2. In almost all cases the standard method exhibits local linear
convergence with constant near 2, which is consistent with the theoretical analysis. The local
convergence rate of the tensor method is faster with a typical final ratio of around 0.01. Whether
this is a superlinear convergence remains to be determined. We have done similar experiments for
problems with rank(r 2 f(x and the tensor method did not show a faster-than-linear
convergence rate, because it did not have enough information since
The tensor method solved a total of four nonsingular problems, five rank
and 7 rank that Newton's method failed to solve. The reverse never occurred.
This clearly indicates that the tensor method is most likely to be more robust than Newton's
method.
The overall results show that having some extra information about the function and gradient
in the past step direction is quite useful to achieve the advantages of tensor methods.
8. Summary and Future Research
In this paper we presented efficient algorithms for solving large sparse unconstrained optimization
using tensor methods. We described new methods for minimizing the tensor model, that are
efficient for problems where the Hessian matrix is large and sparse. Implementations using these
tensor methods have been shown to be considerably more efficient especially on problems where
Rank tensor/standard pbs solved Average Ratio-Tensor/standard
Table
1: Summary of the CUTE and MINPACK-2 test problems using line search
Iteration (k) Standard method Tensor method
9 0.600 0.126
14 0.969
22 0.896
26 0.667
28 0.666
Table
2: Speed of convergence on the BRYBND problem with rank(r 2 f(x modified
by (7.2), started from x 0 . The ratios in second and third columns are defined by
(7.
the Hessian matrix has a small rank deficiency at the solution. Typical gains over standard
Newton methods range from 40% to 50% in function and gradient evaluations, and in computer
time. The size and consistency of the efficiency gains indicate that the tensor method may be
preferable to Newton's method for solving large sparse unconstrained optimization problems
where analytic gradients and/or Hessians are available. To firmly establish such a conclusion,
additional testing is required, including test problems of very large size.
On sparse problems where the function or the gradient is expensive to evaluate, the finite
difference approximation of the Hessian matrix by the graph coloring algorithm [8] may be
very costly. Hence, Quasi-Newton methods may be preferable to use in this case. These are
methods, which involve low-rank corrections to a current approximate Hessian matrix. We are
currently attempting to extend our tensor methods to Quasi-Newton methods for large sparse
unconstrained minimization problems.
We also considered solving large sparse structured unconstrained optimization problems using
tensor methods. In this variant, we explore the possibility of using exact third and fourth
order derivative information. The calculation of these derivatives is simplified using the concept
of partial separability, a structure which has already proven to be useful when building quadratic
models for large scale nonlinear problems [15]. However, the calculation of the minimizer of this
exact tensor model is more problematic because we need to solve a sparse system of nonlinear
equations. An obvious approach to solve these equations is to use a Newton-like method. Such
a method is characterized by the approximation of the Jacobian used in the Newton process. A
simple idea is to use a fixed Jacobian at each step. This has the advantage that the Jacobian
will have already been obtained in the current tensor iteration. However, potential slow convergence
of such a scheme may make the cost of a tensor iteration prohibitive. We are currently
investigating other possible approaches such as a modified Newton's method in which the approximated
Jacobian matrix will incorporate more useful information, or an iterative method
such as a nonlinear GMRES. This work, a cooperation with Nick Gould [5], will be reported in
the near future.
We are almost done with the implementation and testing of the two-dimensional trust region
global strategy described in x5. This work will be reported in a forthcoming paper.
We are also implementing the algorithms discussed in this paper in a software package. This
package uses one past point in the formation of the tensor terms, which makes the additional
cost and storage of the tensor method over the standard method very small. The package will
be available soon.
Acknowledgments
. We thank Professor Bobby Schnabel for his suggestions on how to
minimize the tensor model when the Hessian is rank deficient, Nick Gould for discussing a number
of implementation issues, Ta-Tung Chow for reviewing the first draft of the paper, and my
CERFACS colleage Jacko Koster for his numerous suggestions.
--R
The Minpack-2 test problem col- lection
CUTE: Constrained and Unconstrained Testing Environment.
Solving large sparse systems of nonlinear equations and nonlinear least squares problems using tensor methods on sequential and parallel computers.
Tensor methods for large-scale unconstrained opti- mization
TENSOLVE: a software package for solving systems of nonlinear equations and nonlinear least squares problems using tensor methods.
Approximation solution of the trust region problem by minimization over two-dimensional subspaces
Estimating sparse hessian matrices.
Numerical methods for unconstrained optimization and nonlinear equations.
A set of Fortran subroutines for solving sparse symmetric sets of linear equations.
Practical method of optimization
Preconditioners for indefinite systems arising in optimization and nonlinear least squares problems.
Practical Optimization.
Analysis of Newton's method at irregular singularities.
On the unconstained optimization of partially separable functions.
The Levenberg-Marquardt algorithm: implementation and theory
A new algorithm for unconstrained optimization.
Tensor methods for unconstrained optimization using second derivatives.
Tensor methods for nonlinear equations.
--TR | large-scale optimization;tensor methods;unconstrained optimization;sparse problems;singular problems |
589234 | A Superlinearly Convergent Primal-Dual Infeasible-Interior-Point Algorithm for Semidefinite Programming. | A primal-dual infeasible-interior-point path-following algorithm is proposed for solving semidefinite programming (SDP) problems. If the problem has a solution, then the algorithm is globally convergent. If the starting point is feasible or close to being feasible, the algorithm finds an optimal solution in at most $O(\sqrt{n}L)$ iterations, where n is the size of the problem and L is the logarithm of the ratio of the initial error and the tolerance. If the starting point is large enough, then the algorithm terminates in at most O(nL) steps either by finding a solution or by determining that the primal-dual problem has no solution of norm less than a given number. Moreover, we propose a sufficient condition for the superlinear convergence of the algorithm. In addition, we give two special cases of SDP for which the algorithm is quadratically convergent. | Introduction
In this paper we consider the semidefinite programming (SDP) problem:
and its associated dual problem:
are given data, and
are the primal and dual variables, respectively. By G ffl H we
denote the trace of (G T H). Without loss of generality, we assume that the matrices C and
are symmetric (otherwise, replace C by (C+C T )=2 and A i by
Also, for simplicity we assume that A i are linearly independent.
Throughout this paper we assume that both (1.1) and (1.2) have finite solutions and
their optimal values are equal. Under this assumption, X and (y ; S ) are solutions of (1.1)
and (1.2) if and only if they are solutions of the following nonlinear system:
Some primal-dual interior-point methods for linear programming have been successfully
extended to solve the SDP problems (1.3). For a survey of results obtained before 1993 in
this field see the paper of Alizadeh [1]. More recent results can be found in [4, 2, 3, 7, 9, 15].
Kojima, Shindoh and Hara [9], Nesterov and Todd [13], and Monteiro [12] extended some
interior-point methods for LP to SDP. In the latter paper Monteiro developed a new formulation
of the primal-dual search direction originally introduced in [9]. All above mentioned
methods, with the exception of the infeasible-interior-point potential-reduction method
of Kojima, Shindoh and Hara [9], require a strictly feasible starting point and therefore
are feasible-interior-point methods. More recently, Zhang [16], Kojima, Shida and Shindoh
[8] and the present authors (in the first version of the paper) independently proposed new
path-following algorithms for SDP. In this version, we have corrected
some flaws in the local convergence analysis contained in the first version. Our algorithm is
a predictor-corrector method generalizing the interior-point method for linear programming
proposed by Mizuno, Todd and Ye [11]. We note that the algorithm of [11] has been also
generalized for linear complementarity problems with feasible starting points in [6] and with
infeasible starting points in [10, 14]. We also mention that the Mizuno-Todd-Ye predictor-corrector
method has been extended to self-scaled cones, which includes SDP, by Nesterov
and Todd[13] under the assumption that the starting point is strictly feasible.
The algorithm to be presented in the present paper is globally convergent whenever the
problem (1.3) has a solution. The starting point does not have to be feasible. In particular
we can take as starting point any positive multiples of the identity matrix.
If the starting point is feasible or close to being feasible our algorithm finds a solution
in at most O(
iterations. If the starting point is large enough, then the algorithm
terminates in at most O(nL) steps either by finding a solution or by determining that
the primal-dual problem has no solution of norm less than a given number. Moreover, we
propose a sufficient condition for the superlinear convergence of the algorithm. The sufficient
condition is satisfied under conditions (A), (B) and (C) of Kojima, Shida and Shindoh
[8]. In addition, we give two special cases of SDP for which the algorithm is quadratically
convergent. Superlinearly convergent algorithms tend to perform in practice much better
than indicated by their iteration complexity which is based on global linear convergence
estimates. Indeed, superlinear convergence has been observed experimentally in efficient
practical algorithms.
The following notation and terminology are used throughout the paper:
the p-dimensional Euclidean space;
nonnegative orthant of IR
the positive orthant of IR
the set of all p \Theta q matrices with real entries;
the set of all p \Theta p symmetric matrices;
: the set of all p \Theta p symmetric positive semidefinite matrices;
: the set of all p \Theta p symmetric positive matrices;
the (i; j)-th entry of a matrix M;
Tr(M the trace of a p \Theta p matrix, equals
0: M is positive semidefinite;
0: M is positive definite;
n: the eigenvalues of M 2 S
the largest, smallest, eigenvalue of M 2 S
Euclidean norm of a vector and the corresponding norm of a matrix, i.e.,
Frobenius norm of a matrix;
An infeasible-interior-point algorithm
We denote the feasible set of the problem (1.3) by
and its solution set by F , i.e.,
The residues of (1.3a) and (1.3b) are denoted by:
For any given ffl ? 0 we define the set of ffl-approximate solutions of (1.3) as
In what follows we present an algorithm that finds a point in this set in a finite number of
steps (provided the problem has a solution). The algorithm will perform in a neighborhood
of the infeasible central
d g:
In our algorithm the positive parameter - will be driven to zero and therefore the residues
will also be driven to zero at the same rate as - . We use the following neighborhood of the
above central
++ \Theta IR m \Theta S n
where fl is a constant such that 1. Throughout the paper we also use the notation:
The algorithm depends on two positive parameters ff; fi satisfying the inequalities
For example, verify (2.3). At a typical step of our algorithm we are given
(X; and obtain a predictor direction (U; w; V ) 2 S n \Theta IR m \Theta S n by solving
the linear system
We will see later on that the above linear system has a unique solution, which we call the
affine scaling direction. If we take a steplength ' along this direction we obtain the points
Theoretically we would like to compute the step length
However this involves computing the root of a complicated nonlinear equation. In Lemma 2.5
we will show that
(2.
where
and
In what follows we assume that a steplength ' satisfying
is computed, and we consider the predicted points
In case of (which is very unlikely), it is easily seen that a solution (X; is at
hand and the algorithm terminates. Now suppose that ' ! 1. Then X and S are symmetric
positive definite matrices since - i (X(')S(') -
Therefore we can define the corrector direction (U ; w; V ) as the solution of the following
linear system
We will prove later on that the above linear system has a unique solution. By taking a unit
steplength along the corrector direction we obtain a new point
Correspondingly, we define
Le us note that in setting up the linear systems (2.4) and (2.11) it is not necessary to compute
square roots of matrices. Indeed, as pointed out by Monteiro [12], it is easily seen that an
equation of the form
can be written equivalently under the form
Summarizing, we can formally define our algorithm as follows:
Algorithm 2.1 Choose (X
For do A1 through A5:
A3 Find the solution U; w; V of the linear system (2.4), define X; as in (2.10), and
set
terminate.
Find the solution U of the linear system (2.11) and define as in
(2.12) and (2.14).
In analysing our algorithm we need the following technical results.
Lemma 2.2 Suppose that M 2 IR p\Thetap is a nonsingular matrix and E 2 IR p\Thetap has at least
one real eigenvalue. Then,
Proof. See Lemma 2.6 of Monteiro [12].
Lemma 2.3 (Monteiro [12], Lemma 2.2) Let (X;
n\Thetan \Theta IR m \Theta IR n\Thetan is a solution of the linear system:
\Deltay
for some H 2 IR n\Thetan , and let
s
F
The following corollary is essentially Corollary 2.3 of Monteiro [12].
Corollary 2.4 Let (X;
the system of linear equations (2.18) has a unique solution (D
Therefore, both linear systems (2.4) and (2.11) have unique solutions.
Lemma 2.5 If (X;
defined by (2.5) satisfies (2.6) where b
' is
given by (2.7) and (2.8).
Proof. By definition, we have
If we set
then, in view of (2.19) and (2.4a), we obtain
Therefore,2
Hence, for any given parameter - 2 [0; 1) for all ' 2 [0; min( b
'; -)), we must have X(') -
-)). Otherwise, there must exist
such that X(' 0 )S(' 0 ) is singular, which means
However, using (2.16) with
(from (2:20))
which contradicts (2.21). Since X(') - 0, its square root X(') 1=2 exists and is uniquely
defined. Applying (2.17) of Lemma 2.2 with
noting that P
(from (2:20))
Therefore,
choose
which gives ' - b
'. Finally, if b
for all ' 2 [0; 1), which implies X(1) - 0;
'.
Before stating our main result let us note that the standard choice of starting points
is perfectly centered and satisfies (X required in the algorithm.
We will see that if the problem has a solution, then for any ffl ? 0 Algorithm 2.1 terminates
in a finite number (say K ffl ) of iterations. If then the algorithm is likely to generate an
infinite sequence. However it may happen that at a certain iteration (let us say at iteration
which implies that an exact solution is obtained, and therefore the
algorithm terminates at iteration K 0 . If this (unlikely) phenomenon does not happen we set
Theorem 2.6 For any integer 0 - k ! K 0 , Algorithm 2.1 defines a triple
and the corresponding residuals satisfy
where
and ' j is defined by (2.9).
Proof. The proof is by induction. For are clearly satisfied. Suppose
they are satisfied for some k - 0. As in Algorithm 2.1 we will omit the index k. Therefore
we can write
(X;
The fact that (2.23) and (2.24) hold for immediately from (2.13) and (2.14).
From (2.12) and (2.11a) we have
Then, recalling (2.26) and (2.11a), we obtain
Hence by applying Lemma 2.3 we
deduce
Using Lemma 2.3 again, we have
which implies that I
exists. Using (2.26), (2.27), applying Lemma 2.2 with
noting that
(from (2:27))
The above inequality implies that
Hence which gives S In view of (2.29), this shows that (2.22)
holds Finally, (2.25) is an immediate consequence of (2.22).
3 Global convergence and polynomial complexity
In this section we assume that F is nonempty. Under this assumption we will prove that
Algorithm 2.1, with globally convergent in the sense that
lim
lim
In the sequel, we will frequently use the following inequality: for any M 1 ; M 2 2 IR n\Thetan ,
(see exercise 20 in section 5.6 of [5]).
Lemma 3.1 For any ( f
Proof. Let
From (2.1), (2.23), it is easily seen that (X
which implies X 0 ffl S
By expanding (3.2) we obtain the desired result.
Lemma 3.2 Assume that F is nonempty. Then for any (X ; y
N (ff; -) we have
where
Proof. The results follow by using Lemma 3.1 with ( f
Theorem 2.6
and the fact that S ffl X
Lemma 3.2 shows that the pair (X k generated by Algorithm 2.1 is bounded. More
precisely, we have the following corollary, which can easily be deduced from Lemma 3.2 and
Theorem 2.6.
Corollary 3.3
(3.
Lemma 3.4 Suppose ( f
Then the quantity ffi defined by (2.8) satisfies the inequality:
!/
Proof. It is easily seen that (U
Hence, according to Lemma 2.3, we have
Therefore,
and the lemma follows by virtue of (2.8).
Lemma 3.5 Under the hypothesis of Lemma 3.4 we have
where
Proof. Using the notation of Lemma 3.4, and Lemma 3.3, we can write
Also,
and
Then (3.11) follows from Lemma 3.4.
According to Lemma 2.5 and Lemma 3.5, it follows that if F is not empty, then the
step length ' k defined by (2.9) is bounded away from 0. This implies global convergence as
shown in the following theorem.
Theorem 3.6 If F is not empty, then Algorithm 2.1 is globally convergent at a linear rate.
Moreover, the iteration sequence (X bounded and every accumulation point of
belongs to F (i.e., is a primal dual optimal solution of the SDP problem).
Using Lemma 3.5, we can easily deduce the following result.
Theorem 3.7 Suppose that F is nonempty and that the starting point is chosen such that
there is a constant - independent of n satisfying the inequality
Then Algorithm 2.1 terminates in at most O(
d k; jR 0
Theorem 3.8 Suppose constant such that kX k -
. Then the step length ' k defined by (2.9) satisfies
the inequality
with
Proof. According to Lemma 3.2, we have
i.e.,
In the sequel we will frequently use the fact that ff ! 0:5. Since X ffl S we get the
relation
which implies
Applying (3.16), (3.17), Lemma 3.3, and Lemma 3.4 with ( f
F
F
In view of (3.18), (3.19) and Lemma 3.3, we obtain
F
F
Therefore,
!/
Consequently, ' ? 1=(!n).
In the following corollary we summarize the complexity results for standard starting point
of the form X
Corollary 3.9 Assume that in Algorithm 2.1 we choose a starting point of the form X
constant. Let ffl 0 be given by (3.12) and let ffl ? 0 be arbitrary.
Then the following statements hold:
(i) If F 6= ;, then the algorithm terminates with an ffl-approximate solution (X
F ffl in a finite number of steps
(iii) For any choice of ae ? 0 there is an index
such that either
or,
and in the latter case there is no solution (X kg.
4 Superlinear convergence
The next two lemmas are well known and can be easily proved.
Lemma 4.1 Let A 1
. Then A 1 ffl A
. Then, there exists an orthogonal matrix
such that Q T X Q and Q T S Q are diagonal matrices. In other words, q are eigen-vectors
of X and S .
Definition 4.3 A triple (X ; y is called a strictly complementary solution of
In this section we investigate the asymptotic behavior of Algorithm 2.1. We will propose a
sufficient condition for the superlinear convergence of Algorithm 2.1.
Assumption 1. The SDP problem has a strictly complementary solution (X
be an orthogonal matrix such that q are eigenvectors of X
and S , and define
It is easily seen that IB [ ng. For simplicity, let us assume that
where B and N are diagonal matrices. Here and in the sequel, if we write a matrix M in
the block form
then we assume that the dimensions of M 11 and M 22 are jIBj \Theta jIBj and jINj \Theta jINj, respectively.
In the next lemma we use the following notations:
Lemma 4.4 Under Assumption 1,
ks
ks
Proof. Because the sequence f(X k ; S k )g is bounded, we have
ks
In view of (3.3b), we get
Tr(X
where
Tr(X
and q T
Similarly, ks
From
Theorem 2.6, we obtain
which implies
i.e.,
Therefore,
Hence, for any i 2 IB we can write
ks
Also, for any i 2 IN,
ks
Therefore, kb x
Similarly, by considering
we can show that kb s
Using Lemma 4.4, we can write
O(
-) O(
O(
O(
Let us define a linear manifold:
It is easily seen that if (X
Lemma 4.5 Under Assumption 1, F ae M.
Proof. For any (X ; y
Hence,
which implies
4.1, we have
which implies
i.e.,
If i or j 2 IB, then q T
is positive, which implies q T
according
to (4.4). Similarly, we can show that q T
Therefore,
which gives F ae M.
Lemma 4.6 Under Assumption 1, every accumulation point of (X strictly
complementary solution of (1.3).
Proof. Suppose (X "; y"; S") 2 F is an accumulation point. Let us assume, without loss
of generality, that (X k ; y k according to Lemma 4.5,
for some symmetric positive semidefinite matrices MB and MN . In order to show
0, it remains to prove that MB and MN are nonsingular and therefore positive definite. From
Lemma 4.4 or (4.1), we have
O(
it has an accumulation
point. Without loss of generality, we may assume
we obtain
which implies
Hence f
must be nonsingular. Obviously f
so that MB is nonsingular. Similarly,
we can show that MN is nonsingular.
In the next theorem, we propose a sufficient condition for the superlinear convergence of
Algorithm 2.1. Let us define
is the solution of the following minimization problem:
and \Gamma is a constant such that k(X k ; S k )k F - \Gamma; 8k. Note that every accumulation point of
belongs to the feasible set of the above minimization problem and the feasible
set is bounded. Therefore ( -
exists for each k.
Theorem 4.7 Under Assumption 1, if Algorithm 1.3 is superlinearly
convergent. Moreover, if there exists a constant oe ? 0 such that
k ), then the
convergence has Q-order at least 1 oe in the sense that -
Proof. By Lemma 2.5, it remains to prove that us
omit the index k. It is easily seen that (U
with
where
Here we have used the relation -
Denoting
and applying Lemma 2.3, we obtain
which implies
Similarly,
By Lemma 4.4 and the fact that ( -
Similarly,
Let us observe that
Then from (4.8), (4.9), (4.10), (4.11), (4.12) and Corollary 3.3, we have
Hence, Finally, if
k ) for some constant oe ? 0, then we have
Therefore,
Recalling (2.25), we obtain -
We mention that our sufficient condition in the above theorem is satisfied under conditions
(A), (B) and (C) of Kojima, Shida and Shindoh [8].
We end this paper by giving two special cases of SDP for which Algorithm 2.1 is quadratically
convergent.
Proposition 4.8 If ng or Algorithm 2.1 is quadratically
convergent.
Proof. Let
S) be the solution of the minimization problem:
min
We will show that
We will prove (4.14) only for ng and a similar proof applies to the case
ng. The manifold M reduces to
Assume by contradiction that (4.14) does not hold, i.e., there exists a subsequence such that
Let us define
It is easily seen that
(\Deltay k
\Deltay k depends linearly on \DeltaS k (cf. (4.18)), we deduce that
there exists a convergent subsequence of (\DeltaX k ; \Deltay k ; \DeltaS k ). Without loss of generality we
can write
Letting k !1 in (4.18), we obtain
(\Deltay 0
From Lemma 4.4, we have for each i 2
ks
which implies that \DeltaS
Since
is not the solution of the minimization problem (4.13) for sufficiently large k,
which is a contradiction.
It is easily seen from (4.14) that ( -
Hence we can choose \Gamma in (4.6) such
that
Therefore,
From Lemma 4.4 we deduce that if ng or
In view of (4.14), (4.20) and (4.21), we have and the result follows from Theorem
4.7.
Acknowledgment
The authors would like to thank Professor Masakazu Kojima, Masayuki Shida, and
Susumu Shindoh for sending us their paper and kindly pointing out some errors in the
first version of the present paper.
--R
Interior point methods in semidefinite programming with applications to combinatorial optimization.
Complexity and nondegeneracy in semidefinite programming.
An interior-point method for semidefinite programming
Matrix Analysis.
A predictor-corrector method for linear complementarity problems with polynomial complexity and superlinear convergence
Linear algebra for semidefinite programming.
Global and local convergence of predictor-corrector infeasible-interior-point algorithms for semidefinite programs
A unified approach to infeasible-interior-point algorithms via geometrical linear complementarity problems
On adaptive-step primal-dual interior-point algorithms for linear programming
A modified O(nL) infeasible-interior-point algorithm for LCP with quadratic convergence
Positive definite programming.
On extending primal-dual interior-point algorithms from linear programming to semidefinite programming
--TR
--CTR
S. J. Li , S. Y. Wu , X. Q. Yang , K. L. Teo, A relaxed cutting plane method for semi-infinite semi-definite programming, Journal of Computational and Applied Mathematics, v.196 n.2, p.459-473, 15 November 2006
Zhensheng Yu, Solving semidefinite programming problems via alternating direction methods, Journal of Computational and Applied Mathematics, v.193 n.2, p.437-445, 1 September 2006
Stefania Bellavia , Sandra Pieraccini, Convergence Analysis of an Inexact Infeasible Interior Point Method for Semidefinite Programming, Computational Optimization and Applications, v.29 n.3, p.289-313, December 2004 | path-following;superlinear convergence;infeasible-interior-point algorithm;polynomiality;semidefinite programming |
589237 | BFGS with Update Skipping and Varying Memory. | We give conditions under which limited-memory quasi-Newton methods with exact line searches will terminate in n steps when minimizing n-dimensional quadratic functions. We show that although all Broyden family methods terminate in n steps in their full-memory versions, only BFGS does so with limited-memory. Additionally, we show that full-memory Broyden family methods with exact line searches terminate in at most n steps when p matrix updates are skipped. We introduce new limited-memory BFGS variants and test them on nonquadratic minimization problems. | Introduction
. The quasi-Newton family of algorithms remains a standard
workhorse for minimization. Many of these methods share the properties of finite
termination on strictly convex quadratic functions, a linear or superlinear rate of
convergence on general convex functions, and no need to store or evaluate the second
derivative matrix. In general, an approximation to the second derivative matrix is
built by accumulating the results of earlier steps. Descriptions of many quasi-Newton
algorithms can be found in books by Luenberger [16], Dennis and Schnabel [7], and
Golub and Van Loan [11].
Although there are an infinite number of quasi-Newton methods, one method surpasses
the others in popularity: the BFGS algorithm of Broyden, Fletcher, Goldfarb,
and Shanno; see, e.g., Dennis and Schnabel [7]. This method exhibits more robust behavior
than its relatives. Many attempts have been made to explain this robustness,
but a complete understanding is yet to be obtained [23]. One result of the work in this
paper is a small step toward this understanding, since we investigate the question of
how much and which information can be dropped in BFGS and other quasi-Newton
methods without destroying the property of quadratic termination.
We answer this question in the context of exact line search methods, those that
find a minimizer on a one-dimensional subspace at every iteration. (In practice,
inexact line searches that satisfy side conditions such as those proposed by Wolfe, see
x4.3, are substituted for exact line searches.) We focus on modifications of well-known
quasi-Newton algorithms resulting from limiting the memory, either by discarding
the results of early steps (x2) or by skipping some updates to the second derivative
approximation (x3). We give conditions under which quasi-Newton methods will
terminate in n steps when minimizing quadratic functions of n variables. Although
all Broyden family methods (see x2) terminate in n steps in their full-memory versions,
we show that only BFGS has n-step termination under limited-memory. We also show
that the methods from the Broyden family terminate in n steps even if p updates
are skipped, but termination is lost if we both skip updates and limit the memory.
y Applied Mathematics Program, University of Maryland, College Park, MD 20742.
gibson@math.umd.edu. This work was supported in part by the National Physical Science Con-
sortium, the National Security Agency, and the University of Maryland.
z Department of Computer Science and Institute for Advanced Computer Studies, University of
Maryland, College Park, MD 20742. oleary@cs.umd.edu. This work was supported by the National
Science Foundation under grant NSF 95-03126.
x Department of Pure and Applied Mathematics, Washington State University, Pullman, WA
99164. nazareth@amath.washington.edu.
T. Gibson, D. P. O'Leary, L. Nazareth
In x4, we report the results of experiments with new limited-memory BFGS variants
on problems taken from the CUTE [3] test set, showing that some savings in
time can be achieved.
Notation. Matrices and vectors are denoted by boldface upper-case and lower-case
letters respectively. Scalars are denoted by Greek or Roman letters. The superscript
"T" denotes transpose. Subscripts denote iteration number. Products are always
taken from left to right. The notation spanfx 1 denotes the subspace
spanned by the vectors x Whenever we refer to an n-dimensional strictly
convex quadratic function, we assume it is of the form
where A is a positive definite n \Theta n matrix and b is an n-vector.
2. Limited-Memory Variations of Quasi-Newton Algorithms. In this
section we characterize full-memory and limited-memory methods that terminate in
n iterations on n-dimensional strictly convex quadratic minimization problems using
exact line searches. Most full-memory versions of the methods we will discuss are
known to terminate in n iterations. Limited-memory BFGS (L-BFGS) was shown by
Nocedal [22] to terminate in n steps. The preconditioned conjugate gradient method,
which can be cast as a limited-memory quasi-Newton method, is also known to terminate
in n iterations; see, e.g., Luenberger [16]. Little else is known about termination
of limited-memory methods.
Let f(x) denote the strictly convex quadratic function to be minimized, and let
g(x) denote the gradient of f . We define is the kth iterate. Let
denote the change in the current iterate and
denote the change in gradient.
Let x 0 be the starting point, and let H 0 be the initial inverse Hessian approximation.
For
1. Compute
2. Choose ff k ? 0 such that f(x k
3. Set s
4. Set x
5. Compute
7. Choose H k+1 .
Fig. 2.1. General Quasi-Newton Method
We present a general result that characterizes quasi-Newton methods, see Figure 2.1,
that terminate in n iterations. We restrict ourselves to methods with an update of
the form
mk
Here,
L-BFGS Variations 3
1. H 0 is an n \Theta n symmetric positive definite matrix that remains constant for
all k, and fl k is a nonzero scalar that can be thought of as an iterative rescaling of
2. P k is an n \Theta n matrix that is the product of projection matrices of the form
where is an n \Theta n matrix
that is the product of projection matrices of the same form where u is any n-vector
3. m k is a nonnegative integer, w ik n-vector, and z ik
vector in spanfs g.
We refer to this form as the general form. The general form fits many known
quasi-Newton methods, including the Broyden family and the limited-memory BFGS
method. We do not assume that these quasi-Newton methods satisfy the secant
condition,
nor that H k+1 is positive definite and symmetric. Symmetric positive definite updates
are desirable since this guarantees that the quasi-Newton method produces descent
directions. Note that if the update is not positive definite, we may produce a d k such
that d T
which case we choose ff k over all negative ff rather than all positive
ff.
Example. The method of steepest descent [16] fits the general form (2.1). For
each k we define
Note that neither w nor z vectors are specified since m
Example. The 1)st update for the conjugate gradient method with preconditioner
fits the general form (2.1) with
Example. The L-BFGS update, see Nocedal [22], with limited-memory constant
m can be written as
k\Gammam k+1;k
Y
L-BFGS fits the general form (2.1) if at iteration k we choose
4 T. Gibson, D. P. O'Leary, L. Nazareth
Observe that P k ; Q k and z ik all obey the constraints imposed on their construction.
BFGS is related to L-BFGS is the following way: if we were to use every (s; y)
pair in the formation of each update (i.e. we have unlimited memory), we would be
creating the same updates as BFGS. In practice, however, one would never do that
because it would take more memory than storing the BFGS matrix.
Example. We will define limited-memory DFP (L-DFP). Our definition is consistent
with the definition of limited-memory BFGS given in Nocedal [22]. Let m - 1
mg. In order to define the L-DFP update we need to create
a sequence of auxiliary matrices for
where
U DFP (H; s;
ss T
The matrix -
k+1 is the result of applying the DFP update m k times to the matrix
H 0 with the m k most recent (s; y) pairs. Thus, the 1)st L-DFP matrix is given
by
To simplify our description, note that -
k+1 can be rewritten as
k\Gammam k+i
k\Gammam k+i
k\Gammam k+i
k\Gammam k+i
y k\Gammam k+i
k\Gammam k+j
k\Gammam k+j
y k\Gammam k+j
Y
y k\Gammam k+l
k\Gammam k+l
Thus H k+1 can be written as
mk
ik
k\Gammam k+i
k\Gammam k+i
y k\Gammam k+i
where
mk
Y
y k\Gammam k+j
H (j \Gamma1)
k\Gammam k+j
H (j \Gamma1)
L-BFGS Variations 5
Equation (2.7) looks very much like the general form given in (2.1). L-DFP fits the
general form with the following choices:
k\Gammam k+i
y k\Gammam k+i ); and z
Except for the choice of P k , it is trivial to verify that the choices satisfy the general
form. To prove that P k satisfies the requirements, we need to show
Proposition 2.1. For limited-memory DFP, the following two conditions hold
for each value of k:
Proof. We will prove this via induction. Suppose We have
(Recall that spanfs 0 g is trivially equal to spanfH 0 g 0 g.) Furthermore,
So we can conclude,
and so the base case holds.
Assume that
We will use induction on i to show (2.10) for the 1)st case. For
Using the induction assumptions from the induction on k, we get that
6 T. Gibson, D. P. O'Leary, L. Nazareth
Assume that -
(induction assumption for i). Next,
For values of
maps any vector v into
and so -
is in
Using the induction assumptions for both i and k, we get
and we can continue the induction on i. If
so
Hence the induction on i is complete and this proves (2.10) in the (k 1)st case.
consider
mk
k\Gammam k+i
k\Gammam k+i
Using the structure of V jk and (2.10) we see that
Hence, (2.11) also holds in the (k 1)st case.
Example. The Broyden Class or Broyden Family is the class of quasi-Newton
methods whose matrices are linear combinations of the DFP and BFGS matrices:
see, e.g., Luenberger [16, Chap. 9]. The parameter OE is usually restricted to values
that are guaranteed to produce a positive definite update, although recent work with
SR1, a Broyden Class method, by Khalfan, Byrd and Schnabel [14] may change this
practice. No restriction on OE is necessary for the development of our theory. The
Broyden class update can be expressed as
Variations 7
We sketch the explanation of how the full-memory version fits the general form
given in (2.1). The limited-memory case is similar. We can rewrite the Broyden Class
update as follows:
Hence,
where
hi
y
s
y
It is left to the reader to show that H k y k is in spanfs thus the
Broyden Class updates fit the form in (2.1).
2.1. Termination of Limited-Memory Methods. In this section we show
that methods fitting the general form (2.1) produce conjugate search directions (The-
orem 2.2) and terminate in n iterations (Corollary 2.3) if and only if P k maps
spanfy into spanfy for each n. Furthermore, this
condition on P k is satisfied only if y k is used in its formation (Corollary 2.4).
Theorem 2.2. Suppose that we apply a quasi-Newton method (Figure 2.1) with
an update of the form (2.1) to minimize an n-dimensional strictly convex quadratic
function. Then for each k before termination (i.e. g k+1 6= 0),
As
if and only if
Proof. (() Assume that (2.15) holds. We will prove (2.12)-(2.14) by induction.
Since the line searches are exact, g 1 is orthogonal to s 0 . Using the fact that P 0 y
8 T. Gibson, D. P. O'Leary, L. Nazareth
from (2.15), and the fact that z i0 2 spanfs 0 g implies g T
see that s 1 is conjugate to s 0 since
z i0 w T
0:
Lastly, spanfs 0 g, and so the base case is established.
We will assume that claims (2.12)-(2.14) hold for
that they also hold for
The vector g - k+1 is orthogonal to s - k since the line search is exact. Using the
induction hypotheses that g -
k is orthogonal to fs
is conjugate to
g, we see that for
Hence, (2.12) holds for
To prove (2.13), we note that
As
so it is sufficient to prove that g T
We will use the
following facts:
- k+1 since the v in each of the projections used to form Q - k is in
k+1 is orthogonal to that span.
since each z i - k is in spanfs
is orthogonal to that span.
(iii) Since we are assuming that (2.15) holds true, for each
there
exists can be expressed as
P-
(iv) For is orthogonal to H 0 y i because g -
k+1 is orthogonal
to spanfs
from (2.14).
Thus,
Variations 9
0:
Thus, (2.13) holds for
k.
Lastly, using (i) and (ii) from above,
maps any vector v into spanfv; s by construction, there exist
Hence,
so
To show equality of the sets, we will show that H 0 g - k+1 is linearly independent of
g. (We already know that the basis fH 0 is linearly
independent since it spans the same space as the linearly independent set fs
and has the same number of elements.) Suppose that H 0 g - k+1 is not linearly indepen-
dent. Then there exist OE
k , not all zero, such that
Recall that g - k+1 is orthogonal to fs
g. By our induction assumption, this
implies that g -
k+1 is also orthogonal to fH 0 g. Thus for any j between 0
and - k,
positive definite and g j is nonzero, we conclude that OE j must be zero.
Since this is true for every j between zero and k, we have a contradiction. Thus, the
set fH 0 is linearly independent. Hence, (2.14) holds for
k.
Assume that (2.12)-(2.14) hold for all k such that g k+1 6= 0 but that (2.15)
does not hold; i.e., there exist j and k such that g k+1 6= 0, j is between 0 and k, and
(2.
T. Gibson, D. P. O'Leary, L. Nazareth
This will lead to a contradiction. By construction of P k , there exist -
that
By assumption (2.16), - k must be nonzero. From (2.13), it follows that g T
Using facts (i), (ii), and (iv) from before, (2.14) and (2.17), we get
mk
z ik w T
ik
mk
Thus since neither fl k nor - k is zero, we must have
but this is a contradiction since H 0 is positive definite and g k+1 was assumed to be
nonzero.
When a method produces conjugate search directions, we can say something about
termination.
Corollary 2.3. Suppose we have a method of the type described in Theorem 2.2
satisfying (2.15). Suppose further that H j Then the scheme
reproduces the iterates from the conjugate gradient method with preconditioner H 0 and
terminates in no more than n iterations.
Proof. Let k be such that are all nonzero and such that H i
we have a method of the type described in Theorem 2.2 satisfying
(2.15), conditions (2.12) - (2.14) hold. We claim that the (k 1)st subspace of
search directions, spanfs is equivalent to the 1)st Krylov subspace,
g.
From (2.14), we know that spanfs g. We will
show via induction that spanfH 0 g.
This base case is trivial, so assume that
for some
L-BFGS Variations 11
and from (2.14) and the induction hypothesis,
which implies that
Hence, the search directions span the Krylov subspace. Since the search directions
are conjugate (2.13) and span the Krylov subspace, the iterates are the same as those
produced by conjugate gradients with preconditioner H 0 .
Since we produce the same iterates as the conjugate gradient method and the
conjugate gradient method is well-known to terminate within n iterations, we can
conclude that this scheme terminates in at most n iterations.
Note that we require that H j g j be nonzero whenever g j is nonzero; this requirement
is necessary since not all the methods produce positive definite updates and it
is possible to construct an update that maps g j to zero. If this were to happen, we
would have a breakdown in the method.
The next corollary defines the role that the latest information (s k and y k ) plays
in the formation of the kth H-update.
Corollary 2.4. Suppose we have a method of the type described in Theorem
2.2 satisfying (2.15). Suppose further that at the kth iteration P k is composed
of p projections of the form in (2.2). Then at least one of the projections must have
is a single projection (p = 1), then v
must be of the form
Proof. Consider the case of p = 1. We have
where k+1g. We will assume that
for some scalars oe i and ae i . By (2.15), there exist -
Then
and so
(2.
12 T. Gibson, D. P. O'Leary, L. Nazareth
From (2.13), the set fs is conjugate and thus linearly independent. Since we
are working with a quadratic, y As i for all i; and since A is symmetric positive
definite, the set fy is also linearly independent. So the coefficient of the y k
on the left-hand side of (2.18) must match that on the right-hand side, thus
Hence,
and y k must make a nontrivial contribution to P k .
Next we will show that ae Assume that j is between 0
As j
As j
Now s j As j is nonzero because A is positive definite. If ae j is nonzero then the coefficient
of u is nonzero and so y k must make a nontrivial contribution to P k y j , implying
that g. This is a contradiction. Hence, ae
To show that ae k 6= 0, consider P k y k . Suppose that ae
As k+1
and so
This contradicts P k y k 2 spanfy must be nonzero.
Now we will discuss that p ? 1 case. Label the u-components of the p projections
as
for some scalars fl 1 through fl p . We know that
L-BFGS Variations 13
and that
Thus
and since u we can conclude that at least one u i
must have a nontrivial contribution from y k .
2.2. Examples of Methods that Reproduce the CG Iterates. Here are
some specific examples of methods that fit the general form, satisfy condition (2.15)
of Theorem 2.2, and thus terminate in at most n iterations.
Example. The conjugate gradient method with preconditioner H 0 , see (2.4),
satisfies condition (2.15) of Theorem 2.2 since
Example. Limited-memory BFGS, see (2.6), satisfies condition (2.15) of Theorem
2.2 since
ae 0 for
Example. DFP (with full memory), see (2.8), satisfies condition (2.15) of Theorem
2.2. Consider P k in the full memory case. We have
Y
For full-memory DFP, H i y 1. Using this fact, one can easily
verify that P k y Therefore, full-memory DFP satisfies condition
(2.15) of Theorem 2.2. The same reasoning does not apply to the limited-memory
case as we shall show in x2.3.
The next corollary gives some ideas for other methods that are related to L-BFGS
and terminate in at most n iterations on strictly convex quadratics.
Corollary 2.5. The L-BFGS (2.5) method will terminate in n iterations on
an n-dimensional strictly convex quadratic function even if any combination of the
following modifications is made to the update:
1. Vary the limited-memory constant, keeping m k - 1.
2. Form the projections used in V k from the most recent along with
any set of other pairs from f(s
3. Form the projections used in V k from the most recent along with
linear combinations of pairs from f(s 0 ; y
4. Iteratively rescale H 0 .
Proof. For each variant, we show that the method fits the general form in (2.1),
satisfies condition (2.15) of Theorem 2.2 and hence terminates by Corollary 2.3.
1. Let m ? 0 be any value which may change from iteration to iteration, and
define
Y
14 T. Gibson, D. P. O'Leary, L. Nazareth
Choose
These choices fit the general form. Furthermore,
so this variation satisfies condition (2.15) of Theorem 2.2.
2. This is a special case of the next variant.
3. At iteration k, let (- s (i)
y (i)
k ) denote the ith choice of any
linear combination from the span of the set
and let (- s (m)
Y
(- y (i)
Choose
These choices satisfy the general form (2.1). Furthermore,
ae
k for some i; and
Hence, this variation satisfies condition (2.15) of Theorem 2.2.
4. Let fl k in (2.1) be the scaling constant, and choose the other vectors and
matrices as in L-BFGS (2.6).
Combinations of variants are left to the reader.
Remark. Part 3 of the previous corollary shows that the "accumulated step"
method of Gill and Murray [10] terminates on quadratics.
Remark. Part 4 of the previous corollary shows that scaling does not affect
termination in L-BFGS. In fact, for any method that fits the general form, it is easy
to see that scaling will not affect termination on quadratics.
2.3. Examples of Methods that Do Not Reproduce the CG Iterates.
We will discuss several methods that fit the general form given in (2.1) but do not
satisfy the conditions of Theorem 2.2.
L-BFGS Variations 15
Example. Steepest descent, see (2.3), does not satisfy condition (2.15) of Theorem
2.2 and thus does not produce conjugate search directions. This fact is well-
known; see, e.g., Luenberger [16].
Example. Limited-memory DFP, see (2.8), with does not satisfy the condition
on P k (2.15) for all k, and so the method will not produce conjugate directions.
For example, suppose that we have a convex quadratic with
Using a limited-memory constant of exact arithmetic, it can be seen that
the iteration does not terminate within the first 20 iterations of limited-memory DFP
with I. The MAPLE notebook file used to compute this example is available
on the World Wide Web [9].
Remark. Using the above example, we can easily see that no limited-memory
Broyden class method except limited-memory BFGS terminates within the first n
iterations.
3. Update-Skipping Variations for Broyden Class Quasi-Newton Algo-
rithms. The previous section discussed limited-memory methods that behave like
conjugate gradients on n-dimensional strictly convex quadratic functions. In this sec-
tion, we are concerned with methods that skip some updates in order to reduce the
memory demands. We establish conditions under which finite termination is preserved
but delayed for the Broyden Class.
3.1. Termination when Updates are Skipped. It was shown by Powell [26]
that if we skip every other update and take direct prediction steps (i.e. steps of length
one) in a Broyden class method, then the procedure will terminate in no more than
2n+1 iterations on an n-dimensional strictly convex quadratic function. An alternate
proof of this result is given by Nazareth [21].
We will prove a related result. Suppose that we are doing exact line searches using
a Broyden Class quasi-Newton method on a strictly convex quadratic function and
decide to "skip" p updates to H (i.e. choose H occasions). Then, the
algorithm terminates in no more than n iterations. In contrast to Powell's result,
it does not matter which updates are skipped or if multiple updates are skipped in a
row.
Theorem 3.1. Suppose that a Broyden Class method using exact line searches
is applied to an n-dimensional strictly convex quadratic function and p updates are
skipped. Let
the update at iteration j is not skippedg:
Then for all
As
Furthermore, the method terminates in at most n iterations at the exact minimizer
Proof. We will use induction on k to show (3.1) and
T. Gibson, D. P. O'Leary, L. Nazareth
Then (3.2) follows easily since for all j 2 J(k),
As
0:
be the least value of k such that J(k) is nonempty; i.e., J(k 0 g.
Then g k0+1 is orthogonal to s k0 since line searches are exact, and H k0+1 y
since all members of the Broyden Family satisfy the secant condition. Hence, the base
case is true. Now assume that (3.1) and (3.3) hold for all values of
We will show that they also hold for
Case I. Suppose that - k 62 J( - k). Then H -
k and J( -
any j 2 J( - k),
As j
and
Case II. Suppose that - k 2 J( - k). Then H - k+1 satisfies the secant condition and
kg. Now g - k+1 is orthogonal to s k since the line searches are exact,
and it is orthogonal to the older s j by the argument in (3.4). The secant condition
guarantees that H - k+1 y
k , and for
we have
!/
As j
!/
As j
In either case, the induction result follows.
Suppose that we skip p updates. Then the set J(n cardinality n.
Without loss of generality, assume that the set fs i g i2J(n\Gamma1+p) has no zero elements.
From (3.2), the vectors are linearly independent. By (3.1),
and so gn+p must be zero. This implies that xn+p is the exact minimizer of f .
L-BFGS Variations 17
3.2. Loss of Termination for Update Skipping with Limited-Memory.
Unfortunately, updates that use both limited-memory and repeated update-skipping
do not produce n conjugate search directions for n-dimensional strictly convex qua-
dratics, and the termination property is lost. We will show a simple example, limited-memory
skipping every other update. Note that according to
Corollary 2.4, we would still be guaranteed termination if we used the most recent
information in each update.
Example. Suppose that we have a convex quadratic with
We apply limited-memory BFGS with limited-memory constant
and skip every-other update to H. Using exact arithmetic in MAPLE, we observe
that the process does not terminate even after 100 iterations [9].
4. Experimental Results. The results of x2 and x3 lead to a number of ideas
for new methods for unconstrained optimization. In this section, we motivate, de-
velop, and test these ideas. We describe the collection of test problems in x4.2. The
test environment is described in x4.3. Section 4.4.1 outlines the implementation of the
L-BFGS method (our base for all comparisons) and xx4.4.2-4.4.7 describe the varia-
tions. Pseudo-code for L-BFGS and its variations is given in Appendix B. Complete
numerical results, many graphs of the numerical results, and the original FORTRAN
code are available [9].
4.1. Motivation. So far we have only given results for convex quadratic func-
tions. While termination on quadratics is beautiful in theory, it does not necessarily
yield insight into how these methods will do in practice.
We will not present any new results relating to convergence of these algorithms
on general functions; however, many of these can be shown to converge using the
convergence analysis presented in x7 of [15]. In [15], Liu and Nocedal show that
a limited-memory BFGS method implemented with a line search that satisfies the
strong Wolfe conditions (see x4.3 for a definition) is R-linearly convergent on a convex
function that satisfies a few modest conditions.
4.2. Test Problems. For our test problems, we used the Constrained and Unconstrained
Testing Environment (CUTE) by Bongartz, Conn, Gould and Toint. The
package is documented in [3] and can be obtained via the world wide web [2] or via ftp
[1]. The package contains a large collection of test problems as well as the interfaces
necessary for using the problems. The test problems are stored as "SIF" files. We
chose a collection of 22 unconstrained problems. The problems ranged in size from
to 10,000 variables, but each took L-BFGS with limited-memory constant
at least 60 iterations to solve. Table 4.1 enumerates the problems, giving the SIF file
name, the dimension (n), and a description for each problem. The CUTE package
also provides a starting point
4.3. Test Environment. We used FORTRAN77 code on an SGI Indigo 2 to
run the algorithms, with FORTRAN BLAS routines from NETLIB. We used the
compiler's default optimization level.
Figure
2.1 outlines the general quasi-Newton implementation that we followed.
For the line search, we use the routines cvsrch and cstep written by Jorge J. Mor'e
T. Gibson, D. P. O'Leary, L. Nazareth
No. SIF Name n Description & Reference
Extended Rosenbrock function (nonseparable
version) [30, Problem 10].
problem [17, Problem 20].
3 TOINTGOR 50 Toint's operations research problem [29].
4 TOINTPSP 50 Toint's PSP operations research problem [29].
5 CHNROSNB 50 Chained Rosenbrock function [29].
6 ERRINROS 50 Nonlinear problem similar to CHNROSNB [28].
7 FLETCHBV 100 Fletcher's boundary value problem [8, Problem
1].
8 FLETCHCR 100 Fletcher's chained Rosenbrock function [8, Problem
2].
9 PENALTY2 100 Second penalty problem [17, Problem 24].
Problem 5].
11 BDQRTIC 1000 Quartic with a banded Hessian with band-
diagonal variant of the Broyden tridiagonal
system with a band away from diagonal [29].
First penalty problem [17, Problem 23].
14 POWER 1000 Power problem by Oren [25].
MSQRTALS 1024 The dense matrix square root problem by Nocedal
and Liu (case 0) seen as a nonlinear equation
problem [4, Problem 204].
MSQRTBLS 1025 The dense matrix square root problem by Nocedal
and Liu (case 1) seen as a nonlinear equation
problem [4, Problem 201].
17 CRAGGLVY 5000 Extended Cragg & Levy problem [30, Problem
test problem [5, Problem
57].
19 POWELLSG 10000 Extended Powell singular function [17, Problem
13].
Another function with nontrivial groups and repetitious
elements [12].
tridiagonal matrix square root
problem [4, Problem 151].
22 TRIDIA 10000 Shanno's TRIDIA quadratic tridiagonal problem
[30, Problem 8].
Table
Test problem collection. Each problems was chosen from the CUTE package.
and David Thuente from a 1983 version of MINPACK. This line search routine finds an
ff that meets the strong Wolfe conditions,
see, e.g., Nocedal [23]. We used 0:9. Except for the first
iteration, we always attempt a step length of 1.0 first and only use an alternate value
if 1.0 does not satisfy the Wolfe conditions. In the first iteration, we initially try a
step length equal to kg . The remaining line search parameters are detailed in
Appendix
A.
We generate the matrix H k by either the limited-memory update or one of the
variations described in x4.4, storing the matrix implicitly in order to save both memory
and computation time.
We terminate the iterations if any of the following conditions are met at iteration
L-BFGS Variations 19
k:
1. The inequality
is satisfied,
2. the line search fails, or
3. the number of iterations exceeds 3000.
We say that the iterates have converged if the first condition is satisfied. Otherwise,
the method has failed.
4.4. L-BFGS and its variations. We tried a number of variations to the standard
L-BFGS algorithm. L-BFGS and these variations are described in this subsection
and summarized in Table 4.2.
4.4.1. L-BFGS: Algorithm 0. The limited-memory BFGS update is given in
(2.5) and described fully by Byrd, Nocedal and Schnabel [22]. Our implementation
and the following description come essentially from [22].
Let H 0 be symmetric and positive definite and assume that the m k pairs
each satisfy s T
We will let
and m is some positive integer. We will assume that
I and that H 0 is iteratively rescaled by a constant fl k as is commonly done
in practice. Then, the matrix H k obtained by k applications of the limited-memory
BFGS update can be expressed as
\GammaU
where U k and D k are the m k \Theta m k matrices given by
ae
and
We will describe how to compute d k in the case that k ? 0. Let x k
be the current iterate. Let m Given s , the matrices
and the vectors S T
1. Update the n \Theta m k\Gamma1 matrices S k\Gamma1 and Y k\Gamma1 to get the n \Theta m k matrices
using s k\Gamma1 and y
2. Compute the m k -vectors S T
3. Compute the m k -vectors S T
by using the fact that
We already know components of S k g k\Gamma1 from S k\Gamma1 g k\Gamma1 , and likewise for
. We need only compute s T
and do the subtractions.
20 T. Gibson, D. P. O'Leary, L. Nazareth
No. Reference Brief Description
x4.4.1 L-BFGS with no options.
Allow m to vary iteratively basing the choice of m of kgk
and not allowing m to decrease.
Allow m to vary iteratively basing the choice of m of kgk
and allowing m to decrease.
Allow m to vary iteratively basing the choice of m of kg=xk
and not allowing m to decrease.
Allow m to vary iteratively basing the choice of m of kg=xk
and allowing m to decrease.
5 x4.4.3 Dispose of old information if the step length is greater than
one.
6 x4.4.4, Variation 1 Back-up if the current iteration is odd.
7 x4.4.4, Variation 2 Back-up if the current iteration is even.
8 x4.4.4, Variation 3 Back-up if a step length of 1.0 was used in the last iteration.
9 x4.4.4, Variation 4 Back-up if kg k k ? kg
Back-up if a step length of 1.0 was used in the last iteration
and we did not back-up on the last iteration.
and we did not back-up on the
last iteration.
neither of the two vectors to be merged is itself
the result of a merge and the 2nd and 3rd most recent steps
taken were of length 1.0.
13 x4.4.5, Variation 2 Merge if we did not do a merge the last iteration and there
are at least two old s vectors to merge.
14 x4.4.6, Variation 1 Skip update on odd iterations.
update on even iterations.
Alg. 5 & Alg. 8 Dispose of old information and back-up on the next iteration
if the step length is greater than one.
Alg. 13 & Alg. 1 Merge if we did not do a merge the last iteration and there
are at least two old s vectors to merge, and allow m to vary
iteratively basing the choice of m of kgk and not allowing
m to decrease.
19 Alg. 13 & Alg. 3 Merge if we did not do a merge the last iteration and there
are at least two old s vectors to merge, and allow m to
vary iteratively basing the choice of m of kg=xk and not
allowing m to decrease.
Alg. 13 & Alg. 2 Merge if we did not do a merge the last iteration and there
are at least two old s vectors to merge, and allow m to vary
iteratively basing the choice of m of kgk and allowing m to
decrease.
Alg. 13 & Alg. 2 Merge if we did not do a merge the last iteration and there
are at least two old s vectors to merge, and allow m to vary
iteratively basing the choice of m of kg=xk and allowing m
to decrease.
Table
Description of Numerical Algorithms
4. Compute
k . Rather than recomputing U
k , we update the matrix from
the previous iteration by deleting the leftmost column and topmost row if m
and appending a new column on the right and a new row on the bottom. Let ae
1=s T
be the (m lower right submatrix of U
and let (S T
be the upper
L-BFGS Variations 21
Note that s T
so is already computed.
5. Assemble Y T
We have already computed all the components.
6. Update D k using D k\Gamma1 and s T
7. Compute
Note that both y T
8. Compute two intermediate values
9. Compute
The storage costs for this are very low. In order to reconstruct H k , we need to
store
diagonal matrix) and a few m-vectors. This requires
only 2mn Assuming m !! n, this is much less storage than
the n 2 storage required for typical implementation of BFGS.
Step Operation Count
9
Table
Operations Count for Computation of H k g k . Steps with no operations are not shown.
The computation of Hg takes at most O(mn) operations assuming n ?? m. (See
Table
4.3.) This is much less than the O(n 2 normally needed to compute Hg
when the whole matrix H is stored.
We are using L-BFGS as our basis for comparison. For information on the performance
of L-BFGS see Liu and Nocedal [15] and Nash and Nocedal [19].
4.4.2. Varying m iteratively: Algorithms 1-4. In typical implementations
of L-BFGS, m is fixed throughout the iterations: once m updates have accumulated,
m updates are always used. We considered the possibility of varying m iteratively,
preserving finite termination on convex quadratics. Using an argument similar to that
presented in [15], we can also prove that this algorithm has a linear rate of convergence
on a convex function that satisfies a few modest conditions.
We tried four different variations on this theme. All were based on the following
linear formula that scales m in relation to the size of kgk. Let m k be the number of
iterates saved at the kth iteration, with Here, think of m as the maximum
allowable value of m k . Let the convergence test be given by kg k k=kx k k ! ffl. Then
the formula for m k at iteration k is
ae
log
oe
22 T. Gibson, D. P. O'Leary, L. Nazareth
Alg. No.
Table
The number of failures of the algorithms on the 22 test problems. An algorithm is said to have
"failed" on a particular problem if a line search fails or the maximum allowable number of iterations
(3000 in our case) is exceeded.
We have two choices for ffi k , and a choice of whether or not we will allow m k to decrease
as well as increase. The four variations are
1.
2.
3.
4.
We used four values of m: 5,10,15 and 50, for each algorithm. The results are
summarized in Tables 4.4 - 4.8. More extensive results can be obtained [9].
Table
4.4 shows that these algorithms had roughly the same number of failures
as L-BFGS.
Table
4.5 compares each algorithm to L-BFGS in terms of function evaluations.
For each algorithm and each value of m, the number of times that the algorithm
used as few or fewer function evaluations than L-BFGS is listed relative to the total
number of admissible problems. Problems are admissible if at least one of the two
methods solved it. We observe that in all but three cases, the algorithm used as few
or fewer function evaluations than L-BFGS for over half the test problems.
Table
4.6 compares each algorithm to L-BFGS in terms of time. The entries are
similar to those in Table 4.5. Observe that Algorithms 1-4 did very well in terms of
time, doing as well or better than L-BFGS in nearly every case.
For each problem in each algorithm, we computed the ratio of the number of
function evaluations for the algorithm to the number of function evaluations for L-
BFGS.
Table
4.7 lists the means of these ratios. A mean below 1.0 implies that
the algorithm does better than L-BFGS on average. The average is better for the
algorithms in 6 out of 16 cases for the first four algorithms. Observe, however, that
all the means are close to one.
L-BFGS Variations 23
Alg. No. m= 5
5 19/22 20/22 20/22 21/22
7 8/22 12/22 10/22 10/22
8 12/22 14/22 12/22 15/22
9 6/22 13/22 12/22 16/22
13 3/22 4/22 4/22 4/22
14 2/21 2/22 2/22 2/21
19 2/22 3/22 4/22 4/22
Table
Function Evaluations Comparison. The first number in each entry is the number of times the
algorithm did as well as or better than normal L-BFGS in terms of function evaluations. The second
number is the total number of problems solved by at least one of the two methods (the algorithm
and/or L-BFGS).
We experience savings in terms of time for the first four algorithms. These algorithms
will tend save fewer vectors than L-BFGS since m k is typically less than m;
and so less work is done computing H k g k in these algorithms. Table 4.8 gives the
mean of the ratios of time to solve for each value of m in each algorithm. Note that
most of the ratios are far below one in this case.
These variations did particularly well on problem 7. See [9] for more information.
4.4.3. Disposing of old information: Algorithm 5. We may decide that we
are storing too much old information and that we should stop using it. For example,
we may choose to throw away everything except for the most recent information
whenever we take a big step, since the old information may not be relevant to the
new neighborhood. We use the following test: If the last step length was bigger than
1, dispose of the old information.
The algorithm performed nearly the same as L-BFGS. There was substantial
deviation on only one or two problems for each value of m, and this seemed evenly
divided in terms of better and worse. From Table 4.4, we see that this algorithm
successfully converged on every problem. Table 4.5 shows that it almost always did
as well or better than L-BFGS in terms of function evaluations. However, Table 4.7
shows that the differences were minor. In terms of time, we observe that the algorithm
generally was faster than L-BFGS (Table 4.6), but again, considering the mean ratios
of time (Table 4.8), the differences were minor. The method also does particularly
well on problem 7 [9].
4.4.4. Backing Up in the Update to H: Algorithms 6-11. As discussed
in x2.2, if we always use the most recent s and y in the update, we preserve quadratic
termination regardless of which older values of s and y we use.
T. Gibson, D. P. O'Leary, L. Nazareth
Alg. No.
5 15/22 13/22 14/22 15/22
7 11/22 11/22 10/22 7/22
9 9/22 10/22 7/22 8/22
13 5/22 10/22 13/22 17/22
14 2/21 2/22 2/22 3/21
19 11/22 11/22 17/22 19/22
Table
Time Comparison. The first number in each entry is the number of times the algorithm did as
well as or better than normal L-BFGS in terms of time. The second number is the total number of
problems solved by at least one of the two methods (the algorithm and/or L-BFGS).
Using this idea, we created some algorithms. Under certain conditions, we discard
the next most recent values of s and y in the H although we still use the most recent
s and y vectors and any other vectors that have been saved from previous iterations.
We call this "backing up" because it as if we back-up over the next most recent values
of s and y. These algorithms used the following four tests to trigger backing up:
1. The current iteration is odd.
2. The current iteration is even.
3. A step length of 1.0 was used in the last iteration.
4. kg k k ? kg
In two additional algorithms, we varied situations 3 and 4 by not allowing a back-up
if a back-up was performed on the previous iteration.
The backing up strategy seemed robust in terms of failures. In 4 out of the 6
variations we did for this algorithm, there were no failures at all. See Table 4.4 for
more information.
It is interesting to observe that backing up on odd iterations (Algorithm
backing up on even iterations (Algorithm 7) caused very different results. Backing
up on odd iterations seemed to have almost no effect on the number of function
evaluations (Table 4.7) and little effect on the time (Table 4.8). However, backing up
on even iterations causes much different behavior from L-BFGS. It does worse than
L-BFGS on most problems, but better on a few.
Algorithms were two variations of the same idea: backing up if the
previous step length was one. This wipes out the data from the previous iteration
after it has been used in one update. Both show improvement over L-BFGS in terms
of function evaluations; in fact, these two algorithms have the best function evaluation
ratio for the case (Table 4.7). Unfortunately, these algorithms did not
compete with L-BFGS in terms of time (Table 4.8). There is little difference between
L-BFGS Variations 25
Alg. No. m= 5
6 1.000 1.000 1.000 1.000
9 1.035 1.371 1.005 0.947
14 7.521 7.917 8.288 8.502
19 1.212 1.959 1.242 1.387
Table
Mean function evaluations ratios for each algorithm compared to L-BFGS. Problems for which
either method failed are not used in this mean.
Algorithms probably because there were rarely many steps of length one
is a row.
Algorithms 9 and 11 are also two variations of the same idea: back-up on iteration
the norm of g k is bigger than the norm of g k+1 . There is a larger difference
between the results of 9 and 11 than there was between 8 and 10. In terms of function
evaluation ratios (Table 4.7), Algorithm 11 did better, indicating that it may not be
wise to back-up twice in a row. Both of these did poorly in terms of time as compared
with L-BFGS (Table 4.8).
4.4.5. Merging s and y information in the update: Algorithms 12 and
13. Yet another idea is to "merge" s data so that it takes up less storage and computation
time. By merging, we mean forming some linear combination of various s
vectors. The y vectors would be merged correspondingly. Corollary 2.5 shows that
as long as the most recent s and y are used without merge, old s vectors may be
replaced by any linear combination of the old s vectors in L-BFGS.
We used this idea in the following way: if certain criteria were met, we replaced
the second and third newest s vectors in the collection by their sum, and did similarly
for the y vectors. We used various tests to determine when we would do a merge:
1. Neither of the two vectors to be merged is itself the result of a merge and
the second and third most recent steps taken were of length 1.0.
2. We did not do a merge the last iteration and there are at least two old s
vectors to merge.
The first variation (Algorithm 12) performs almost identically to L-BFGS, especially
in terms of time (Table 4.5). Occasionally it did worse in terms of time
Table
4.6). These observations are also reflected in the other results in Table 4.7 and
Table
4.8. It is likely that very few vectors were merged.
The second variation (Algorithm 13) makes gains in terms of time, especially for
26 T. Gibson, D. P. O'Leary, L. Nazareth
Alg. No.
6 1.007 0.983 0.977 0.995
9 1.032 1.220 1.043 1.173
14 4.585 3.703 3.228 2.417
Table
Mean time ratios for each algorithm compared to L-BFGS. Problems for which either method
failed are not used in this mean.
the larger values of m (Table 4.6 and Table 4.8). Unfortunately, this reflects only a
saving in the amount of linear algebra required. The number of function evaluations
generally is larger for this algorithm than L-BFGS (Table 4.5 and Table 4.7).
4.4.6. Skipping Updates to H: Algorithms 14-16. If every other update
to H is skipped and a step length of one is always chosen, BFGS will terminate in
2n iterations on a strictly convex quadratic function. The same holds true when
doing an exact line search. (See x3.) Unfortunately, neither property holds in the
limited-memory case. We will, however, try some algorithms motivated by this idea.
The idea is that, every so often, we do not use the current s and y to update H,
and instead just use the old H. There are three variations on this theme.
1. Skip update on odd iterations.
2. Skip update on even iterations.
3. Skip update if kg k+1k ? kg k k.
As with the algorithms that did back-ups, the results of the skipping on odd or
even iterations were quite different. Skipping on odd updates (Algorithm 14) did
extremely well for every value of m on only two problems: 1 and 12. Otherwise, it did
very badly. Skipping on even updates (Algorithm 15) performed somewhat better. It
did extremely well on problem 7 but not on problems 1 and 12. It also did better than
L-BFGS in terms of time on more occasions than Algorithm 14 (Table 4.6). Neither
did well in terms of function evaluations, but the mean ratios for function evaluations
Table
4.7) and time (Table 4.8) were usually far greater than one.
Skipping the update if the norm of g increased (Algorithm 16) did not do well at
all. It only did better in terms of function evaluations for one problem for each value
of m (
Table
4.5) and rarely did better in terms of time (Table 4.6). It ratios were
very bad for function evaluations (Table 4.7) and time (Table 4.8)
L-BFGS Variations 27
4.4.7. Combined Methods: Algorithms 17-21. We did some experimentation
with combinations of methods described in the previous sections.
In Algorithm 17, we combined Algorithms 5 and 8: we dispose of old information
and back-up on the next iterations if the step length is greater than one. Essentially
we are assuming that we have stepped out of the region being modeled by the quasi-Newton
matrix if we take a long step and we should thus rid the quasi-Newton matrix
of that information. This algorithm did well in terms of function evaluations, having
mean ratios of less than one for three values of m (Table 4.7), but it did not do as
well in terms of time.
In Algorithms 19-21, we combined merging and varying m. These algorithms did
well in terms of time for larger m (Table 4.8) but not in terms of function evaluations
Table
4.7).
5. Conclusions. There is a spectrum of quasi-Newton methods, ranging from
those that require the storage of an n \Theta n approximate Hessian (e.g. the Broyden fam-
ily) to those that require only the storage of a few vectors (e.g. conjugate gradients).
Limited-memory quasi-Newton methods fall in between these extremes in terms of
performance and storage. There are other methods that fall into the middle ground;
for example, conjugate gradient methods such as those proposed by Shanno [27] and
Nazareth [20], the truncated-Newton method [24, 6] and the partitioned quasi-Newton
method [13].
We have characterized which limited-memory quasi-Newton methods fitting a general
form (2.1) have the property of producing conjugate search directions on convex
quadratics. We have shown that limited-memory BFGS is the only Broyden family
member that has a limited-memory analog with this property. We also considered
update-skipping, something that may seem attractive in a parallel environment. We
show that update skipping on quadratic problems is acceptable for full-memory Broyden
class members in that it only delays termination, but that we lose the property
of finite termination if we both limit memory and skip updates.
We have also introduced some simple-to-implement modifications of the standard
limited-memoryBFGS algorithm that seem to behave well on some practical problems.
Appendix
A. Line Search Parameters. Table A.1 give the line search parameters
used for our code. Note that in the first iteration, the initial steplength is
rather than 1.0.
Variable Value Description
STP 1.0 Step length to try first.
\Gamma4 Value of ! 1 in Wolfe conditions.
GTOL 0.9 Value of ! 2 in Wolfe conditions.
Relative width of interval of uncertainty.
Maximum number of function evaluations.
Table
A.1
Line Search Parameters
Appendix
B. Pseudo-Code.
B.1. L-BFGS: Algorithm 0. The pseudo-code for the computation of d
\GammaH k g k at iteration k for L-BFGS is given in Figure B.2. The update of H is also
handled implicitly in this computation.
28 T. Gibson, D. P. O'Leary, L. Nazareth
if (sze ==
else
% This is needed for Step 3 before we overwrite Stg and Ytg
Fig. B.1. MATLAB pseudo-code for the computation of d = Hg in L-BFGS. sze is the number
of s vectors available for the update this iteration and oldsze is the number of s vectors that were
available the previous iteration. For L-BFGS, sze is chosen as the minimum of oldsze
(the limited-memory constant).
B.2. Varying m iteratively: Algorithms 1-4. Suppose that m k denotes the
number of (s; y) pairs to be used in the kth update. Then simply chose sze as the
minimum of oldsze computing d k .
B.3. Disposing of old information: Algorithm 5. If the disposal criterion
is met at iteration k, set oldsze to zero and sze to one before computing d k .
B.4. Backing Up in the Update to H: Algorithms 6-11. If we are to
back-up at iterations k, set oldsze to the one less than the previous value of sze and
set sze as the minimum of oldsze m, as usual.
B.5. Merging s and y information in the update: Algorithms 12 and 13.
Merging is the most complicated variation to handle. Before we determine the newest
sze and before we compute d k , we execute the pseudo-code given in Figure B.1. We
then set oldsze to one less than the previous value of sze and set sze as the minimum
of oldsze m, as usual. We are assuming we are at iteration k, but that the
Variations 29
newest values of s and y have not yet been added to S and Y.
Execute before choosing new value for sze and before computing d
Fig. B.2. MATLAB pseudo-code for the merge variation. This fixes the values of the components
that are used in the computation of d k .
B.6. Skipping Updates to H: Algorithms 14-16. To skip the update at
iteration k, set sze to oldsze. Compute Stg and Ytg before Step 0 and then skip to
Step 8 and continue.
--R
ftp://thales.
Test functions for unconstrained minimization
Performance of a multifrontal scheme for partially separable optimization
Numerical Methods for Unconstrained Optimization and Nonlinear Equations
An optimal positive definite update for sparse Hessian matrices
http://www.
Matrix Computations
Private communication to authors of
Partitioned variable metric updates for large structured optimization problems
A theoretical and experimental study of the symmetric rank one update
On the limited memory BFGS method for large scale optimization
Linear and Nonlinear Programming
A numerical study of the limited memory BFGS method and the truncated-Newton method for large scale optimization
A relationship between BFGS and conjugate gradient algorithms and its implications for new algorithms
Updating quasi-Newton matrices with limited storage
A discrete Newton algorithm for minimizing a function of many variables
Quadratic termination properties of minimization algorithms I.
Conjugate gradient methods with inexact line searches
An error in specifying problem CHNROSNB.
--TR
--CTR
Sun Linping, Updating the Self-Scaling Symmetric Rank One Algorithm with Limited Memory for Large-Scale Unconstrained Optimization, Computational Optimization and Applications, v.27 n.1, p.23-29, January 2004
Adi Ditkowski , Gadi Fibich , Nir Gavish, Efficient Solution of A, Using A-1, Journal of Scientific Computing, v.32 n.1, p.29-44, July 2007
M. Al-Baali, Extra-updates criterion for the limited memory BFGS Algorithm for large scale nonlinear optimization, Journal of Complexity, v.18 n.2, p.557-572, June 2002 | update skipping;limited-memory;broyden family;BFGS;minimization;quasi-Newton |
589240 | Interior Point Trajectories in Semidefinite Programming. | In this paper we study interior point trajectories in semidefinite programming (SDP) including the central path of an SDP. This work was inspired by the seminal work of Megiddo on linear programming trajectories [ Progress in Math. Programming: Interior-Point Algorithms and Related Methods, N. Megiddo, ed., Springer-Verlag, Berlin, 1989, pp. 131--158]. Under an assumption of primal and dual strict feasibility, we show that the primal and dual central paths exist and converge to the analytic centers of the optimal faces of, respectively, the primal and the dual problems. We consider a class of trajectories that are similar to the central path but can be constructed to pass through any given interior feasible or infeasible point, and study their convergence. Finally, we study the derivatives of these trajectories and their convergence. | Introduction
The purpose of this paper is to study properties of the trajectories associated
with interior point methods for semidefinite programming (SDP) prob-
lems. Since many aspects of semidefinite programming find close analogs
in linear programming, several interior point methods designed for linear
programming (LP) have been successfully extended to apply to semidefinite
programming (e.g., see [2], [4], [10], [12], [17], [19], [20], [21], [25]). Many
Research supported in part by NSF Grants DMS 91-06195, DMS 94-14438 and DMS
95-27124 and DOE Grant DE-FG02-92ER25126
y This author was supported in part by an IBM Cooperative Fellowship
of these aspects have also been studied in the more general framework of
self-scaled cones in [20], [21].
Many interior point methods can be viewed as iterative approximations
to continuous path-following methods. Our aim is to provide a theoretical
basis for such methods for SDP by describing the limiting behavior of the
continuous central path and related trajectories for such problems.
This work is an extension of the linear programming results in [15] to
semidefinite programming. We characterize the optimal face of an SDP
problem and prove that the central path converges to the analytic center
of the optimal face. Unlike LP problems, an SDP problem does not always
have a strictly complementary primal-dual pair of solutions (e.g, see [3],
[12]). Thus the SDP central path cannot be guaranteed to converge to such
a pair as it does in LP. However, we show that it converges to a "least
nonstrictly complementary" pair, in the sense that the sum of the ranks of
the primal and the dual solutions (viewed as matrices) is as large as possible.
Another issue that makes SDP different from LP is the absence (at least
as far as we know) of a suitable concept of a weighted central path. Given
that it is difficult in practice to obtain a point on the central path, it is
important to have a class of trajectories that have properties similar to the
properties of the central path and that pass through any given pair of interior
primal and dual solutions. Such trajectories for linear programming are introduced
in [6] and [1] and are called primal affine scaling (PAS) trajectories
due to the fact that they correspond to continuous versions of primal affine
scaling iterative algorithms. We study the SDP analogs of PAS-trajectories
and prove that the main convergence results of [1] hold.
We show that under the assumptions of primal and dual nondegeneracy
and strict complementarity defined in [3], the first order derivatives of the
central path are bounded in the limit. We also provide formulae for the
limit of these derivatives and show that the factorization of only one matrix
is required to compute these and all higher order derivatives of a solution
on the central path.
The paper is organized as follows. In Section 2 we describe the central
path for a primal-dual pair of SDP problems and introduce our basic assumptions
and some notation. In Section 3 we characterize the optimal faces of
the primal and the dual SDP problems, and prove our main convergence
result for the primal-dual central path in Section 4. We extend the results
of Section 4 to the shifted central path (an analog of the PAS-trajectory)
in Section 5. Finally, in Section 6 we analyze the limiting properties of the
derivatives of the central path and show that computation of the derivatives
requires factorizing a single matrix for all orders of the derivatives.
2 The Central Path
In this paper we consider the semidefinite programming problem, henceforth
referred to as the primal problem,
n\Thetan denotes the space of real
. The inner product on S n\Thetan is
we mean that X
is positive semidefinite (positive definite).
The problem dual to (P ) is the semidefinite programming problem:
(D) s:t:
Throughout the paper the following are assumed to hold:
Assumption 2.1 The matrices A are linearly independent; i.e.,
implies that u
Assumption 2.2 Both the primal and the dual problem have interior feasible
solutions, i.e.
and
Under Assumption 2.2 both primal and dual problems have finite optimal
solutions, -
X and (-y; -
Z), and the duality gap -
[19]. The optimal
solutions also satisfy -
The central path for the problem (P ) is a trajectory of the solutions
n\Thetan to the following parametric family of problems for values of
the parameter - ? 0 ([13],[19],[25]):
From Assumption 2.2 and the strict convexity of the logarithmic barrier
objective function for any - ? 0, problem (P - ) has a unique solution that
satisfies the Karush-Kuhn-Tucker optimality conditions for (P -
The central path for the dual problem can be defined in an analogous
manner and is the trajectory (y(-); Z(-)) n\Thetan whose points satisfy
the same system (CP - ) as the points X(-) on the primal central path.
Hence it makes sense to refer to the trajectory (X(-); y(-); Z(- ? 0 of
solutions to (CP - ) as the primal-dual central path. Under Assumption 2.2,
not only does this path exist, but also it converges to an optimal primal-dual
solution (e.g, see [13], [19], [26]).
To conclude this section we introduce some notation that we will use
later in the paper. First, we note that the variables X and Z can be viewed
both as symmetric matrices and as vectors (obtained from these matrices by
stacking their columns one after the other), lying in a n(n+1)=2-dimensional
subspace of R n 2
. Whenever we refer to the matrix X as a vector, we denote
it by vec(X). By the constraint matrix A we denote the m \Theta n 2 matrix, the
i-th row of which equals . Note that C ffl
the usual inner product.
The Kronecker product
M\Omega N of matrices M 2 R n\Thetan and N 2 R n\Thetan
is defined as
There are two properties of the Kronecker product that we will need later:
T\Omega M)vec(N) and
(M\Omega P
(MN)\Omega (PS) [7].
If X is a positive semidefinite symmetric matrix, then X has a spectral
is an orthogonal matrix of eigenvectors
of X and is a diagonal matrix with the eigenvalues of X on the diagonal.
Throughout this paper the upper case letter Q will always denote a matrix
with orthonormal columns and
and\Omega will always denote diagonal matrices
of eigenvalues.
Lastly, from properties of the trace we have
Property 2.3 Let A 2 R n\Thetan , X 2 R r\Thetar and P 2 R n\Thetar . Then A ffl
Property 2.4 Let A 2 R n\Thetan , A - 0 and B 2 R n\Thetan , B - 0. Then
3 Optimal Faces of the Primal and Dual Problem
Properties of the faces of the cone of positive semidefinite matrices, are
studied in [5]. The facial structure of semidefinite programming problems
(i.e., the intersection of the cone of positive semidefinite matrices with an
affine subspace) is studied in general terms in [22], [23]. Here we derive a
particular system which describes the optimal face of an SDP problem.
Let us introduce some more notation and recall some well-known facts.
Let R(X) denote the range (column space) of X . If X is a positive semidefinite
symmetric matrix, it can be factorized as
where is a diagonal matrix whose diagonal elements are the positive eigen-values
of X and Q is a matrix with orthogonal columns that are eigenvectors
corresponding to these eigenvalues. Clearly, span(Q), the subspace
spanned by the columns of Q, and the dimension of this subspace (i.e., the
number of positive eigenvalues of X) equals the rank of X .
Z be an optimal primal-dual pair of solutions. It is well
known that they can be represented as -
and\Omega are diagonal matrices with the positive eigenvalues of -
Z,
respectively, on their diagonals and Q T
and R( -
X).
O P denote the primal optimal face, i.e., the set of primal optimal
solutions, and let OD denote the dual optimal face. Note that both O P and
OD are convex subsets of affine subspaces of S n\Thetan . By ri O P (ri OD ) we
denote the relative interior of O P (OD ). Then the following lemma holds
[5]:
Lemma 3.1 For any -
any ~
riO D ), R( -
Z)).
This lemma shows that any ~
riO P is an optimal solution of maximum
rank. Moreover, if both -
X and ~
X are in ri O P , it follows from Lemma 3.1
that R( -
X). Let us denote this subspace by R P . Analogously,
let RD be the subspace spanned by the eigenvectors corresponding to the
positive eigenvalues of Z for any dual solution (y; Z) in the relative interior
of OD .
Let dim R From the complementarity of any
primal-dual pair of optimal solutions R P ?RD . Hence, r
we say that the primal-dual pair of problems does not satisfy strict
complementarity. Note that this can never happen in linear programming.
If we define
we have a partition of R n into three mutually
orthogonal subspaces.
be any n \Theta r matrix whose columns form an orthonormal basis
for R P . Then any solution X O P can be written as
so the optimal face of (P ) is given by the set of the solutions to the following
system:
Indeed, for any U feasible for (1), Q P
is feasible for (P ), and since
and Z satisfy
complementary slackness. Therefore, Q
is an optimal solution to (P ).
Similarly, let the columns of Q D
form an orthonormal basis for RD .
Then any optimal dual solution can be written as
the optimal face of (D) is given by the set of solutions to the system:
Notice that the definitions of the primal and dual optimal faces are
invariant with respect to the choices of Q P
and Q D
as long as their columns
form orthonormal bases for the subspaces R P and RD , respectively.
The following lemma shows that under Assumptions 2.1 and 2.2 both
primal and dual optimal faces are bounded. Thus their analytic centers are
well defined, which is important for the results of the next section.
Lemma 3.2 Let Assumptions 2.1 and 2.2 hold, then the optimal sets of the
primal and the dual problems are bounded.
Proof. Suppose that the set of optimal dual solutions is unbounded. That
is, there exists a nonzero direction (u; V ), satisfying:
Multiplying the second equation by an interior feasible primal solution X
which exists by Assumption 2.2, we obtain
Therefore
It then follows that
which by Assumption 2.1 implies
that
The boundedness of the set of optimal primal solutions can be proved in
a similar manner. 2
4 Convergence of the Central Path
We prove in this section that the primal central path converges to the analytic
center of the optimal face O P . First we show that the limit -
X of the
central path is in the relative interior of the optimal face. Then we show
that -
is, in fact, the analytic center of the optimal face. We then extend
these results to the dual central path.
In [27] it is shown that in the case of convex homogeneous self-dual
cones, which includes the case of the cone of positive semidefinite matrices,
the central path converges to a strictly complementary solution provided
that one exists. In [14], under the assumption of strict complementarity, it
is shown that the primal-dual central path of an SDP problem converges to
the analytic center of the optimal face. We obtain the same results without
assuming strict complementarity.
X be the limit of the primal central path as - ! 0.
Lemma 4.1 There exists a spectral factorization -
and a sequence
such that X(- k
, where
spectral factorization of X(- k ).
Proof. The proof follows trivially from the compactness of the set of the
orthogonal matrices. Notice that the limit -
is uniquely defined by -
the limit -
Q, generally speaking, depends on the sequence f- k g. 2
We know that -
X and (-y; -
Z) are optimal solutions to the primal and dual
problems, respectively. We first want to prove that each is in the relative
interior of the optimal face for its respective problem.
Lemma 4.2 -
Z)) belongs to the relative interior of the primal (dual)
optimal face.
Proof. Let (X(-); y(-); Z(-)) be a point on the central path. Let ~
and (~y; ~
riO D . It is trivial to verify that
and since ~
~
Now both terms on the left side of (3) are nonnegative by Proposition 2.4;
hence
~
Consider the sequence f- k g as defined in Lemma 4.1, such that X(- k
X and the spectral factorizations
~
~
(The columns
of ~
Q are eigenvectors of ~
X that span R P .) Let us order the columns of
Q and partition -
into two parts [ -
QND ] so that -
Q P has r columns
and -
~
Q P is nonsingular. This is always possible since -
Q P has full
column rank. Let us order the columns of Q(- k ) and the columns and
rows of -
and (- k ) and partition them according to the column order and
partitioning of -
Q. Then -
ND (- k ), and from (4)
~
~
~
~
~
Since both terms in this sum are nonnegative by Proposition 2.4, it follows
from Property 2.3 that
~
~
~
~
~
U P
~
~
~
U P
~
~
It then follows
from (5) that
r
the sum of the ratios ( -
finite, it follows that
X has rank r proving
that -
ri O P . Similarly, it can be shown that (-y; -
ri OD . 2
From Lemma 3.2 it follows that the analytic center of the optimal face
O P is well defined. We now show that -
is, in fact, this analytic center.
Theorem 4.3 Let -
X be the limit of the primal central path as - ! 0. Then
U depends on he choice of the orthonormal basis Q P
for R P and is the unique solution to the problem
X is the analytic center of the primal optimal face.
Proof. Problem (6) can be rewritten in an equivalent form:
where c(0) is the optimal objective function value. The solution of this
problem is unique and satisfies the following system of optimality conditions:
For any fixed - ? 0, let C ffl is the solution to
the problem (CP - ). Then X(-) is a solution to the following problem:
Using notation of Lemma 4.2 -
QND ] is a matrix of eigenvectors of -
P is a diagonal
matrix of positive eigenvalues of -
X (r of them) and -
4.2 it follows that -
As in Lemma 4.2 consider the convergent sequence X(- k
to -
Q and (- k ) converges to -
as
and (- k ) is diagonal, requiring X in (9) to be of the form
where U - 0 and V is equal to ND (- k ), does not affect the solution
of problem (9). We restrict V and not U because, as we have already
shown, the sequence of the solutions converges to an
optimal solution, where and U is a positive definite matrix. Also,
. Therefore from (9) and Property 2.3 we obtain
the following maximization problem:
The unique optimal solution satisfies the following
system:
As
Q and the system (11) converges to (8) with Q
Since P (- k ) is the solution to (11), then the limit -
has to satisfy
(8). This proves that -
X is the analytic center of the primal optimal face. 2
As in LP, problems (P ) and (D) can be written in a "symmetric" form.
Specifically, we can use the "conic" formulation given in Chapter 3 of [19]
(see also [25]). Let L be the subspace of S n\Thetan spanned by A
and D 2 S n\Thetan be such that A i ffl
and (D) can be formulated as
and
(D
Consequently, all the results in this section extend to the dual problem, and
in particular, in terms of formulation (D) of the dual, we have the following:
Theorem 4.4 Let (-y; -
Z) be a limit point of the dual central path. Then
W depends on the choice of the orthogonal basis Q D
for RD and is the unique solution to the problem
Z) is the analytic center of the dual optimal face.
5 Shifted Central Paths
In this section we present a class of primal affine scaling trajectories analogous
to those introduced by Bayer and Lagarias [6] and later studied by
Adler and Monteiro [1]. Affine scaling vector fields associated with semidefinite
programs are studied in [8] and [9]. Here we analyze the limiting
behavior of affine scaling trajectories in SDP.
As far as we know there is no suitable concept of a scaled or weighted
central path, defined as a trajectory of solution of a class of minimization
problems, passing through any given pair of primal and dual interior solu-
tions. Therefore, we do not consider weighted trajectories as in [15]. How-
ever, we can consider "shifted" central paths, or primal affine scaling (PAS)
trajectories, as they are called in [1] or A-trajectories as they are called in
[6]. We study the properties and convergence of these trajectories, using the
same techniques that we used for the central path.
In [1] it is shown that the tangent to a PAS trajectory at any given point
has the same direction as the primal affine scaling step. The same is true in
semidefinite programming [9].
Consider the family of problems dependent on a parameter - ? 0
n\Thetan is some arbitrary fixed symmetric matrix. If problem (SP - )
has a solution for some - ? 0 then that solution is unique and satisfies the
Karush-Kuhn-Tucker necessary conditions
The trajectory of dual solutions (y(-); Z(-)) defined by the system (SCP - ),
parametrized by -, is generally different from the dual shifted central path
associated with T . Thus, when referring to the shifted central path, we
mean the primal and dual trajectories defined by (SCP - ).
For any given - ? 0 and T 2 S n\Thetan if there exists a (y n\Thetan
such that
-T and Z 0 - 0 then (SCP - ) has a unique
solution. Using the notation of [1], let Y (T ;
and I(T ;g. By Assumption 2.2, Y (T ;
T , and it is easy to see that I(T ) is an open interval, which is nonempty
as long as C \Gamma -T is not spanned by the matrices A i m). Thus
Lemma 5.1 If the feasible set of problem (P ) is bounded, then I(T
(0; 1) for any T 2 S n\Thetan ; i.e., (SCP - ) has a unique solution for all
1.
This lemma is proved in [24] and a similar, but more general, result is
proved as Theorem 2.4 in [9]. We want to choose T so that the shifted
central path passes through an arbitrary given primal-dual pair of interior
solutions. Specifically, given (X
then it is easy to verify that I(T consequently, that I(T ) 3
In other words this choice guarantees the existence of a trajectory
passing through the primal-dual point (X
It is shown in [1] (Proposition 2.4) that in linear programming the choice
of the initial dual solution (y does not affect the PAS trajectory. The
same is true in the case of the shifted central path (i.e., PAS trajectory) for
a semidefinite program. The proofs are identical.
We are ready to discuss the limiting behavior of the shifted central path.
Z) be a limit point of the solution (X(-); y(-); Z(-)) to (SCP - )
In [9] it is shown that -
X is an optimal solution to (P ) and (-y; -
is an optimal solution to (D). As in the case of the central path, we can
show that -
X is in the relative interior of the primal optimal face and (-y; -
is in the relative interior of the dual optimal face. The proof is analogous
to the proof of Lemma 4.2 with the exception that X(-) ffl Z(- N- for
some large number N .
More importantly, it is trivial to extend the proof of Theorem 4.3 to give
Theorem 5.2 -
U is the unique solution to the problem
X is the "shifted" analytic center of the primal optimal face.
Just as in the case of LP [1], the dual solutions of the shifted central
path converge to the analytic center (not shifted) of the dual optimal face.
Theorem 5.3 Let (-y; -
Z) be a limit dual point of the shifted central path,
then -
W is the unique solution to the problem
Z) is the analytic center of the dual optimal face.
Proof. First we observe that the dual solution (y(-); Z(-)) associated
with the system of optimality condinions (SCP - ) is the unique optimal
solution to the system
s:t:
Given a sequence Z(- k
Z, we know that Z(- k
Z. Let
be a sequence of dual solutions on the shifted central path,
which converges to (-y; -
Z) and for which the sequence of spectral factorizations
converges. By a similar argument
to that used in the proof of Theorem 4.3,
=\Omega D (- k ) is a
solution to the following problem
When the solution to the above problem converges to the solution
to the problem
which is equivalent to Problem (14) defining the analytic center of the dual
optimal face. Thus (-y; -
W ) is the unique solution of Problem (14). 2
Thus, the choice of the initial point (X affects the limit of the
trajectory of the primal solutions (obviously, only if the optimal face is
of dimension greater than zero), but does not affect the limit of the dual
trajectory.
Remark. Notice, that the dual problem (15) is in fact a shifted barrier
problem for the original dual (D).
We now consider the tangent to a shifted central path at an arbitrary
point on it. Our results apply to the central path as a special case
Let (X; (we omit the argument - for simplicity) be on the shifted
central path corresponding to a given shift T . Differentiating the system
respect to - for any - ? 0, yields
In [9] it is shown that this system of differential equations is generated
by the generalized primal affine scaling vector field. In our terms, this is
equivalent to the fact that we can rewrite the above system as
y, -
X\Omega X and H
(i.e., the rows of -
A equal
and vec( -
can be viewed as an orthogonal
projection of a scaled objective vector onto the kernel of a scaled constraint
matrix, where the scaling depends only on the primal solution. The direction
of the tangent is
and the directions -
y and -
Z can be calculated from the dual estimates y E
and
Z , which in turn, can be computed from the projection
operator.
Remark. We would like to study the limiting behavior of the dual estimates
Z that are computed at every step of an algorithm that
uses an affine scaling direction. In the next section we show that under
assumptions of strict complementarity and primal and dual nondegeneracy
the limit of (y the limit of (y(-); Z(-)) as - ! 0.
6 Derivatives of the Central and Shifted Central
We begin this section by showing that as in case of linear programming
[15] the computation of derivatives of any order of solutions on the central
path or a shifted central path involves inverting a single matrix. In contrast
with LP, the Schur complement of this matrix, which we must factorize, is
fully dense, even if the constraint matrices are sparse. Consequently, this
factorization step is very expensive in SDP and it is desirable to use as
much information (e.g. higher order derivatives) as possible from it, as in
the interior point LP methods proposed in [18] and [16].
Let us consider solutions X and (y; Z) on the shifted central path corresponding
to a shift T . ( X , y and Z depend on -, but we omit the argument.)
As shown in the previous section (see (17)) the derivatives -
y in vector
form satisfy
y
For the second derivatives we have
y
and it can be easily shown by induction that the k-th derivatives X (k) and
y (k) on the central path must satisfy a system of equations of the form
y
where R is a function of (T ; X; -
We now turn to the limiting properties of the first order derivatives of
the central path. As earlier, let ( -
Z) be the limit of the central path.
We need the following assumption:
Assumption 6.1 i) The primal and dual solutions -
X and (-y; -
are
strictly
ii) The primal solution -
X is nondegenerate; i.e., the matrices
QD
are linearly independent in S n\Thetan .
iii) The dual solution (-y; -
Z) is nondegenerate; i.e., the matrices
span S r\Thetar .
For these and other equivalent definitions of primal and dual nondegeneracy
see Alizadeh, Haeberly, and Overton [3]. They also prove that under
Assumption 6.1 the optimal primal-dual solution is unique. Hence if Assumption
6.1 holds it makes sense to say that Problems (P ) and (D) are
nondegenerate and strictly complementary. For the remainder of this section
we shall assume that Assumption 6.1 holds.
To show that the first order derivatives of the central path converge to
finite limits as - ! 0, we consider the following system which, as shown in
[4], is equivalent to (CP - ):2 (XZ
Viewing all symmetric matrices in the above system as vectors in R n(n+1)=2
and differentiating, we obtain the following system6 4
Z I 0 X I
y
where svec maps a matrix in S n\Thetan into a vector in R n(n+1)=2 and denotes
"symmetric" Kronecker product. See [4] for definitions and properties of
svec and . Alizadeh, Haeberly, and Overton in [4] prove that under Assumption
6.1 the coefficient matrix of (21) is nonsingular at the limit
Therefore, -
y and -
Z are bounded and converge as - ! 0. Geometrically
this means that the central path approaches the boundary of the feasible
region at a strictly positive angle; i.e., the angle between the tangent to
the central path and the tangent to the boundary at the optimal solution is
strictly positive. From
X, and the boundedness of the derivative of X
we conclude that -
X, where -
X(0) is
the right derivative of X(-) at
Similar arguments prove that the derivatives of the primal and dual
solutions on the shifted central path are bounded and converge as - ! 0.
We obtain system similar to (21), with the same coefficient matrix and the
right hand side, which depends on the shift T and on X and is uniformly
bounded as - ! 0.
Let us consider the derivatives of the eigenvalues of X and Z on the
central path. Since is symmetric and is a smooth function of a
single parameter, its eigenvalues can be ordered so that - of X is
in C 1 [11]. If - i is an eigenvalue of multiplicity k then -
the vector of eigenvalues of the matrix
where Q i is a matrix of eigenvectors of X corresponding to - i . If
and -
does not generally converge; however, the
subspace spanned by the columns of Q i converges as - ! 0 and therefore
any accumulation point of the matrices (22) has the same eigenvalues. Thus
the limit of -
exists. The same can be shown for the derivatives -
i of the
of the the dual slack matrix Z.
Let us consider the eigenvalues of X and Z that converge to zero. The
multiplicity of the zero eigenvalue of -
s. The multiplicity of the
zero eigenvalue of -
Z is r. We know that for X and Z on the central path,
and therefore
Z as - ! 0,
Considering the above system in more detail, we obtain
QD
QD
QD
QD
From the diagonal blocks of this matrix equa-
tion, it then follows that
and
Therefore, we can conclude that there are orderings of the eigenvalues of -
and -
Z , for which -
D and
P . This generalizes the result on
the limits of the derivatives of nonbasic variables (i.e., variables converging
to zero) in linear programming. Complete information on -
Z can be
obtained by solving the system (21).
We can now conclude, that the dual estimates y
y and Z
Z that appear in (18) converge to the same limits as y and Z, since -
y
and -
Z are bounded as - ! 0.
Remark. To prove the convergence of the derivatives of the central
path as - ! 0 we assumed primal and dual nondegeneracy and strict com-
plementarity. These assumptions imply the uniqueness of the primal and
dual solutions [3]. However, we conjecture that it is sufficient to only assume
strict complementarity. Note that strict complementarity is necessary
for boundedness of the derivatives, since if there is an index i such that both
primal and dual eigenvalues - i (-
then from
hence that
both -
cannot be finite at the limit as - ! 0.
In [14] it is shown that assuming strict complementarity,
O(-) and
which implies that the derivatives of the
central path are bounded as - ! 0.
Acknowledgement
. The authors are grateful to Yurii Nesterov for
bringing the question of the convergence of the central path in SDP to their
attention, to Michael Overton for many stimulating discussions on SDP, and
to Jean-Pierre Haeberly and two anonymous referees for helpful comments
on earlier versions of the paper.
--R
"Limiting behavior of the affine scaling continuous trajectories for linear programming problems"
"Interior point methods in semidefinite programming with applications to combinatorial optimization"
"Complementarity and nondegeneracy in semidefinite programming"
"Primal-Dual Interior-Point Methods for Semidefinite Programming: Convergence Rates, Stability and Numerical results"
"Cones of diagonally dominant matri- ces"
"The nonlinear geometry of linear pro- gramming. I. Affine and projective scaling trajectories"
Kronecker Products and Matrix Calculus: with Applica- tions
"On a matrix generalization of affine-scaling vector fields"
"A Hamiltonian structure of generalized affine scaling vector fields"
"An interior-point method for semidefinite programming"
A Short Introduction to Perturbation Theory for Linear Op- erators
"Local convergence of predictor-corrector Infeasible-Interior-Point Algorithms for SDPs and SDLCPs"
"Interior-point methods for monotone semidefinite linear complementarity problem in symmetric matri- ces"
"Superlinear convergence of a symmetric primal-dual path following algorithms for semidefinite program- ming"
"Pathways to the optimal set in linear programming"
"Higher order methods and their performance"
"Primal-dual path following algorithms for semidefinite programming"
"A polynomial time primal-dual affine scaling algorithm for linear and convex quadratic programming and its power series extension"
Interior point Methods in Convex Programming: Theory and Application
"Self-scaled cones and interior-point methods in nonlinear programming"
"Primal-dual interior point algorithms for self-scaled cones"
"On the facial structure of cone-LP's and semi-definite pro- grams"
"Strong duality for semidefinite programming"
"Issues in interior point methods in semidefintie and linear programming"
"A primal-dual potential reduction method for problems involving matrix inequalities"
"Positive-definite programming"
"Convergence behavior on central paths for convex homogeneous self-dual cones"
"On Extending primal-dual interior-point algorithms from linear programming to semidefinite programming"
--TR
--CTR
Anthony Man-Cho So , Yinyu Ye, Theory of semidefinite programming for sensor network localization, Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms, January 23-25, 2005, Vancouver, British Columbia | interior point methods;central path;semidefinite programming |
589254 | Optimality Conditions for Optimization Problems with Complementarity Constraints. | Optimization problems with complementarity constraints are closely related to optimization problems with variational inequality constraints and bilevel programming problems. In this paper, under mild constraint qualifications, we derive some necessary and sufficient optimality conditions involving the proximal coderivatives. As an illustration of applications, the result is applied to the bilevel programming problems where the lower level is a parametric linear quadratic problem. | Introduction
. The main purpose of this paper is to derive necessary and
su#cient optimality conditions for the optimization problem with complementarity
constraints (OPCC) defined as follows:
y, u)
s.t. #u, #(x, y, y, u) # 0
y, y, u) # 0, (x, y, u)
and# is a nonempty subset of R n+m+q .
(OPCC) is an optimization problem with equality and inequality constraints.
However, due to the complementarity constraint (1.1), the Karush-Kuhn-Tucker
(KKT) necessary optimality condition is rarely satisfied by (OPCC) since it can be
shown as in [9, Proposition 1.1] that there always exists a nontrivial abnormal multi-
plier. This is equivalent to saying that the usual constraint qualification conditions,
such as the Mangasarian-Fromovitz condition, will never be satisfied (see [8, Proposition
3.1]). The purpose of this paper is to derive necessary and su#cient optimality
conditions under mild constraint qualifications that are satisfied by a large class of
OPCCs.
To motivate our main results, we formulate problem (OPCC),
as the following optimization problem with a generalized equation constraint:
(GP) min f(x, y, u)
y, u) +N(u,R q
y, y, u) # 0,
where
N(u, C) := # the normal cone of C at y if
# Received by the editors May 26, 1997; accepted for publication (in revised form) May 4, 1998;
published electronically March 17, 1999. This work was supported by the Natural Sciences and
Engineering Research Council of Canada and a University of Victoria internal research grant.
http://www.siam.org/journals/siopt/9-2/32188.html
Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3P4,
Canada (janeye@uvic.ca).
OPTIMIZATION PROBLEMS WITH COMPLEMENTARITY CONSTRAINTS 375
is the normal cone operator in the sense of convex analysis.
y, -
u) be a solution of (OPCC),
single-valued and smooth, then the generalized equation constraint (1.2) would reduce
to an ordinary equation. Using the KKT condition, we could deduce that if a
constraint qualification is satisfied for (GP) and the problem data are smooth, then
there exist KKT multipliers # R l , # R d , # R q such that
y, - u) +#L(-x, -
y, -
y, -
y, -
(-u) #,
y, - u)# 0,
where # denotes the usual gradient, M # denotes the transpose of the matrix M , and
NC denotes the map y # N(y, C). However, u # N(u, R q
in general a set-valued
map. Naturally, we hope to replace #N R q
(-u) # by the image of some derivatives
of the set-valued map u # N(u, R q
acting on the vector #. The natural candidate
for such a derivative of set-valued maps is the Mordukhovich coderivative (see Definition
2.3) since the Mordukhovich coderivatives have a good calculus, and in the case
when the set-valued map is single-valued and smooth, the image of the Mordukhovich
coderivative acting on a vector coincides with the usual gradient operator acting on
the vector (see [6, Proposition 2.4]). Indeed, as in [7], we can show that if (-x, -
y, - u) is
an optimal solution of (OPCC) and a constraint qualification holds, then there exist
q such that
y, -
y, -
y, -
y, -
y, - u))(#),
y, - u)# 0,
where D # denotes the Mordukhovich coderivative (see Definition 2.3). Recall from [7,
Definition 2.8] that a set-valued
R q with a closed graph is said to be
pseudo-upper-Lipschitz continuous at (-z, - v) with - v #(-z) if there exist a neighborhood
U of -
z, a neighborhood V of - v, and a constant - > 0 such that
The constraint qualification for the above necessary condition involving the Mor-
dukhovich coderivative turns out to be the pseudo-upper-Lipschitz continuity of the
set-valued map
y, u)+N(u,R q
y, u)+v 3 # 0}
at (-x, -
y, - u, 0). This constraint qualification is very mild since the pseudo-upper-
Lipschitz continuity is weaker than both the upper-Lipschitz continuity and the pseudo-
Lipschitz continuity (the so-called Aubin property). However, the Mordukhovich
normal cone involved in the necessary condition may be too large sometimes. For ex-
ample, in [7, Example 4.1], both (0, 0) and (1, 1) satisfy the above necessary conditions,
but only (1, 1) is the unique optimal solution. Can one replace the Mordukhovich
normal cone involved in the necessary condition by the potentially smaller proximal
normal cone? The answer is negative in general, since the proximal coderivative as
defined in Definition 2.3 usually has only a "fuzzy" calculus. Consider the following
376 J. J. YE
optimization problem:
min -y
s.t. y -
The unique optimal solution (0, does not satisfy the KKT condition but satisfies
the necessary condition involving the Mordukhovich coderivatives. It does not satisfy
the necessary condition with the Mordukhovich normal cone replaced by the proximal
normal cone. This example shows that some extra assumptions are needed for the
necessary condition involving the proximal coderivatives to hold. In this paper such a
condition is found. Moreover, we show that the proximal normal cone involved in the
necessary condition can be represented by a system of linear and nonlinear equations,
and the necessary optimality conditions involving the proximal coderivatives turn out
to be su#cient under some convexity assumptions on the problem data.
Although the optimization problems with complementarity constraints are a class
of optimization problems with independent interest, the incentive to study (OPCC)
mainly comes from the following optimization problem with variational inequality
constraints (OPVIC), where the constraint region of the variational inequality is a
system of inequalities:
R,# is a nonempty subset of R m+n and S(x) is the solution set
of a variational inequality with parameter x; i.e.,
. The recent monograph [4] by Luo, Pang,
and Ralph has an extensive study for (OPVIC). The reader may find the references
for the various optimality conditions for (OPVIC) from [4].
(OPCC) is closely related to OPVICs and bilevel programming problems. Indeed,
if is C 1 and quasi convex in y and a certain constraint qualification condition holds
at - y for the optimization problem
min #F (-x, -
then by the KKT necessary and su#cient optimality condition, (-x, -
y) is a solution
of (OPVIC) if and only if there exists - u # R q such that (-x, -
y, - u) is a solution of the
following optimization problem:
(KS) min f(x, y)
s.t. #u, (x,
which is a special case of (OPCC).
In the case where F (x,
# R is di#erentiable and
pseudoconvex in y, (KS) is equivalent to the following bilevel programming problem
(BLPP), or so-called Stackelberg game:
(BLPP) min f(x, y)
OPTIMIZATION PROBLEMS WITH COMPLEMENTARITY CONSTRAINTS 377
where S(x) is the set of solutions of the problem (P x
We organize the paper as follows. Section 2 contains background material on
nonsmooth analysis and preliminary results. In section 3 we derive the necessary and
su#cient optimality conditions for (OPCC). As an illustration of applications, we also
apply the result to (BLPP), where the lower level is a linear quadratic programming
problem.
2. Preliminaries. This section contains some background material on non-smooth
analysis and preliminary results which will be used later. We give only concise
definitions that will be needed in the paper. For more detailed information on the
subject, our references are Clarke [1, 2], Loewen [3], and Mordukhovich [6].
First we give some concepts for various normal cones and subgradients.
Definition 2.1.
Let# be a nonempty subset of R n . Given -
z #
cl# , the closure
of
set# , the convex cone
#z
is called the proximal normal cone to
set# at point - z, and the closed cone
N(-z, := { lim
is called the limiting normal cone
to# at point -
z.
Remark 2.1. It is known that
if# is convex, then the proximal normal cone
and the limiting normal cones coincide with the normal cone in the sense of convex
analysis.
Definition 2.2. Let f : R n
R #} be lower semicontinuous and finite at
z # R n . The limiting subgradient of f at -
z is defined to be the set
N(-z,
where denotes the epigragh of f .
Remark 2.2. It is known that if f is a convex function, the limiting subgradient
coincides with the subgradient in the sense of convex analysis. For a locally Lipschitz
function f ,
#f(x), where # denotes the Clarke generalized gradient and co
denotes the convex hull. Hence the limiting subgradient is in general a smaller set
than the Clarke generalized gradient.
For set-valued maps, the definition for limiting normal cone leads to the definition
of coderivative of a set-valued map introduced by Mordukhovich (see, e.g., [6]).
Definition 2.3. Let # : R n
R q be an arbitrary set-valued map (assigning to
each z # R n a set #(z) # R q which may be empty) and (-z, - v) # cl Gr#, where Gr#
denotes the graph of #; i.e., (z, v) # Gr# if and only if v #(z). The set-valued
maps from R q into R n defined by
v), Gr#)}
are called the proximal and Mordukhovich coderivatives of # at point (-z, - v), respectively
378 J. J. YE
Proposition 2.4. Suppose B is closed, -
# B. Then
Proof. Since -
is closed, there exists a neighborhood of -
x that is not
contained in B. Therefore, from the definition of the proximal normal cone, we have
In the following proposition we show that the proximal normal cone of a union of
a finite number of sets is the intersection of the proximal cones.
Proposition 2.5.
are closed. Then
Proof. Let # N # (-x, . Then, by definition, there exists a constant M > 0 such
that
#, x - x# M #x -
#x
the above inequality implies that # m
Conversely, suppose # m
there exists
#, x -
#x
That is, there exists
#, x - x# M #x -
#x
which implies that # N #
The above decomposition formula for calculating the proximal normal cones turns
out to be very useful, since when a set can be written as a union of some convex sets,
the task of calculating the proximal normal cones is reduced to calculating the normal
cone to convex sets which are easier to calculate. The following proposition is a nice
application of the decomposition formula and will be used to calculate the proximal
normal cone to the graph of the set-valued map N R q
for general q in Proposition 2.7.
Proposition 2.6.
Proof. It is easy to see that GrNR+
We discuss the following three cases.
Case 1. -
In this case, (-x, - y)
closed, by Proposition 2.4
we have in this case
OPTIMIZATION PROBLEMS WITH COMPLEMENTARITY CONSTRAINTS 379
Case 2. -
y < 0.
In this case, (-x, - y)
closed, by Proposition 2.4
we have in this case
Case 3. -
In this case, (-x, -
. By Proposition 2.5 we have
(R - [0, #))
Now we are in a position to give an expression for the proximal normal cone to
the graph of the set-valued map N R q
for general q.
Proposition 2.7. For any (-x, - y) #GrN R q
, define
I
I 0 := I 0 (-x, -
Then
y), GrN R q
Proof. Since
we have
if and only if
GrNR+ .
Hence from the definition, it is clear that
if and only if
The rest of the proof follows from Proposition 2.6.
It turns out that we can express any element of N # ((-x, - y),GrN R q
by a system
of nonlinear equations as in the following proposition.
J. J. YE
Proposition 2.8.
if and only if there exist # R 2q
such that
Proof. By Proposition 2.7, (# N # ((-x, -
y), GrN R q
if and only if
By the definition for the index sets I 0 , I + , L in Proposition 2.7, we have
Since for any (-x, - y) # GrN R q
y # 0, for nonnegative vectors # and #, (2.1) is
equivalent to
Hence the existence of nonnegative vectors # and # satisfying (2.1)-(2.2) is equivalent
to the following condition:
Consequently, it is equivalent to
The proof of the proposition is therefore complete.
Finally, we would like to recall the following definition of a very mild constraint
qualification called "calmness," introduced by Clarke [1].
Definition 2.9. Let -
x be a local solution to the following mathematical programming
problem:
minimize f(x)
OPTIMIZATION PROBLEMS WITH COMPLEMENTARITY CONSTRAINTS 381
and C is a closed subset of R d . The above
mathematical programming problem is said to be calm at -
x provided that there exist
positive # and M such that for all (p, q) #B, for all x in -
x+#B satisfying g(x)+p #
where B is the open unit ball in the appropriate space.
It is well known that the calmness condition is a constraint qualification for the
existence of a KKT multiplier and the su#cient conditions for the calmness condition
include the linear independence condition, the Slater condition, and the Mangasarian-
Fromowitz condition. Moreover, the calmness condition is satisfied automatically in
the case where the feasible region is a polyhedron.
3. Optimality conditions for OPCC. Let (-x, -
y, -
# and g(-x, -
y, -
y, -
I
y,
y, - u) < 0},
I 0 (-x, -
y,
y,
Where there is no confusion, we simply use L, I + , I 0 instead of L(-u), I
y, -
u),
I 0 (-x, -
y, -
u), respectively. It is clear that {1, 2, . ,
y, -
y, -
u).
Let
y, u)
y, y, u) # 0
#u, #(x, y, y, u) # 0
be the feasible region of (OPCC). For any I # {1, 2, . , q}, let
F I :=
y, u)
y, y, u) # 0
y,
y, u) # 0 #i # {1, 2, . , q}\I
denote a piece of the feasible region F .
Taking the "piecewise programming" approach in the terminology of [4], as in
Corollary 2 of [5], we observe that the feasible region of the problem (OPCC) can
be rewritten as a union of all pieces I . Therefore, a local solution
y, -
u) for (OPCC) is also a local solution for each subproblem of minimizing the
objective function f over a piece which contains the point (-x, - y, -
u). Moreover, if
y, -
u) is contained in all pieces and all subproblems are convex, then it is a global
minimum for the original problem (OPCC). Hence the following proposition follows
from this observation.
Proposition 3.1. Let (-x, - y, -
u) be a local optimal solution to (OPCC). Suppose
that f , g, , L are locally Lipschitz near (-x, - y, -
and# is closed. If for any given
index set # I 0 , the problem of minimizing f over F#L is calm in the sense of
Definition 2.9 at (-x, - y, -
u), then there exist # R l , # R d , # R q , # R q such that
y, -
l
y,
d
y,
y, -
y, -
(3.
J. J. YE
y, -
Conversely, let (-x, - y, -
u) be a feasible solution for (OPCC), and for all index sets
I 0 , there exist # R l , # R d , # R q , # R q such that (3.1)-(3.3) are satisfied.
If f is either convex or pseudoconvex, g is convex, , L are a#ne,
and# is convex,
then (-x, - y, -
u) is a minimum of f over all (x, y, u) #I0 F#L. If in addition to the
above assumptions I
y, - u) is a global solution for (OPCC).
Proof. It is obvious that the feasible region of (OPCC) can be represented as
the union of pieces I . Since -
y, - u) < 0
y, -
u), and
y, u)
y, y, u) # 0
y,
y,
y, u) # 0 #i # I 0 \#
we have
y, - u) #I0 F#L
and
y, -
Hence if (-x, -
y, - u) is optimal for (OPCC), then for any given index set # I 0 , (-x, -
y, -
is also a minimum for f over F#L . Since this problem is calm, by the well-known
nonsmooth necessary optimality condition (see, e.g., [1, 2, 3]), there exist # R l ,
q such that (3.1)-(3.3) are satisfied. Conversely, suppose
that for each # I 0 there exist # R l , # R d , # R q , # R q such that (3.1)-
are satisfied and the problem is convex. By virtue of Remarks 2.1 and 2.2, the
limiting subgradients and the limiting normal cones coincide with the subgradients
and the normal cone in the sense of convex analysis, respectively. Hence, by the
standard first-order su#cient optimality conditions, (-x, -
y, -
u) is a minimum of f over
F#L for each # I 0 and hence is a minimum of f over #I0 F#L . In the case when
I and the feasible region
y, -
is a global optimal for (OPCC) in this case. The proof of the proposition is now
complete.
Remark 3.1. The necessary part of the above proposition with smooth problem
data is given by Luo, Pang, and Ralph in [4] under the so-called "basic constraint
qualification."
Note that the multipliers in Proposition 3.1 depend on the index set # through
(3.3). However, if for some pair of index sets # I 0 ) and I 0 \#, the components
of the multipliers are the same, then we would have a necessary condition
that does not depend on the index set #. In this case the necessary condition turns
out to be the necessary condition involving the proximal coderivatives as in (b) of the
following theorem.
Theorem 3.2. Suppose f, g, L, are continuously di#erentiable. Then the following
three conditions are equivalent:
OPTIMIZATION PROBLEMS WITH COMPLEMENTARITY CONSTRAINTS 383
(a) There exist # R l , # R d , # R q such that
y,
l
y, -
d
y, - u)
y,
y, -
(b) There exist # R l , # R d , # R q such that
y, -
l
y,
d
y, -
y, -
y, - u))(#),
y, -
(c) There exist # R l , # R d , # R q , # R 2q
such that (3.4) and (3.5)
are satisfied and
y, -
y, -
Let (-x, - y, -
u) be a local optimal solution to (OPCC),
that there exists an index set # I 0 such that the problem of minimizing f over F#L
and the problem of minimizing f over F (I 0 \#L are calm. Furthermore, suppose that
l
y, -
d
y, -
y,
y, -
implies that # the three equivalent conditions (a)-(c) hold.
Conversely, let (-x, -
y, - u) be a feasible solution to (OPCC),
let f be pseudoconvex, g be convex, #, L be a#ne. If one of the equivalent conditions
(a)-(c) holds, then (-x, -
y, -
u) is a minimum of f over all (x, y, u) #I0 F#L . If in
addition to the above assumptions I
y, -
u) is a global solution
for (OPCC).
Proof. By the definition of the proximal coderivatives (Definition 2.3),
y, - u))(#)
if and only if
384 J. J. YE
Hence the equivalence of condition (a) and condition (b) follows from Proposition 2.7.
The equivalence of condition (b) and condition (c) follows from Proposition 2.8.
Let (-x, - y, -
u) be a local optimal solution to (OPCC),
it is also a local optimal solution to the problem of minimizing f over F#L and
the problem of minimizing f over F (I 0 \#L . By the calmness assumption for these
two problems, there exist # i
(3.1)-(3.3), which implies that
l
y, -
d
y, - u)
y, -
y, -
By the assumption we arrive at # 1
I0 . Since by (3.3), # 1
I0 \# 0, we have
That is, condition (a) holds.
The su#cient part of the theorem follows from the su#cient part of Proposition
3.1.
As observed in [4, Proposition 4.3.5], the necessary optimality conditions (3.4)-
(3.6) happen to be the KKT condition for the relaxed problem
(RP) minf(x, y, u)
s.t. y,
y, -
u),
y, -
u),
y, y, u) # 0,
and (#) satisfies (3.4)-(3.6) if and only if it satisfies the KKT condition for the
subproblem of minimizing f over the feasible region F#L , i.e., (3.1)-(3.3) with the
smooth problem data
y, - u). Conse-
quently, if the strict Mangasarian-Fromovitz constraint qualification (SMFCQ) holds
for problem (RP) at (#) which satisfies (3.4)-(3.6), then (#) is the unique
multiplier which satisfies (3.4)-(3.6). Since the index sets # only a#ect the (# I0 , # I0 )
components of the multiplier (#), we observe that the existence of multipliers
satisfying (3.4)-(3.6) is equivalent to the existence of multipliers satisfying (3.1)-(3.3)
for all index sets # I 0 (-x, -
y, -
u) with the components (# I0 , # I0 ) having the same sign.
From the proof of Theorem 3.2, it is easy to see that the condition that no nonzero vectors
satisfy (3.9)-(3.10) is a su#cient condition for the existence of common (# I0 , # I0 )
components of the multiplier (#) for all index sets # I 0 (-x, -
y, -
u). Hence this
condition refines the su#cient condition of a unique multiplier such as the SMFCQ
for the relaxed problem proposed in [4, Proposition 4.3.5].
We now give an example which does not have a unique multiplier satisfying (3.4)-
but does satisfy the condition proposed in Theorem 3.2.
OPTIMIZATION PROBLEMS WITH COMPLEMENTARITY CONSTRAINTS 385
Example 3.1 (see [4, Example 4.3.6]). Consider the following OPCC:
s.t.
x 2 are any real numbers, are obviously solutions
to the above problem. As pointed out in [4, Example 4.3.6], SMFCQ does not hold
for this problem. However, we can verify that it satisfies our condition. Indeed, the
equation (3.9) for this problem is
which implies that
Moreover, the calmness condition is satisfied since the constraint region for each
subproblem F#L is a polyhedron due to the fact that # and g are both a#ne. Hence
by Theorem 3.2, if (-x, -
u) is a local minimum to the above problem, then there exist
# such that
which implies #
is a global optimal solution according to Theorem 3.2 and (-x, 0, 0) with
are local optimal solutions.
To illustrate the application of the result obtained, we now consider the following
bilevel programming problem (BLQP), where the lower level problem is linear
quadratic:
(BLQP) min f(x, y)
s.t. y # S(x),
where G and H are l - n and l - m matrices, respectively, a # R l , and S(x) is the
solution set of the quadratic programming problem with parameter x:
where Q # R m-m is a symmetric and positive semidefinite matrix, p # R n , q
are q - n and q -m matrices, respectively, and b # R q .
Replacing the bilevel constraint by the KKT condition for the lower level problem,
it is easy to see that (BLQP) is equivalent to the problem
(KKT) min f(x, y)
386 J. J. YE
which is an OPCC. Let (-x, - y) be an optimal solution of (BLQP) and -
u a corresponding
u,
Then
I
The feasible region of problem (KKT) is
and for any I # {1, 2, . , q},
y, u) # R
Since F#L for any index set # I 0 has linear constraints only, the problem of
minimizing f over F#L is calm. Hence the following result follows from Proposition
3.1.
Corollary 3.3. Let (-x, - y) be an optimal solution of (BLQP) and - u a corresponding
multiplier. Suppose that f is locally Lipschitz near (-x, -
y). Then for each
I 0 , there exist # R m , # R d , # R q such that
If f is either convex or pseudoconvex, then the above necessary condition is also
su#cient for a feasible solution (-x, -
y, -
u) of (KKT) to be a minimum of f over all
y, u) #I0 F#L . In particular, if f is either convex or pseudoconvex and I
{1, 2, . , q}, then the above condition is su#cient for a feasible solution (-x, - y) to be
a global optimum for (BLQP).
The following result follows from Theorem 3.2.
Corollary 3.4. Let (-x, - y) be an optimal solution of (BLQP) and - u a corresponding
multiplier. Suppose that f is C 1 and
implies # there exist # R m , # R d , # R q such that
OPTIMIZATION PROBLEMS WITH COMPLEMENTARITY CONSTRAINTS 387
Equivalently, there exist # R l , # R d , # R q such that (3.16)-(3.17) are satisfied
and
Equivalently, there exist # R l , # R d , # R q , # R 2q
such that (3.16)-(3.17)
are satisfied and
Conversely, let (-x, -
y) be any vector in R n+m satisfying the constraints G-x+H -
be pseudoconvex. If there exists -
(3.11)-(3.12) such that one of the above equivalent conditions holds, then (-x, - y, -
u) is a
minimum of f over all (x, y, u) #I0 F#L . In addition to the above assumptions,
if I
y) is a global minimum for (BLQP).
Acknowledgments
. The author would like to thank Dr. Qing Lin for a helpful
discussion of Proposition 2.8.
--R
Optimization and Nonsmooth Analysis
Optimal Control via Nonsmooth Analysis
Mathematical Programs with Equilibrium Constraints
Piecewise Sequential Quadratic Programming for Mathematical Programs with Nonlinear Complementarity Constraints
Generalized di
Necessary optimality conditions for optimization problems with variational inequality constraints
Optimality conditions for bilevel programming problems
Exact penalization and necessary optimality conditions for generalized bilevel programming problems
--TR
--CTR
Jin-Bao Jian, A Superlinearly Convergent Implicit Smooth SQP Algorithm for Mathematical Programs with Nonlinear Complementarity Constraints, Computational Optimization and Applications, v.31 n.3, p.335-361, July 2005
Houyuan Jiang , Daniel Ralph, Extension of Quasi-Newton Methods to Mathematical Programs with Complementarity Constraints, Computational Optimization and Applications, v.25 n.1-3, p.123-150 | optimality conditions;optimization problems;bilevel programming problems;complementarity constraints;proximal normal cones |
589256 | Solving the Trust-Region Subproblem using the Lanczos Method. | The approximate minimization of a quadratic function within an ellipsoidal trust region is an important subproblem for many nonlinear programming methods. When the number of variables is large, the most widely used strategy is to trace the path of conjugate gradient iterates either to convergence or until it reaches the trust-region boundary. In this paper, we investigate ways of continuing the process once the boundary has been encountered. The key is to observe that the trust-region problem within the currently generated Krylov subspace has a very special structure which enables it to be solved very efficiently. We compare the new strategy with existing methods. The resulting software package is available as HSL_VF05 within the Harwell Subroutine Library. | Introduction
Trust-region methods for unconstrained minimization are blessed with both strong theoretical
convergence properties and a good reputation in practice. The main computational step in these
methods is to find an approximate minimizer of some model of the true objective function within
a "trust" region for which a suitable norm of the correction lies within a given bound. This
restriction is known as the trust-region constraint, and the bound on the norm is its radius. The
radius is adjusted so that successive model problems mimic the true objective within the trust
region.
The most widely-used models are quadratic approximations to the objective function, as these
are simple to manipulate and may lead to rapid convergence of the underlying method. From
a theoretical point of view, the norm which defines the trust region is irrelevant so long as it
"uniformly" related to the ' 2 norm. From a practical perspective, this choice certainly affects the
subproblem, and thus the methods one can consider when solving it. The most popular practical
choices are the ' 2 - and ' 1 -norms, and weighted variants thereof. In our opinion, it is important
that the choice of norm reflects the underlying geometry of the problem; simply picking the
may not be adequate when the problem is large, and the eigenvalues of the Hessian of
the model widely spread. We believe that weighting the norm is essential for many large-scale
problems.
In this paper, we consider the solution of the quadratic-model trust-region subproblem in a
weighted ' 2 -norm. We are interested in solving large problems, and thus cannot rely solely on
factorizations of the matrices involved. We thus concentrate on iterative methods. If the model of
the Hessian is known to be positive definite and the trust-region radius sufficiently large that the
trust-region constraint is inactive at the unconstrained minimizer of the model, the obvious way
to solve the problem is to use the preconditioned conjugate-gradient method. Note that the role
of the preconditioner here is the same as the role of the norm used for the trust-region, namely to
change the underlying geometry so that the Hessian in the rescaled space is better conditioned.
Thus, it will come as no surprise that the two should be intimately connected. Formally, we shall
require that the weighting in the ' 2 -norm and the preconditioning are performed by the same
matrix.
When the radius is smaller than a critical value, the unconstrained minimizer of the model
will no longer lie within the trust-region, and thus the required solution will lie on the trust-region
boundary. The simplest strategy in this case is to consider the piecewise linear path connecting
the conjugate-gradient iterates, and to stop at the point where this path leaves the trust region.
Such a strategy was first proposed independently by Steihaug (1983) and Toint (1981), and
we shall refer to the terminating point as the Steihaug-Toint point. Remarkably, it is easy to
establish the global convergence of a trust-region method based on such a simple strategy. The
key is that global convergence may be proved provided that the accepted estimate of the solution
has a model value no larger than at the Cauchy point (see Powell, 1975). The Cauchy point
is simply the minimizer of the model within the trust-region along the preconditioned steepest-descent
direction. As the first segment on the piecewise-linear conjugate-gradient path gives
N. I. M. Gould, S. Lucidi, M. Roma and Ph. L. Toint
precisely this point, and as the model value is monotonically decreasing along the entire path,
the Steihaug-Toint strategy ensures convergence.
If the model Hessian is indefinite, the solution must also lie on the trust-region boundary.
This case may also be simply handled using preconditioned conjugate gradients. Once again
the piecewise linear path is followed until either it leaves the trust-region, or a segment with
negative curvature is found (a vector p is a direction of negative curvature if the inner product
is the model Hessian). In the latter case, the path is continued downhill
along this direction of negative curvature as far as the constraint boundary. This variant was
proposed by Steihaug (1983), while Toint (1981) suggests simply returning to the Cauchy point.
As before, global convergence is ensured as either of these terminating points, as the objective
function values there are no larger than at the Cauchy point. For consistency with the previous
paragraph, we shall continue to refer to the terminating point in Steihaug's algorithm as the
Steihaug-Toint point, although strictly Toint's point in this case may be different.
The Steihaug-Toint method is basically unconcerned with the trust region until it blunders
into its boundary and stops. This is rather unfortunate, particularly as considerable experience
has shown that this frequently happens during the first few, and often the first, iteration(s) when
negative curvature is present. The resulting step is then barely, if at all, better than the Cauchy
direction, and this may lead to a slow but globally convergent algorithm in theory and a barely
convergent method in practice. In this paper, we consider an alternative which aims to avoid
this drawback by trying harder to solve the subproblem when the boundary is encountered, while
maintaining the efficiencies of the conjugate gradient method so long as the iterates lie interior.
The mechanism we use is the Lanczos method.
The paper is organized as follows. In Section 2 we formally define the problem and any
notation that we will use. The basis of our new method is given in Section 3, while in Section 4,
we will review basic properties of the preconditioned conjugate-gradient and Lanczos methods.
Our new method is given in detail in Section 5. Some numerical experiments demonstrating the
effectiveness of the approach are given in Section 6, and a number of conclusions and perspectives
are drawn in the final section.
Solving the trust-region subproblem using the Lanczos method 3
2 The trust-region subproblem and its solution
Let M be a symmetric positive-definite easily-invertible approximation to the symmetric matrix
H. Furthermore, define the M-norm of a vector as
where h\Delta; \Deltai is the usual Euclidean inner product. In this paper, we consider the M-norm trust-region
problem
minimize
subject to ksk M - \Delta; (2.1)
for some vector g and radius \Delta ? 0.
A global solution to the problem is characterized by the following result.
Theorem 2.1 (Gay, 1981, Sorensen, 1982) Any global minimizer s M of q(s) subject to
satisfies the equation
positive semi-definite, - M - 0 and - M (ks
is positive definite, s M is unique.
This result is the basis of a series of related methods for solving the problem which are appropriate
when forming factorizations of H(-) j H +-M for a number of different values of - is realistic.
For then, either the solution lies interior, and hence - g, or the solution lies
on the boundary and - M satisfies the nonlinear equation
denotes the pseudo-inverse of H. Equation (2.3) is straightforward to solve using a
safeguarded Newton iteration, except in the so-called hard case for which g lies in the null-space of
H(- M ). In this case, an additional vector in the range-space of H(- M ) may be required if a solution
on the trust-region boundary is sought. Goldfeldt, Quandt and Trotter (1966), Hebden (1973)
and Gay (1981) all proposed algorithms of this form. The most sophisticated algorithm to date,
by Mor'e and Sorensen (1983), is available as subroutine GQTPAR in the MINPACK-2 package, and
guarantees that a nearly optimal solution will be obtained after a finite number of factorizations.
While such algorithms are appropriate for large problems with special Hessian structure -
such as for band matrices - the demands of a factorization at each iteration limits their applicability
for general large problems. It is for this reason that methods which do not require
factorizations are of interest.
Throughout the paper, we shall denote the k by k identity matrix by I k , and its j-th column
by e j . A set of vectors fq i g are said to be M-orthonormal if hq the Kronecker delta,
4 N. I. M. Gould, S. Lucidi, M. Roma and Ph. L. Toint
and the matrix Q formed from these vectors is an M-orthonormal matrix. The set
of vectors fp i g are H-conjugate (or H-orthogonal) if hp
3 A new algorithm for large-scale trust-region subproblems
To set the scene for this paper, we recall that the Cauchy point may be defined as the solution
to the problem
minimize
subject to ksk M - \Delta; (3.1)
that is as the minimizer of q within the trust region where s is restricted to the 1-dimensional
subspace span \Phi
. The dogleg methods (see, Powell, 1970, Dennis and Mei, 1979) aim to
solve the same problem over a one-dimensional arc, while Byrd, Schnabel and Schultz (1985) do
the same over a two-dimensional subspace. In each of these cases the solution is easy to find as
the search space is small. The difficulty with the general problem (2.1) is that the search space
R n is large. This leads immediately to the possibility of solving a compromise problem
minimize
subject to ksk M - \Delta; (3.2)
where S is a specially chosen subspace of R n .
Now consider the Steihaug-Toint algorithm at an iteration k before the trust-region boundary
is encountered. In this case, the point s k+1 is the solution to (3.2) with the set
span
the Krylov space generated by the starting vector M \Gamma1 g and matrix M \Gamma1 H. That is, the
Steihaug-Toint algorithm gradually widens the search space using the very efficient preconditioned
conjugate gradient method. However, as soon as the Steihaug-Toint algorithm moves
across the trust-region boundary, the terminating point s k+1 no longer necessarily solves the
problem in (3.3), indeed it is very unlikely to do so when k ? 0. (As the iterates generated by
the method increase in M-norm, once an iterate leaves the trust region, the solution to (3.3),
and thus (2.1), must lie on the boundary. See, Steihaug, 1983, Theorem 2.1, for details). Can
we do better? Yes, by recalling that the preconditioned conjugate gradient and Lanczos methods
generate different bases for the same Krylov space.
Solving the trust-region subproblem using the Lanczos method 5
4 The preconditioned conjugate-gradient and Lanczos methods
The preconditioned conjugate-gradient and Lanczos methods may be viewed as efficient techniques
for constructing different bases for the same Krylov space, K k . The conjugate gradient
method aims for an H-conjugate basis, while the Lanczos method obtains an M-orthonormal
basis.
Algorithm 4.1: The preconditioned conjugate gradient method
perform the iteration,
Algorithm 4.2: Preconditioned Lanczos method
perform the iteration,
The conjugate gradient method generates the basis
from Algorithm 4.1, while the Lanczos method generates the basis
6 N. I. M. Gould, S. Lucidi, M. Roma and Ph. L. Toint
with Algorithm 4.2. The Lanczos iteration is often written in the more compact form
k+1 and (4.14)
is the matrix (q and the matrix
@
A
is tridiagonal. It then follows directly that
The two methods are intimately related. In particular, so long as the conjugate-gradient
iteration does not break down, the Lanczos vectors may be recovered from the conjugate-gradient
iterates as
while the Lanczos tridiagonal may be expressed as
ff
ff k\Gamma2
ff
A
The conjugate gradient iteration may breakdown if hp which can only occur if H is
not positive definite, and will stop if hg On the other hand, the Lanczos iteration can
only fail if K j is an invariant subspace for M \Gamma1 H.
If q(s) is convex in the manifold K j+1 , the minimizer s j+1 of q in this manifold satisfies
so long as the initial value s chosen. Thus this estimate is easy to recur from the conjugate-gradient
iteration. The minimizers in successive manifolds may also be easily obtained using the
Lanczos process, although the conjugate-gradient iteration is slightly less expensive, and thus to
be preferred.
Solving the trust-region subproblem using the Lanczos method 7
The vector g j+1 in the conjugate gradient method gives the gradient of q(s) at s j+1 . It is
quite common to stop the method as soon as this gradient is sufficiently small, and the method
naturally records the M \Gamma1 -norm of the gradient, kg This norm is also available
in the Lanczos method as
solves the tridiagonal linear system T k The last component, he k+1
of h k is available as a further by-product.
5 The truncated Lanczos approach
Rather than use the preconditioned conjugate gradient basis fp for S, we shall use
the equivalent Lanczos M-orthonormal basis fq g. The Lanczos basis has previously
been used by Nash (1984) - to convexify the quadratic model - and Lucidi and Roma (1997)
- to compute good directions of negative curvature - within linesearch based method for unconstrained
minimization. We shall consider vectors of the form
and seek
solves the problem
minimize
s 2S
subject to ksk M - \Delta: (5.2)
It then follows directly from (4.15), (4.17) and (4.18) that h k solves the problem
minimize
subject to khk 2 - \Delta: (5.3)
There are a number of crucial observations to be made here. Firstly, it is important to note that
the resulting trust-region problem involves the two-norm rather than the M-norm. Secondly,
as T k is tridiagonal, it is feasible to use the Mor'e-Sorensen algorithm to compute the model
minimizer even when n is large. Thirdly, having found h k , the matrix Q k is needed to recover
thus the Lanczos vectors will either need to be saved on backing store or regenerated.
As we shall see, we only need Q once we are satisfied that continuing the Lanczos process will
give little extra benefit. Fourthly, one would hope that as a sequence of such problems may be
solved, and as T k only changes by the addition of an extra diagonal and superdiagonal entry,
solution data from one subproblem may be useful for starting the next. We consider this issue in
Section 5.2.
The basic trust-region solution classification theorem, Theorem 2.1, shows that
8 N. I. M. Gould, S. Lucidi, M. Roma and Ph. L. Toint
I k+1 is positive semi-definite, What does this tell
us about s k ? Firstly, using (4.17), (4.18) and (5.4) we have
and additionally that
Comparing these with the trust-region classification theorem, we see that s k is the Galerkin
approximation to s M from the space spanned by Q k .
We may then ask how good the approximation is. In particular, what is the error (H
g? The simplest way of measuring this error would be to calculate h k and - k by
solving (5.3), then to recover s k as Q k h k and finally to substitute s k and - k into (H +-M)s+ g.
However this is inconvenient as it requires that we have easy access to Q k . Fortunately there is
a far better way.
Theorem 5.1
and
Proof. We have that
iw k+1 from (4.14)
iw k+1 from (5.4)
This then directly gives (5.6), and (5.7) follows from the M \Gamma1 -orthonormality of w k+1 . 2
Therefore we can indirectly measure the error (in the M \Gamma1 -norm) knowing simply fl k+1 and the
last component of h k , and we do not need s k or Q k at all. Observant readers will notice the strong
similarity between this error estimate and the estimate (4.22) for the gradient of the model in
the Lanczos method, but this is not at all surprising as the two methods are aiming for the same
point if the trust-region radius is large enough. An interpretation of (5.7) is also identical to that
of (4.22). The error will be small when either of fl k+1 or the last component of h k is small.
We now consider the problem (5.3) in more detail. We say that a symmetric tridiagonal matrix
is degenerate if one or more of its off-diagonal entries is zero; otherwise it is non-degenerate. We
then have the following preliminary result.
Solving the trust-region subproblem using the Lanczos method 9
Lemma 5.2 (See also, Parlett, 1980, Theorem 7.9.5) Suppose that the tridiagonal matrix
T is non-degenerate, and that v is an eigenvector of T . Then the first component of v is
nonzero.
Proof. By definition
for some eigenvector '. Suppose that the first component of v is zero. Considering the first
component of (5.8), we have that the second component of v is zero as T is tridiagonal and
non-degenerate. Repeating this argument for the i-th component of (5.8), we deduce that the
the 1-st component of v is zero for all i, and hence that contradicts the
assumption that v is an eigenvector, and so the first component of v cannot be zero. 2
This immediately yields the following useful result.
Theorem 5.3 Suppose that T k is non-degenerate. Then the hard case cannot occur for the
subproblem (5.3).
Proof. Suppose the hard case occurs. Then, by definition, fl 0 e 1 is orthogonal to v k , the
eigenvector corresponding to the leftmost eigenvalue, \Gamma' k , of T k . Thus, the first component of
v k is zero, which, following Lemma 5.2, contradicts the assumption that v k is an eigenvector.
Thus the hard case cannot occur. 2
This result is important as it suggests that the full power of the Mor'e and Sorensen (1983)
algorithm is not needed to solve (5.3). We shall return to this in Section 5.2. We also have an
immediate corollary.
Corollary 5.4 Suppose that T n\Gamma1 is non-degenerate. Then the hard case cannot occur for
the original problem (2.1).
Proof. When the columns of Q forms a basis for R n , Thus the problems (2.1)
and (5.2) are identical, and (5.2) and (5.3) are related through a nonsingular transformation.
The result then follows directly from Theorem 5.3 in the case
Thus, if the hard case occurs for (2:1), the Lanczos tridiagonal must degenerate at some stage.
N. I. M. Gould, S. Lucidi, M. Roma and Ph. L. Toint
Theorem 5.5 Suppose that T k is non-degenerate, that h k and - k satisfy (5.4) and that
I k+1 is positive semi-definite. Then I k+1 is positive definite.
Proof. Suppose that T k I k+1 is singular. Then there is a nonzero eigenvector v k for
which Hence, combining this with (5.4) reveals that
and hence that the first component of v k is zero. But this contradicts Lemma 5.2. Hence
I k+1 is both positive semi-definite and nonsingular, and thus positive definite. 2
This result implies that (5.4) has a unique solution. We now consider this solution.
Theorem 5.6 Suppose that he k+1
Proof. Suppose that T k is not degenerate. As the k 1-st component of h k is zero, then the
non-degeneracy of T k and the k 1-st equation of (5.4), we deduce that the k-th component
of h k is zero. Repeating this argument for the 1-st equation of (5.4), we deduce that the
i-th component of h k is zero for 1 - i - k, and hence that h contradicts the
first equation of (5.4), and thus T k must be degenerate. 2
Thus we see that of the two possibilities suggested by Theorem 5.1 for obtaining an s k for which
will be the possibility fl that occurs before he k+1
Theorem 5.7 Suppose that the hard case does not occur for (2.1), and that fl
Proof. If the Krylov space K k is an invariant subspace M \Gamma1 H, and by construction
the first basis element of this space is M \Gamma1 g. As the hard case does not occur for (2.1), this
space must also contain at least one eigenvector corresponding to the leftmost eigenvalue, \Gamma',
of M \Gamma1 H. Thus one of the eigenvalues of T k must be \Gamma', and - k - ' as T k +- k I k+1 is positive
semi-definite. But this implies that H positive semi-definite, which combines with
(5.1), (5.5) and Theorem 5.1 with to show that s k satisfies the optimality conditions
shown in Theorem 2.1. 2
Thus we see that in the easy case, the required solution will be obtained from the first non-degenerate
block of the Lanczos tridiagonal. It remains for us to consider the hard case. In
Solving the trust-region subproblem using the Lanczos method 11
view of Corollary 5.4, this case can only occur when T k is degenerate. Suppose therefore that T k
degenerates into ' blocks of the form
@
A
where each of the T k i
defines an invariant subspace for M \Gamma1 H and where the last block T k '
is
the first to yield the leftmost eigenvalue, \Gamma', of M \Gamma1 H. Then there are two cases to consider.
Theorem 5.8 Suppose that the hard case occurs for (2.1), that T k is as described by (5.9),
and the last block T k '
is the first to yield the leftmost eigenvalue, \Gamma', of M \Gamma1 H. Then,
1. if ' - k1 , a solution to (2.1) is given by s k1 solves the positive-definite
system
2. if ' ? - k1 , a solution to (2.1) is given by s
@
A
h is the solution of the nonsingular tridiagonal system
u is an eigenvector of T k '
corresponding to \Gamma', and ff is chosen so that
Proof. In case 1, H+- positive semi-definite as - k 1 - ', and the remaining optimality
conditions are satisfied as k1 +1 is positive
definite follows from Theorem 5.5. In case 2, H+'M is positive semi-definite. Furthermore, as
is easy to show that khk 2 ! kh k1 k 2 - \Delta, and hence that there is a root ff for which
ks \Delta. Finally, as each Q k i
defines an invariant subspace, HQ k i
Writing
u, we therefore have
N. I. M. Gould, S. Lucidi, M. Roma and Ph. L. Toint
and
all of the optimality conditions for (5.2). 2
Notice that to obtain s k as described in this theorem, we only require the Lanczos vectors corresponding
to blocks one and, perhaps, ' of T k .
We do not claim that to solve the problem as outlined in Theorem 5.8 is realistic, as it relies
on our being sure that we have located the left-most eigenvalue of M \Gamma1 H. With Lanczos-type
methods, one cannot normally guarantee that all eigenvalues, including the leftmost, will be
found unless one ensures that all invariant subspaces have been investigated, and this may prove
to be very expensive for large problems. In particular, the Lanczos algorithm, Algorithm 4.2,
terminates each time an invariant subspace has been determined, and must be restarted using
a vector q which is M-orthonormal to the previous Lanczos directions. Such a vector may be
obtained from the Gram-Schmidt process by re-orthonormalizing a suitable vector - a vector
with some component M-orthogonal to the existing invariant subspaces, perhaps a random vector
- with respect to the previous Lanczos directions, which means that these directions will have to
be regenerated or reread from backing store. Thus, while this form of the solution is of theoretical
interest, it is unlikely to be of practical interest if a cheap approximation to the solution is all
that is required.
5.1 The algorithm
We may now outline our algorithm, Algorithm 5.1, the generalized Lanczos trust region (GLTR)
method. We stress that, as our goal is merely to improve upon the value delivered by the
Steihaug-Toint method, we do not use the full power of Theorem 5.8, and are content just
to investigate the first invariant subspace produced by the Lanczos algorithm. In almost all
cases, this subspace contains the global solution to the problem, and the complications and costs
required to implement a method based on Theorem 5.8 are, we believe, prohibitive in our context.
Solving the trust-region subproblem using the Lanczos method 13
Algorithm 5.1: The generalized Lanczos trust region method
. Set the flag INTERIOR as
true. For convergence, perform the iteration,
using (4.20)
If INTERIOR is true, but ff k - 0 or ks k reset INTERIOR to false.
If INTERIOR is true
else
solve the tridiagonal trust-region subproblem (5.3) to obtain h k
If INTERIOR is true
test for convergence using the residual kg
else
test for convergence using the value fl k+1 jhe
If INTERIOR is false, recover s by rerunning the recurrences or obtaining Q k from
backing store.
When recovering s by rerunning the recurrences, economies can be made by saving the
during the first pass, and reusing them during the second. A potentially bigger saving
may be made if one is prepared to accept a slightly inferior value of the objective function. The
idea is simply to save the value of q at each iteration. On convergence, one looks back through
this list to find an iteration, ' say, for which a required percentage of the best value was obtained,
recompute h ' and then accept s as the required estimate of the solution. If the required
percentage occurs at an iteration before the boundary is encountered, both the final point before
the boundary and the Steihaug-Toint point are suitable and available without the need for the
second pass.
We note that we have used the conjugate-gradient method (Algorithm 4.1) to generate the
Lanczos vectors. If the inner-product hp k ; Hp k i proves to be tiny, it is easy to continue using the
Lanczos method (Algorithm 4.2) itself; the vectors
14 N. I. M. Gould, S. Lucidi, M. Roma and Ph. L. Toint
required to continue the Lanczos recurrence (4.11) are directly calculable from conjugate-gradient
method.
At each stage of both the Steihaug-Toint algorithm and our GLTR method (Algorithm 5.1),
we need to calculate ks k . This issue is not discussed by Steihaug as it is implicitly
assumed that M is available. However, it may be the case that all that is actually available is a
procedure which returns M \Gamma1 v for a given input v, and thus M is unavailable. Fortunately this
is not a significant drawback as it is possible to calculate ks k +ffp k k M from available information.
To see this, observe that
ks
ks
and thus that we can find ks k+1 k 2
M from ks k k 2
M so long as we already know hs k ; Mp k i and
M . But it is straightforward to show that these quantities may be calculated from the pair
of recurrences
and (5.12)
where, of course, hg k ; v k i has already been calculated as part of the preconditioned conjugate-gradient
method.
5.2 Solving the nondegenerate tridiagonal trust-region subproblem
In view of Theorem 5.3, the nondegenerate tridiagonal trust-region subproblem (5.3) is, in theory,
easier to solve than the general problem. This is so both because the Hessian is tridiagonal (and
thus very inexpensive to factorize), and because the hard case cannot occur. We should be
cautious here, because the so-called "almost" hard case - which occurs when g only has a tiny
component in the range-space of H(- M ) - may still happen, and the trust-region problem in
this case is naturally ill conditioned and thus likely to be difficult to solve.
The Mor'e and Sorensen (1983) algorithm is based on being able to form factorizations of the
model Hessian (which is certainly the case here as is tridiagonal), but does not try
to calculate the leftmost eigenvalue of the pencil H + -M . In the tridiagonal case, computing
the extreme eigenvalues is straightforward, particularly if a sequence of related problems are to
be solved. Thus, rather than using the Mor'e and Sorensen algorithm, we prefer the following
method.
We restrict ourselves to the case where the solution lies on the trust-region boundary - we
will only switch to this approach when the conjugate gradient iteration leaves the trust region.
The basic iteration is identical to that proposed by Mor'e and Sorensen (1983), namely to apply
Newton's method to
where
Solving the trust-region subproblem using the Lanczos method 15
Recalling that we denote the leftmost eigenvalue of T k by \Gamma' k , the main difference between
our approach and Mor'e and Sorensen's is that we always start from some point in the interval
this interval is characterized by both being positive definite and
as then the resulting Newton iteration is globally linearly, and asymptotically
quadratically, convergent without any further safeguards. The Newton iteration is performed
using Algorithm 5.2.
Algorithm 5.2: Newton's method to solve
1. Factorize are unit bidiagonal and diagonal
matrices, respectively.
2. Solve BDB T
3. Solve
4. Replace - by -
The Newton correction in Step 4 of this algorithm is given by
while the exact form given is obtained by using the identity
where w is as computed in Step 3. It is slightly more efficient to pick B to be unit upper-
bidiagonal rather than unit lower-bidiagonal, as then the Step 2 simplifies to B T
because of the structure of the right-hand side.
To obtain a suitable starting value, two possibilities are considered. Firstly, we attempt
to use the solution value - k\Gamma1 from the previous subproblem. Recall that T k is merely T
with an appended row and column. As we already have a factorization of T
trivial to obtain that of T and thus to determine if the latter is positive definite.
If turns out to be positive definite, h k (- k\Gamma1 ) is computed from (5.15) and if
is used to start the Newton iteration.
Secondly, if - k\Gamma1 is unsuitable, we monitor T k to see if it is indefinite. This is trivial, as for
instance, the matrix is positive definite so long as all of the ff i , generated by the
conjugate-gradient method are positive. If T k is positive definite, we start the Newton iteration
with the value which by assumption gives kh k (0)k 2 - \Delta as the unconstrained solution lies
outside the trust region. Otherwise, we determine the leftmost eigenvalue, \Gamma' k , of T k , and start
N. I. M. Gould, S. Lucidi, M. Roma and Ph. L. Toint
with positive number chosen to make T k numerically
"just" positive definite. By this we mean, that its BDB T factorization should exist, but that ffl
should be as small as possible. We have found that a value (1
is the unit
roundoff, is almost always suitable, but have added the precaution of multiplying this value by
increasing powers of 2 so long as the factorization fails.
If we need to compute the leftmost eigenvalue of T k , we use an iteration based upon the
last-pivot function proposed by Parlett and Reid (1981). The last-pivot function, ffi k ('), is simply
the value of the last diagonal entry of the BDB T factor D k (-) of T k \Gamma 'I k+1 . This value will be
zero, and the other diagonal entries positive, when
of uncertainty [' l ; ' u ] is placed around the required root. The initial interval is given by the
Gersgorin bounds on the leftmost eigenvalue. When it is known, the leftmost eigenvalue, \Gamma'
of T k\Gamma1 may be used to improve the lower bound, because of the Cauchy interlacing property
of the eigenvalues of T k\Gamma1 and T k (see, for instance, Parlett, 1980, Theorem 10.1.2). Given an
initial estimate of ' k , an improvement may be sought by applying Newton's method to ffi k ('); the
derivative of ffi k is easy to obtain by recurrence. However, as Parlett and Reid point out,
and thus has a pole at Hence it is better to choose the new point by fitting the model
to the function and derivative value at the current ', and then to pick the new iterate as the
larger root of ffi M
('). If the new iterate lies outside the interval of uncertainty, it is replaced by
the midpoint of the interval. The interval is then contracted by computing ffi k at the new iterate,
and replacing the appropriate endpoint by the iterate. The iteration is stopped if the length of
the interval or the value of
If ' k\Gamma1 is known, the initial iterate chosen as ' positive ffl -
successive iterates generated from (5.16), the iterates convergence globally, and asymptotically
superlinearly, from the left. If the Newton iteration is used, the required root is frequently
obscured, and the scheme resorts to interval bisection. Thus the Parlett and Reid scheme is to
be preferred.
Other means of locating the required eigenvalue, based on using the determinant det(T
were tried, but proved to be less reliable because of the huge numerical
range (and thus potential overflow) of the determinant.
Solving the trust-region subproblem using the Lanczos method 17
6 Numerical experiments
The algorithm sketched in Sections 5.1 and 5.2 has been implemented as a Fortran 90 module,
HSL VF05, within the Harwell Subroutine Library (1998).
As our main interest is in using the methods described in this paper within a trust-region
algorithm, we are particularly concerned with two issues. Firstly, can we obtain significantly
better values of the model by finding better approximations to its solution than the Steihaug-
Toint method? And secondly, do better approximations to the minimizer of the model necessarily
translate into fewer iterations of the trust-region method? In this section, we address these
outstanding questions.
Throughout, we will consider the basic problem of minimizing an objective f(x) of n real
variables x. We shall use the following standard trust-region method.
Algorithm 6.1: Standard Trust-Region Algorithm
An initial point x 0 and an initial trust-region radius \Delta 0 are given, as are constants ffl g ,
are required to satisfy the conditions
1. Stop if kr x f(x k )k 2 - ffl g .
2. Define a second-order Taylor series model q k and a positive-definite preconditioner
. Compute a step s k to "sufficiently reduce the model" q k within the trust-region
3. Compute the ratio
4. Set
Increment k by one and go to Step 1.
We choose the specific values ffl set an
upper limit of n iterations. The step s k in step 2 is computed using either Algorithm 5.1 or the
Steihaug-Toint algorithm. Convergence in both algorithms for the subproblem occurs as soon as
N. I. M. Gould, S. Lucidi, M. Roma and Ph. L. Toint
or if more than n iterations have been performed. In addition, of course, the Steihaug-Toint
algorithm terminates as soon as the boundary is crossed.
All our tests were performed on an IBM RISC System/6000 3BT workstation with 64 Mega-bytes
of RAM; the codes are all double precision Fortran 90, compiled under xlf90 with -O
optimization, and IBM library BLAS are used. The test examples we consider are the larger
examples from the CUTE test set (see Bongartz, Conn, Gould and Toint, 1995) for which negative
curvature is frequently encountered. Tests were terminated if more than thirty CPU minutes
elapsed.
6.1 Can we get much better model values than Steihaug-Toint?
We first consider problems of the form (2.1). Our test examples are generated by running Algorithm
6.1 on the CUTE set for 10 iterations, and taking the trust-region subproblem at iteration
as our example. The idea here is to simulate the kind of subproblems which occur in practice,
not those which result at the starting point for the algorithm as such points frequently have
special (favourable) properties.
Our aim is to see whether there is any significant advantage in continuing the minimization of
the trust-region subproblem once the boundary of the trust region has been encountered. We ran
HSL VF05 to convergence, stopping when kg more than
iterations had been performed.
In all of the experiments reported here, the best value found was in fact the optimum value
- a factorization of H was used to confirm that the matrix was positive semi-definite,
while the algorithm ensured that the remaining optimality conditions hold - although, of course,
there is no guarantee that this will always be the case. We measured the iteration (ST) and the
percentage (ratio) of the optimal value obtained at the point at which the Steihaug-Toint method
left the trust region, as well as the number of iterations taken to achieve 10%, 90% and 99% of
the optimal reduction (10%, 90%, 99% respectively).
The results of these experiments are summarized in Table 6.1. In this table we give the
name of each example used, along with its dimension n, and the statistics "ratio"(expressed in
the form x(y) as a shorthand for x \Theta 10 y ), "ST", "10%", "90%" and "99%" as just described.
Some of the problems had interior solutions, in which case the "ratio" and "ST" statistics are
absent (as indicated by a dash). We considered both the unpreconditioned method
and a variety of standard preconditioners - a band preconditioner with semi-bandwidth of 5,
and modified incomplete and sparse Cholesky factorizations, with the modifications as proposed
by Schnabel and Eskow (1991) - used by the LANCELOT package (see, Conn, Gould and Toint,
1992, Chapter 3). The Cholesky factorization methods both failed for the problem MSQRTALS for
which the Hessian matrix required too much storage.
We make a number of observations.
1. On some problems, the Steihaug-Toint point gives a model value which is a good approximation
to the optimal value.
Solving the trust-region subproblem using the Lanczos method 19
no preconditioner 5-band
example
BRYBND 1000 3(-5) 23 24 28
CHAINWOO 1000 4(-5) 15
COSINE 1000
GENROSE 1000
MSQRTALS 1024 1(-1) 12 11 23
Incomplete Cholesky Modified Cholesky
example
COSINE 1000
GENROSE 1000
MSQRTALS 1024 factorization failure factorization failure
Table
6.1: A comparison of the number of iterations required to achieve a given percentage of
the optimal model value for a variety of preconditioners. See the text for a key to the data.
N. I. M. Gould, S. Lucidi, M. Roma and Ph. L. Toint
2. On other problems, a few extra iterations beyond the Steihaug-Toint point pay handsome
dividends.
3. Getting to within 90% or even 99% of the best value very rarely requires many more
iterations than to find the Steihaug-Toint point.
In conclusion, based on these numbers, we suggest that a good strategy would be to perform
a few (say 5) iterations beyond the Steihaug-Toint point, and only accept the improved point
if its model value is significantly better (as this will cost a second pass to compute the Lanczos
vectors). We shall consider this further in the next section.
6.2 Do better values than Steihaug-Toint imply a better trust-region method?
We now consider how the methods we have described for approximately solving the trust-region
subproblem perform within a trust-region algorithm. Of particular interest is the question as to
whether solving the subproblem more accurately reduces the number of trust-region iterations,
or more particularly the cost of solving the problem - the number of iterations is of concern if
the evaluation of the objective function and its derivatives is the dominant cost as then there is
a direct correlation between the number of iterations and the overall cost of solving the problem.
In
Tables
6.2 and 6.3, we compare the Steihaug-Toint scheme with the GLTR algorithm
(Algorithm 5.1) run to high accuracy. We exclude the problem HYDC20LS for our reported results
as no method succeeded in solving the problem in fewer than our limit of n iterations, and
the problems BROYDN7D and SPMSRTLS as a number of different local minima were found. In
these tables, in addition to the name and dimension of each example, we give the number of
objective function ("#f") and derivative ("#g") values computed, the total number of matrix-vector
products ("#prod") required to solve the subproblems, and the total CPU time required
in seconds. We compare the same preconditioners M as we used in the previous section. We
indicate those cases where one or other method performs at least 10% better than its competitor
by highlighting the relevant figure in bold.
We observe the following.
1. The use of different M leads to radically different behaviour. Different preconditioners
appear to be particularly suited to different problems. Surprisingly, perhaps, the unpreconditioned
algorithm often performs the best overall.
2. In the unpreconditioned case, the model-optimum variant frequently requires significantly
fewer function evaluations than the Steihaug-Toint method. However, the extra algebraic
costs per iteration often outweigh the reduction in the numbers of iterations. The advantage
in function calls for the other preconditioners is less pronounced.
Ideally, one would like to retain the advantage in numbers of function calls, while reducing
the cost per iteration. As we noted in Section 6.1, one normally gets a good approximation to
the optimal model value after a modest number of iterations. Moreover, while the Steihaug-Toint
point often gives a significantly sub-optimal value, a few extra iterations usually suffices to give
Solving the trust-region subproblem using the Lanczos method 21
no preconditioner Steihaug-Toint model optimum
example
iterations 865 577 34419 145.02
DQRTIC 1000 43 43 83 0.3 43 43 91 0.3
FREUROTH 1000 17 17 34 0.4 17 17 34 0.4
GENROSE 1000 859 777 6092 28.8 773 642 24466 82.2
MSQRTALS 1024 44 34 7795 486.0
NONCVXUN 1000 492 466 177942 1017.9 ? 1800 seconds
SENSORS 100 20 19 37 6.4 20 19 140 8.8
SINQUAD 5000 182 114 363 24.3 161 106 382 24.6
5-band Steihaug-Toint model optimum
example
CHAINWOO 1000 146 99 145 4.8 191 123 196 6.3
COSINE 1000 21 15 20 0.4 21 15
CRAGGLVY 1000 22 22 21 1.1 22 22 21 1.1
DQRTIC 1000 54 54 53 0.9 54 54 53 1.0
EIGENALS 930 56 43 171 75.2 53 42 222 75.8
FREUROTH 1000 20
iterations ? n iterations
MANCINO 100 91 72 90 87.2 52 43 90 52.2
MSQRTALS 1024 88 62 9793 700.2 73 52 19416 1292.2
22 N. I. M. Gould, S. Lucidi, M. Roma and Ph. L. Toint
Incomplete Cholesky Steihaug-Toint model optimum
example
CHAINWOO 1000 174 115 173 8.1 183 121 309 10.3
COSINE 1000 22 17 26 0.8 22 19 49 1.2
CRAGGLVY 1000 22 22 21 1.5 22 22 21 1.5
DQRTIC 1000 54 54 53 0.9 54 54 53 1.1
iterations ? n iterations
GENROSE 1000 948 629 951 35.5 496 322 847 23.5
28
MSQRTALS 1024 factorization failure factorization failure
28 150 41.2
iterations ? n iterations
iterations ? n iterations
SINQUAD 5000 77 52 89 542.6 78 50 121 526.7
Modified Cholesky Steihaug-Toint model optimum
example
BRYBND 1000 15 15 14 2.2 59 37 61 7.7
CHAINWOO 1000 178 119 177 7.6 183 121 309 10.3
COSINE 1000
CRAGGLVY 1000 23 23 33 1.4 22 22 21 1.6
DQRTIC 1000 54 54 53 1.2 54 54 53 1.1
iterations ? n iterations
GENROSE 1000 462 332 463 16.5 496 322 847 23.4
MANCINO 100 31 28 28
MSQRTALS 1024 factorization failure factorization failure
Solving the trust-region subproblem using the Lanczos method 23
a large percentage of the optimum. Thus, we next investigate both of these issues in the context
of an overall trust-region method.
In
Tables
6.4 and 6.5, we compare the number of function evaluations (#f), and the CPU time
taken to solve the problem for the Steihaug-Toint ("ST") method with a number of variations
on our basic GLTR method (Algorithm 5.1). The basic requirement is that we compute a model
value which is at least 90% of the best value found during the first pass of the GLTR method. If
this value is obtained by an iterate before that which gives the Steihaug-Toint point, the Steihaug-
Toint point is accepted. Otherwise, a second pass is performed to recover the first point at which
90% of the best value was observed. The other ingredient is the choice of the stopping rule for
the first pass. One possibility is to stop this pass as soon as the test (6.4) is satisfied. We denote
this strategy by "90%best". The other possibility is to stop when either (6.4) is satisfied or at
most a fixed number of iterations beyond the Steihaug-Toint point have occurred. We refer to
this as "90%(ST+k)", where k gives the number of additional iterations allowed. We investigate
the cases Once again, we compare the same preconditioners M as we used in
the previous section. We highlight in bold those entries which are at least 10% better than the
competition.
The conclusions are broadly as before. Each method has its successes and failures, and there
is no clear overall best method or preconditioner, although the unpreconditioned version performs
surprisingly well. Restricting the number of iteration allowed after the Steihaug-Toint point has
been found appears to curb the worst behaviour of the unrestricted method.
N. I. M. Gould, S. Lucidi, M. Roma and Ph. L. Toint
no preconditioner ST 90%(ST+1) 90%(ST+5) 90%(ST+10) 90%best
example
CRAGGLVY 1000 19 1.0 19 0.9 19 0.9 19 0.9 19 1.0
DQRTIC 1000 43 0.3 43 0.3 43 0.3 43 0.3 43 0.3
GENROSE 1000 859 28.8 748 38.9 721 48.1 738 57.3 728 60.0
MSQRTALS 1024 44 486.0
NONCVXUN 1000 492 1017.9 368 861.3 ? 1800 secs. ? 1800 secs. 433 1198.6
SENSORS 100 20 6.4 23 7.3 21 8.1 21 8.0 21 8.1
SINQUAD 5000 182 24.3 152 20.8 152 21.7 152 21.4 152 21.5
5-band ST 90%(ST+1) 90%(ST+5) 90%(ST+10) 90%best
example
CHAINWOO 1000 146 4.8 159 5.1 159 5.1 159 5.2 159 5.1
COSINE 1000 21 0.4 21 0.5 21 0.4 21 0.4 21 0.5
CRAGGLVY 1000 22 1.1 22 1.0 22 1.1 22 1.1 22 1.1
DQRTIC 1000 54 0.9 54 0.9 54 1.0 54 1.0 54 1.0
MANCINO 100 91 87.2 52 51.8 52 51.8 52 52.0 52 51.8
MSQRTALS 1024 88 700.2 97 756.7 73 704.9 74 844.7 79 981.5
Solving the trust-region subproblem using the Lanczos method 25
Incomplete Cholesky ST 90%(ST+1) 90%(ST+5) 90%(ST+10) 90%best
example
BRYBND 1000 55 3.9 56 4.2 56 4.3 56 4.3 56 5.0
CHAINWOO 1000 174 8.1 199 9.7 199 10.1 199 10.2 199 10.1
CRAGGLVY 1000 22 1.5 22 1.6 22 1.6 22 1.5 22 1.6
DQRTIC 1000 54 0.9 54 1.0 54 1.1 54 1.1 54 1.1
EIGENALS 930 76 94.6 77 97.2 74 97.2 74 97.3 74 96.8
GENROSE 1000 948 35.5 500 22.4 499 23.0 499 23.0 499 23.0
MSQRTALS 1024 fact. failure fact. failure fact. failure fact. failure fact. failure
SINQUAD 5000 77 542.6 68 484.2 68 484.1 68 485.4 68 489.0
Modified Cholesky ST 90%(ST+1) 90%(ST+5) 90%(ST+10) 90%best
example
BRYBND 1000 15 2.2 15 2.2 15 2.3 15 2.2 15 2.2
CHAINWOO 1000 178 7.6 176 7.9 176 7.9 176 7.8 176 8.0
COSINE 1000 41 1.1 41 1.3 41 1.3 41 1.3 41 1.3
DQRTIC 1000 54 1.2 54 1.2 54 1.3 54 1.3 54 1.3
GENROSE 1000 462 16.5 434 18.8 434 19.3 434 19.1 434 19.1
MANCINO 100 31 129.3 64 232.3 77 275.9 77 275.5 77 275.6
MSQRTALS 1024 fact. failure fact. failure fact. failure fact. failure fact. failure
26 N. I. M. Gould, S. Lucidi, M. Roma and Ph. L. Toint
7 Perspectives and conclusions
We have considered a number of methods which aim to find a better approximation to the solution
of the trust-region subproblem than that delivered by the Steihaug-Toint scheme. These methods
are based on solving the subproblem within a subspace defined by the Krylov space generated
by the conjugate-gradient and Lanczos methods. The Krylov subproblem has a number of useful
properties which lead to its efficient solution. The resulting algorithm is available as a Fortran
90 module, HSL VF05, within the Harwell Subroutine Library (1998).
We must admit to being slightly disappointed that the new method did not perform uniformly
better than the Steihaug-Toint scheme, and were genuinely surprised that a more accurate approximation
does not appear to significantly reduce the number of function evaluations within
a standard trust-region method. While this may limit the use of the methods developed here,
it also calls into question a number of other recent eigensolution-based proposals for solving the
trust-region subproblem (see Rendl, Vanderbei and Wolkowicz, 1995, Rendl and Wolkowicz, 1997,
Sorensen, 1997, Santos and Sorensen, 1995). While these authors demonstrate that their methods
provide an effective means of solving the subproblem, they make no effort to evaluate whether
this is actually useful within a trust-region method. The results given in this paper suggest that
this may not in fact be the case. This also leads to the interesting question as to whether it is
possible to obtain useful low-accuracy solutions with these methods.
We should not pretend that the formulae given in this paper are exact or even accurate
in floating-point arithmetic. Indeed, it is well-known that the floating-point matrices Q k from
the Lanczos method quickly loose M-orthonormality (see, for instance, Parlett, 1980, Section
13.3). Despite this, the method as given appears to be capable of producing usable approximate
solutions to the trust-region subproblem. We are currently investigating why this should be so.
One further possibility, which we have not considered so far, is to find an estimate - using
the first pass of Algorithm 5.1, and then to compute the required s by minimizing the unconstrained
model
using the preconditioned conjugate gradient method.
The advantage of doing this is that any instability in the first pass does not necessarily reappear
in this auxiliary calculation. The disadvantages are that it may require more work than simply
using (5.1), and that - must be computed sufficiently large to ensure that H + -M is positive
semi-definite.
Acknowledgement
We would like to thank John Reid for his helpful advice on computing eigenvalues of tridiagonal
matrices, and Jorge Mor'e for his useful comments on the Mor'e and Sorensen (1983) method.
We are grateful to the British Council-MURST for a travel grant (ROM/889/95/53) which made
some of this research possible.
Solving the trust-region subproblem using the Lanczos method 27
--R
CUTE: Constrained and unconstrained testing environment.
A family of trust-region-based algorithms for unconstrained minimization with strong global convergence properties
LANCELOT: a Fortran package for large-scale nonlinear optimization (Release
Two new unconstrained optimization algorithms which use function and gradient values.
Computing optimal locally constrained steps.
Maximization by quadratic hill-climbing
A catalogue of subroutines (release 13).
An algorithm for minimization using exact second derivatives.
Numerical experience with new truncated Newton methods in large scale unconstrained optimization.
Computing a trust region step.
The Symmetric Eigenvalue Problem.
Tracking the progress of the Lanczos algorithm for large symmetric eigenproblems.
Convergence properties of a class of minimization algorithms.
A semidefinite framework for trust region subproblems with applications to large scale minimization.
A new matrix-free algorithm for the large-scale trust-region subproblem
A new modified Cholesky factorization.
Newton's method with a model trust modification.
Minimization of a large-scale quadratic function subject to a spherical constraint
SIAM Journal on Optimization
The conjugate gradient method and trust regions in large scale optimization.
Towards an efficient sparsity exploiting Newton method for minimization.
--TR
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Nicholas I. M. Gould , Philippe L. Toint, FILTRANE, a Fortran 95 filter-trust-region package for solving nonlinear least-squares and nonlinear feasibility problems, ACM Transactions on Mathematical Software (TOMS), v.33 n.1, p.3-es, March 2007
Giovanni Fasano , Massimo Roma, Iterative computation of negative curvature directions in large scale optimization, Computational Optimization and Applications, v.38 n.1, p.81-104, September 2007
Nicholas I. M. Gould , Dominique Orban , Philippe L. Toint, GALAHAD, a library of thread-safe Fortran 90 packages for large-scale nonlinear optimization, ACM Transactions on Mathematical Software (TOMS), v.29 n.4, p.353-372, December
Nicholas I. M. Gould , Dominique Orban , Philippe L. Toint, CUTEr and SifDec: A constrained and unconstrained testing environment, revisited, ACM Transactions on Mathematical Software (TOMS), v.29 n.4, p.373-394, December
Nicholas I. M. Gould , Philippe L. Toint, An iterative working-set method for large-scale nonconvex quadratic programming, Applied Numerical Mathematics, v.43 n.1-2, p.109-128, October 2002 | trust-region subproblem;conjugate gradients;preconditioning;lanczos method |
589272 | Superlinearly Convergent Algorithms for Solving Singular Equations and Smooth Reformulations of Complementarity Problems. | We propose a new algorithm for solving smooth nonlinear equations in the case where their solutions can be singular. Compared to other techniques for computing singular solutions, a distinctive feature of our approach is that we do not employ second derivatives of the equation mapping in the algorithm and we do not assume their existence in the convergence analysis. Important examples of once but not twice differentiable equations whose solutions are inherently singular are smooth equation-based reformulations of the nonlinear complementarity problems. Reformulations of complementarity problems serve both as illustration of and motivation for our approach, and one of them we consider in detail. We show that the proposed method possesses local superlinear/quadratic convergence under reasonable assumptions. We further demonstrate that these assumptions are in general not weaker and not stronger than regularity conditions employed in the context of other superlinearly convergent Newton-type algorithms for solving complementarity problems, which are typically based on nonsmooth reformulations. Therefore our approach appears to be an interesting complement to the existing ones. | Introduction
. In this paper we are interested in solving nonlinear equations
in the case where their solutions can be singular, and smoothness requirements are
weaker than those usually assumed in this context. Our development is partially
motivated by the nonlinear complementarity problem, which we consider in detail,
and for which our method takes a particularly simple and readily implementable
form.
be a given mapping, where V is a neighborhood of a point
x in
x being a solution of the system of equations
In the sequel, F is assumed to be once (but not necessarily twice) dierentiable on
V . In this setting
x is referred to as singular solution if the linear operator F 0 (x) is
singular, i.e.,
or, equivalently,
In other cases x is referred to as regular solution.
Computing Center of the Russian Academy of Sciences, Vavilova Str. 40, Moscow, GSP-1, Russia
(izmaf@ccas.ru). Research of this author is supported by the Russian Foundation for Basic Research
Grants 99-01-00472 and 01-01-00810. The rst author also thanks IMPA, where he was a visiting
professor during the completion of this work.
y Instituto de Matematica Pura e Aplicada, Estrada Dona Castorina 110, Jardim Bot^anico, Rio
de Janeiro, RJ 22460-320, Brazil (solodov@impa.br). Research of this author is supported in part
by CNPq Grant 300734/95-6, by PRONEX{Optimization, and by FAPERJ.
A. F. IZMAILOV AND M. V. SOLODOV
gives rise to numerous di-culties. It is well-known that for Newton-type
methods at best one can guarantee linear convergence rate to a singular solution
[6, 7, 9]. Moreover, it is not su-cient to choose a starting point only close enough
to a solution (usually the set of appropriate starting points does not contain a full
neighborhood of the solution, although this set is normally rather \dense" [18]). We
refer the reader to the survey [19] and references therein. Another di-culty typical
in this context is related to possible instability of a singular solution with respect
to perturbations of F [27]. Certain special approaches to overcome those di-culties
have been developed in the last two decades, but they employ second derivatives of
F . Concerning methods for computing singular solutions, we cite [8, 20, 19, 43, 14, 1],
and the more recent proposals in [26, 22, 21, 2, 27, 4] (of course, this list does not
mention all contributions in this eld).
One of the motivations for our new approach to solving singular equations lies in
applications to the classical nonlinear complementarity problem (NCP) [37, 12, 13],
which is to nd an x 2 R n such that
smooth. One of the most useful approaches to numerical and
theoretical treatment of the NCP consists in reformulating it as a system of smooth
or nonsmooth equations [35, 29, 46]. One possible choice of a smooth reformulation
is given by the following function (for other choices, see Section 5.1):
It is easy to check that for this mapping the solution set of the system of equations
coincides with the solution set of the NCP (1.2) [29, 47]. If
x is a solution of the
NCP, by direct computations (see Section 3), we obtain that
denotes the standard basis in R n , and the index sets I 0 , I 1 and I 2
are dened by
I 0 :=
I 1 :=
I 2 :=
It is immediately clear that F 0 (x) cannot be nonsingular, unless the index set I 0 is
empty. The latter strict complementarity assumption is regarded rather restrictive.
Therefore, smooth NCP reformulation provided by (1.3) gives rise to inherently singular
solutions of the corresponding system of equations. In fact, it is known that
any other smooth NCP reformulation has the same singularity properties [31] (see
also Section 5.1). Furthermore, it is clear that F is once dierentiable with Lipschitz-continuous
derivative (if g is twice continuously dierentiable), but F is not twice
dierentiable when I 0 6= ;. This is also a common property shared by all useful
smooth reformulations, e.g., see the collection [16]. Thus NCP reformulations provide
an interesting example of once dierentiable nonlinear equations whose solutions
are inherently singular. As discussed above, application of standard numerical techniques
(e.g., Newton methods) in this context is prone to di-culties (and even failure)
ALGORITHMS FOR SINGULAR EQUATIONS AND COMPLEMENTARITY 3
because of singularity. On the other hand, known special approaches to computing
singular solutions are inapplicable, since these require second derivatives of F . This is
the apparent reason why superlinearly convergent Newton-type algorithms for solving
the NCP are typically based on nonsmooth equation reformulations and nonsmooth
Newton methods (see [13] for a discussion and some references). In this paper we
show that it is, in fact, possible to devise superlinearly convergent algorithms based
on the smooth NCP reformulations. Specically, we propose an alternative approach
based on computing singular solutions of the smooth reformulation stated above, and
show that conditions needed for convergence of our method are principally dierent
from those required for convergence of known nonsmooth algorithms. Thus the two
can be considered as a complement to each other.
We complete this section with some notation, which is fairly standard. We denote
by L n the space of linear operators from R n to R n . For A 2 L n , let ker
stand for its kernel (null space), and im stand for
its image (range space). For a bilinear mapping and an element
the linear operator B[p] 2 L n by Recall that
symmetric bilinear mappings and linear operators of the form p
are in isometrically isomorphic correspondence to each other, i.e., the correspondence
is one-to-one, linear, and it preserves the norm. Therefore, in the sequel we shall not
be making a formal distinction between those objects. Given a set S in a vector space,
by conv S we denote its convex hull, and by span S its linear hull. Finally, by E we
denote the identity operator in R n .
2. A General Approach to Solving Singular Equations. We start with describing
an approach to computing singular solutions of twice dierentiable nonlinear
equations, which was developed in [26, 22, 27]. We then extend it to the setting of
once dierentiable mappings, and in the next section show how it applies to solving
complementarity problems.
A solution
x of (1.1) being regular is equivalent to saying that im F 0
while singularity means that im F 0 (x) 6= R n . In this situation, one possibility to
\regularize" a singular solution
x is to add to the left-hand side of (1.1) another term,
which vanishes at
x (so that
x remains a solution), and such that its Jacobian at x
\compensates" for the singularity of F 0 (x) (so that to complement im F 0 (x) in R n ).
It is natural to base this extra term on the information about the rst derivative of
F .
To this end, dene the mappings
and consider the equation
Suppose that P () is dened in such a way that for
holds that
Then, by the structure of , solution
x of (1.1) is also a solution for (2.2). Furthermore,
if F is su-ciently smooth (at least twice dierentiable at
x), then under appropriate
assumptions on the rst two derivatives of F at
x, and on P () and h(), it is possible
to ensure that is dierentiable at x, and x is a regular solution of (2.2). As these
assumptions will not be used in this paper, we omit the details, referring the reader
4 A. F. IZMAILOV AND M. V. SOLODOV
to [26, 27]. The regular solution x of (2.2) can be computed by means of eective
special methods [26, 22, 27], or by conventional numerical techniques (the latter would
typically require stronger assumptions, in order to ensure dierentiability of not
only at
x but also in its neighborhood). There exist certain general techniques to
dene P () and h() with necessary properties (see [26, 27]). However, when one has
additional information about the structure of singularity of F at
x (e.g., recall (1.4) for
the NCP reformulation), it can often be used to choose P () and h() in a particularly
simple and constructive way. One such application is precisely the NCP, where the
subspace im F 0 (x) can be identied (locally, but without knowing x), and so the two
mappings can be chosen constant (see Section 3).
In this paper, we shall focus exclusively on the case where it is possible to choose
We emphasize that, of course,
P should be determined without knowing the exact solution x. The simplest case
when this is possible is when we know that corank F 0
when we are interested in determining a solution specically with this particular type
of singularity. In that case, it is natural to take
In Section 3, we show
how appropriate
P for the NCP reformulation can be determined using information
available at any point close enough to a solution (but without knowing the solution
itself). In general, if P () is dened as a constant
satisfying (2.3), one also can
usually take h() p, with p 2 R n n f0g being an arbitrary element. Indeed, with
those choices the function dened by (2.1) takes the form
and x is still a solution of (2.2), due to (2.3). If F is twice dierentiable at
x, then it
is clear that is dierentiable at this point, and
Therefore,
x is a regular solution of (2.2) if the linear operator in the right-hand side
of (2.5) is nonsingular. This is possible under appropriate assumptions. Since the
case of twice dierentiable F is not the subject of this paper, we shall not discuss here
technical details. We only note that nonsingularity of (2.5) subsumes the condition
Observe that the latter relation implies that (2.3) must hold as equality. Summarizing,
we obtain the following assumptions on the choice of
ker
These assumptions clearly hold if, for example,
P is the projector onto some complement
of im F 0 (x) in R n parallel to im F 0 (x). With this choice, nonsingularity of (2.5)
formally coincides with the notion of 2-regularity of at
x with respect to p 2 R n ,
in the sense of [23, 3, 27]. We note, however, that this connection does not seem conceptually
important, and in fact, appears to be in some sense a coincidence. Indeed,
in the case of once dierentiable mappings considered below, nonsingularity condition
that would be required no longer has any direct relation to 2-regularity for mappings
with Lipschitzian derivatives, as dened in [24, 25].
As a nal note, we remark that it can be shown (by simple argument, see [26, 27])
that if there exists at least one element p 2 R n such that the operator (2.5) is
nonsingular, then it will be so for almost every p 2 R n .
ALGORITHMS FOR SINGULAR EQUATIONS AND COMPLEMENTARITY 5
We conclude the discussion of the twice dierentiable case by the following ex-
ample, which is very simple but serves to illustrate the basic idea.
Example 2.1. Let R be twice continuously dierentiable
on V , where V is a neighbourhood of
x 2 R which is a singular solution of (1.1).
The latter means here that F Taking
we obtain the following regularized equation: Obviously,
which is distinct from zero for any p 2 R n f0g,
This shows that in this example, if F 00 (x) 6= 0, singularity can
be easily dealt with by using the second-order information.
In the approach outlined above, F is assumed to be twice dierentiable. Suppose
now that F is once (but not twice) dierentiable, and its rst derivative is Lipschitz-continuous
on V . Then dened by (2.4) is also Lipschitz-continuous on V , and it
is natural to try to apply to the corresponding equation (2.2) the generalized (non-
smooth) Newton method [32, 33, 41, 42, 40, 28]. We emphasize that we shall use
the nonsmooth Newton method to solve a (nonsmooth) regularization of a smooth
equation. In the context of NCP, this should be compared to the more traditional
approach of solving an inherently nonsmooth reformulation by the nonsmooth Newton
method. As we shall show in Section 4, the two dierent approaches lead to two
dierent regularity conditions, neither of which is weaker or stronger than the other.
Let @(x) denote the Clarke's generalized Jacobian [5] of at x 2 V . That is
where @B (x) stands for the B-subdierential [45] of at x, which is the set
with D V being the set of points at which is dierentiable. With this notation,
the nonsmooth Newton method is the following iterative procedure:
It is well-known [42, 40, 28] that if
(i) is semismooth [36] at x, and
(ii) all the linear operators comprising @(x) are nonsingular,
then the process (2.7) is locally well-dened and superlinearly convergent to x. More-
over, if is strongly semismooth [36] then the rate of convergence is quadratic.
The regularity condition (ii) can be relaxed if a more specic rule of determining
employed. For example, if one chooses
is enough to assume BD-regularity, i.e., that all elements in @B (x) are nonsingular
[41].
In applications, usually has some special (tractable) structure, and at each
iterate x k we are interested in obtaining just one, preferably easily computable,
This would be precisely the case here. The choice of an element
in @(x) that we suggest to use in the nonsmooth Newton method for solving (2.2)
with given by (2.4), is the following:
denotes the usual directional derivative of the mapping
at with respect to a direction p 2 R n . In Section 3, we show that this
6 A. F. IZMAILOV AND M. V. SOLODOV
H(x) is explicitly and easily computable for the NCP reformulations. The validity
of the choice suggested in (2.8) for an element of @(x) is actually not so obvious.
The possibility of choosing the directional derivative (
as an element in the
generalized Jacobian of
is based on the following fact. At a point x
its derivative is in fact the second derivative of
PF (). Due to
can be considered as a symmetric bilinear mapping. This symmetry
will be essential in the proof of Lemma 2.1 below. For a mapping x ! Q(x)p, where
is an arbitrary Lipschitzian mapping, the inclusion
can be in general invalid.
Lemma 2.1. Suppose that F : V ! R n has a Lipschitzian derivative on V , where
V is an open set in R n . Assume that for some
the mapping
is directionally dierentiable at a point x 2 V with respect to a direction p 2 R n .
are dened in (2.8) and (2.4), respectively.
Proof. Since
clearly Lipschitz-continuous, using further the assumption
that
PF 0 is directionally dierentiable at x with respect to p, it follows that there
exists a linear operator
The above conclusion can be deduced from [42, Lemma 2.2(ii)] after identifying the
space L n with the equivalent space R m , using the equivalence of the
norms in nite-dimensional spaces.
By the denition of the generalized Jacobian, B 2 @(
means that there
exist an integer m, sequences fx i;k g V and numbers i , with the
following properties: i
is dierentiable at each x i;k , and
where the limits in the right-hand side of the second equality exist for each
Note that dierentiability of
at each x i;k means that the mapping
twice dierentiable at these points. Taking into account the symmetry of
the bilinear mapping representing the second derivative, we conclude that
Therefore,
where the second equality follows from (2.10), and the third from (2.9). Using the
denition of the generalized Jacobian, we conclude that H(x) 2 @(x).
Remark 2.1. There exists another way to construct the regularized equation
which can have advantages in certain situations over the one described
above. Specically, the mapping dened by (2.4) can be modied as follows:
ALGORITHMS FOR SINGULAR EQUATIONS AND COMPLEMENTARITY 7
It is clear that with this denition,
x is still a solution of Modifying H
accordingly, we have
Furthermore, it is clear that Lemma 2.1 is still valid with and H dened by (2.11)
and (2.12). Finally, it is easy to see that since
the possible limits of H(x)
as
x are the same, whether H is dened by (2.8) or (2.12). Hence, the regularity
condition at
x that would be needed for the superlinear convergence of our method
is again the same, whether the method is applied to one regularized equation or the
other.
The possible advantage of the modied equation is the following. If singularity
of F 0 (x) has a certain structure, then not all the components of F may need to be
computed in (2.11). Furthermore, (2.12) can also take a simpler form in that case.
For example, suppose that F 0 (x) is such that
satisfying (2.3) can be chosen as
the orthogonal projector onto the subspace span fe is the
standard basis in R n and I
P is the orthogonal projector
onto span fe ng n Ig. It is easy to see that in this case, (2.11) would not
require computing the function values F i I . Furthermore, the derivatives of
would not appear in (2.12), and so this part would also be simplied. This
feature would be further illustrated in the context of NCP in Section 3.
In the sequel, we shall also consider the following modication of the Newton
algorithm (2.7), which will be useful for solving the NCP reformulation in Section 3:
This modication is essentially motivated by the idea of \truncating" elements of the
(generalized) Jacobian by omitting the terms which vanish at the solution
x. These
terms typically involve some higher-order derivatives of the problem data (in the
context of NCP (1.2), the second derivatives of g), and so it can be advantageous not
to compute them, if possible.
Note that the regularity condition which is typically employed in nonsmooth
Newton methods consists of saying that every element in the generalized Jacobian
@(x) (or the B-subdierential @B (x)) is nonsingular (recall condition (ii) stated
above). This seems to be unnecessarily restrictive, because in most implementable
algorithms some specic rule to choose H(x k used. We shall therefore
replace the traditional condition by a weaker one. Specically, we shall assume that
all the possible limits of H(x k ) as x k !
x are nonsingular, where H(x k ) is precisely
the element given by (2.8) (or by (2.12)). To this end, we shall dene the set
Hg:
Our regularity assumption would be that elements in p (x) are nonsingular. We
remind the reader that this set is the same for both choices of H , i.e., (2.8) and (2.12).
We point out that unlike in the twice dierentiable case, this regularity condition
cannot be related to the notion of 2-regularity [24, 25] of at
x.
Lemma 2.1 in hand, convergence of algorithms (2.7) and (2.13), with the
data dened in (2.4) and (2.8) or (2.11) and (2.12), can be established similarly to
[41], but taking into account the modied nonsingularity assumption.
8 A. F. IZMAILOV AND M. V. SOLODOV
Theorem 2.2. Suppose has a Lipschitz-continuous derivative on
is a neighborhood of a solution x of (1.1). Let
Assume further that the mapping
directionally dierentiable with
respect to a direction p 2 R n at any point in V , and the mapping
is semismooth at
x. Let and H be dened by (2.4) and (2.8), or (2.11) and (2.12).
Assume further that all linear operators comprising p (x) are nonsingular.
Then the iterates given by (2.7) or (2.13) are locally well-dened, and converge
to x superlinearly. If, in addition, the mapping
strongly semismooth at x
then the rate of convergence is quadratic.
Proof. It is easy to see that under our regularity assumption, (H()) 1 is locally
uniformly bounded. Indeed, assume the contrary, i.e., that there exists a sequence
x, and the sequence f(H(x k is unbounded
(this subsumes the possibility that some elements of the latter sequence are not even
well-dened). Recall that the generalized Jacobian is locally bounded [5]. Since,
by Lemma 2.1, H(x k ) 2 @(x k ) for every k, it follows that the sequence fH(x k )g is
bounded. Hence, we can assume that fH(x k )g converges to some
the inclusion is by the very denition of the set p (x). But then
H is nonsingular,
which is in contradiction with the earlier assumption that f(H(x k is unbounded.
Consider rst algorithm (2.7), and suppose that the iterates are well-dened up
to some index k 0. We have that
where M > 0. Note that when
semismooth, so is (). It is
known [40, Proposition 1] that semismoothness of at x implies that
sup
(the latter property was introduced in the context of the nonsmooth Newton methods
in [33]). Using Lemma 2.1 and combining the last two relations, well-denedness of
the whole sequence fx k g and its superlinear convergence to x follow by a standard
argument.
In the strongly semismooth case, one has that
sup
and so convergence is quadratic.
Consider now the iterates fx k g generated by (2.13). By our regularity assumption
and the classical results of linear analysis, the condition
implies that
Hence,
ALGORITHMS FOR SINGULAR EQUATIONS AND COMPLEMENTARITY 9
where the Lipschitz-continuity of was also used. It follows that the dierence between
the original and modied steps is of the second order. By the obvious argument,
it can now be easily seen that the modied algorithm has the same convergence rate
as the original one.
Note that in principle, our regularity condition depends not only on the structure
of singularity of F at x, but also on the choice of p. Implementation of this approach
presumes that there exists at least one p 2 R n for which this condition is satised.
Furthermore, a way to choose such p should be available. Fortunately, a typical
situation is the following. The existence of one suitable p can usually be established
under some reasonable regularity assumption. Then, given the existence of one such
it can further be proven that the set of appropriate elements is, in fact, open and dense
in the whole space. Hence, p can be chosen arbitrary, with the understanding that
almost any is suitable. We shall come back to this issue in Section 4, where regularity
conditions for NCP are discussed. In the computational experience of [22, 26], where
conceptually related methods for smooth operator equations are considered, a random
choice of p does the job. Even though this choice certainly aects the rate and range
of convergence, the dierences between dierent choices are usually not dramatic.
Finally, we remark that the development presented above can be extended to the
case when P () is not necessarily constant, but it is a Lipschitzian mapping satisfying
with
In that case, we would have to provide a technique to dene
such P () in the general setting. Such techniques are possible, but they go beyond the
scope of the present paper. Here we are mainly concerned with a specic application
of our approach to the nonlinear complementarity problem, which we consider next.
3. Algorithm for the Nonlinear Complementarity Problem. Consider
the nonlinear complementarity problem (1.2), and its reformulation as a system of
smooth equations (1.1), given by (1.3). For convenience, we re-state the associated
function F , which is
We choose a specic reformulation for the clarity of presentation. In Section 5.1, we
show that our analysis is intrinsic and extends to other smooth reformulations.
Let
x 2 R n be a solution of NCP. Suppose that g is twice continuously dier-
entiable in some neighborhood V of
x in R n . Then it is easy to see that F has a
Lipschitz-continuous derivative on V , which is given by
is the standard basis in R n . Recalling the three index sets
I 0 :=
I 1 :=
I 2 :=
from (3.1) we immediately obtain that
(3.
A. F. IZMAILOV AND M. V. SOLODOV
As already discussed in Section 1, the Jacobian F 0 (x) is necessarily singular whenever
I 0 6= ;, the latter being the usual situation for complementarity problems of interest.
Furthermore, F is not twice dierentiable. Hence, smooth NCP reformulations fall
precisely within the framework of Section 2. Such equations cannot be eectively
solved by previously available methods, and so our approach comes into play. We
next show that in the setting of NCP, the general algorithm introduced in Section 2
takes a simple implementable form.
Given the structure of F 0 (x), we have that
Then the natural choice of
condition (2.3)), is
the operator with the matrix representation consisting of rows
At the end of this section, we shall show how to dene
knowing the solu-
tion
x (clearly, this task reduces to identifying the index set I 0 ). This is possible by
means of error bound analysis. A su-cient condition for our error bound is weaker
than b-regularity [39], which is currently the weakest assumption under which Newton
methods for nonsmooth NCP reformulations are known to be (superlinearly) convergent
[30, 34].
Once
P is dened according to (3.3), we x p 2 R n n f0g arbitrarily. Then the
function dened by (2.4) takes the form
According to Section 2,
x is a solution of which is our \regularized" equa-
tion. We proceed to derive explicit forms for iterations of algorithms (2.7) and (2.13),
and the regularity condition needed for their convergence.
First, by (2.8) and (3.3), the matrix representation of H(x), which is the element
of @(x) employed in algorithm (2.7), consists of rows
Furthermore, the directional derivatives employed in (3.5) exist, and can be obtained
explicitly from (3.1):
where
ALGORITHMS FOR SINGULAR EQUATIONS AND COMPLEMENTARITY 11
Note that according to (3.5), one has to compute
Another
useful observation which would suggest truncation of the Jacobian to be discussed
later, is that for i 2 I 0 all the terms in (3.6) involving the second derivatives of g
vanish at
x.
Furthermore, taking into account (3.5), (3.2), (3.6) and (3.7), we conclude that
the matrix representation of an arbitrary limit point
H of H(x) as x ! x consists of
rows
Hence, we can state the following su-cient condition for nonsingularity of every linear
operator in p (x). Denote by J the collection of pairs of index sets (J
that Our regularity condition consists of saying that for
every pair of index sets (J holds that
are linearly independent in R
In Section 4, we shall discuss the relation between this condition and other regularity
conditions for the NCP, as well as compare convergence results of our algorithm with
convergence results of other locally superlinearly convergent equation-based methods
for solving NCP.
Under our assumptions, semismoothness of
readily from (3.1)
and standard calculus of semismooth mappings [36, Theorem 5]. Moreover, under
the additional assumption of Lipschitz-continuity of g 00 () on V ,
strongly
semismooth, which follows from results on the superposition of strongly semismooth
mappings [15, Theorem 19]. Hence, () is (strongly) semismooth.
Note that all the elements involved in the iteration scheme (2.7) are computed in
this section by explicit formulas. In principle, computing H via (3.5)-(3.7) involves
second derivatives of g. However, as already noted above, the terms containing second
derivatives of g tend to zero as x ! x. This suggests the idea to modify the process
by omitting these terms, which leads to the method represented by (2.13). We shall
also take into account the structure of
P , and make use of Remark 2.1.
Note that for
given by (3.3), we have that (E
P ) is the orthogonal projector
onto span fe g. According to (2.11), we can therefore re-dene
Taking into account (2.12) and omitting further the terms that vanish at
x, we can
take
~
Comparing expressions (3.9) and (3.10) with (3.4) and (3.5), one can easily observe
that the former are simpler and require less computations.
A. F. IZMAILOV AND M. V. SOLODOV
Furthermore, under our smoothness assumptions, it is easy to see that
and so the modied Newton method given by (2.13) is applicable.
We next give a formal statement of the convergence result for our methods applied
to NCP, which is a corollary of Theorem 2.2.
Theorem 3.1. Let be a twice continuously dierentiable mapping
on being a neighborhood of a solution
x of the NCP (1.2). Assume that for some
condition (3.8) is satised for every pair of index sets (J
Then the iterates given by (2.7) or (2.13) (with all the objects as dened in this
section) converge to
x locally superlinearly. If, in addition, the second derivative of g
is Lipschitz-continuous on V , then the rate of convergence is quadratic.
We next show how to construct
knowing the solution x. Given the
structure of
P , see (3.3), it is clear that this task reduces to correct identication of
the degenerate set I 0 . This can be done with the help of error bounds, as described
next (our approach is in the spirit of the technique developed in [10] for identication
of active constraints in nonlinear programming). To our knowledge, the weakest
condition under which a local error bound for NCP is currently available is the 2-
regularity of F given by (1.3) at the NCP solution x [25]. Specically, if F is 2-regular
at x then there exist a neighborhood U of x in R n and a constant M 1 > 0 such that
We shall not introduce the notion of 2-regularity formally here, as this would require
an extensive discussion. We only emphasize that the bound (3.11) may hold when the
so-called natural residual minfx; g(x)g does not provide an error bound, and always
holds when it does (see [25], and in particular [25, Example 1]). Hence, 2-regularity
of F is a weaker assumption than the R 0 type property or semistability, which in case
of NCP are both equivalent to an error bound in terms of the natural residual [38].
And it is further weaker than b-regularity, see [25].
We note that in Lemma 3.2 below we could also use other error bounds for
identifying I 0 . However, they would require either stronger local assumptions, or
global assumptions.
Lemma 3.2. Suppose that
x is a solution of NCP, g is Lipschitz-continuous on
is a neighborhood of x. Suppose nally that the local error bound (3.11)
holds. Then for any 2 (0; 1) there exists a neighborhood U of
x such that
ng
Proof. It is easy to observe that there exist some M 2 > 0 and some neighborhood
U of
x such that
where the inequality follows from the Lipschitz-continuity of the functions involved.
Therefore, by (3.11) (possibly adjusting the neighborhood U ), for an arbitrary xed
we have that
ALGORITHMS FOR SINGULAR EQUATIONS AND COMPLEMENTARITY 13
In particular, the quantity in the left-hand side of the inequality above tends to zero
as x tends to
x. On the other hand, it is clear that there exists " > 0 such that
(U should be adjusted again, if necessary). Combining those facts, we obtain (3.12)
for U su-ciently small.
By Lemma 3.2, the index set I 0 , and hence the mapping
P , are correctly identied
by (3.12), provided one has a point close enough to the solution. We note that this
requirement of closedness to solution is completely consistent with the setting of the
paper, since the subject under consideration are superlinearly convergent Newton-like
methods, which are local by nature.
Finally, we mention other considerations that can also be useful for identifying
I 0 . Sometimes the cardinality r of I 0 may be known from a priori analysis of the
problem, or one can be interested in nding an NCP solution with a given cardinality
of I 0 . Then for any x 2 R n su-ciently close to
x, the set I 0 coincides with the set
of indices corresponding to the r smallest values of j maxfg i (x); x i gj. In this case, no
error bound is needed to identify I 0 . We note that in the present setting, cardinality
of I 0 is closely related to corank of singularity. In the literature on numerical methods
for solving singular equations, the assumption that corank of singularity is known is
absolutely standard [20, 19, 43, 14, 1, 2]. In the complementarity literature, on the
other hand, assumptions about cardinality of I 0 are not common, except possibly for
I
4. Regularity Conditions. The weakest condition under which there exists
a locally superlinearly convergent Newton-type algorithm for solving a (nonsmooth)
equation reformulation of the NCP, is the b-regularity assumption, which can be stated
as follows: for every pair of index sets (J holds that
are linearly independent in R
Under this assumption, the natural residual mapping x !
is BD-regular at
x. Furthermore, it is also (strongly) semismooth under standard
assumptions on g. Hence, the nonsmooth Newton method (2.7) based on it converges
locally superlinearly [30, 34]. Note that Newton methods applied to another
popular reformulation based on the Fischer-Burmeister function [17, 11], require for
convergence the stronger R-regularity [44] assumption, see [34].
In what follows, we compare our regularity condition (3.8) with b-regularity, and
show that they are essentially dierent. In general, neither is weaker or stronger than
the other. This implies that our approach based on the smooth NCP reformulation
is a complement to nonsmooth reformulations, and vice versa, as each approach can
be successful in situations when the other is not.
The next result is important to obtain an insight into the nature of our regularity
condition (3.8). We start with the following denition.
Definition 4.1. A solution x of the NCP (1.2) is referred to as quasi-regular, if
for every pair of index sets (J there exists an element
such that (3.8) is satised.
Proposition 4.2. Suppose that the solution x of the NCP (1.2) is quasi-regular.
Then there exists a universal p 2 R n which satises (3.8) for every pair (J
J . Moreover, the set of such p is open and dense in R n .
14 A. F. IZMAILOV AND M. V. SOLODOV
Proof. Fix a pair (J consider the determinant of the system
of vectors in (3.8) as a function of p. This function is a polynomial on R n , and this
polynomial is not everywhere zero, since it is not zero at
But then the set where the polynomial is not zero is obviously open and dense in R n .
Moreover, the intersection of such sets corresponding to pairs (J
open and dense, since it is a nite intersection of open and dense sets.
It follows that if x is a quasi-regular solution of NCP in the sense of Denition
4.1, then even picking a random p 2 R n one is extremely unlikely to pick a \wrong"
(as the set of wrong elements is \thin"). Hence, under the assumption of quasi-
regularity, for the implementation of the algorithm described in Section 3 we can
choose arbitrarily, with the understanding that almost every p 2 R n is
appropriate. In particular, for all practical purposes, we can think of quasi-regularity
as the regularity condition needed for superlinear convergence of our algorithm. We
next investigate the relationship between quasi-regularity and b-regularity.
First, we show that if the cardinality of I 0 is equal to one, then quasi-regularity
is in fact weaker than b-regularity.
Proposition 4.3. Suppose that x is a b-regular solution of the NCP, and the
cardinality of I 0 is equal to one. Then
x is quasi-regular.
Proof. Let I and denote
g. In this
setting, b-regularity clearly means that
corresponding to the two possible choices of (J It follows that
Assume for a contradiction that x is not quasi-regular. Then by Denition 4.1, there
exists a pair (J such that for every p 2 R n condition (3.8) is violated. This
means that is either
or
Taking any q 2 L ? n f0g, we deduce that for every p 2 R n either
or
Setting we then obtain that either
or
ALGORITHMS FOR SINGULAR EQUATIONS AND COMPLEMENTARITY 15
which contradicts b-regularity, because of (4.1).
It is easy to see that in the setting of Proposition 4.3, the quasi-regularity condition
can be satised without b-regularity. For example, let g 0
Then b-regularity is violated. On the other hand, quasi-regularity here is equivalent
to saying that there exist elements (corresponding to the two possible
choices of (J
which is satised for almost any p 1 and p 2 , provided
It is also quite clear
that just choosing randomly should do the job.
In general, i.e., in the cases of higher cardinality of I 0 , b-regularity and quasi-regularity
become dierent, not directly related conditions. In particular, neither is
stronger or weaker than the other, as illustrated by the following examples.
Example 4.1. Let
Then b-regularity is obvious, but
This means that for J does not hold for any p, and so the
quasi-regularity condition is not satised.
Example 4.2. Let
Then b-regularity does not hold (the linear independence condition is violated for
but quasi-regularity is satised, which can be shown by straight-forward
computations. We omit the details, as they do not provide any further insight
We complete our discussion with a su-cient condition for quasi-regularity of x,
which is meaningful when the cardinality of I 0 is not greater than n=2, half dimensionality
of the space. Specically, suppose that
are linearly independent in R n ;
and
are linearly independent in R
It is clear that (4.3) is subsumed by b-regularity (where it must hold for all partitions
of I 0 ). It is also not di-cult to see that (4.3) is necessary for quasi-regularity of x.
Hence, this assumption does not introduce any additional restrictions with respect to
regularity conditions under consideration. Furthermore, for non-pathological problems
the cardinality of I 0 should not be too large compared to the dimensionality of
the space, and so condition (4.2) should not be di-cult to satisfy. Therefore, (4.2)
and (4.3) appear to be not restrictive.
Proposition 4.4. Suppose that (4.2) and (4.3) hold. Then x is a quasi-regular
solution of NCP.
A. F. IZMAILOV AND M. V. SOLODOV
Proof. Take any pair of index sets (J consider the system of (twice
the cardinality of I 0 ) linear equations
in the variable p 2 R n . Under the assumption (4.2), this system has a solution
further that substituting this p into (3.8) reduces the system
of vectors appearing in (3.8) precisely to the system of vectors appearing in (4.3),
which is linearly independent by the hypothesis.
Again, it is easy to see that the latter su-cient condition for quasi-regularity of
x can hold without b-regularity. On the other hand, in general it is not implied by
b-regularity. In particular, b-regularity need not imply (4.2).
Summarizing the preceding discussion, we conclude that the regularity assumption
required for the algorithm proposed in Section 3 for solving the NCP is dier-
ent from b-regularity, which is the typical assumption in the context of nonsmooth
Newton-type methods for solving nonsmooth NCP reformulations. In fact, the two
assumptions are of a rather distinct nature. This is not surprising, considering that
they result from approaches which are also quite dierent.
5. Some Further Applications. The general approach presented in Section
2 can be also useful in other problems where complementarity is present. Below
we outline applications to another class of smooth reformulations of NCP (dierent
from (1.3)), and to the mixed complementarity problems. We limit this discussion to
exhibiting the structure of singularity associated with the smooth equation reformulations
of those problems. Deriving the resulting regularity conditions and comparing
them to known ones requires too much space. Without going into detail, we claim
that regularity assumptions needed for our approach would again be dierent from
assumptions of Newton methods for nonsmooth equations.
5.1. Other NCP Reformulations. The analysis presented in Sections 3 and
4 for NCP is intrinsic in the sense that it is also applicable to smooth reformulations
other than the one given by (1.3). Indeed, following [35], consider the family of
functions
strictly increasing function such that It can be
checked that the NCP solution set coincides with zeroes of F . As an aside, note that
reformulation (1.3) cannot be written in the form stated above, so the two are really
dierent.
Suppose further that is dierentiable on R with 0
t > 0. For example, we could take
Let
x be some solution of NCP, and V be its neighborhood. If g is twice continuously
dierentiable on V and 0 is Lipschitz-continuous, then the derivative of F is Lipschitz-continuous
near
x, and it is given by
ALGORITHMS FOR SINGULAR EQUATIONS AND COMPLEMENTARITY 17
As is easy to see,
Since 0 (t) > 0 for any t > 0, we conclude that the structure of singularity here is
absolutely identical to that for F given by (1.3) (recall (1.4)). In particular,
and all the objects and the analysis in Sections 3 and 4 can be derived in a similar
fashion.
5.2. Mixed Complementarity Problems. The mixed complementarity problem
(MCP) is a variational inequality on a (generalized) box
ug, where l i 2 [1;+1) and u are such that l i < u
Specically, the problem is to nd
It can be seen that this is equivalent to x 2 R n satisfying the
following conditions: for every
if
if
if
NCP is a special case of MCP corresponding to l
We claim that solutions of MCP coincide with zeroes of the function F
whose components are given by
We omit the proof, which can be carried out by direct verication. Let x be some
solution of MCP, and V be its neighborhood. If g is twice continuously dierentiable
on V then the derivative of F is Lipschitz-continuous near
x. Dening
I 0 :=
I 1 :=
I 2 :=
it can be veried that
A. F. IZMAILOV AND M. V. SOLODOV
where
In particular, i . Observing the structure of F 0 (x),
further analysis can now follow the ideas of Sections 3 and 4.
6. Concluding Remarks. We have presented a new approach to solving smooth
singular equations. Unlike previously available algorithms, our method is applicable
when the equation mapping is not necessarily twice dierentiable. Important examples
of once dierentiable singular equations are reformulations of the nonlinear
complementarity problems, which we have studied in detail. In particular, we have
demonstrated that in the case of NCP our method takes a readily implementable
simple form. Furthermore, the structure of singularity can be completely identied
by means of local error bound analysis, without knowing the solution itself. It was
further shown that the regularity condition required for the superlinear convergence of
the presented algorithm is dierent from conditions needed for the nonsmooth Newton
methods applied to nonsmooth NCP reformulations. Thus the two approaches should
be regarded as complementing each other. Finally, it was demonstrated that the main
ideas of this paper should be also applicable to other problems where complementarity
structures are present.
--R
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Convergence acceleration for Newton's method at singular points.
Convergence rates for Newton's method at singular points.
On the accurate identi
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A geometric framework for the numerical study of singular points.
Solution of monotone complementarity problems with locally Lipschitzian functions.
On the resolution of monotone complementarity problems.
Starlike domains of convergence for Newton's method at singularities.
On solving nonlinear equations with simple singularities or nearly singular solutions.
Characterization and computation of generalized turning points.
Augmented systems for computation of singular points in Banach space problems.
Stable methods for
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bounds for 2-regular mappings with Lipschitzian derivatives and their applications
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Semismoothness and superlinear convergence in non-smooth optimization and nonsmooth equations
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--TR | singularity;superlinear convergence;complementarity;reformulation;nonlinear equations;regularity |
589276 | A Polynomial Time Algorithm for Shaped Partition Problems. | We consider the class of shaped partition problems of partitioning n given vectors in d-dimensional criteria space into p parts so as to maximize an arbitrary objective function which is convex on the sum of vectors in each part, subject to arbitrary constraints on the number of elements in each part. This class has broad expressive power and captures NP-hard problems even if either d or p is fixed. In contrast, we show that when both d and p are fixed, the problem can be solved in strongly polynomial time. Our solution method relies on studying the corresponding class of shaped partition polytopes. Such polytopes may have exponentially many vertices and facets even when one of d or p is fixed; however, we show that when both d and p are fixed, the number of vertices of any shaped partition polytope is $O(n^{d{p\choose 2}})$ and all vertices can be produced in strongly polynomial time. | Introduction
The Partition Problem concerns the partitioning of vectors A in d-space into p parts
so as to maximize an objective function which is convex on the sum of vectors in each part;
see [3]. Each vector A i represents d numerical attributes associated with the ith element
of the set ng to be partitioned. Each ordered partition -
is then associated with the d \Theta p matrix A
whose jth column
represents the total attribute vector of the jth part. The problem is to find an admissible
partition - which maximizes an objective function f given by
a real convex functional on IR d\Thetap . Of particular interest is the Shaped Partition Problem,
Department of Applied Mathematics, Chiaotung University, Hsinchu, 30045, Taiwan. Email address:
fhwang@math.nctu.edu.tw.
y William Davidson Faculty of Industrial Engineering and Management, Technion-Israel Institute of
Technology, 32000 Haifa, Israel. Email address: onn@ie.technion.ac.il. Research supported in part by the
Mathematical Sciences Research Institute at Berkeley California through NSF Grant DMS-9022140, by the
N. Haar and R. Zinn Research Fund at the Technion, and by the Fund for the Promotion of Research at
the Technion.
z William Davidson Faculty of Industrial Engineering and Management, Technion-Israel Institute of
Technology, 32000 Haifa, Israel. Email address: rothblum@ie.technion.ac.il. Research supported in part by
the E. and J. Bishop Research Fund at the Technion and by ONR Grant N00014-92-J1142.
Shaped partition problems
where the admissible partitions are those - whose shape (j- 1 lies in a prescribed
set of admissible shapes. In this article we concentrate on this later situation.
The Shaped Partition Problem has applications in diverse fields that include circuit
layout, clustering, inventory, splitting, ranking, scheduling and reliability, see [5, 9, 14, 15]
and references therein. Further, as we demonstrate later on, the problem has expressive
power that captures NP-hard problems such as the Max-Cut problem and the Traveling
Salesman problem, even when the number p of parts or attribute dimension d are fixed.
Our first goal in this article is to demonstrate, constructively, that a polynomial time
algorithm for the Shaped Partition Problem does exist when both p and d are fixed. This
result is valid when the set of admissible shapes and the function C are presented by
oracles. So, our first result (formally stated and proved in Section 4) is:
ffl Theorem 4.2: Any Shaped Partition Problem is solvable in polynomial oracle time
using O(n dp 2
operations and queries.
Our solution method is based on the observation that since C is convex, the Shaped Partition
Problem can be embedded into the problem of maximizing C over the Shaped Partition
A defined to be the convex hull of all matrices A - corresponding to partitions of
admissible shapes. The class of Shaped Partition Polytopes is very broad and generalizes
and unifies classical permutation polytopes such as Birkhoff's polytope and the Permuto-
hedron (see e.g. [8, 19, 21]). Its subclass of bounded shaped partition polytopes with lower
and upper bounds on the shapes was previously studied in [3], under the assumption that
the vectors A are distinct. Therein a polynomial procedure for testing whether
a given A - is a vertex of P
A was obtained. This procedure is simplified and extended
in [11]. A related but different generalization of classical permutation polytopes, arising
when algebraic (representation-theoretical) constraints, rather than shape constraints, are
imposed on the permutations involved, was studied in [19] and references therein.
Since a Shaped Partition Polytope is defined as the convex hull of an implicitly presented
set whose size is typically exponential in the input size even when both p and d are fixed,
an efficient representation as the convex hull of vertices or as the intersection of half-spaces
is not readily expected. Our second objective is to prove that, nevertheless, for fixed p and
d, the number of vertices of Shaped Partition Polytopes is polynomially bounded in n, and
that it is possible to explicitly enumerate all vertices in polynomial time. So, our second
result (formally stated and proved in Section 4) is:
ffl Theorem 4.3: Any Shaped Partition Polytope P
A has O(n
vertices which can be
produced in polynomial oracle time using O(n d 2 p 3
operations and queries.
An immediate corollary of Theorem 4.3 is that, for fixed d; p, the number of facets of P
A
is polynomially bounded as well and that all facets can be produced in polynomial oracle
time (Corollary 4.4). Theorem 4.3 shows, in particular, that it is possible to compute the
number of vertices efficiently. This might be extendible to the situation of variable d and
counting vertices is generally a hard task (cf. [16]), as is counting partitions with
various prescribed properties (see [4, 10]). The vertex counting problem for variable d and
p will be addressed elsewhere.
Frank K. Hwang, Shmuel Onn and Uriel G. Rothblum 3
A special role in our development is played by separable partitions, defined as partitions
where vectors in distinct sets are (weakly) separable by hyperplanes. In the special case
partitions had been studied quite extensively (see e.g. [2, 5, 7, 17]). The case
has also been considered quite recently in [6]. Here we study such partitions
for all d; p, as well as a class of generic partitions, and provide an upper bound on their
number and an algorithm for producing them. In our recent related work [1], the precise
extremal asymptotical behavior of such partitions is determined.
The embedding of the partition problem into the problem of maximizing the convex
function C over the partition polytope is useful due to the optimality of vertices in the
latter problem. When consists of a single shape, the optimality of vertices holds for the
more general class of Asymmetric Schur convex function, introduced in [13]; see [8]. All of
our results apply with C as any asymmetric Schur convex function and consisting of a
single shape.
The article is organized as follows. In the next section we formally define the Shaped
Partition Problem and Shaped Partition Polytope. We demonstrate the expressive power of
this problem by giving four examples. For the first two examples, in which the parameters
d; p are typically small and fixed, our Theorem 4.2 provides a polynomial time solution.
The last two examples show that the Max-Cut problem and Traveling Salesman problem
can be modeled as Shaped Partition Problems with fixed respectively, and
that the corresponding polytopes have exponentially many vertices. In Section 3 we study
separability properties of vertices of Shaped Partition Polytopes and discuss separable and
generic partitions. In the final Section 4 we use our preparatory results of Section 3 to
establish Theorems 4.2 and 4.3 and Corollary 4.4.
Shaped Partition Problems and Polytopes
A p-partition of [n] := ng is an ordered collection -
sets (possibly empty) whose union is [n]. A p-shape of n is a tuple
nonnegative integers
n. The shape of a p-partition - is the
p-shape of n given by j-j := (j- 1 j). If is a set of p-shapes of n then a -partition
is any partition - whose shape j-j is a member of .
Let A be a real d \Theta n matrix; for to denote the ith column of A.
For each p-partition - of [n] we define the A-matrix of - to be the d \Theta p matrix
with
We consider the following algorithmic problem.
Shaped Partition Problem. Given positive integers d;
set of p-shapes of n, and objective function on -partitions given by
C convex on IR d\Thetap , find a -partition - which maximizes f , namely satisfies
4 Shaped partition problems
Of course, the complexity of this problem depends on the presentation of and C. But, we
will construct algorithms that work in strongly polynomial time and can cope with minimal
information on and C. Specifically, we assume that the set of admissible p-partitions
can be represented by a membership oracle which on query - answers whether or not - 2 .
The convex functional C on IR d\Thetap can be presented by an evaluation oracle that on query
A - with - a -partition returns
Since C is convex, the Shaped Partition Problem can be embedded into the problem of
maximizing C over the convex hull of A-matrices of feasible partitions, defined as follows.
Shaped Partition Polytope. For a matrix A 2 IR d\Thetan and nonempty set of p-shapes
of n we define the Shaped Partition Polytope P
A to be the convex hull of all A-matrices of
-partitions, that is,
A := conv
We point out that for any A, the polytope P
A is the image of the Shaped Partition Polytope
I , with I the n \Theta n identity, under the projection X 7! AX . In [12] this is exploited, for
the situation where the function C is linear and is a set of bounded
shapes, to solve the corresponding Shaped Partition Problem for all n; d; p in polynomial
time by linear programming over P
I .
We now demonstrate the expressive power of the Shaped Partition Problem. In par-
ticular, we show that even if one of d or p is fixed, the Shaped Partition Problem may be
NP-hard, and the number of vertices of the Shaped Partition Polytope may be exponen-
tial. Therefore, polynomial time algorithms for optimization and vertex enumeration are
expected to (and, as we show, do) exist only when both d and p are fixed. We start with
two examples in which it is natural to have d; p small and fixed.
Example 2.1 Splitting. The n assets of a company are to be split among its p owners
as follows. For the jth owner prescribes a nonnegative vector A
entries represent the relative values of the various
assets to this owner. A partition - which splits the assets
among the owners and maximizes the l q -norm (
q of the total value vector
whose jth entry
A j;i is the total relative value of the assets allocated to the jth
owner by -. An alternative interpretation of the splitting problem concerns the division
of an estate consisting of n assets among p inheritors having equal rights against the es-
tate. With 2, the model captures a problem of a divorcing couple dividing their joint
Frank K. Hwang, Shmuel Onn and Uriel G. Rothblum 5
For fixed p, Theorem 4.2 asserts that we can find an optimal partition in polynomial time
O(n
while the number p n of -partitions is exponential. We note that other (convex)
functions C can be used within our framework. In particular, if C is linear on IR p\Thetap
when our results of [12] apply and y ield a polynomial time solution even when p is
variable.
Example 2.2 Balanced Clustering. Given are objects represented by attribute
vectors A . The objects are to be grouped in p clusters, each containing
points, so as to to minimize the sum of cluster variance of a partition - given by
, where jj \Delta jj denotes the l 2 -norm and -
A j is
the barycenter of the ith cluster.
with
d
Here, we use the fact that
. For fixed d; p,
by Theorem 4.2 we can find an optimal balanced clustering in polynomial time O(n dp 2
while the number of -partitions is
The next two examples show that unless both d; p are fixed, the Shaped Partition Problem
may be NP-hard. The idea is simple: the formulation is such that every -partition -
gives a distinct vertex A - of the Shaped Partition Polytope P
A . Then, any function f on
-partitions factors as f(-) := convex C on P
A , say the one given by
In the following examples, the membership oracle for and the evaluation oracle for f(-) :=
restricted to A-matrices, are easily polynomial time realizable from the natural data
for the problem.
Example 2.3 Max-Cut Problem and Unit Cube. Find a cut with maximum number
of crossing edges in a given graph E).
Here, the A-matrices of -partitions are precisely all (0; 1)-valued n \Theta 2 matrices with each
row sum equals 1; in particular, each such matrix is determined by its first column. It
follows that the Shaped Partition Polytope P
A has 2 n vertices which stand in bijection
with -partitions, and is affinely equivalent to the n-dimensional unit cube by projection
of matrices onto their first column. So, each A - is a distinct vertex of P
A and there is a
convex C on IR d\Theta2 such that
6 Shaped partition problems
Example 2.4 Travelling Salesman Problem and Permutohedron. Find a shortest
Hamiltonian path on n sites under a given symmetric nonnegative matrix D, where D i;j
represents the distance between sites i and j.
where we regard a partition simply as the corresponding permutation. The matrices A -
in this case are simply all permutations of A. The Shaped Partition Polytope P
A has n!
vertices which stand in bijection with -partitions, and is the so-called Permutohedron.
Since each A - is a distinct vertex of P
A , there is again a convex C on IR n such that
3 Vertices and generic partitions
In this section we show that every vertex of any Shaped Partition Polytope P
A equals the
A-matrix A - of some A-generic partition, a notion that we introduce and develop below.
The convex hull of a subset U in IR d will be denoted conv(U ). Two finite sets U; V of
points in IR d are separable if there is a vector h 2 IR d such that h T
and with u 6= v; in this case, we refer to h as a separating vector of U and V . The
proof of the following characterization of separability is standard and is left to the reader.
It implies, in particular, that if U and V are separable then
Lemma 3.1 Let U and V be finite sets of IR d . Then U and V are separable if and only if
their convex hulls are either disjoint or intersect in a single point which is a common vertex
of both.
Let A be a given d \Theta n matrix. For a subset S ' [n] let A
the set of columns of A indexed by S (with multiple copies of columns identified). A p-
partition A-separable if the sets A -r and A -s are separable for each pair
that is, for each there is a vector h r;s 2 IR d such that
r;s A j for all We have the following lemma which
generalizes a result of [3] from matrices with no zero columns and no repeated columns.
Lemma 3.2 Let A be a matrix in IR d\Thetan , let be a nonempty set of p-shapes of n, and let
- be a -partition. If A - is a vertex of P
A then - is an A-separable partition.
Proof. The claim being obvious for suppose that p - 2. Let A - be a vertex of
A . Then there is a matrix C 2 IR d\Thetap such that the linear functional on IR d\Thetap given by
the inner product hC;
i is uniquely maximized over P
A at A - . Pick
any pair 1 . Suppose there are
with A i 6= A j (otherwise A -r and A -s are trivially separable). Let - 0 be the -partition
obtained from - by switching i and j, i.e. taking - 0
Frank K. Hwang, Shmuel Onn and Uriel G. Rothblum 7
. By
choice of C we have hC; A - 0
r;s
This proves A -r ,A -s are separable for each pair 1 - r ! s - p, hence - is A-separable.
We need some more terminology. Let A 2 IR d\Thetan . A p-partition -
[n] is A-disjoint if are disjoint for each pair 1 - r ! s - p. As
the convex hulls of finite sets are disjoint if and only if the sets can be strictly separated
by a hyperplane, we have that - is A-disjoint if and only if for each
there exists a vector h r;s 2 IR d such that (h r;s
Of course, A-disjointness implies A-separability, and the two properties coincide when the
columns of A are distinct.
For the vector obtained by appending a first coordinate 1
to v. For a matrix A 2 IR d\Thetan and indices 1 -
sign A (i
A matrix A is generic if its columns are in affine general position, that is, if any set of d+ 1
vectors or less from among f -
ng are linearly independent; in particular, if
? d this is the case if and only if all signs sign A (i
are nonzero. Also, the columns of a generic matrix are distinct.
We next provide a representation of the set of A-disjoint 2-partitions for generic matrices
A. The case where n - d is simple.
Lemma 3.3 Let A 2 IR d\Thetan be generic, p - 2 and n - d. Then every p-partition of [n] is
A-disjoint.
Proof. It suffices to consider the case 2. A standard result from linear algebra shows
that as -
A n are linearly independent, the range of [ -
a 2-partition - of [n], there is a vector - 2 IR d+1 with - T A i ? 0 for each
for each obtained from - by truncating its first coordinate - 1 we then have
proving that - is A-disjoint.
Let A 2 IR d\Thetan be generic with n - d. For any d-subset I of [n] with
Of course, fI \Gamma
A g is a 2-partition of [n] n I . Let I ' [n] be a d-set and
2-partition of I . The 2-partitions of [n] associated with A; I and are defined to be
either of the two 2-partitions
Lemma 3.4 Let A 2 IR d\Thetan be generic with n - d. Then the set of A-disjoint 2-partitions
is the set of all 2-partitions associated with A, d-sets I ' [n] and 2-partitions
I.
8 Shaped partition problems
Proof. We will show that for each d-set I ' [n] and 2-partition I , the
two 2-partitions associated with A; I and are A-disjoint and that each A-disjoint
2-partition is generated in this way.
First, let I ' [n] have d-elements, say
of I . Then H
is a hyperplane that contains the
columns of A indexed by I ; this hyperplane can be written as fx 2
Thus, h T A
A . We next observe that
assures that the 2-partition
of [d] is B-disjoint. Thus, there exists a vector d 2 IR d with d T A i ? d T A j
for all . For sufficiently small positive t we then have that (C
proving that
A-disjoint. It follows immediately that
proving that
the two 2-partitions of [n] associated with A; I and 2-partition of I are A-disjoint.
Next assume that - is an A-disjoint 2-partition. Then there exists a hyperplane strictly
separating A - 1 and A - 2 . Any such hyperplane can be perturbed to a hyperplane that is
spanned by d columns of A and weakly separates A - 1 and A - 2 (the details of constructing
such a perturbation are left to the reader). In particular, if A span the hyperplane
A [ I
or
A [ I . In the former case we have
I and in the latter case
and
2. With each list [-
2-partitions of [n]
associate a p-tuple - of subsets of [n] as follows: for
of the p-tuple associated
with the given list are pairwise disjoint. If [ p
holds as well then - is a p-partition
which will be called the partition associated with the given list.
Lemma 3.5 For A 2 IR d\Thetan and p - 2, the set of A-disjoint p-partitions equals the set of
p-partitions associated with lists of
A-disjoint 2-partitions.
Proof. First, consider a p-partition - associated with a list of
Then for each
so - is A-disjoint. Conversely, let - be an A-disjoint p-partition. Consider
any pair 1 are disjoint, there is a hyperplane
H r;s which contains no column of A and defines two corresponding half spaces
r;s and H
r;s
that satisfy A -r ae H \Gamma
r;s and A -s ae H
r;s . Let - r;s := (- r;s
2 ) be the A-disjoint 2-partition
defined by - r;s
r;s g and - r;s
r;s g. Let - 0 be the
Frank K. Hwang, Shmuel Onn and Uriel G. Rothblum 9
p-tuple associated with the constructed - r;s 's. Then the sets of - 0 are pairwise disjoint and
we have
is the p-partition associated with the
constructed list of
A-disjoint 2-partitions.
For each ffl ? 0 define the ffl-perturbation A(ffl) 2 IR d\Thetan of A as follows: for
let the ith column of A(ffl) be A(ffl) i := A i
is the image of
i on the moment curve in IR d . Consider any 1 - n. Then the determinant
d
is a polynomial of degree d in ffl, with D d being the Van der Monde determinant
d ] which is known to be nonzero. So for all sufficiently small ffl ? 0,
sign A(ffl) (i the sign of the first nonzero coefficient among
d and is either \Gamma1 or 1 and independent of ffl. We define the generic sign of
A at (i as the common value of sign A(ffl) (i
sufficiently small positive ffl.
Lemma 3.6 Let A 2 IR d\Thetan and p - 1. For all sufficiently small ffl ? 0, A(ffl) is generic
and the set of A(ffl)-disjoint p-partitions is the same. Further, for every d-set I 2 [n], the
sets I \Gamma
A(ffl) and I
are independent of ffl.
Proof. By Lemma 3.5, the set of A(ffl)-disjoint p-partitions is entirely determined by the
set of A(ffl)-disjoint 2-partitions. Thus, it suffices to consider only 2.
First assume that n ! d. In this case augment A with vectors to obtain
a matrix A 0 2 IR d\Theta(d+1) . The above arguments show that for sufficiently small positive ffl,
det -
A 0 (ffl) is nonzero, implying that -
A(ffl) n are linearly independent. From Lemma
3.3 it follows that for such ffl, the set of A(ffl)-disjoint 2-partitions of [n] is the set of all
2-partitions of [n].
Next assume that n ? d. As explained above, for all sufficiently small ffl ? 0, sign A(ffl)
the nonzero generic sign -A (i It
follows that for all sufficiently small ffl, the matrix A(ffl) is generic and for every d-set I , the
sets I \Gamma
A(ffl) and I
are independent of ffl. By Lemma 3.4, the set of A(ffl)-disjoint 2-partitions
is the set of all pairs of 2-partitions of [n] associated with A; d-sets I ' [n] and 2-partitions
but each such pair depends only on I \Gamma
. Hence is the
same for all sufficiently small ffl ? 0.
Let A 2 IR d\Thetan . A p-partition of [n] is A-generic if it is A(ffl)-disjoint for all sufficiently
A the set of A-generic p-partitions.
Lemma 3.6 shows that for all sufficiently small ffl ? 0, the set of A(ffl)-disjoint partitions is
the same and equals \Pi p
A . The final lemma of this section links vertices of Shaped Partition
Polytopes with generic partitions.
Shaped partition problems
Lemma 3.7 Let A 2 IR d\Thetan and let be a nonempty set of p-shapes of [n]. Then every
vertex of the polytope P
A has a representation as the A-matrix A - of some A-generic -
partition.
Proof. Let d\Thetap be a vertex of P
A and let C 2 IR d\Thetap be a matrix such that hC; \Deltai
is uniquely maximized over P
A at B. Let \Pi \Lambdag be the set of -partitions
and let \Pi := f- Bg. Then there is a sufficiently small ffl ? 0 such that
in addition, as guaranteed by
Lemma 3.6, A(ffl) is generic and the set of A(ffl)-disjoint p-partitions equals \Pi p
A . For such
ffl hC; \Deltai is maximized over the perturbed polytope P
A(ffl) at a vertex of the form A(ffl) -
for
some - 2 \Pi . By Lemma 3.2, - is A(ffl)-separable. Since A(ffl) is generic it has distinct
columns and therefore - is also A(ffl)-disjoint. We conclude that - is A-generic, proving
that - contains a generic partition.
4 Optimization and Vertex Enumeration
We now use the facts established in the previous section to prove our main results. Our
computational complexity terminology is fairly standard (cf. [20]). In all our algorithms,
the positive integer n will be input in unary representation, whereas all other numerical
data such as the matrix A will be input in binary representation. An algorithm is strongly
polynomial time if it uses a number of arithmetic operations polynomially bounded in n,
and runs in time polynomially bounded in n plus the bit size of all other numerical input.
Lemma 4.1 Let d; p be fixed. For any A 2 IR d\Thetan , the set \Pi p
A of A-generic p-partitions has
Further, there is an algorithm that, given n 2 IN and A 2 l
produces
A in strongly polynomial time using O(n dp 2
Proof. If n - d, the set of A-generic p-partitions is the set of all partitions, of which
there are p n - p d . Henceforth we assume that n ? d. If
A := f([n])g consists
of the single p-partition ([n]). Suppose now p - 2. For each choice 1 -
compute the generic sign -A (i Evaluate the polynomial
d
at Each evaluation involves the computation
of the determinant of a matrix of order d+1 and can be done, say by Gaussian elimination,
using O(d 3 ) arithmetic operations and, for rational A, in strongly polynomial time. Then,
solve the following linear system of equations
d
to obtain the indeterminates D . This can be done by inverting the nonsingular Van
der Monde matrix of coefficients of this system, again by Gaussian elimination. The generic
Frank K. Hwang, Shmuel Onn and Uriel G. Rothblum 11
sign -A (i is then the sign of the first nonzero D i . So, for fixed d the number of
operations needed to compute all
signs is O
By Lemma 3.6, for sufficiently small positive ffl, for each d-set I ' [n], I \Gamma
A(ffl) and I
are
independent of ffl. For a d-set I ' [n] and such ffl, I \Gamma
A(ffl) and I
are available from the above
signs that determine det[ -
permutation that puts -
into the right location may be applied). Further, from Lemma 3.6 and 3.4, \Pi 2
A equals the
common set of A(ffl)-disjoint partitions for sufficiently small positive ffl, and this set is the
set of partitions of [n] of the form
is a d-subset of [n] and 2-partition of I . For each d-set I ' [n], the common
2-partitions
sufficiently small positive ffl has been determined; hence a list of
the 2-partitions in \Pi 2
A is available (the construction may contain duplicates). As there are
d
d-subsets I and 2 d 2-partitions of each I , we have j\Pi 2
d
all partitions in \Pi 2
A can be obtained from the generic signs using again O(n d+1 ) operations.
For sufficiently small positive ffl, \Pi p
A is the common set of A(ffl)-disjoint p-partitions and
A is the common set of A(ffl)-disjoint 2-partitions. It follows from Lemma 3.5 that \Pi p
A is
the set of all p-partitions associated with lists of
2-partitions from \Pi 2
A . This shows that
To construct \Pi p
A , produce all such lists of
2-partitions from \Pi 2
A ; for each list form the
associated p-tuple -, and test if it is a partition (i.e. if [ p
[n]). As there are O(n
lists, all this work can be easily done using O(n dp 2
operations which subsumes
the work for computing the generic signs and constructing \Pi 2
A , and is the claimed bound.
We can now provide the solution of the Shaped Partition Problem. The set of admissible
p-partitions can be represented by a membership oracle which on query - answers whether
or not - 2 . The convex functional C on IR d\Thetap can be presented by an evaluation oracle
that on query A - with - a -partition returns The oracle for C will be called
M-guaranteed if guaranteed to be a rational number whose absolute value is no
larger than M for any -partition -. The algorithm is then strongly polynomial oracle time
if it uses a number of arithmetic operations and oracle queries polynomially bounded in n,
and runs in time polynomially bounded in n plus the bit size of A and M .
Theorem 4.2 For every fixed d; p there is an algorithm that, given n; M 2 IN, A 2 l
oracle presented nonempty set of p-shapes of n, and M-guaranteed oracle presented convex
functional C on l
solves the Shaped Partition Problem in strongly polynomial oracle
time using O(n dp 2
operations and oracle queries.
Proof. Use the algorithm of Lemma 4.1 to construct the set \Pi p
A of A-generic p-partitions
in strongly polynomial time using O(n dp 2
test shapes of the
partitions in the list to obtain the subset \Pi := f- 2 \Pi p
\Lambdag of A-generic -
partitions by querying the -oracle on each of the j\Pi p
partitions in \Pi p
A . Since
C is convex, it is maximized over the Shaped Partition Polytope P
A at a vertex of P
A .
Shaped partition problems
By Lemma 3.7, this vertex equals the A-matrix A - of some partition in \Pi . Therefore,
any - 2 \Pi achieving
is an optimal solution to
the Shaped Partition Problem. To find such - compute for each - 2 \Pi the matrix
, query the C-oracle for the value pick the best.
The number of operations involved and queries to the C-oracle is again O(n dp 2
). The bit
size of the numbers manipulated throughout this process is polynomially bounded in the
bit size of M and A hence the algorithm is strongly polynomial oracle time.
Recall that the Shaped Partition Polytope is defined as P
\Lambdag.
The number of matrices in the set fA \Lambdag is typically exponential in n, even
for fixed d; p. Therefore, although the dimension of P
A is bounded by dp, this polytope
can potentially have exponentially many vertices and facets as well. But, Lemmas 3.7 and
4.1 yield the following theorem which shows that, in fact, Shaped Partition Polytopes are
exceptionally well behaved.
Theorem 4.3 Let d; p be fixed. For any A 2 IR d\Thetan and nonempty set of p-shapes of n,
the number of vertices of the Shaped Partition Polytope P
A is O(n d( p) ). Further, there is
an algorithm that, given n 2 IN, A 2 l
presented , produces all vertices of
A in strongly polynomial oracle time using O(n d 2 p 3
operations and queries.
Proof. By Lemma 3.7, each vertex of P
A equals the A-matrix A - of some partition in
A . Therefore, the number of vertices of P
A is bounded above by j\Pi p
A j hence, by Lemma
4.1, is O(n
To construct the set of vertices given a rational matrix A, proceed as fol-
lows. Use the algorithm of Lemma 4.1 to construct the set \Pi p
A of A-generic p-partitions in
strongly polynomial time using O(n dp 2
operations. Test the shapes of the partitions
in the list to obtain its subset \Pi := f- 2 \Pi p
\Lambdag of A-generic -partitions by
querying the -oracle on each of the j\Pi p
partitions in \Pi p
A . Construct the set
of matrices U := fA - 2 \Pi g with multiple copies identified. This set U is contained in
A and by Lemma 3.7 contains the set of vertices of P
A . So u 2 U will be a vertex precisely
when it is not a convex combination of other elements of U . This could be tested using
any linear programming algorithm, but to obtain a strongly polynomial time procedure,
we proceed as follows. By Carath'eodory's theorem, u will be a vertex if and only if it is
not in the convex hull of any affine basis of U n fug. So, to test if u 2 U is a vertex of P
A ,
compute the affine dimension a of U n fug. For each (a of U n fug,
test if it is an affine basis of U n fug, and if it is compute the unique -
P a
P a
in the convex hull of fu if and only
a vertex of P
A if and only if for each affine basis we get some
Computing the affine dimension a, testing if an (a+1)-subset of U nfug is an affine
basis and computing the - i , can all be done by Gaussian elimination in strongly polynomial
time. Since we have to perform the entire procedure for each of the jU j - j\Pi
elements for each such u the number of affine bases of U nfug is at most
the number of arithmetic operations involved is O(jU j
which absorbs
the work for constructing \Pi and obeys the claimed bound.
Frank K. Hwang, Shmuel Onn and Uriel G. Rothblum 13
As an immediate corollary of Theorem 4.3, we get the following polynomial bound on
the number of facets of any Shaped Partition Polytope and a strongly polynomial oracle
time procedure for producing all facets (by which we mean finding, for each facet F , a
supporting P
A at F ).
Corollary 4.4 Let d; p be fixed. For any A 2 IR d\Thetan and nonempty set of p-shapes of n,
the number of facets of the Shaped Partition Polytope P
A is O(n d 2 p 3
Further, there is an
algorithm that, given n 2 IN, A 2 l
presented , produces all facets of P
A
in strongly polynomial oracle time using O(n d 2 p 3
operations and queries.
Proof. By the well known Upper Bound Theorem [18], the number of facets of any
k-dimensional polytope with m vertices is O(m k
Applying this to P
A with k - dp and
get the bound on the number of facets of P
A . To construct the facets,
construct first the set V of vertices using the algorithm of Theorem 4.3. Compute the
dimension a of aff(P (possibly empty) set S of dp \Gamma a points
that together with V affinely span IR d\Thetap . For each affinely independent a-subset T of V ,
compute the hyperplane fX 2 spanned by S [ T . This hyperplane
supports a facet of P
A if and only if all points in V lie on one of its closed half-spaces.
Clearly, all facets of P
A are obtained that way, in strongly polynomial time and number of
arithmetic operations and oracle queries bounded as claimed.
Acknowledgment
Shmuel Onn thanks the Mathematical Sciences Research Institute at Berkeley for its support
while part of this research was done.
--R
Discrete Applied Mathematics
A simple on-line randomized incremental algorithm for computing higher order (Voronoi) diagrams
Optimal partitions having disjoint convex and conic hulls
Mathematics of Operations Research
Consecutive optimizers for a partitioning problem with applications to optimal inventory groupings for joint replenish- ment
Cutting dense point sets in half
The Art of Counting (edited by Joel Spencer)
Partition polytopes over 1- dimensional points
The Pareto set of the partition bargaining game
Journal of Combinatorial Theory Ser.
Representations and characterizations of the vertices of bounded-shape partition polytopes
Linear programming over partitions
Partitions: Clustering and Optimality
SIAM Journal on Algebraic and Discrete Mathematics
Discrete and Computational Geometry.
On the number of halving lines
The maximum numbers of faces of a convex polytope.
Theory of Linear and Integer Programming
Lectures in Polytopes
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F. K. Hwang , J. S. Lee , Y. C. Liu , U. G. Rothblum, Sortability of vector partitions, Discrete Mathematics, v.263 n.1-3, p.129-142, 28 February | polynomial time;optimization;polytope;convex;programming;enumeration;partition;separation;cluster |
589281 | Cut Size Statistics of Graph Bisection Heuristics. | We investigate the statistical properties of cut sizes generated by heuristic algorithms which solve the graph bisection problem approximately. On an ensemble of sparse random graphs, we find empirically that the distribution of the cut sizes found by "local" algorithms becomes peaked as the number of vertices in the graphs becomes large. Evidence is given that this distribution tends toward a Gaussian whose mean and variance scales linearly with the number of vertices of the graphs. Given the distribution of cut sizes associated with each heuristic, we provide a ranking procedure that takes into account both the quality of the solutions and the speed of the algorithms. This procedure is demonstrated for a selection of local graph bisection heuristics. | Introduction
. Algorithms for tackling combinatorial optimization problems [27] may be divided
into two classes. Exact algorithms such as exhaustive search, branch-and-bound, or branch-and-cut,
form the first class; they determine (exactly) the optimum of the cost function which is to be minimized.
However, for NP-hard problems, they require large computation ressources, and in particular, large
computation times. The second class consists of "heuristic" algorithms; these are not guaranteed to find
the optimal (lowest cost) solution, nor even a solution very close to the optimum, but in practice they
find good approximate solutions very fast. For problems in science, one's main interest is in the optimal
solution, so an exact algorithm is required. However, for many engineering applications, the heuristic
approach may be preferable. There are several reasons for this: (i) The computational ressources are
simply insufficient to solve the instances of interest by exact methods; (ii) The cost function one wants to
minimize is computationally very demanding, and limited resources force one to use an approximate cost
function instead. This is the rule rather than the exception with very complex systems such as VLSI.
If the true cost function cannot be used, there is little point in finding the true optimum for the wrong
problem. (iii) Heuristic algorithms typically generate numerous "good enough" solutions, thus providing
information about the statistical properties of low cost solutions. This information can in turn be used
for generating better heuristics, or for finding new criteria for guiding the branching in exact algorithms
such as branch-and-bound.
For almost any combinatorial optimization problem, it is very easy to devise heuristic algorithms
which perform quite well; this is probably why so many such algorithms have been proposed to date.
Usually they fall into just a few families, the most popular of which are local search, simulated annealing,
tabu search, and evolutionary computation. The practitioner is frequently confronted with the problem
of choosing which method to use. Thus he would like to rank these algorithms and determine which one
is best for his "instance" (the set of parameters which completely specify the cost function). A difficulty
then arises because most heuristic algorithms are stochastic, so that they can give many different solutions
for a single instance. In general, the distributions of solution costs generated by the different heuristics
overlap, so that the winning algorithm varies from one trial to another. Furthermore, it is necessary
to balance the quality of the solutions found against the time necessary to find them since in practice
heuristics run at very different speeds. The final goal of this paper is to do just this kind of balancing:
in Section 8 we shall introduce a generally applicable ranking method which is based on the possibility
of performing multiple runs from random starts for each algorithm until an allotted amount of computer
time is exhausted. Our ranking method then determines whether it is better to have a fast heuristic
which gives not so good solutions or a slower heuristic which can give better solutions.
Establishing a ranking on a single instance may be what is needed for a real world problem, but it is
not a useful prediction tool. It is preferable to consider the effectiveness of a heuristic when it is applied to
a family of instances. Since a detailed knowledge of the distribution of costs is necessary for our ranking
procedure, the major part of this paper is an in depth study of the statistics of costs found by several
classes of heuristics. The NP-hard [9] combinatorial optimization problem chosen for our study is the
graph bisection problem, hereafter simply called the graph partitioning problem (GPP). This choice is
justified by the wide range of practical applications of the GPP. These include host scheduling [3], memory
paging and program segmentation [17], load balancing [21], and numerous aspects of VLSI-design such
as logic partitioning [12] and placement [6, 19]. Because of these applications, the GPP has been used as
a testing ground for many heuristics. For our work, a selection had to be made; in view of the previous
studies by Johnson et al. [13], Lang and Rao [20], and Berry and Goldberg [4], we have restricted our
study to iterative improvement heuristics based on local search and to simulated annealing. Having
made a choice of optimization problem and algorithms, it remains to define the class of instances for the
testbeds. Ideally, this family of instances should reflect the structure of the actual instances of interest
to the practitioner. Since we do not have a particular application in mind, we shall follow the studies
of [13, 20, 4], and consider an ensemble of sparse random graphs. From our numerical study, we have
found that all of the heuristics tested share the following properties when the random graphs become
large: (i) each algorithm can be characterized by a fixed percentage excess above the optimum cost; (ii)
the partitions generated have a distribution of costs which becomes peaked, both within a given graph
and across all graphs; (iii) these distributions tend towards Gaussians. Because of these properties, our
ranking of heuristics on large graphs is largely determined by the mean and variance of the costs found,
and thus a constant speed-up factor has only a very small effect on the ranking. We expect this property
to hold for most problems and heuristics of practical interest, leading to a very robust ranking.
The paper is organized as follows. In Section 2 we define the GPP as well as the ensemble of random
graphs used for our testbed. Section 3 derives properties of random partitions, and shows that the
Cut Size Statistics of Graph Bisection Heuristics 3
distribution of cut sizes has a relative width which goes to zero as the instance size grows. In Section 4
we argue why this property should hold also for the distribution of costs found by heuristic algorithms
based on local iterative processes. In Section 5 we discuss the heuristic algorithms we have included in
our tests. Section 6 gives the mean and standard deviation of the costs found as a function of graph size;
the distribution for the costs is indeed found to be peaked. This leads to a first ranking which, however,
does not take into account computation times. To implement our speed-dependent ranking, we must
determine the distribution of cut sizes found by the different algorithms. This is the subject of Section 7,
where evidence is given that the distribution on any typical graph tends towards a Gaussian in the limit
of large graphs. In Section 8 we present our ranking method which takes into account both the quality
of the solutions as well as the speed of the heuristics. In Section 9, finally, we discuss the results and
conclude.
2. Minimum cuts. The graph partitioning (or graph "bisection") problem (GPP) can be defined
as follows. Consider a graph E) which consists of a set of N vertices and
a set of (non-oriented) edges connecting pairs of vertices. It is convenient to introduce the
called the connectivity matrix, given by
is connected to v j
Since the edges are non-oriented, . (Some of what will be discussed applies to weighted graphs;
represent the weight of the ij edge.) A partition of G is given by dividing the vertices of G
into two disjoint subsets V 1 and V 2 such that . The number of edges connecting V 1 to V 2 is
called the cut of the partition, and will be denoted by C. It is given by
The GPP (or "Min-cut" problem) consists of finding the partition which the cost (2.1) is
minimum subject to given constraints on the sizes of V 1 and V 2 . The GPP is NP-hard [9]. In the
standard formulation to which we shall restrict ourselves in this work, V 1 and V 2 have equal sizes.
For our study, it is necessary to fix an ensemble of graphs for the testbed. We have chosen G(N; p)
the ensemble of random graphs of N vertices where each edge is present with probability p. The choice of
G(N; p) is justified by its tractable mathematical properties and by the fact that many workers [13, 20, 4]
have used graphs in this ensemble to test heuristics. The problem of finding the properties of the minimum
cut size when the graphs belong to such an ensemble is sometimes called the stochastic GPP. Let us review
some of the known results for this problem; this will serve to motivate our conjectures for the behavior
of cuts obtained from heuristics. For each graph G i , call C 0 its minimum cut size. Taking G i from the
ensemble G(N; p), C 0 is a random variable. Following derivations now standard in a number of other
stochastic combinatorial optimization problems (COP), it is possible to show using Azuma's inequality
[1] that the distribution of C 0 becomes peaked as N ! 1. This means that as N becomes large,
, the relative fluctuations about the mean, tend to zero. This property, often referred
to as "self-averaging", is typical of processes to which many terms contribute. For certain stochastic
COP, it is possible to show further that the mean minimum cost satisfies a power scaling law in N , so
that C 0 =N fl converges in probability to a limiting value as N ! 1. In the case of the stochastic GPP,
there is no proof that such property hold. Nevertheless, it is believed that such a scaling holds: within
the G(N; p) ensemble at p fixed, calculations show that C 0 =N 2 ! p=4 with probability one as
[8]. As will be shown in the next section, this is also the limiting behavior of random cuts, and so the
ensemble at p fixed is not a challenging one for heuristics. The reason for this "uninteresting" scaling is
the high number of edges connecting to any vertex. Thus we consider in this work the ensemble G(N; p),
ff is the mean connectivity (number of neighbors of a vertex) of the graphs.
These graphs are sparse, in contrast to the dense graphs obtained by taking p to be independent of N .
Consider the optimal partition. At a typical vertex in V 1 , some finite fraction of its edges will connect
to vertices in V 2 . With each vertex contributing an O(1) amount to the cut size, C 0 is expected to grow
linearly with N . Since C 0 =N is known to be peaked at large N , it is natural to conjecture the stronger
property that C 0 =N tends towards a constant with probability one as N ! 1. A major motivation for
this work is our expectation that an identical scaling law should hold if we replace C 0 by the cost found
4 G.R. Schreiber and O.C. Martin
by a heuristic algorithm, albeit that the limiting constant depends on the heuristic. To motivate such
a property, the next section analyzes the cut sizes of random partitions; then in Section 4 we consider
the "statistical physics" of the GPP so as to interpolate between the case of minimum cuts and that of
random cuts.
3. Cuts of random partitions. Here we show explicitly that a large N scaling law holds for the
cut sizes of random partitions, and that asymptotically these random cuts have a Gaussian distribution
with a relative variance proportional to 1=N .
Consider any graph in G(N; p). One can always write the cut size of a random partition as
where X is the mean (random) cut size for the graph under consideration, and hY is the average
over the random partitions.) Averaging explicitly over all balanced partitions of the fixed graph, we find
1)]. The interpretation of this formula is very simple: any edge of weight E ij has a
probability being cut.
In the ensemble G(N; p) of random graphs, it is easy to calculate the first few moments of X . In
particular, we find
denotes the average
over the ensemble G(N; p).) We also see that X is the sum of independent random
this implies that the kth cumulant (connected moment) of the distribution of X statisfies
c
c
At large N , we then have
c
in the constant p ensemble, and
c
- ffN in the p - ff=N
ensemble.
The random variable Y is more subtle as it is the sum of M correlated variables. Nevertheless, for
any graph, it is possible to compute the moments of Y , and we have done this explicitly for the second
and third moments. (The expressions are too long to be given here.) If we average Y 2 both over random
partitions and over G(N; p), we obtain:
The calculations get significantly more complicated for the higher moments. In order to keep to simple
expressions, we limit ourselves to the ensemble with 1). Then we find:
Furthermore, the graph to graph fluctuations of hY 2 i become negligible in relative magnitude, so that
the ratio of a typical variance to the mean variance goes to 1 at large N . This however is not true for the
higher moments; for instance, we find that the typical value of hY 3 i grows as N 1=2 , but taking in addition
the mean over graphs leads to a N independent behavior. Finally, one can show that hY k i c =hY 2
with probability one. This shows that as N ! 1, Y has a Gaussian distribution, of zero mean, and of
variance growing linearly with N , whose coefficient is graph independent.
Coming back to the cut size of a random partition, we find that the normalized correlation
coefficients between powers of X and Y tend to zero at large N , and thus X and Y become independent
random variables in that limit. This, along with the results previously derived, shows that at large N , C
itself has a Gaussian distribution. From these results, we deduce the large N behavior:
so that relative deviations from the mean go to zero. Thus the distribution of C becomes peaked, and
probability one as N ! 1. The convergence of the distribution of C=N to a "delta"
function is referred to as the self-averaging of C.
Cut Size Statistics of Graph Bisection Heuristics 5
The scaling of the variances can be summarized at large N by writing
Y
y
where x and y are independent Gaussian random variables of zero mean and unit variance; oe
is the standard deviation (rescaled by 1=
N) of X , and oe
ff=8 that of Y . Thus oe
Y describes the
fluctuations of the cut sizes within a graph, and oe
X describes the fluctuations of the mean cut size from
graph to graph.
We have used these analytical results to test the validity of our computer programs. The first two
moments of X allowed us to test our generation of random graphs in G(N; p). Similarly, a check on our
random number generator was obtained by verifying on several graphs that the second moment of Y
found by the numerics was in agreement with our formulae. Finally, we also checked that random cut
sizes have a limiting Gaussian distribution, with a third moment which scales to zero at large N . (For
this check, we performed random partitions on 100 000 graphs for
4. Statistical physics of the GPP. We saw that cut sizes of random partitions in G(N; p) have a
self-averaging property; we conjectured that this property also holds for the minimum cut. It is possible
to interpolate between these two kinds of partitions (random and min-cut) by following the formalizm of
statistical physics. For any given graph, consider the "Boltzmann" probability distribution pB , defined
for an arbitrary partition P of cut size C(P
e \GammaC(P )=T
Z
Z is chosen so that pB is normalized (a probability distribution) and T is an arbitrary positive parameter
called the temperature. When T !1, we recover the ensemble of random partitions where all partitions
are equally probable, while when T ! 0, the ensemble reduces to the partitions of minimum cut size. For
intermediate values of the temperature, the partitions are weighted according to an exponential of their
cut size. In this "Boltzmann" ensemble, one can define the moments of the cut sizes just as was done
in the case of random partitions. In most statistical physics problems, it is possible to show that the
quantity in the exponential of Eq. (4.1) (here, the cut size) is self-averaging. For random graphs, however,
the proofs are inapplicable; nevertheless, other evidence indicates that the cut size is self-averaging at
any temperature [26]. This self-averaging can be understood qualitatively at low temperature as follows.
The number N (C) of partitions of cut size C is a sharply increasing function of C, whereas the Boltzmann
factor is a sharply decreasing function of C. Note that the probability distribution P (C) of C is given by
the product of these two functions. Using naive but standard statistical physics arguments for N (C), one
finds that P (C) has a peak at C (T ) which grows linearly with N and that the width of the distribution
is O(
N ), which gives the self-averaging property for C. In addition, this kind of argument says that
becomes Gaussian at large N , a result which is usually correct in statistical physics systems.
A number of statistical physics results have been obtained for the GPP in the ensemble of dense
random graphs, i.e., for G(N; p) at p fixed. In particular, highly technical calculations [26, 8] indicate
that the cut sizes are self-averaging at all temperatures, that is as N ! 1, relative fluctuations within
a fixed graph become negligible, as well as those from graph to graph. The mean cut size is given by
as N ! 1. (If the mean over graphs is not performed, the formula remains valid for "almost all"
sequences of graphs with N !1.) In this equation, U(T ) is a function of temperature only, there is no
dependence on p as long as p is independent of N . The limit T ! 0 gives the expected (and typical) value
of the minimum cut, with 0:3816:. Although there is no proof yet that these calculations are
exact, there is general agreement in the statistical physics community that the results are correct.
The case of sparse random graphs (p - 1=N) has also been studied within the statistical physics
approach [2, 5]. So far, however, the problem has proven to be intractable with no plausible solution in
sight. Nevertheless, it is expected that the cut sizes are self-averaging at any temperature and that the
mean of the distribution scales linearly with N at large N .
6 G.R. Schreiber and O.C. Martin
The property of self-averaging seems quite generic. The reason it should hold in these systems is that
the cut size of a partition is the sum of a large number of random variables which are not too correlated.
It is very plausible that the cut size is self-averaging whenever partitions are generated by an iterative
process involving just a few vertices at a time. All local search methods, and modifications thereof such
as simulated annealing, fall into this category. Thus our claim is that any heuristic algorithm which
generates partitions iteratively according to local (in vertex space) criteria will lead to cut sizes which
are self-averaging. Thus the distribution of cut sizes found by any such heuristic should become peaked
as N ! 1. Furthermore, in this limit, the distribution should converge towards a Gaussian in the way
given by the central limit theorem. We will see in the sections to follow that this is indeed born out
empirically for all of the heuristics which we have investigated.
The arguments we have presented are not specific to the graph partitioning problem, so we expect
them to apply to most stochastic COPs having many variables in their cost function. Surprisingly, there
has been very little research on this topic. In the context of the "NK" model with binary variables, a
study by Kauffman and Levin [16] found that the costs of local minima became peaked towards the value
of a random cost as N grew. (This peculiar property is due to the structure of the energy landscape
in that model.) However, concerning the behavior of heuristic solutions, research has almost exclusively
focused on the case of the Euclidean traveling salesman problem where points are laid out on the plane.
Most practitioners in that field know that local search heuristics give rise to costs whose relative variance
decreases as the number of points increases. Furthermore, it was observed by Johnson and McGeoch [14]
among others that the costs tend towards a fixed percentage excess above the optimum. Our purpose
here is to show how this convergence occurs, albeit in a different combinatorial optimization problem,
and to provide a theoretical framework for understanding where this behavior comes from. Also, we
pay special attention to the distinction between fluctuations within an instance and from one instance
to another. We believe our findings are quite general, and in particular that the ensemble of instances
considered need not be based on points in a physical space.
5. Algorithms used in the testbed. In view of the previous arguments, we have restricted ourselves
to local heuristics. Without trying to be complete nor representative, we have studied the statistics
of cut sizes for three types of local search and four versions of simulated annealing algorithms. In this section
we sketch the workings of these heuristics. In Sections 6 and 7, we show that the same self-averaging
properties hold for all these algorithms in spite of their significant differences. There is thus no reason
to believe that our claims are affected by the details of such algorithms; rather, the properties are most
likely generic to dynamics which are local.
5.1. Kernighan-Lin (KL). In simple local search, one performs elementary transformations to a
feasible solution of the COP as long as they decrease the cost, a procedure sometimes called -opting
[22]. A more sophisticated version consists in using "variable depth" search: one builds a sequence of
elementary transformations, usually according to a greedy criterion. p is not set ahead of time, and
depends on the sequence of costs found. The elementary transformations are not imposed to decrease
the cost, but the sequence of length p must do so if it is to be applied to the current solution. Such a
procedure was first proposed by Kernighan and Lin [18], in fact in the framework of the GPP. Hereafter
we will refer to their algorithm as "KL". The elementary transformation they use is the exchange of a
pair of vertices, one vertex in V 1 being exchanged for one in V 2 . A sequence of such exchanges is built up
in a greedy and tabu fashion by performing a "sweep" of all the vertices: at each step of the sweep, one
finds the best (largest cost gain) pair to exchange among those vertices which have not yet been moved in
the sweep (tabu condition). The sweep has length N=2. When the sweep is finished one finds the position
along the sequence of exchanges generated where the cut size is minimum. If this minimum leads to an
improved partition, the transformation of p exchanges is performed on the partition and another sweep
is initiated; otherwise the search is stopped and the partition is "KL-opt", i.e., it is a local minimum
under KL.
The KL algorithm is deterministic although it is possible to introduce stochasticity to break degeneracies
in selecting the best pair to exchange. Its computational complexity is not easy to estimate
because the number of sweeps is not known in advance. (This is a generic difficulty in estimating the
speed of iterative improvement heuristics.) However, in practice, one finds that KL finishes in a "small"
number of sweeps. Thus the computational complexity is estimated to be a few times that of performing
the last sweep, known as the check-out sweep. For our study, we have used our own implementation of
KL [24], which uses heaps to find the best pair to exchange at each step. For sparse graphs, this leads to
O(N operations per sweep. A nearly identical KL is provided in the Chaco software package, which
Cut Size Statistics of Graph Bisection Heuristics 7
gives sensibly identical results. A faster implementation of the algorithm has been given by Fiduccia and
Mattheyses [7] whenever the use of a radix sort is possible; then the time for each sweep is O(N ).
In terms of quality of solutions found, KL is quite good. What is surprising is that although Kernighan
and Lin proposed their method over 20 years ago, KL remains relatively unchallenged, at least as a general
purpose method applicable to any kind of graph, regardless of its structure. Of course, for special kinds
of graphs, such as meshes, other heuristics (e.g., spectral bisection) perform better [4, 11, 13, 15].
5.2. A multilevel KL-algorithm: CHACO. The Chaco software package includes a number
of heuristics for partitioning graphs. (For information about this package, see the Chaco user's guide
[10].) For our purposes, we have used only its "multilevel" generalization of KL, hereafter referred to
simply as CHACO. The CHACO algorithm is based on a coarse graining or "compactification" of the
graph to be partitioned. At each level, vertices are paired using a matching algorithm, and paired
vertices are then considered as the vertices of the next higher level of compactification. Because of this
process, it is necessary to have weighted edges; the weights are also propagated to the higher level. The
compactification is repeated until a sufficiently small graph is obtained to which spectral bisection is
applied to get a first partition. Then this partition is used as the starting partition in KL for the graph
at the level below it. This process is recursive, until one obtains a KL-opt partition of the original graph.
(Note that this construction is deterministic, and does not require an initial "random" partition.) Such
a multilevel strategy has been very successful for unstructured 2 and 3 dimensional meshes [11, 15], both
in terms of solution quality (much better than for KL alone), and in terms of speed (much faster than
KL because of the hierarchical nature). However, the usefulness of CHACO on random graphs is not a
priori obvious, both in terms of speed and quality of solutions.
5.3. Simulated Annealing algorithms. We have choosen as a third comparative algorithm simulated
annealing (SA). SA is based on a set of elementary moves, just like local search, but now moves
which increase the cost are accepted with (low) probability. Because of this, it is sometimes appropriate
to consider SA as a noisy local search method. Simulated annealing is really a family of algorithms.
To include some of the different bells and whistles proposed for this algorithm, we have considered four
variations. These are: (i) the SA as first introduced by Kirkpatrick et al. [19] (referred to as FSA) where
the initial and final temperatures are fixed ahead of time by the user and where a predetermined number
of trial moves are performed at each temperature; (ii) Kirkpatrick et al. also proposed to determine
the initial and final temperatures of the schedule dynamically. They set the initial temperature at the
beginning of the run using the criterion that about 80% of the trial moves are accepted at that tempera-
ture. Similarly, they stop the cooling if for 5 cooling steps the energy does not decrease. We will refer to
this method as KSA. (iii) Johnson et al. [13] improved the speed of this algorithm by allowing an early
exit to the next temperature of the schedule; the condition they proposed for exiting is having accepted
a minimum number of moves. Also they modified the termination criterion to having an acceptance
rate less than a threshold value. We will refer to this version as JSA. All three of these SA methods
use an exponential cooling schedule with a cooling factor of 0:95. (iv) The last SA variation consists in
using an adaptive schedule whereby the next temperature value is determined on the fly according to the
energy fluctuations at the current temperature. We have choosen for this variation the implementation
of van Laarhoven and Aarts [28, 29]. To obtain good results one would have to spend a long time in the
"freezing" phase of the cooling. Since this would increase the computation times significantly we have
choosen not to use a fine-tuned adaptive schedule but one which provides a cooling factor of the same
magnitude as in the other SA algorithms presented. This allows us to have similar computation times
for all the simulated annealing algorithms investigated.
In SA, one can use the same elementary moves as in local search, i.e., for the GPP, pair exchanges.
However, once a low cost partition is obtained, it will take a long time (or a lot of luck) to find further good
exchanges. Finding a good pair is best done by finding the first vertex to transfer and then the second,
i.e., by using a sequential process. This suggests relaxing the constraint of having balanced partitions,
and replacing it by a penalty function which keeps the sizes of V 1 and V 2 nearly equal (small off-balance).
We have followed a slightly different approach where each move destroying the balance must be followed
by a move restoring the balance. Then the Markov chain explores the partitions which are balanced and
those with "off-balance" of \Sigma1. It is easy to see that this method is equivalent to having the cost of all the
other partitions equal infinity; at fixed temperature and for long chains, one generates partitions with cut
sizes given by the Boltzmann factor, within the constraint for the "off-balance". Indeed, the succession
of accept/reject decisions makes the global probability distribution Boltzmannian in this enlarged space,
so that we guarantee the same convergence properties as in the standard case.
8 G.R. Schreiber and O.C. Martin
Some remarks concerning our implementations are in order. First, at fixed temperature, we perform
a certain number of "sweeps". In each sweep, every vertex is sequentially considered as a candidate
for changing sides of the partition; if the move were to violate our limit on the "off-balance", the move
is rejected (in fact, it simply is not considered). A sweep thus requires O(N) operations. Our sweeps
use random permutations rather than a fixed or random ordering of the vertices. The use of random
permutations should - according to certain authors [13, 28, 29] - result in a enhancement of the quality
of the solutions found. Second, the maximum number of sweeps at any temperature is set to ff-, with
all of our implementations. For FSA and KSA, this is in fact the (actual) number of sweeps, so
that their computational complexity is O(ff-N) times the number of temperature steps used. The cases
of JSA and ASA are more difficult to evaluate. In practice we find that JSA is faster than KSA, but
not by more than a constant factor. ASA on the other hand spends quite a lot of time at intermediate
temperatures, all the more so that N increases; empirically, we have found an O(N 3=2 ) complexity.
In terms of quality, we are aware of no systematic study on sparse random graphs. In a previous
SA work on the GPP, Van Laarhoven and Aarts used an adaptive decrement rule [28, 29] and claim a
gain of about 13% over simpler non-adaptive algorithms. They also compared their results to those from
the algorithm used by Johnson et al. for the GPP, who claimed an enhancement of about 5% for JSA
over the Kernighan-Lin algorithm. The small gain found by Johnson [13] is, according to van Laarhoven
and Aarts [28, 29], due to the use of a non-adaptive choice of the temperature decrement rule. However,
we have found for sparse random graphs that the different variants of simulated annealing are nearly
indistinguishable in terms of quality of solutions. This may be due to our not using a penalty term or to
the different nature of the graphs used in the present study.
5.4. Chained-Local-Optimization (CLO). The chained-local-optimization (CLO) strategy is a
synthesis of local search and of simulated annealing [25]. The essential idea is to have simulated annealing
sample not all solutions, but only locally optimal solutions. This strategy is guaranteed to be at least as
good as local search, and has been successfully applied to the traveling salesman problem [23] and to the
partitioning of unstructured meshes [24].
In this work, we use KL as the local search engine. Given any initial KL-opt partition P i , the simplest
implementation of CLO will: (i) apply a perturbation or "kick" to modify significantly the partition (in
practice this means exchanging clusters of vertices); (ii) run KL on the modified partition so as to reach
a new KL-opt partitionP f ; (iii) apply the accept/reject procedure for going from the initial partition
to the final one (P f ). This defines the analogue of one move of a simulated annealing algorithm,
except that many modifications to the partition have occured in this single step. The temperature may
be modified according to a schedule if desired, but for simplicity, we have set the temperature to zero in
all of our runs.
As was discussed in the context of simulated annealing, it is inefficient to exchange vertices or clusters
simultaneously, it is better to do it sequentially. Our present CLO algorithm thus proceeds as follows.
Given P i an initial balanced KL-opt partition, choose a (connected) cluster of p vertices in V 1 (or
and move them into V 2 (respectively V 1 ). KL-optimize this partition to generate an intermediate (off-
balanced) partition. Now choose a cluster of p vertices in V 2
this modified partition to generate P f , the final (and balanced) partition. This whole procedure is our
"simulated annealing" step, and we apply the accept/reject criterion for going from P i to P f .
When runing CLO on irregular meshes [24], it was possible to perform large kicks, exchanging many
vertices at once. Unfortunately, for sparse random graphs, we find that the acceptance when doing so
becomes low. We have thus used "small" kicks, creating clusters of sizes varying randomly between 3
and 13. Given such small kicks, KL usually terminates in just 2 sweeps, and the speed of CLO per kick
is about half that of KL.
Consider now the limit of large N . Using the analogy with simulated annealing, if a fixed (N -
independent) number of small kicks are used, it can be expected that CLO will perform no better than
KL itself. We have thus chosen to use a number of kicks which scales linearly in N , namely -N with
This choice of course influences the quality of the solutions generated, a larger value of - giving
a priori better results. The computational complexity of this algorithm is then of order N 2 log(N ).
6. Self-averaging of the cut size. In the rest of this paper, we study the statistical properties of
the cut sizes generated by the algorithms described in Section 5 when applied to random initial partitions.
The ensemble of graphs used is that of random graphs with mean connectivity
Section 2). This value was chosen because at much larger connectivities, the ratio between the best
and worst cut size approaches 1, and at lower connectivities, algorithms taking explicit advantage of
Cut Size Statistics of Graph Bisection Heuristics 9
disconnected parts of the graph will outperform general purpose heuristics. In order to minimize effects
associated with our finite sample of graphs in the ensemble, we have benchmarked all the algorithms on
the same graphs. The number of graphs used during the production runs was 10 000 with values of N
ranging between 50 and 200; however, because the CHACO algorithm was fast, we have also performed
runs on 100 000 graphs for that heuristic.
The purpose of this section is to give numerical evidence that the distribution of cut sizes becomes
peaked in the limit of large graphs, for each of the heuristics considered. (Further properties of the
distribution will be given in Section 7.) We find that each algorithm generates cut sizes for which both
the mean and variance scale linearly in N . From this behavior, it is clear that the distribution of cut
sizes becomes peaked at large N , i.e., that the cut sizes are self-averaging. Also, assuming (cf. Section
2) that the minimum (i.e., optimum) cut size scales linearly with N at large N , we then see that each
heuristic algorithm leads to a fixed percentage excess above the true optimum. (Note that the worst cut
size also has a linear scaling in N .) This percentage excess provides a first ranking of the algorithms,
which, however, does not take into account the speed of execution.
If C(i; m) is the cut obtained by a heuristic for the graph G i and an initial partition m, define the
mean cut per vertex
by:
m)
where the averages are over initial partitions and over the ensemble of graphs studied (cf. Section 3 for
the notation). We compute these ensemble averages numerically using the standard estimator (hereafter,
overlines refer to numerical averages):
The approximation is due to a statistical error e associated with fluctuations of C(i; m) both with m and
i. It is not difficult to see that for our problem, one does not need to perform an average over m; using any
finite number R of partitions for each graph G i provides an unbiassed estimator of
. Furthermore,
the statistical error e is not very sensitive to R, making it numerically inefficient to take a large value
for R. Because of this, we have performed the numerical averages with this leads to a simple
expression for e, the statistical error on c:
ci
Figure
6.1 shows the dependence of c on 1=N . (The error bars are too small to be visible. Also, in order
to avoid cluttering the figure, we have included among the simulated annealing algorithms only KSA; the
other implementations of simulated annealing give nearly identical results.)
For all algorithms, the figure suggests that there is a limiting large N value for c and that the convergence
to this limit is linear in 1=N . We have thus fitted the data to a linear function:
The values of the A and B coefficients obtained from the fits are given in Table 6.1, and the - 2 values
show that the fits are good.
An identical analysis can be performed on the variance of the cuts found by the different algorithms.
Figure
6.2 shows the dependence on N for the rescaled quantity The scaling in N is apparent,
just as it was for c.
In summary, our data lead us to conclude that the mean and variance of C scale linearly with N
at large N . Then the relative width of the distribution of C is proportional to 1=
showing that the
distribution for the cut sizes becomes peaked for all the algorithms investigated. (One can also say that
the distribution of C(i; m)=N tends towards a delta function as N !1, which is what we mean by self-
averaging.) Since the fluctuations of C(i; m) include both graph to graph fluctuations and fluctuations
G.R. Schreiber and O.C. Martin
CHACO
KSA
KL
CLO
Fig. 6.1. Scaled mean cut sizes for the different algorithms.
algorithm A B % excess
KSA 0.4485 4.95 0.00
ASA 0.4499 4.96 0.32
JSA 0.4513 4.88 0.63
CLO 0.4568 4.85 1.8
CHACO 0.4802 5.81 7.1
KL
Table
Estimates for the large N value and slope of the mean
cut size per vertex and percentage excess relative to the KSA
heuristic.
within a graph, we can conclude that the relative fluctuations within a fixed typical graph necessarily
also go to zero. (N.B.: although for our runs we use our observable is an unbiassed
estimator for
ci
which includes both types of fluctuations.) Thus in the large N limit,
each algorithm will give a fixed percentage excess above the minimum for almost all graphs and almost
all random initial partitions.
A speed independent ranking. Since each algorithm is characterized by a percentage excess, we
can introduce a ranking of the different heuristics according to their excess in the large N limit. (Of
course, this ranking does not take into account the speed of the algorithms!) For our graphs and our
implementation of the different heuristics, the winners are in the class of simulated annealing. The best
is KSA; using this as the reference rather than the true min cut size (which is unknown), JSA has an
excess of 0:63%, ASA an excess of 0:32%, and FSA an excess of 0:08%. The next best heuristic is the
CLO-algorithm, followed by CHACO, and finally KL. (The results for the excesses are given in Table
6.1.) We have also included for general interest the excess obtained by a zero temperature "simulated
annealing": 18:21%; note that it gives much less good results than KL, while true simulated annealing
gives much better results than KL.
As a comment, let us remark that the relative solution quality of the algorithms is determined
to higher precision than the absolute quality. Simply put, the cut sizes we obtain for the different
algorithms are correlated because they are performed on the same graphs, so that the statistical error on
for instance is 3:2 times smaller than the statistical error on
alone. This is
why it is possible to give reliable values for the excesses of the different simulated annealing algorithms
even though their solution quality is very similar. Nevertheless the ranking for the simulated annealing
algorithms is not without ambiguity. The FSA algorithm is, for larger N , within the statistical error of
the KSA algorithm, and hence we have no strong evidence that one is better than the other.
The other algorithms are easily ranked. KL and CHACO are 9:6% and 7:1% worse than KSA, but
CLO is only 1:8% worse. The comparison with KL is qualitatively (though not quantitatively) similar
Cut Size Statistics of Graph Bisection Heuristics 11
KSA
KL
CLO
CHACO
Fig. 6.2. Scaled variance of the cut sizes for the different algorithms.
to that given by Johnson et al. [13] and by van Laarhoven and Aarts [29]. Both claimed a gain of the
SA-algorithm over the KL-algorithm of about 5% and 13%, respectively. The differences with our results
have several origins. First, we have performed an average over an ensemble of graphs. Second, our graphs
have slightly different characteristics from the ones they use. Third we have not introduced a penalty
term in our implementation of simulated annealing; this probably affects the quality of the solutions
found.
7. Distribution of cut sizes. In this section we deepen our statistical study of C. As shown in
the previous section, the distribution of C=N tends towards a delta function; it is natural to ask how this
limit is reached, and to understand the nature of intra- and inter-graph fluctuations. It is convenient to
use the framework introduced in Section 3 but where random partitions are replaced by the partitions
found by applying one of our heuristics to a random start. For each graph G i , and each initial partition
m, we define
where hY (i; so that X(i) is the average cut size found on graph G i , and Y (i; m) gives the
fluctuation of the cut size about its mean for that graph. For each of our heuristics, our study indicates
that for a large random graph G i , Y has a nearly Gaussian distribution, and that the width of this
distribution is essentially independent of i. We study this distribution at large N and show that its width
is self-averaging and that its relative asymmetry goes to zero. Finally, we have evidence that X and Y
become independent variables at large N . These properties will lead to a fast and robust ranking of the
heuristics in Section 8.
Figure
7.1 shows the distribution of cut sizes found by KL on one graph chosen at random
from G(N; p) with 1). Superposed is a Gaussian with the same mean and variance. The
figure gives good evidence that the distribution of Y for that graph is very close to a Gaussian. Then
an obvious question is whether the distribution of Y is similar across different graphs. For each of our
heuristics, we find that the answer is yes, as indicated by the following study of the moments of Y . (Note
that for the CHACO algorithm, the default parameter setting generates the initial starting partition
deterministically by application of the coarse graining strategy, then a spectral method is applied. Since
there is no "random" initial partition, there are no fluctuations in the cut size as a function of m and so
little in this section applies to CHACO with these parameter settings.)
To quantify how oe 2
m)i varies from graph to graph, we measured its mean and variance
over i. First, we measured the ensemble averages
Y (i)
=N . For each heuristic, the data extrapolates
to a limiting value as N becomes large. Comparing with the results for the mean cut size, we find that
the algorithms which lead to the best cut sizes also have the smallest widths for the Y distribution.
Second, we studied the variance of oe 2
Y (i), i.e., oe 2
Y (i)
\Delta . This study requires high statistics, and so
was performed to high accuracy only for KL, the fastest of our algorithms; however the other algorithms
show qualitatively the same behavior. Figure 7.2 displays for KL the 1=N dependence of the relative
variance of oe 2
Y (i), i.e., the inter-graph variance of oe 2
Y (i) divided by the square of its mean. As can be
seen from the figure, the ratio goes to zero at large N , showing that oe 2
Y (i) is self-averaging. Simply put,
G.R. Schreiber and O.C. Martin
frequency
of
cut
sizes
Fig. 7.1. Histogram of KL cut sizes for one graph with overlaid Gaussian.
this means that the width (over m) of the Y distribution has relative fluctuations from graph to graph
which dissapear as N ! 1. (Our lower statistics data for the other heuristics are consistent with this
Y
Fig. 7.2. Relative variance of the intra-graph cut size variance oe 2
Y
Following the statistical physics analogy given in Section 4, there is reason to believe that the distribution
of Y tends towards a Gaussian as in the case of random partitions. To test this conjecture, we
have measured the asymmetry of the distribution of Y on numerous graphs for KL. First, we find that
the typical asymmetry is small, and that the mean of the third moment of Y satisfies
Y (i)
as N !1. Second, we have checked that the average of the squared asymmetry is also small, i.e.,
Y (i)
0:
These properties give strong evidence that the distribution of Y for any graph tends towards a Gaussian
of zero mean and of variance AN as N ! 1, where A depends on the heurisitic but not on the actual
graph.
The distribution of X(i) can be studied similarly. The previous section gave its mean as a function of
N and also showed that it is self-averaging. It is of interest to quantify the decrease with N of its relative
variance. We have found that the distribution of X is roughly compatible with a Gaussian distribution
of width proportional to
N for each of the algorithms. (Unfortunately, a quantitive test of this requires
very high statistics.) However, the distribution of X(i) is not essential for our ranking procedure as will
be clear in the next section, so we have not studied it in greater depth.
Cut Size Statistics of Graph Bisection Heuristics 13
Finally, to completely specify the statistics of C(i; m), it is necessary to describe the correlations
between X(i) and Y (i; m). We have found numerically that these variables are nearly uncorrelated, with
in particular the correlation between X(i) and oe 2
Y (i) tending towards zero as N ! 1. Assuming that
this holds and that X has a Gaussian distribution, then the distribution of C(i; m) is also Gaussian. Our
measurement of the asymetry (jointly over i and m) of C(i; m) is compatible with this property at large
N . (The total variance is then given by the sum of the variances of X and Y .) This can be summarized
mathematically by introducing two Gaussian random variables x and y of zero mean and unit variance,
and modeling the rescaled cut size as the following sum:
m) -
oe
oe
Y
This equation is then the exact analogue of what was derived for the cut sizes of random partitions (see
Eq. 3.5).
8. A speed dependent ranking of heuristics. In this section we come back to the initial motivation
for this work, namely the necessity of comparing heuristics of very different speeds. The possibility
of doing so is very relevant, as for most combinatorial optimization problems local search is quite fast
and simulated annealing notoriously slow. Any meaningful ranking must determine whether it is better
to have a fast heuristic which gives not so good solutions, or a slower heuristic giving better solutions.
We now show how to introduce such a ranking when considering first just one graph, and then generalize
to an ensemble of graphs. Finally, we illustrate what this ranking gives in the case of the heuristics in
our testbed when applied to sparse random graphs.
The case of one graph. Consider a single graph G on which one is to provide a ranking of a number
of heuristics which give various cut sizes and run at different speeds. To take into account both the speed
of the algorithms and the quality of the solutions they generate, we fix the amount of computation time
allotted per algorithm. Call this time - (measured for instance in CPU seconds on a given machine). Each
heuristic then generates (non-optimal) solutions during that time using multiple random initial starts.
Suppose that the speed of the algorithm of interest is such that k independent starts can be performed in
the allotted time - . (We shall assume that the execution time is insensitive to the random initial start,
as this is the case in practice with our heuristics. Knowledge of the speed of the algorithm then gives the
value of k which can be used.) For each start, there is an output or "best-found" cost. The output at
the end of the k starts is the best of these k costs, hereafter called "best-of-k". The different algorithms
are then ranked on the basis of the ensemble mean of their "best-of-k" (the value of k depending on -
and on the algorithm). This ensemble average is the average over the random numbers used both for the
random initial starts and for running the algorithms (if any). This establishes a ranking for a particular
graph and for a given amount of computation time - .
It is inefficient to perform the average just mentioned in a "direct" way, i.e., by extracting values of
"best-of-k" over many multiple runs; it is far better to compute the average starting with the distribution
of the "best-found" cut sizes associated with single random starts. Call P (C) the probability of finding
a "best-found" cut size of value C, and Q(C) the associated cumulative distribution, i.e., the probability
of finding a cut size (strictly) smaller than C. Since the cut sizes are integer valued, we then have
Q(C). Introducing the analogous probabilities ~
for the "best-of-k" values,
one has:
The distribution for "best-of-k" can thus be generated from that of "best-found", and then C , the
mean of "best-of-k", is easily extracted. (This construction explains why we studied the distribution of
single cut sizes in Section 7.) Note also that it is possible to extract C for a whole range of - values
with essentially no extra work since - affects only k and the determination of the mean of "best-of-k"
represents a negligible amount of work once the distribution of "best-found" is known.
The quantity C is in effect a quantitative measure of the effectiveness of the algorithm. Of course,
C depends on the amount of computation ressources allotted, i.e. As - increases, k increases (in
jumps of unity), and C decreases. The broader the distribution of "best-found", the faster the decrease
of C and the more useful it is to perform multiple runs.
To establish the ranking, simply order the algorithms according to their C . In general, this ranking
may depend on - , and clearly it is sensitive to the lower tail of the distribution of "best-found". Let us
14 G.R. Schreiber and O.C. Martin
illustrate this by considering for instance two heuristics H 1 and H 2 having two overlapping distributions
for "best-found", with averages satisfying hC H1 In the mean, H 1 seems better than H 2 , but if
H 2 is significantly faster, and if the tail of its distribution extends well into the domain of CH1 , then one
can have C
H1 . H 2 may then be the more effective algorithm, assuming of course that - is large
enough so that indeed H 2 can be run multiple times. Some general properties may be derived assuming
for instance that CH1 and CH2 are described by the same distribution but are shifted with respect to one
another. Then if the tail of the distribution falls off as an exponential or faster, H 2 will not become more
effective than H 1 as - !1.
Ranking on an ensemble of graphs. The extension of this ranking to an ensemble of graphs is
straight-forward. Assume that C is known for each graph G and for each heuristic. C is a (real number)
measure of the effectiveness of the heuristic on that graph, given an amount of computation time - . We
can then generalize this measure from one graph to an ensemble of graphs by considering
, the mean
of C over the relevant ensemble. The final ranking is then simply given by the ordering of the algorithms
according to their mean effectiveness.
Our expectation is that in a relatively homogeneous ensemble, the effectiveness (and thus the ranking)
will be nearly the same for essentially all sufficiently large graphs and so the average behavior is also the
typical behavior. We can expect this to happen whenever the distribution of cut sizes associated with the
different heuristics do not overlap too much and have the same pattern regardless of the graph. This is
what occurs in the case of our ensemble of random graphs: indeed, we saw that each algorithm leads to a
fixed percentage excess cost at large N and that the distribution of costs is peaked. Then two algorithms
have non overlapping distributions as give rise to the same percentage excess). It
is then clear that at large N , the mean ranking is the same as the typical ranking. It is also clear that
increasing the amount of computer resources (- and thus speeding up an algorithm while keeping
the quality of its solutions the same does very little to improve its ranking.
Illustration. For each value of N and - , we can follow the procedure just given to obtain C for the
different heuristics of interest for any given graph G, and repeat this for many graphs in G(N; p). There
are, however, a number of possible speed-ups in our case because of the statistical properties derived in
the previous sections. First, although in principle the "best-of-k" construction has to be repeated for
each graph, the results of Section 7 provide a short-cut. Since the distribution for "best-found" is (to
high accuracy) Gaussian, it is possible to map the mean of "best-found" to that of "best-of-k" once and
for all: the mapping is just a shift by a k-dependent number of standard deviations. Second, noting that
at fixed N , the variance of this Gaussian as well as the speed of the algorithm is essentially constant from
graph to graph, we can calculate
(the average over graphs) in terms of: (i) the CPU time necessary
to find one "best-found"; (ii) the mean cut size,
(iii) the variance of the intra-graph cut sizes,
m)i, which is graph independent at large N . These quantities were measured for a number of
values of N , and then fits were performed to interpolate to arbitrary values of N . From these fits, it
is possible to compute analytically the values of
for any values of N and - , and in particular the
"winning" algorithm (the first in our ranking). From this, define regions in (N ,- ) space where a given
heuristic is the winner, leading to a "diagram" as in Figure 8.1.
In our construction of this diagram, we have included JSA in our ranking but not FSA, KSA, nor
ASA. This is because for our choice of parameters, all of the simulated annealing algorithms tested give
very similar quality solutions, but JSA is slightly faster. Although the effectiveness of all these SA
algorithms are nearly identical, their ranking depends on N and - because of the discrete jumps in k.
(Whenever one algorithm increases its k before the others, it may change its ranking.) In the diagram of
Figure
8.1, we have labeled the different regions according to the associated "winner", and have indicated
the boundaries separating them. (Again, because of the discrete nature of k, we have smoothed these
curves.) The labeling "SA" in fact corresponds to JSA. The CPU time is expressed in multiples of CPU-
cycles. To give these units a machine independent and less technical meaning, it is enough to say that
the lower boundary of the CHACO region corresponds to the time CHACO needs to run once.
From this diagram, we see that at large N , given enough CPU time, the best algorithm is simulated
annealing, simply because its mean excess cost is lower than that of the other algorithms. In this limit, the
distributions for the cut sizes overlap very little, so the ranking is relatively insensitive to the algorithm's
speed: using multiple random starts does very little to improve the quality of the solutions found as
fluctuations about the mean become negligible. At smaller values of N , the fluctuations arising from
Cut Size Statistics of Graph Bisection Heuristics 150
CLO SACHACO
KL0
CPU
TIME5000Fig. 8.1. Ranking diagram
different random starts are not negligible, so faster algorithms can outperform simulated annealing by
using the best of k runs. If we compare KL, CHACO, and CLO, we see that CLO is a bit slower but
leads to substantially better solutions, and so is the winner if the amount of CPU time is enough for it
to run. The other algorithms are competitive only if neither CLO nor simulated annealing can terminate
a run. This explains why the KL region is nearly invisible, squeezed under the CHACO region, itself
below the CLO and SA region. (Note: (i) on our random graphs, CHACO is slower than KL; (ii) the
initial partition is set deterministically within the default settings of CHACO, so that its "best-found"
and "best-of-k" values are identical.)
9. Discussion and Conclusions. We have studied the statistics of cut sizes generated by graph
partitioning heuristics, both within a given graph and over an ensemble of graphs. Motivated by a
statistical physics analogy and by what happens for random partitions (Section 3), we obtained strong
numerical evidence that the cut sizes generated on sparse random graphs are self-averaging, i.e., that their
distribution becomes peaked as the number of vertices N becomes large. (Quantitatively, this simply
means that the relative fluctuations about the mean tend tend to zero as N ! 1.) For the mean cut
size, we found a linear dependence on N , indicating that each heuristic leads to a fixed percentage excess
cut size above the true minimum. We expect analogous properties to hold for all local heuristics applied
to any combinatorial optimization problem in which each variable is coupled to just a few others.
We also investigated how the distribution of cut sizes approaches its limiting large N behavior, and
gave evidence that on typical graphs the distribution of cut sizes generated becomes Gaussian as N !1.
In that limit, each heuristic is then characterized by a mean cut size (over all graphs) and a variance
describing the fluctuations in the cut sizes on any typical graph. This variance seems to scale linearly
with N in the large N limit and to be self-averaging also.
The principal motivation for this work was to introduce a method to rank heuristics while taking
into account both the quality of the solutions found and the speed of the algorithms. Knowledge of the
distribution of cut sizes allows one to establish a meaningful ranking of the heuristics by assuming that
the algorithms may be applied to k different random starts, with the best of the k runs giving the final
cost. Although this ranking can be done by brute force, we have used the properties just described to
demonstrate it on the heuristics in our testbed. At "large" values of N (N ? 700), the winner is almost
always simulated annealing. In fact, at large N , the distributions associated with the algorithms we have
tested do not overlap significantly, so that the use of multiple runs to explore the tail of the distributions
is not effective. For smaller values of N , the faster algorithms are more competitive, and we find that the
winner is CLO except when the allotted time is too short for running even one run of CLO. Since the
graph to graph fluctuations in the variance of the cut sizes found are small, this ranking "in the mean"
is also in almost all cases the ranking on individual graphs; it is thus very robust.
G.R. Schreiber and O.C. Martin
A number of questions remain open. How can one characterize the distribution of X(i), the mean
cut size on graph i? To what extent do similar properties hold for heuristics which are manifestly not
local? Can the information found help generate better heuristics? Concerning this last question, it is
worth pointing out that although simulated annealing is a general purpose method, it outperforms the
other heuristics which were specifically developped for the graph partitioning problem. This suggests
that some improvements in these methods might be obtainable by suitable modifications.
10.
Acknowledgement
. We are indebted to Bruce Hendrickson and Robert Leland for providing
us with their software package Chaco 2.0. We also thanks S. W. Otto and N. Sourlas for stimulating
discussions. G.R.S. acknowledges support from an Individual EC research grant under contract number
ERBCHBICT941665, and O.C.M. acknowledges support from the Institut Universitaire de France. Fur-
thermore, G.R.S. would like to express his gratitude to Professor J.M. G'omez G'omez for his generous
hospitality at the Department of Theoretical Physics of the Universidad Complutense de Madrid, where
part of this work was accomplished.
--R
Weighted sums of certain dependent random variables
Graph bipartitioning and statistical mechanics
A partitioning strategy for non-uniform problems on multiprocessors
Path optimization for graph partitioning problems.
Replica symmetry breaking in finite connectivity systems: a large connectivity expansion at finite and zero temperature
A procedure for placement of standard-cell VLSI circuits
A linear-time heuristic for improving network partitions
Application of statistical mechanics to NP-complete problems in combinatorial optimization
Computers and Intractability: A Guide to the Theory of NP-Completeness
The Chaco user's guide: Version 2.0
Partitioning of vlsi circuits and systems
Optimization by simulated annealing: An experimental evaluation
The traveling salesman problem: A case study in local optimization
A fast and high quality multilevel scheme for partitioning irregular graphs
Towards a general theory of adaptive walks on rugged landscapes
Some Graph Partitioning Problems Related to Program Segmentation
An efficient heuristic procedure for partitioning graphs
Optimization by Simulated Annealing
an empirical evaluation
An empirical study of static load balancing algorithms
Computer solutions of the traveling salesman problem
Partitioning of unstructured meshes for load balancing
Glass Theory and Beyond
Algorithms and Complexity
A general approach to combinatorial optimization problems
--TR
--CTR
Andrew E. Caldwell , Igor L. Markov, Toward CAD-IP Reuse: A Web Bookshelf of Fundamental Algorithms, IEEE Design & Test, v.19 n.3, p.72-81, May 2002
Angel , Vassilis Zissimopoulos, On the Hardness of the Quadratic Assignment Problem with Metaheuristics, Journal of Heuristics, v.8 n.4, p.399-414, July 2002
Andrew E. Caldwell , Andrew B. Kahng , Andrew A. Kennings , Igor L. Markov, Hypergraph partitioning for VLSI CAD: methodology for heuristic development, experimentation and reporting, Proceedings of the 36th ACM/IEEE conference on Design automation, p.349-354, June 21-25, 1999, New Orleans, Louisiana, United States
Andrew E. Caldwell , Andrew B. Kahng , Igor L. Markov, Design and implementation of move-based heuristics for VLSI hypergraph partitioning, Journal of Experimental Algorithmics (JEA), 5, p.5-es, 2000
Olivier C. Martin , Rmi Monasson , Riccardo Zecchina, Statistical mechanics methods and phase transitions in optimizationproblems, Theoretical Computer Science, v.265 n.1_2, p.3-67, 08/28/2001 | graph partitioning;heuristics;ranking;self-averaging |
589287 | On the Local Convergence of a Predictor-Corrector Method for Semidefinite Programming. | We study the local convergence of a predictor-corrector algorithm for semidefinite programming problems based on the Monteiro--Zhang unified direction whose polynomial convergence was recently established by Monteiro. Under strict complementarity and nondegeneracy assumptions superlinear convergence with Q-order 1.5 is proved if the scaling matrices in the corrector step have bounded condition number. A version of the predictor-corrector algorithm enjoys quadratic convergence if the scaling matrices in both predictor and corrector steps have bounded condition numbers. The latter results apply in particular to algorithms using the Alizadeh--Haeberly--Overton (AHO) direction since there the scaling matrix is the identity matrix. | Introduction
The study of superlinear convergence of interior-point methods for linear programming (LP)
was initiated in the early 90s in an effort to explain the fact that interior point methods
tend to perform significantly better in practice than indicated by the polynomial complexity
bounds. This discrepancy is due to the limitation of the worst case analysis used in deriving
polynomial complexity bounds and reflects the inherent conflict between the requirements
of global convergence and fast local convergence. Superlinear convergence is especially important
for semidefinite programming (SDP) since no finite termination schemes exist for
such problems. As predicted by theory and confirmed by numerical experiments the condition
number of the linear systems defining the search directions increases as 1=-, where
- is the normalized duality gap, so that the respective systems become very ill conditioned
as we approach the solution. Therefore an interior point method that is not superlinearly
convergent is unlikely to obtain high accuracy in practice in spite of its theoretical "polyno-
mial complexity". On the other hand a superlinearly convergent interior point method will
achieve good accuracy (e.g. 10 \Gamma10 or better) in substantially fewer iterations than indicated
by its worse case global linear convergence rate that is related to polynomial complexity.
The local convergence analysis for interior point algorithms for SDP is much more challenging
than those for LP as shown by a relatively smaller number of papers addressing
this subject. The first two papers investigating superlinear convergence of interior point
algorithms were written independently by Kojima, Shida and Shindoh [4] and by Potra and
Sheng [13]. The algorithm investigated in these papers is an extension of Mizuno-Todd-
Ye predictor-corrector algorithm for LP and uses the KSH/HRVW/M search direction (see
the next section for a definition of this search direction). Kojima, Shida and Shindoh [4]
established the superlinear convergence under the following three assumptions:
(A) SDP has a strictly complementary solution;
nondegenerate in the sense that the Jacobian matrix of its KKT system is
(C) the iterates converge tangentially to the central path in the sense that the size of the
neighborhood containing the iterates must approach zero, namely,
lim
Here k:k F denotes the Frobenius norm of a matrix and "ffl" denotes the corresponding
scalar product (see the next section for precise definitions). In [13] we have not used assumptions
(B) and (C). Instead we proposed a sufficient condition for superlinear convergence that
is implied by the above assumptions. In [14] we improved this result and obtained superlinear
convergence under assumption (A) and the following condition:
(D) lim
which is clearly weaker than (C). Of course both (C) and (D) can be enforced by the al-
gorithm, but the practical efficiency of such an approach is questionable. However, from
a theoretical point of view it is proved in [14] that the modified algorithm in [4] that uses
several corrector steps in order to enforce (C) has polynomial complexity and is superlinearly
convergent under assumption (A) only. It is well known that assumption (A) is necessary for
superlinear convergence of standard interior point methods even in the QP case (see [10]).
Kojima, Shida and Shindoh [4] also gave an example suggesting that interior point algorithms
for SDP based on the KSH/HRVW/M search direction are unlikely to be superlinearly
convergent without imposing a condition like (C). In [5] the same authors showed that
a predictor-corrector algorithm using the AHO direction is quadratically convergent under
assumptions (A) and (B) (see the next section for a definition of the AHO search direction).
They also proved that the algorithm is globally convergent but no polynomial complexity
bounds have been found for this algorithm. It is shown that condition (C) is automatically
satisfied by the iteration sequence generated by the algorithm. It appears that the use of
the AHO direction in the corrector step has a strong effect on centering. We exploited this
property in [15] where we showed that a direct extension of Mizuno-Todd-Ye algorithm,
based on the KSH/HRVW/M direction in the predictor step and the AHO direction in the
corrector step, has polynomial complexity and is superlinearly convergent with Q-order 1:5
under assumptions (A) and (B).
An interesting superlinearly convergent predictor-corrector algorithm based on the NT
search direction was proposed by Luo, Sturm and Zhang [7]. The algorithm depends on a
parameter ffl ? 0. It produces points (X
is defined in (2.7), ffl=4. The
algorithm starts from a feasible point (X and for any given ~ ffl - ffl=4
finds a feasible point (X in at most O(
iterations. However
this bound on the number of iterations is not proved to hold for hence the
algorithm is not polynomial in the usual sense. The algorithm is superlinearly convergent
under assumption (A). It turns out that (C) is enforced by the algorithm since it is proved
in [7] that for sufficiently large k
It is also proved that if one uses one predictor and r correctors per iteration, then - k converges
to zero with Q-order 2=(1
In this paper we investigate the local behavior of the predictor-corrector algorithm considered
by Monteiro [9] for SDP using the MZ-family of search directions. We show that
the sufficient condition of Potra and Sheng [13] for superlinear convergence applies for this
algorithm. The sufficient condition is independent of scaling matrices. In particular we show
that the algorithm is superlinearly convergent if (A) and (D) are satisfied. More specifically,
we show that under the assumptions (A) and (B), superlinear convergence with Q-order
1.5 is obtained if the scaling matrices in the corrector step have bounded condition num-
ber. Finally, we propose a new version of the predictor-corrector algorithm which enjoys
quadratic convergence if the scaling matrices in both predictor and corrector steps have
bounded condition numbers and (A) and (B) are satisfied.
The following notation and terminology are used throughout the paper:
the p-dimensional Euclidean space;
nonnegative orthant of IR
the positive orthant of IR
the set of all p \Theta q matrices with real entries;
the set of all p \Theta p symmetric matrices;
: the set of all p \Theta p symmetric positive semidefinite matrices;
: the set of all p \Theta p symmetric positive matrices;
the (i; j)-th entry of a matrix M;
Tr(M the trace of a p \Theta p matrix, equals
0: M is positive semidefinite;
0: M is positive definite;
n: the eigenvalues of M 2 S
the largest, smallest, eigenvalue of M 2 S
Euclidean norm of a vector and the corresponding norm of a matrix, i.e.,
Frobenius norm of a matrix;
k(G;
G;
2 The predictor-corrector algorithm for SDP
We consider the semidefinite programming (SDP) problem:
and its associated dual problem:
are given data, and
are the primal and dual variables, respectively. By G ffl H we
denote the trace of (G T H). Also, for simplicity we assume that A i are linearly
independent.
Throughout this paper we assume that both (2.1) and (2.2) have finite solutions and
their optimal values are equal. Under this assumption, X and (y ; S ) are solutions of (2.1)
and (2.2) if and only if they are solutions of the following nonlinear system:
We denote the feasible set of the problem (2.3) by
and its solution set by F , i.e.,
We consider the symmetrization operator [17]
Since, as observed by Zhang [17],
for any nonsingular matrix P , any matrix M with real spectrum, and any - 2 IR, it follows
that for any given nonsingular matrix P , (2.3) is equivalent to
A perturbed Newton method applied to the system (2.4) leads to the following linear system:
m \Theta S n is the unknown search direction, - 2 [0; 1] is the centering
parameter, and is the normalized duality gap corresponding to (X;
The search direction obtained through (2.5) is called the Monteiro-Zhang (MZ) unified
direction [17, 11]. The matrix P used in (2.5) is called the scaling matrix for the search
direction. It is well known that taking I results in the Alizadeh-Haeberly-Overton
(AHO) search direction [1], corresponds to the Kojima-Shindoh-Hara/Helmberg-
Rendl-Vanderbei-Wolkowicz/Monteiro (KSH/HRVW/M) search direction [6, 3, 8], and the
case of P T coincides with the Nesterov-Todd (NT) search
direction [12]. Monteiro and Zhang [11] established the polynomiality of a long-step path-following
method based on search directions defined by scaling matrices belonging to the
class
such that
Following [11], Sheng et al. [16] proved the polynomiality of a Mizuno-Todd-Ye type
predictor-corrector algorithm for SDP by imposing the scaling matrices to be chosen from
the class
n\Thetan is nonsingular and PXSP
Moreover, its superlinear convergence was proved under an addtional simple condition. The
primal-dual algorithms considered by Monteiro [9] are based on the centrality measure
\Theta S n
1), we denote by
N (fl) the following neighborhood of the central path:
Monteiro's generalized predictor-corrector algorithm for semidefinite programming based on
the MZ family of directions consists of a predictor step and a corrector step at each iteration.
Starting from a strictly feasible pair (X generates a sequence of iterates
in N (ff). An iteration of Monteiro's generalized predictor-corrector algorithm
can be described as follows.
Algorithm
Given choose nonsingular n \Theta n matrices P k and P k
ffl Predictor Step. Solve the system (2.5) with (X;
Denote the solution (U;
m \Theta S n , and set
Compute the step length
ffl Corrector Step. Solve the system (2.5) with (X;
and be the solution, and set
End of iteration.
Using an elegant analysis, Monteiro [9] proved that the predictor-corrector algorithm
defined above with properly chosen parameters ff and fi (0 well defined and
that it needs at most O(
iterations for producing a pair (X
is the initial gap. More precisely, Monteiro showed that
for all k - 0.
3 Technical results
In analyzing the local behavior of the predictor-corrector algorithm of Monteiro, we need
the following technical result proved in [8, Lemma 2.6] and [9, Lemma 2.1(b)].
Lemma 3.1 Suppose that M 2 IR p\Thetap is a nonsingular matrix and E 2 IR p\Thetap has at least
one real eigenvalue. Then,
The following lemma is part of Lemma 3.5 of Monteiro [9].
Lemma 3.2 Let W 2 IR
n\Thetan be such that GWG \Gamma1 is skew-symmetric for some nonsingular
n\Thetan . Then,
The following technical result will play an important role in our analysis.
Lemma 3.3 Let (X;
1). Suppose that (D x ; \Deltay; D s
n\Thetan \Theta IR m \Theta S n\Thetan is a solution of the linear system:
\Deltay
for some K 2 IR n\Thetan . Then we have
F ,
where
Proof. By denoting
we can write
and
It is easily seen that -
and
Using the notation
it follows that
Using (3.8) and Lemma 3.2 with
On the other hand, using (3.8) again, we obtain
\GammakX \Gamma1=2 D x X \Gamma1=2 (X 1=2 SX
s
s
s
which implies (i). Then (ii) follows from (i), (3.9), and the fact that
It is interesting to note that the inequalities in the above lemma are independent of the
scaling matrix P . In the next lemma we establish a lower bound for the stepsize ' k , which
together with Lemma 3.3 enables us to analyze the asymptotic behavior of the predictor-corrector
algorithm.
be generated by the predictor-corrector algorithm.
Then
where
Proof. For simplicity, let us omit the index k. By (2.8), we have
which together with the linearity of H P (\Delta), the fact that T r[H P (M
and (2.5a) with imply that
Using the fact that U ffl
Therefore,
and
Hence, we have X(') - 0 and S(') - 0 for all ' 2 [0; -
']. Otherwise, there exists a ' 0 2 [0; -
such that X(' 0 )S(' 0 ) is singular, which means
On the other hand, (3.2) with implies that
which contradicts (3.10). Using (3.4) with
Therefore,
'. The result follows from the definition of '.
4 A sufficient condition for superlinear convergence
In this section we will investigate the asymptotic behavior of the predictor-corrector algorithm
and obtain a sufficient condition for superlinear convergence.
Definition 4.1 A triple (X ; y is called a strictly complementary solution of
Throughout the paper we assume that the following condition holds.
Assumption 1. The SDP problem has a strictly complementary solution (X
be an orthogonal matrix such that q are eigenvectors of X
and S , and define
It is easily seen that IB [ ng. For simplicity, let us assume that
where B and N are diagonal matrices. Here and in the sequel, if we write a matrix M in
the block form
then we assume that the dimensions of M 11 and M 22 are jIBj \Theta jIBj and jINj \Theta jINj, respectively.
In the next lemma we use the following notation:
Lemma 4.2 (Potra-Sheng [13, Lemma 4.4]) Under Assumption 1 we have
ks
ks
Using Lemma 4.3, we can write
O(
O(
O(
Using the same techniques, we obtain a similar result for the predicted pair (X k
Lemma 4.3 Let X Assumption 1 is satisfied, then we have
O(
O(
O(
As in [13], let us define a linear manifold:
It is easily seen that if (X
Lemma 4.4 (Potra-Sheng [13, Lemma 4.5]) Under Assumption 1, F ae M.
Lemma 4.5 (Potra-Sheng [13, Lemma 4.6]) Under Assumption 1, every accumulation point
of strictly complementary solution of (2.3).
Let us define
is the solution of the following minimization problem:
and \Gamma is a constant such that k(X k ; S k )k F - \Gamma; 8k. Note that every accumulation point of
belongs to the feasible set of the above minimization problem and the feasible
set is bounded. Therefore ( -
exists for each k.
Theorem 4.6 Under Assumption 1, if then the predictor-corrector
algorithm is superlinearly convergent. Moreover, if there exists a constant oe ? 0 such that
then the convergence has Q-order at least 1+oe in the sense that -
Proof. For simplicity, let us omit the index k. It is easily seen that (U
Here we have used the relation -
clearly satisfies the equation
Denoting
and applying (i) of Lemma 3.3, we obtain
which implies
Similarly,
By Lemma 4.3 and the fact that ( -
In a similar manner we obtain
Let us observe that
Then from (4.6), (4.7), (4.8), (4.9) and (4.10), we get
Hence, Applying (ii) of Lemma 3.3, we obtain
Noting that
we deduce
Finally, if
k ) for some constant oe ? 0, then we have
k ). From Lemma 3.4, it follows that
Therefore,
Lemma 4.6 was originally obtained by Potra and Sheng [14]. Based on Lemma 4.6, we
establish the following generalization of the result of Potra and Sheng [14, Theorem 6.1].
Theorem 4.7 Under Assumption 1, if X k S
!1, then the predictor-corrector
algorithm is superlinearly convergent. Moreover, if X k S
constant oe ? 0, then the convergence has Q-order at least 1 0:5g.
5 Superlinear convergence under strict complementarity
and nondegeneracy
Throughout this section, we will assume that Assumption 1 (strict complementarity) holds.
Let (X ; y ; S ) be a strictly complementary solution of (2.1) and (2.2). We will also assume
the following nondegeneracy condition introduced by Kojima, Shida and Shindoh [4, 5].
First, let us define an affine space G 0 by
Assumption 2. (Nondegeneracy) If X
As remarked in Section 5 of Kojima, Shida and Shindoh [5], under the strict complementarity
assumption, the above nondegeneracy condition is equivalent to the combination of
primal and dual nondegeneracy conditions given by Alizadeh, Haeberly and Overton [2].
Under Assumptions 1 and 2, the solution (X ; S ) is unique. Therefore the iteration
sequence
converges to (X ; S ) and so does the sequence of predicted pairs
Lemma 5.1 (Kojima-Shida-Shindoh [5], Lemma 5.3) Assume that
H I (US +X V
Let R be a nonsingular matrix and
~
(R
It is easily seen that the R-scaled SDP
~
~
also satisfies the strict complementarity and nondegeneracy conditions. Its unique solution
is (RX R T (R
Using Lemma 5.1 and considering the new SDP (5.1), we can easily obtain the following
lemma.
Lemma 5.2 Assume that for some nonsingular matrix R,
In the next lemma cond F denotes the condition number of a matrix B.
Lemma 5.3 If cond F (P k
Proof. Let R
the corrector step of the algorithm, we have
U
it is easily seen that
Suppose (5.2) is not true, i.e., the sequence
is unbounded. Then we can
choose a subsequence such that
(R
and
Obviously, (U . The fact that the matrices A are linearly indepen-
dent, together with (U implies that (U Dividing both sides of (5.3) by
letting k !1 along a subsequence, we obtain
which contradicts Lemma 5.2.
Theorem 5.4 Under the strict complementarity and nondegeneracy assumptions, if cond F (P k
O(1), then the algorithm is superlinearly convergent with Q-order at least 1.5.
Proof. At the predictor step, we have
Thus,
Then, by Lemma 5.3, we obtain
Note that
F
F
Therefore,
which ends the proof by invoking Theorem 4.7.
The above result says that the superlinear convergence of the predictor-corrector algorithm
is independent of the choice of the scaling matrix P k in the predictor step of the
algorithm, while the scaling matrices used in the corrector step need to be "well-conditioned"
for superlinear convergence. Clearly, the family of scaling matrices admissible in the corrector
step for superlinear convergence includes the identity matrix defining the AHO as a
special case. By imposing the same assumption on the scaling matrices used in the predictor
step and a new strategy for the step size, we can improve the order of convergence stated in
Theorem 5.4.
In order to achieve quadratic convergence we need to slightly modify the choice of the
step size. Instead of ' k given by (2.9), we will use:
The predictor-corrector algorithm with this new strategy will be called the modified predictor-corrector
algorithm. It is easily seen that the modified predictor-corrector algorithm still has
polynomial complexity. In what follows we will show that it is also quadratically convergent.
Theorem 5.5 Under the hypothesis of Theorem 5.4, if cond F (P k then the modified
predictor-corrector algorithm is quadratically convergent.
Proof. From the proof of Theorem 5.4 (cf. (5.5)), we have
Using (5.7) and an argument similar to that employed in the proof of Lemma 5.3 we get
Then we can write
g. As in (5.4)-(5.5), we can prove that
Using (5.9) and the same argument as in Lemma 5.3, we get
Observing that
we have
where C 2 is a positive constant. Without loss of generality, we may assume
which, together with (5.6) and (5.13), implies that
and
Let
Evidently,
k ], we have
This means
Therefore
and -
6 Remarks
In this paper we only consider the feasible version of the predictor-corrector method to
keep the presentation simple. However, the analysis used here can be easily extended to the
infeasible predictor-corrector algorithms based on the unified direction proposed by Monteiro
and Zhang. Under the strict complementarity and nondegeneracy assumptions we have
established the superlinear convergence with Q-order 1.5 of the "pure" predictor-corrector
algorithm if the scaling matrices for the corrector step satisfy cond F (P k
superlinear convergence can be obtained under a weaker condition is an interesting topic
for future research. Finally, we mention that quadratic convergence is established for the
predictor-corrector algorithm with a slight modification of the step size selection. It would
be interesting to find out whether quadratic convergence can be proved for the "original"
predictor-corrector algorithm.
--R
Complementarity and nondegeneracy in semidefinite programming.
An interior-point method for semidefinite programming
Local convergence of predictor-corrector infeasiblee-interior-point algorithms for semidefinite programs
A predictor-corrector interior-point algorithm for the semidefinite linear complementarity problem using the Alizadeh-Haeberly-Overton search direction
Superlinear convergence of a symmetric primal-dual path following algorithm for semidefinite programming
Polynomial convergence of primal-dual algorithms for semidefinite programming based on Monteiro and Zhang family of directions
Local convergence of interior-point algorithms for degenerate monotone LCP
A unified analysis for a class of path-following primal-dual interior-point algorithms for semidefinite programming
A superlinearly convergent primal-dual infeasible-interior- point algorithm for semidefinite programming
Superlinear convergence of interior-point algorithms for semidefinite programming
Superlinear convergence of a predictor-corrector method for semidefinite programming without shrinking central path neighborhood
On a general class of interior-point algorithms for semidefinite programming with polynomial complexity and superlinear convergence
On extending primal-dual interior-point algorithms from linear programming to semidefinite programming
--TR
--CTR
Y. B. Zhao, Enlarging neighborhoods of interior-point algorithms for linear programming via least values of proximity measure functions, Applied Numerical Mathematics, v.57 n.9, p.1033-1049, September, 2007 | interior point method;superlinear convergence;semidefinite programming |
589294 | A Spectral Bundle Method for Semidefinite Programming. | A central drawback of primal-dual interior point methods for semidefinite programs is their lack of ability to exploit problem structure in cost and coefficient matrices. This restricts applicability to problems of small dimension. Typically, semidefinite relaxations arising in combinatorial applications have sparse and well-structured cost and coefficient matrices of huge order. We present a method that allows us to compute acceptable approximations to the optimal solution of large problems within reasonable time.Semidefinite programming problems with constant trace on the primal feasible set are equivalent to eigenvalue optimization problems. These are convex nonsmooth programming problems and can be solved by bundle methods. We propose replacing the traditional polyhedral cutting plane model constructed from subgradient information by a semidefinite model that is tailored to eigenvalue problems. Convergence follows from the traditional approach but a proof is included for completeness. We present numerical examples demonstrating the efficiency of the approach on combinatorial examples. | Introduction
. The development of interior point methods for semidefinite
programming [19, 31, 1, 46] has increased interest in semidefinite modeling techniques
in several fields such as control theory, eigenvalue optimization, and combinatorial
optimization. In fact, interior point methods proved to be very useful and reliable
solution methods for semidefinite programs of moderate size. However, if the problem
is defined over large matrix variables or a huge number of constraints interior point
methods grow terribly slow and consume huge amounts of memory. The most efficient
methods of today [15, 23, 2, 32, 45, 29] are primal-dual methods that require, in each
iteration of the interior point method, the factorization of a dense matrix of order
equal to the number of constraints and one to three factorizations of the positive
semidefinite matrix variables within the line search. For a typical workstation this
restricts the number of constraints to 2000 and the size of the matrix variables to
500 if reasonable performance is required. For larger problems time and memory
requirements are prohibitive. It is important to realize that either the primal or the
dual matrix is generically dense even if cost and coefficient matrices are very sparse.
Very recently, a pure dual approach was proposed in [4] which offers some possibilities
to exploit sparsity. It is too early to judge the potential of this method.
In combinatorial optimization semidefinite relaxations where introduced in [27].
At that time they were mainly considered a theoretical tool for obtaining strong
bounds [11, 28, 40]. With the development of interior point methods hopes soared
high that these relaxations could be of practical value. Within short time several
approximation algorithms relying on semidefinite programming were published, most
of them based on the approach by Goemans and Williamson [8]. On the implementational
side [14, 16, 20] cutting plane approaches for semidefinite relaxations of
Konrad-Zuse-Zentrum fur Informationstechnik Berlin, Takustrae 7, 14195 Berlin, Germany.
helmberg@zib.de, http://www.zib.de/helmberg
y Universitat Klagenfurt, Institut f. Mathematik, Universitatsstr. 65-67, A 9020 Klagenfurt, Aus-
tria. franz.rendl@uni-klu.ac.at. Financial support through the Austrian FWF Project P12660-
MAT is greatfully acknowledged.
constrained quadratic 0-1 programming problems proved to yield solutions of high
quality. However, as mentioned above, they were very expensive to compute even for
problems of small size (a few hundred 0-1 variables). Problems arising in practical
applications (starting with a few thousand 0-1 variables) were out of reach. We believe
that the method proposed in this paper will open the door to problems of this
size.
Although combinatorial applications are our primary concern we stress that the
method is not restricted to this kind of problems. In fact it will be a useful alternative
to interior point methods whenever the number of constraints or the order of the
matrices is quite large.
We transform a standard dual semidefinite program into an eigenvalue optimization
problem by reformulating the semidefinite constraint as a non-negativity constraint
on the minimal eigenvalue of the slack matrix variable and lifting this constraint
into the cost function by means of a Lagrange multiplier. The correct value
of the Lagrange multiplier is known in advance if the primal feasible matrices have
constant trace. (This is the case for the combinatorial applications we have in mind.)
In this paper we develop a bundle method for solving the problem of minimizing
the maximal eigenvalue of an affine matrix function with an additional linear objective
term. These functions are well known to be convex and non-smooth. A very general
method for optimizing non-smooth convex functions is the bundle method, see e.g.
[21, 42, 17, 18]. In each step the function value and a subgradient of the function is
computed for some specific point. By means of the collected subgradients a cutting
plane model of the function is formed. The minimizer of the cutting plane model
augmented by a regularization term yields the new point. In the case of eigenvalue
optimization the subgradient is formed by means of an eigenvector to the maximal
eigenvalue. Extremal eigenvalues and associated eigenvectors of large symmetric matrices
can be computed efficiently by Lanczos methods (see e.g. [9]). Lanczos methods
need a subroutine that computes the product of the matrix with a vector. This allows
to exploit any kind of structure present in the matrix.
The polyhedral cutting plane model used in traditional bundle algorithms is up-dated
by new subgradient information such as to approximate well the subdifferential,
and thus the function itself, in the vicinity of the current point. For eigenvalue optimization
problems the subdifferential is generated by a semidefinite set, in particular
by the intersection of a simple affine constraint and a face of the semidefinite cone.
This suggests to use, instead of the traditional polyhedral cutting plane model, a
semidefinite cutting plane model that works with an approximation of this face of
the semidefinite cone. This specialization of the cutting plane model is the main
contribution of the paper.
The semidefinite bundle approach allows for an intriguing interpretation in terms
of the original semidefinite program. The cutting plane model requires that the dual
slack matrix of the semidefinite program is positive semidefinite only with respect to a
subspace of vectors, thus it may be interpreted as a relaxation of the dual semidefinite
program. In general the optimal solution of this relaxed semidefinite problem will
produce an indefinite dual slack matrix. One or more of the negative eigenvalues and
corresponding eigenvectors of the slack matrix are used to update the subspace in
order to improve the relaxation and the process is iterated.
This process trivially provides the optimal solution if the subspace grows to the full
space. However, we show that during the algorithm generically the dimension of the
subspace is bounded by (roughly) the square root of the number of constraints. If this
A SPECTRAL BUNDLE METHOD 3
is still considered too large the introduction of an aggregate subgradient guarantees
convergence for restricted bundle sizes. In the extreme the bundle may consist of one
new eigenvector to the maximal eigenvalue only.
In contrast, the 'classical' algorithms of Cullum, Donath, and Wolfe [6] and Polak
and Wardi [38] require in each iteration the computation of all eigenvectors to
eigenvalues within an "-distance of the maximal eigenvalue, thus close to the optimal
solution this number is at least as large as the multiplicity of the maximal eigenvalue
in the optimal solution. In the quadratically convergent algorithm of Overton [35]
each step is computed from a complete spectral decomposition of the matrix and a
guess of the exact multiplicity of the maximal eigenvalue in the optimal solution. In
recent work [33, 34] Oustry reinterprets the algorithm of Overton within the frame-work
of the U-Lagrangian introduced in [26] and embeds it in a first order method to
ensure global convergence. Again, for global convergence the approach relies on the
spectrum of all eigenvalues within "-distance of the maximal eigenvalue and makes
use of the entire spectral information to obtain local quadratic convergence.
Because of the restricted bundle size quadratic convergence is out of reach for
our algorithm, it is a first order method only. In principle convergence follows from
the traditional approach (see e.g. [21]) but we include a proof for completeness. We
also present a primal-dual interior point code for solving the quadratic semidefinite
programming problems associated with the semidefinite cutting plane models and
discuss efficiency aspects. The properties of the algorithm are illustrated on several
combinatorial examples.
In x2 some basic properties of semidefinite programs are stated. Then we transform
semidefinite programs into eigenvalue optimization problems. Section 3 introduces
the bundle method. The algorithm and the proof of convergence is given in
x4. The quadratic semidefinite subproblems arising in the bundle method can be
solved by interior point methods as explained in x5. Section 6 gives an outline of
the implementation and briefly discusses the computation of the maximal eigenvalue
and an associated eigenvector. Numerical examples on combinatorial problems are
presented in x7. We conclude the paper with a summary and possible extensions and
improvements in x8. For the convenience of the reader an appendix explaining the
notation and the symmetric Kronecker product is included at the end of the paper.
2. Semidefinite programs and eigenvalue optimization. We denote the
set of symmetric matrices of order n by Sn which we regard as a space isomorphic
to R ( n+1
As scalar product of A; B 2 Sn (or more general,
use the trace is the sum of the diagonal elements of a
square matrix. We will often use the same symbol for the canonical scalar product
of vectors a; b a, the appropriate space will be clear from the
context. The subset of positive semidefinite matrices S
n is a full-dimensional, non-
polyhedral convex cone in Sn and defines a partial order on the symmetric matrices
by A B
n . Positive definite matrices are denoted by S ++
n or
A 0.
Consider the standard primal-dual pair of semidefinite programs,
(D)
Z 0:
linear operator and A T its adjoint operator, defined
by hAX; yi
ff for all X 2 Sn and y They are of the form
with A i 2 Sn , is the cost matrix, b 2 R m the right-hand-side
vector.
We assume some constraint qualification to hold, so that these problems satisfy
strong duality in the sense that for any optimal solution X of (P) and any optimal
solution (y ; Z ) of (D) we have
The following assumption allows a simple reformulation of the dual (D) as an
eigenvalue optimization problem. We assume that
for some constant a ? 0. In this case we can add tr a as a redundant constraint
to the primal problem and obtain the following dual equivalent to (D)
Now a ? 0 implies X 6= 0 at the optimum, hence any optimal Z of this dual is
singular. Therefore all dual optimal solutions Z satisfy leading to
Thus we have shown that (D) is equivalent to min y amax
convenience we assume deal with the following problem.
The eigenvalue problem (E) is a convex, non-smooth optimization problem. It is well
studied in the literature. Here we only recall some basic facts. The function
is differentiable if and only if the maximal eigenvalue has multiplicity one. When
optimizing eigenvalue functions, the optimum is generically attained at matrices whose
maximal eigenvalue has multiplicity larger than one. In this case one has to consider
the subdifferential of max at X ,
(see e.g. [35]). In particular, for any v 2 R
n belonging to the eigenspace of the
maximal eigenvalue of X , contained in the subdifferential of max at X .
For the function of interest,
A SPECTRAL BUNDLE METHOD 5
the subdifferential of f at y can be derived by standard rules (see [17]),
Observe that the set of all subgradients is bounded.
Remark 2.1. Even though our assumption (2.2) might look artificial, it does
hold for SDP arising from quadratic 0-1 optimization. It also holds for many other
SDP derived as relaxations of combinatorial optimization problems, see for instance
[1, 12, 24].
3. The bundle method. In this section we develop a new method for minimizing
f . We use two classical ingredients, the proximal point idea, and the bundle
concept. The new contribution lies in the way that we derive the new iterate from
the 'bundle' of subgradient information collected from previous iterates. Since our
approach builds on several subtle ideas, we proceed in small steps and explain first,
how we derive a minorant of f from local information.
3.1. Minorizing f by "
f. Our first goal is to obtain a minorant "
f of f which
approximates f in the neighborhood of the current iterates reasonably well, and which
is easier to handle than f . Introducing the function
we can express f(y) as
This formulation shows that lower approximations of f can be obtained by constraining
W to a subset of all semidefinite matrices with tr
We propose the following choice for this subset. Let P be n \Theta r with P T
n with tr be two matrices. We restrict W to be contained in the
set
c
The
f , defined through P and W , now reads
Wg:
By definition, we have "
f(y) f(y) 8y: If, for some "
W for some
eigenvector v to
f("y). This is e.g. the case if v is a
column of P or v is contained in the range space of P .
The intuitive idea behind our specific choice of c
W is as follows: the matrix P
contains subgradient information from the current point "
y, and perhaps from previous
iterates. We explain below in detail, how we propose to select and update the
matrix P . For computational efficiency, we would like to keep the number r of columns
of P small, independent of the multiplicity of the largest eigenvalue. Therefore we
collect indispensable subgradient information, that has to be removed from P , in an
aggregate subgradient. This aggregation is the final ingredient of our local model of f .
The matrix W plays the role of an aggregate subgradient. Again, we will discuss be-
low, how W is updated during the algorithm. The main point here is that instead of
optimizing over all semidefinite matrices W , we constrain ourselves to a small subset.
Remark 3.1. If we set use for the matrix P a set of eigenvectors
to the r largest eigenvalues at "
y, we would end up with a model closely related to the
approach from [6]. In this case it would be important to select r at least as large as the
multiplicity of the largest eigenvalue. In our present approach this is not necessary.
6 C. HELMBERG AND F. RENDL
3.2. Proximal point idea. The next goal is to minimize "
f instead of f . Since
f is built from local information from a few previous iterates, this model function is
unlikely to be reliable for points far from the current iterate. Therefore we use the
proximal point idea and add a penalty term for the displacement from the current
point. Thus we determine a new candidate y from the current iterate "
y by solving the
following convex problem, referred to as the augmented model. (Here u ? 0 is some
fixed real weight.)
min y
We note that this minimization problem corresponds to the Lagrangian relaxation of
Thus we replace the original function f by its minorant "
f
and minimize locally around "
y. The weight u controls (indirectly) the radius s of
the sphere around " y, over which we minimize. Substituting the definition of "
f , this
problem is the same as
min y
This problem can be simplified, because y is unconstrained. Note that
Therefore we obtain
min y
W2c W; b\GammaAW +u(y\Gamma"y)=0
W2c
The first equality follows from interchanging min and max (see Corollary 37.3.2 of
[41]) and using first order optimality for the inner minimization with respect to y,
The final problem is a semidefinite program with (concave) quadratic cost function.
We will discuss in x5 how problems of this kind can be solved efficiently. Its optimal
solution W k+1 gives the new trial point y by (3.3).
Remark 3.2. The choice of the weight u is somewhat of an art. There are several
clever update strategies published in the literature, see for instance [21, 42].
3.3. One iteration of the algorithm. The main ingredients of our approach
have now been explained, so we can give a formal description of a general iteration
k of the algorithm. To be consistent with the notation of the algorithm given in x4,
let us denote by x k what was called "
y in x3.2. The algorithm may have to compute
several trial points y k+1 , y keeping the same x progress
is not considered satisfactory (null step). For each y k+1 the function is evaluated and
a subgradient (eigenvector) is computed. This information is added to c
W k to form
an improved model c
W k+1 . Therefore, we assume that the current 'bundle'
A SPECTRAL BUNDLE METHOD 7
contains an eigenvector of its span (y k may or may not be equal
to x k ). Other than that, P need only The minorant of f in
iteration k is denoted by "
Here c
represents the current approximation to the set of all semidefinite matrices
of trace one, see (3.1). It will be convenient to introduce also the regularized version
of "
The new trial point y k+1 is obtained by minimizing f k (y) with respect to y. As
described above, this can be done as follows. First, solve by interior point methods
(see x5)
yielding a (not necessarily unique) maximizer W
use (3.3) to compute
To finish an iteration, we have to decide whether enough progress is made to perform
a serious step or not, i.e. whether we are going to set x
how to update P k and W k
If P k does not yet use the maximum number of columns allowed then the update
process is simple: orthogonalize the new eigenvector with respect to P k , add it as a
new column to form P k+1 and continue. In general, however, P k will already use the
maximum number of columns and so we have to make room for the new subgradient
information. Instead of simply eliminating some columns of P k we can do better by
exploiting the information available in ff and V .
Let Q\LambdaQ T be an eigenvalue decomposition of V . Then the 'important' part of
the spectrum of W k+1 (the important subspace within the space spanned by P k ) is
spanned by the eigenvectors associated with the 'large' eigenvalues of V . Thus we
split the eigenvectors of Q into two parts (with corresponding spectra 1
and 2 containing as columns the eigenvectors associated to 'large' eigenvalues
of V and Q 2 containing the remaining columns,
Now the next P k+1 is computed to contain P k Q 1 and at least one eigenvector v k+1
to the maximal eigenvalue of
(The operator orth(.) indicates that we take an orthonormal basis of [P k
The next aggregate matrix is built in such a way that W k+1 2 c
contains only the important part of P k , given by P k Q 1 , we include the remaining
part of P k , given by P k Q 2 in W k+1
(ff W k
Note that W k+1 is scaled to have trace equal one.
Proposition 3.3. Update rules (3.7) and (3.8) ensure that W k+1 2 c
W k+1 .
Proof. Let W k+1 be of the form (3.6). By (3.7) there is an orthonormal matrix
such that P k+1
W k+1 .
We summarize some easy facts, which will be used in the convergence analysis of
the algorithm.
since y k+1 is minimizer of f k . Because f k
Next let
Using the definition of y k+1 from (3.5) it follows easily that
(y
the augmented model of the next iteration will satisfy
(y) 8y:
Remark 3.4. While the choice for the update of P k is fairly natural, we could use
other update formulas, such as W . The main properties
guiding the update are that W k+1 2 c
ensuring (3.11) and that in y k+1 the model
is now supported by a subgradient of f pushing the model towards f in the vicinity of
the last minimizer.
4. Algorithm and convergence analysis. In the previous section we focused
on the question of doing one iteration of the bundle method. Now we provide a formal
description of the method and point out that except for the choice of the bundle, the
nature of the subproblem, and some minor changes in parameters the algorithm and
its proof are identical to the algorithm of Kiwiel as presented in [21]. To keep the
paper self-contained we present and analyze a simplified variant for fixed u. We refer
the reader to [21] for an algorithm with variable choice of u.
Algorithm 4.1.
Input: An initial point y to the maximal
eigenvalue of C \Gamma A T y 0 , an " ? 0 for termination, an improvement parameter mL 2
an upper bound R 1 on the number of columns of P .
1.
2. (Direction finding) Solve (3.4) to get y k+1 from (3.5). Decompose V into
using
(3.8).
3. (Evaluation) Compute and an eigenvector v k+1 . Compute
P k+1 by (3.7).
4. (Termination) If f(x k
A SPECTRAL BUNDLE METHOD 9
5. (Serious step) If
then set x continue with Step 7. Otherwise continue with Step 6.
6. (Null step) Set x
7. Increase k by 1 and go to Step 2.
We prove convergence of the algorithm for If the algorithm stops after a
finite number of iterations then by
and thus by (3.5) 0 2 @f(x k ), so x k is optimal. Assume in the following that the
algorithm does not stop. First consider the case that only null steps occur after some
iteration K.
Lemma 4.2. If there is a K 0 such that (4.1) is violated for all k K, then
Proof. For convenience we set Using the relations (3.10),
and (3.9), we obtain for all k K
(y
Therefore the f k (y k+1 ) converge to some f f(x) and
the computed gradient of f in y k+1 and observe
that the linearization
f of f in y k+1
ff
W k+1 . Thus
\Gamma\Omega
ff
The convergence of the f k (y k+1 ), the boundedness of the gradients and the fact that
imply that the last term goes to zero for k !1. So for all
there is an M 2 N such that for all k ? M
where '!' follows from (4.1) being violated for all k ? K. Thus the sequences f(y k+1 )
both converge to f(x). y k+1 is the minimizer of the regularized function
f k . On the one hand this implies that y k+1 ! x. On the other hand 0 must
be contained in the subgradient @f k (y k+1
Therefore there is a sequence h k 2 @ "
subgradients converging to zero. The
converge to f(x) and the y k+1 converge to x, hence zero
must be contained in @f(x).
We may concentrate on serious steps in the following. In order to simplify notation
we will speak of x k as the sequence generated by serious steps with all duplicates
eliminated. By f k (and the corresponding "
refer to the function whose
minimization gives rise to x k+1 .
The next lemma investigates the case that the f(x k ) remain above some value
f(~x) for some fixed ~
x.
Lemma 4.3. If
fixed ~
m and all k
then the x k converge to a minimizer of f .
Proof. First we prove the boundedness of the x k . To this end denote by g k+1 2
subgradient arising from the optimal solution of the minimization problem
observe that by (3.3)
Therefore the distance of x k+1 to ~
x can be bounded by
2\Omega ~
ff
2\Omega ~
ff
For any k ? K, a recursive application of the bound above yields
uX
By (4.1) the progress of the algorithm in each serious step is at least mL (f(x k
together with (4.2) we obtainX
Therefore the sequence of the x k remains bounded and has an accumulation point
x. By replacing ~ x by
x in (4.4) and choosing K sufficiently large, the remaining sum
can be made smaller than an arbitrary small ffi ? 0, thus proving the convergence of
the x k to x. As the x k+1 converge to
x the g k+1 converge to zero by (4.3), and since
the sequence (f(x k
has to converge to zero as well, we conclude that
x is a minimizer of f .
The lemma also implies that f(x k there are no minimizers. We
summarize the discussion in the following theorem.
Theorem 4.4. [21] If the set of minimizers of f is not empty then the x k converge
to a minimizer of f . In any case f(x k
A SPECTRAL BUNDLE METHOD 11
Remark 4.5. We have just seen that the bundle algorithm works correctly even
if P contains only one column. In this case the use of the aggregate subgradient is
crucial.
To achieve correctness of the bundle algorithm without aggregate subgradients, it
suffices to store in P only the subspace spanning the eigenvectors corresponding to
non-zero eigenvalues of an optimal solution W k+1 of (3.2). Using the bound of [36] it
is not too difficult to show that in this case the maximal number of columns one has to
provide is the largest
plus the number of eigenvectors
to be added in each iteration (this is at least one). In our computational experiments
we found that this upper bound is hardly ever reached. In fact, typical values for the
maximal rank are around half this upper bound.
5. Solving the subproblem. In this section we concentrate on how the minimizer
of f k can be computed efficiently. We have already seen in x3 that this task
is equivalent to solving the quadratic semidefinite program (3.4). Problems of this
kind can be solved by interior point methods, see e.g. [7, 23]. Dropping the iteration
index k and the constants in (3.4) we obtain for
ff
ff 0; V 0:
Expanding into the cost function yields
ff
ff
ff 0; V 0:
Using the svec-operator (see the appendix for a definition and important properties
of svec and the symmetric Kronecker
product\Omega s ) to expand symmetric matrices
from S r into column vectors of length
we obtain the quadratic program (recall
that, for
I svec(V
where (after some technical linear algebra)
ff
ff
+\Omega C; W
At this point it is advisable to spend some thought on W . The algorithm is designed
for very large and sparse cost matrices C. W is of the same size as C. Initially it
might be possible to exploit the low rank structure of W for efficient representations,
but as the algorithm proceeds, the rank of W grows inevitably. Thus it is impossible
to store all the information of W . However, as we can see in (5.2) to (5.6), it suffices to
have available the vector AW 2 R
m and the
scalar\Omega C; W
ff to construct the quadratic
program. Furthermore, by the linearity of A(\Delta) and hC; \Deltai, these values are easily
updated whenever W is changed.
To solve (5.1) we employ a primal-dual interior point strategy. To formulate the
defining equations for the central path we introduce a Lagrange multiplier t for the
equality constraint, a dual slack matrix U 0 as complementary variable to V , a dual
slack scalar fi 0 as complementary variable to ff and a barrier parameter ? 0.
The system reads
ts I \Gamma
I svec(V
The step direction (\Deltaff; \Deltafi; \DeltaU; \DeltaV; \Deltat) is determined via the linearized system
I svec(\DeltaV
In the current context we prefer the linearization
because it makes the system easy to solve for \DeltaV with relatively little computational
work per iteration. The final system for \DeltaV reads
ff
It is not too difficult to see that the system matrix is positive definite (because
suffices to show that Q
using
0). The main work per iteration is the factorization of this matrix
(with v 2 S r this is
it is not possible to do much better since Q 11
has to be inverted at some point. Because of the strong dominance of the factorization
it pays to employ a predictor corrector approach, but we will not delve into this here.
For strictly feasible primal starting point is
a strictly feasible dual starting point can be constructed by choosing t 0 sufficiently
negative such that
A SPECTRAL BUNDLE METHOD 13
Starting from this strictly feasible primal-dual pair we compute the first by
compute the step direction (\Deltaff; \Deltafi; \DeltaU; \DeltaV; \Deltat) as indicated
above , perform a line search with line search parameter
strictly feasible, move to this
new point, compute a new by
ae
oe
with
and iterate. We stop if (hU;
6. Implementation. In our implementation of the algorithm we largely follow
the rules outlined in [21]. In particular u is adapted during the algorithm. The first
guess for u is equal to the norm of the first subgradient determined by v 0 . The
scheme for adapting u is the same as in [21] except for a few changes in parameters.
For example the parameter mL for accepting a step as serious is set to
the parameter mR indicating that the model is so good (progress by the serious step
is larger than mR [f(x k
that u can be decreased is set to
The stopping criterion is formulated in relative precision,
in the implementation.
The choice of the upper bound R on the number of columns r of P and the
selection of the subspace merits some additional remarks. Observe that by Remark 4.5
it is highly unlikely the r violates the bound
even if the number of columns
of P is not restricted.
is also the order of the system matrix in (5.7) and is
usually considerably smaller than the size of the system matrix in traditional interior
point codes for semidefinite programming which is of order m. Furthermore the order
of the matrix variables is r as compared to n for traditional interior point codes.
Thus if the number of constraints m is roughly of the same size as n and a matrix of
order m is still considered factorizable then running the algorithm without bounding
the number of columns of P may turn out to be considerably faster than running an
interior point method. This can be observed in practice, see x7.
For huge n and m primal-dual interior point methods are not applicable any
more, because X , Z \Gamma1 , and the system matrix are dense. In this case the proposed
bundle approach allows to apply the powerful interior point approach at least on an
important subspace of the problem. The correct identification of the relevant subspace
in V is facilitated by the availability of the complementary variable U . U helps to
discern between the small eigenvalues of V (because of the interior point approach
we have V 0!). Eigenvectors v of V that are of no importance for the optimal
solution of the subproblem will have a large value v T U v, whereas eigenvectors, that
are ambiguous, will have both, a small eigenvalue v T V v and a small value v T U v.
In practice we restrict the number of columns of P to 25 and provide room for at
least five new vectors in each iteration (see below). Eigenvectors v that correspond
to small but important eigenvalues of V
are added to W ; important eigenvectors are added to W only if
more room is needed for new vectors.
For large m the computation of (5.2) to (5.6) is quite involved. A central object
appearing in all constants is the projection of the constraint A i on the space spanned
14 C. HELMBERG AND F. RENDL
by P , P T A i P . Since the A i are of the same size as X which we assume to be huge, it
is important to exploit whatever structure is present in A i to compute this projection
efficiently. In combinatorial applications the A i are of the form vv T with v sparse and
the projection can be computed efficiently. In the projection step and in particular in
forming Q 11 the size of r is again of strong influence. If we neglect the computation of
the computation of Q 11 still requires 2m
flops. Indeed, if m is
large then for small r the construction of Q 11 takes longer than solving the associated
quadratic semidefinite program.
The large computational costs involved in the construction and solution of the
semidefinite subproblems may lead to the conviction that this model may not be worth
the trouble. However, the evaluation of the eigenvalue-function is in fact much more
expensive. There has been considerable work on computing eigenvalues of huge, sparse
matrices, see e.g. [9] and the references therein. For extremal eigenvalues of symmetric
matrices there seems to be a general consensus, that Lanczos type methods work best.
Iterative methods run into difficulties if the eigenvalues are not well separated. In our
context it is to be expected that in the course of the algorithm the largest eigenvalues
will get closer and closer till all of them are identical in the optimum. For reasonable
convergence block Lanczos algorithms with blocksize corresponding to the largest
multiplicity of the eigenvalues have to be employed. During the first ten iterations
the largest eigenvalue is usually well separated and the algorithm is fast. But soon
the eigenvalues start to cluster, larger and larger blocksizes have to be used, and
the eigenvalue problem gets more and more difficult to solve. In order to reduce the
number of evaluations it seems worth to employ powerful methods in the cutting plane
model. The increase in computation time required to solve the subproblem goes hand
in hand with the difficulty of the eigenvalue problem because of the correspondence
of the rank of P and the number of clustered eigenvalues.
Iterative methods for computing maximal eigenvectors generically offer approximate
eigenvectors to several other large eigenvalues, as well. The space spanned
by these approximate eigenvectors is likely to be a good approximation of the true
eigenspace. If the maximal number of columns for P is not yet attained it may be
worth to include several of these approximate eigenvectors as well.
In our algorithm we use a block Lanczos code of our own that is based on a
Fortran code of Hua (we guess that this is Hua Dai of [47]). It works with complete
orthogonalization and employs Chebyshev iterations for acceleration. The choice of
the blocksize is based on the approximate eigenvalues produced by previous evaluations
but is at most 30. Four block Lanczos steps are followed by twenty Chebyshev
iterations. This scheme is repeated till the maximal eigenvalue is found to the required
relative precision. The relative precision depends on the distance of the maximal to
the second largest eigenvalue but is bounded by 10 \Gamma6 . As starting vectors we use the
complete block of eigenvectors and Lanczos-vectors from the previous evaluation.
7. Combinatorial applications. The combinatorial problem we investigate is
quadratic programming in f\Gamma1; 1g variables,
In the case that C is the Laplace matrix of a (possible weighted) graph the problem
is known to be equivalent to the max-cut problem.
The standard semidefinite relaxation is based on the identity x T Cx
ff .
For all f\Gamma1; 1g n vectors, xx T is a positive semidefinite matrix with all diagonal elements
equal to one. We relax xx T to X 0 and and obtain the following
A SPECTRAL BUNDLE METHOD 15
primal-dual pair of semidefinite programs,
Z 0:
For non-negatively weighted graphs a celebrated result of Goemans and Williamson
[8] says, that there is always a cut within :878 of the optimal value of the relaxation.
One of the first attempts to approximate (DMC) using eigenvalue optimization
is contained in [39]. The authors use the Bundle code of Schramm and Zowe [42]
with a limited number of bundle iterations, and so do not solve (DMC) exactly. So
far the only practical algorithms for computing the optimal value were primal-dual
interior point algorithms. However these are not able to exploit the sparsity of the
cost function and have to cope with dense matrices X and Z \Gamma1 . An alternative
approach based on a combination of the power method with a generic optimization
scheme of Plotkin, Shmoys, and Tardos [37] was proposed in [22] but seems to be
purely theoretical.
In
Table
7.1 we compare the proposed bundle method to our semidefinite primal-dual
interior point code of [14] (called PDIP in the sequel) for graphs on
nodes that were generated by rudy, a machine independent graph generator written
by G. Rinaldi. Table 7.7 contains the command lines specifying the graphs. Graphs
G 1 to G 5 are unweighted random graphs with a density of 6% (approx. 19000 edges).
G 6 to G 10 are the same graphs with random edge weights from f\Gamma1; 1g. G 11 to G 13
are toroidal grids with random edge weights from f\Gamma1; 1g (1600 edges). G 14 to G 17
are unweighted 'almost' planar graphs having as edge set the union of two (almost
maximal) planar graphs (approx. 4500 edges). G to G 21 are the same almost planar
graphs with random edge weights from f\Gamma1; 1g. In all cases the cost matrix C is the
Laplace matrix of the graph divided by 4, i.e., let A denote the (weighted) adjacency
matrix of G, then
For a description of the code PDIP see [14], the termination criterion requires the
gap between primal and dual optimal solution to be closed to a relative accuracy
of
For the bundle algorithm, (DMC) is transformed into an eigenvalue optimization
problem as described in x2. In addition the diagonal of C is removed so that, in fact,
the algorithm works on the problem
min
with
This does not change problem (PMC) because the
diagonal elements of X are fixed to one. The offset 1e T (Ae \Gamma diag(A)) is added to
the output only and has no influence on the algorithm whatsoever, in particular it
has no influence on the stopping criterion. As starting vector y 0 we choose the zero
vector. All other parameters are as described in x6.
All computation times, for the interior point code PDIP as well as for the bundle
code, refer to the same machine, a Sun sparc Ultra 1 with a Model 140 UltraSPARC
CPU and 64 MB RAM. The time measured is the user time and it is given in the
leading zeros are dropped.
The first column of Table 7.1 identifies the graphs. The second and third refer
to PDIP and contain the optimal objective value produced (these can be regarded as
highly accurate solutions) and the computation time. The fourth and fifth column
give the same numbers for the bundle code.
On these examples the bundle code is superior to PDIP. Although the examples
do belong to the favorable class of instances having small m and relatively large n,
the difference in computation time is astonishing. Note that the termination criterion
used in the bundle code is quite accurate, except for G 11 which seems to be a difficult
problem for the bundle method. This deviation in accuracy is not caused by cancellations
in connection with the offset. The difficulty of an example does not seem to
depend on the number of nonzeros but rather on the shape of the objective function.
For toroidal grid graphs the maximum cut is likely to be not unique, thus the objective
function will be rather flat. This flatness has its effect on the distribution of the
eigenvalues in the optimal solution. Indeed, for G 11 more eigenvalues cluster around
the maximal eigenvalue than for the other problems. We illustrate this in Table 7.2,
which gives the largest eigenvalues of the solution at termination for problems G 1 ,
G 6 , G 11 , G 14 , and G
Table
Comparison of the interior point (PDIP) and the bundle (B) approach. sol gives the computed
solution value and time gives the computation time.
PDIP-sol PDIP-time B-sol B-time
G2 12089.43 1:19:14 12089.45 5:19
G6 2656.16 1:24:53 2656.18 3:57
G 11 629.16 1:28:41 629.21 45:26
G
G 17
G
G 19
Table
7.3 provides additional information on the performance of the bundle algorithm
on the examples of Table 7.1. The second column gives the accumulated time
spent in the eigenvalue computation, it accounts for roughly 90% of the computation
time. serious displays the number of serious steps, iter gives the total number of
iterations including both, serious and null steps. kgk is the norm of the subgradient
arising from the last optimal W k+1 before termination. For G 11 the norm is considerably
higher than for all other examples. Since the desired accuracy was not achieved
for G 11 by the standard stopping criterion it may be worth to consider an alternative
stopping criterion taking into account the norm of the subgradient as well. Column
max-r gives the maximal rank of P attained over all iterations. The rank of P would
A SPECTRAL BUNDLE METHOD 17
Table
The maximal eigenvalues after termination of examples G 1 , G6 , G11 , G14 , and G18 .
13 3.1190 3.2239 0.7651 1.0557 1.4047
14 3.1135 3.2181 0.7650 1.0515 1.4007
19 2.7214 2.7716 0.7647 1.0398 1.3725
22 2.6834 2.6756 0.7644 1.0341 1.3583
26 2.6274 1.8722 0.7636 1.0239 1.3480
28 2.5137 1.7974 0.7633 1.0211 1.3397
29 2.4840 1.4859 0.7630 1.0180 1.3345
have been bounded by 25, but this bound never came into effect for any of these
examples. Aggregation was not necessary. Observe that the theoretic bound allows
for r up to 39, yet the maximal rank is only half this number. The last column gives
the time when the objective value was first within 10 \Gamma3 of the optimum.
For combinatorial applications high accuracy of the optimal solution is of minor
importance. An algorithm should deliver a reasonable bound fast and its solution
should provide some hint on how a good feasible solution can be constructed. The
bundle algorithm offers both. With respect to computation time the bundle algorithm
displays the usual behavior of subgradient algorithms. Initially progress is very fast,
but as the bound approaches the optimum there is a strong tailing off effect. We
illustrate this by giving the objective values and computation times for the serious
steps of example G 6 (the diagonal offset is +77 in this example) in Table 7.4. After
one minute the bound is within 0:1% of the optimum. For the other examples see the
last column of Table 7.3.
With respect to a primal feasible solution observe that P k V k
successively
better and better approximation to the primal optimal solution X . In case
too much information is stored in the aggregate vector AW k (remember that it is not
advisable to store W k itself), P k may be enriched with additional Lanczos-vectors
from the eigenvalue computation. The solution of this enlarged quadratic semidefinite
subproblem will be an acceptable approximation of X . It is not necessary to
Table
Additional information about the bundle algorithm for the examples of Table 7.1. -time gives
the total amount of time spent for computing the eigenvalues and eigenvectors, serious gives the
number of serious steps, iter the total number of iterations including null steps. kgk refers to the
norm of the gradient resulting from the optimal solution of the last semidefinite subproblem. max-r
is the maximum number of columns used in P (the limit would have been 25). 0.1%-time gives the
time when the bound is within 10 \Gamma3 of the optimum in relative precision.
-time serious iter kgk max-r 0.1%-time
G1 3:12 22 33 0.1639
G4 2:38 19 27 0.08235 19 54
G 9
2:59
G13 17:24 43 78 0.218 15 6:17
G
G 19 11:24 41 71 0.1571 15 3:34
construct the whole matrix X . In fact, the factorized form (P k
much more convenient to work with. For example the approximation algorithm of
Goemans and Williamson [8] requires precisely this factorization. A particular x ij
element of X is easily computed by the inner product of row i and j of the n \Theta r
. In principle this opens the door for branch and cut approaches to
improve the initial relaxation. This will be the subject of further work.
Table
7.5 gives a similar set of examples for
A last set of examples is devoted to the Lov'asz #-function [27] which yields an
upper bound on the cardinality of a maximal independent (or stable) set of a graph.
For implementational convenience we use its formulation within the quadratic f\Gamma1; 1g
programming setting, see [24]. For a graph with k nodes and h edges we obtain a
semidefinite program with matrix variables of order
constraints. The examples we are going to consider have more than one thousand
nodes and more than six thousand edges. For these examples interior point methods
are not applicable any more because of memory requirements. It should be clear
from the examples of Table 7.5 that there is also little hope for the bundle method to
terminate within reasonable time. However, the most significant progress is achieved
in the beginning and for the bundle method memory consumption is not a problem.
We run these examples with a time limit of five hours. More precisely, the algorithm
is terminated after the first serious step that occurs after five hours of computation
time.
The graph instances are of the same type as above. The computational results are
displayed in Table 7.6. The new columns n and m give the order of the matrix variable
and the number of constraints, respectively. Observe that the toroidal grid graphs
A SPECTRAL BUNDLE METHOD 19
Table
Detailed account of the serious steps of example G 6 .
iter value time kgk max-r
8 2670.10 28 5.992 15
9 2666.07 34 4.173
14 2656.31 1:57 0.431
19 2656.19 3:33 0.07243
Table
Examples for
B-sol B-time -time serious iter kgk max-r
G22 14135.98 38:11 28:00 26 52 0.0781 23
G26 14132.93 34:45 26:37 31 48 0.3066 23
G 28 4100.81 29:41 21:08 23
G
G
G 36 8006.04 2:56:10 2:31:09 62 115 0.2634 24
G 37
G 38 8015.01 4:03:53 3:39:24 58 155 0.1937 22
G
G 48 and G 49 are perfect with independence number 1500; the independence number
of G 50 is 1440 but G 50 is not perfect. We do not know the independence number of
the other graphs. Except for G 48 and G 49 , which have '(G 48
perfectness, it is hard to judge the quality of the solutions. Tracing the development of
the bounds the last serious steps of examples G 43 to G 47 and G 51 to G 54 still produced
improvements of 0.5% to 1%. This and the rather large norm of the subgradient of
Table
Upper bound on the #-function after five hours of computation time.
serious iter kgk max-r
G 43
G 44 1001 10991 310.13 5:06:31 3:14:25
G48 3001 9001 1526.53 5:11:31 4:59:57 54 94 0.4062 15
G
G 50 3001 9001 1536.12 5:17:51 5:01:25 50 124 0.4728 15
G 53 1001 6915 463.86 5:08:36 4:36:34 41 104 2.593 25
G 51 and G 54 indicate that the values cannot be expected to be 'good' approximations
of the #-function. Also note, that the size of the subspace required for G 48 to G 50 is
still well below 25. In examples G 51 to G 54 the value of ff is almost negligible, but
for G 43 to G 47 the value of ff is roughly 1=3 at termination. Thus for these examples
the restriction to 25 columns became relevant.
The computational results of Table 7.6 demonstrate that the algorithm has its
limits. Nonetheless the bounds obtained are still useful and the primal approximation
corresponding to the subgradient is a reasonable starting point for primal heuristics.
8. Conclusions and extensions. We have proposed a proximal bundle method
for solving semidefinite programs with large sparse or strongly structured coefficient
matrices. The semidefinite constraint is lifted into the objective function by means of a
Lagrange multiplier a whose correct value is not known in general, except for problems
with fixed primal trace. In the latter case a is precisely the value of the trace. The
approach differs from previous bundle methods in that the subproblem is tailored
for semidefinite programming. In fact the whole approach can be interpreted as
semidefinite programming over subspaces where the subspace is successively corrected
and improved till the optimal subspace is identified. The set of subgradients modeled
by the semidefinite subproblem is a superset of the subgradients used in the traditional
polyhedral cutting plane model. Therefore convergence of the new method is a direct
consequence of previous proofs for traditional bundle methods. It is not yet clear
whether the specialized model admits stronger convergence results. The choice of u
is still very much an open problem of high practical importance.
For (constrained) quadratic f\Gamma1; 1g-programming the method offers a good bound
within reasonable time and allows to construct an approximate primal optimal solution
(of the relaxation) in compact representation. To improve the bound by a cutting
plane approach the algorithm must be able to deal with sign constraints on the
y-variables. In principle it is not difficult to model the sign constraints in the semidefinite
subproblem. However, as a consequence the influence of the sign constrained y
variables on the cost coefficients of the quadratic subproblem cannot be eliminated
any longer, rendering the method impractical even for a moderate number of cutting
planes. Alternatively one might consider active set methods but these entail the danger
of destroying convergence. Together with K.C. Kiwiel we are currently working
on alternative methods for incorporating sign constraints on y [13].
The backbone of the method is an efficient routine for computing the maximal
eigenvalue of huge structured symmetric matrices. Although our own implementation
A SPECTRAL BUNDLE METHOD 21
Table
Arguments for generating the graphs by the graph generator rudy.
G1 -rnd graph 800 6 8001
G2 -rnd graph 800 6 8002
G3 -rnd graph 800 6 8003
G4 -rnd graph 800 6 8004
G5 -rnd graph 800 6 8005
G6 -rnd graph 800 6 8001 -random
G7 -rnd graph 800 6 8002 -random
G8 -rnd graph 800 6 8003 -random
G9 -rnd graph 800 6 8004 -random
G12 -toroidal grid 2D 50
G13 -toroidal grid 2D 25
G14 -planar 800 99 8001 -planar 800
G17 -planar 800 99 8007 -planar 800
G22 -rnd graph 2000 1 20001
G23 -rnd graph 2000 1 20002
G24 -rnd graph 2000 1
G25 -rnd graph 2000 1 20004
G26 -rnd graph 2000 1 20005
G27 -rnd graph 2000 1 20001 -random
G28 -rnd graph 2000 1 20002 -random
G29 -rnd graph 2000 1 -times
G31 -rnd graph 2000 1 20005 -random
G32 -toroidal grid 2D 100 20 -random 0 1 -times
G35 -planar 2000 99 20001 -planar 2000
G36 -planar 2000 99 -planar 2000
G37 -planar 2000 99 20005 -planar 2000
G38 -planar 2000 99 20007 -planar 2000
-planar 2000 99 20001 -planar 2000
G40 -planar 2000 99 -planar 2000
-planar 2000 99 20005 -planar 2000 -times 2 -plus
G42 -planar 2000 99 20007 -planar 2000
G43 -rnd graph 1000 2 10001
G44 -rnd graph 1000 2 10002
G45 -rnd graph 1000 2 10003
G46 -rnd graph 1000 2 10004
G47 -rnd graph 1000 2 10005
G48 -toroidal grid 2D 50
-toroidal grid 2D
G50 -toroidal grid 2D 25 120
G52 -planar 1000 100 10003 -planar 1000 100 10004
G53 -planar 1000 100 10005 -planar 1000 100 10006
G54 -planar 1000 100 10007 -planar 1000 100 10008
(based on the code of Hua) seems to work sufficiently stable there is certainly much
room for improvement. A straight forward approach to achieve serious speed-ups is to
implement the algorithm on parallel machines, see for instance [43]. Rather recently
interest in the Lanczos method has risen again, see [25, 3, 5, 10, 30] and references
therein. Most of these papers are based on the concept of an implicit restart proposed
in [44] which is a polynomial acceleration approach that does not require additional
matrix vector multiplications. It will be interesting to test these new ideas within the
bundle framework.
We thank K.C. Kiwiel for fruitful discussions and C. Lemar'echal and an anony-
22 C. HELMBERG AND F. RENDL
mous referee for their constructive critisism that helped to improve the presentation.
Appendix
. Notation.
R
real column vector of dimension n
real matrices
real matrices
positive definite matrices
positive semidefinite matrices
A 0 A is positive definite
A 0 A is positive semidefinite
I , I n identity of appropriate size or of size n
e vector of all ones of appropriate dimension
maximal eigenvalue of A
tr A trace of A 2 M n;n , tr
product in Mm;n ,
dimensional vector representation of A 2 Sn
Kronecker product of A 2 M m;n
diag(A) the diagonal of A 2 Mn as a column vector
diagonal matrix with v on its main diagonal
Sn is isomorphic to R
via the map svec(A) defined by stacking the columns of
the lower triangle of A on top of each other and multiplying the offdiagonal elements
with
2,
a
The factor
for offdiagonal elements ensures that, for
The symmetric Kronecker
product\Omega s is defined for arbitrary square matrices
M n;n by its action on a vector svec(C) for a symmetric matrix C 2 Sn ,
Both concepts were first introduced in [2]. Here we use the notation introduced in
[45]. From the latter paper we also cite some properties of the symmetric Kronecker
product for the convenience of the reader.
1.
B\Omega s A
2.
(A\Omega s B)
3.
A\Omega s I is symmetric if and only if A is.
4.
5.
(A\Omega s
6. If A 0 and B 0 then
(A\Omega s B) 0
7.
--R
Interior point methods in semidefinite programming with applications to combinatorial optimization
Iterative methods for the computation of a few eigenvalues of a large symmetric matrix
Solving large-scale sparse semidefinite programs for combinatorial optimization
An implicitly restarted Lanczos method for large symmetric eigenvalue problems
The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices
Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming
Matrix Computations
A shifted block Lanczos algorithm for solving sparse symmetric generalized eigenproblems
Geometric Algorithms and Combinatorial Optimization
Fixing variables in semidefinite relaxations
Incorporating inequality constraints in the spectral bundle method
An interior-point method for semidefinite programming
Quadratic knapsack relaxations using cutting planes and semidefinite programming
Convex Analysis and Minimization Algorithms I
An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices
Solving graph bisection problems with semidefinite programming
Proximity control in bundle methods for convex nondifferentiable minimization
Efficient approximation algorithms for semidefinite programs arising from MAXCUT and COLORING
Connections between semidefinite relaxations of the max-cut and stable set problems
--TR
--CTR
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Abraham Duarte , ngel Snchez , Felipe Fernndez , Ral Cabido, A low-level hybridization between memetic algorithm and VNS for the max-cut problem, Proceedings of the 2005 conference on Genetic and evolutionary computation, June 25-29, 2005, Washington DC, USA
Gerald Gruber , Franz Rendl, The bundle method for hard combinatorial optimization problems, Combinatorial optimization - eureka, you shrink!, Springer-Verlag New York, Inc., New York, NY,
Tijl De Bie , Nello Cristianini, Fast SDP Relaxations of Graph Cut Clustering, Transduction, and Other Combinatorial Problems, The Journal of Machine Learning Research, 7, p.1409-1436, 12/1/2006
Kazuhide Nakata , Makoto Yamashita , Katsuki Fujisawa , Masakazu Kojima, A parallel primal-dual interior-point method for semidefinite programs using positive definite matrix completion, Parallel Computing, v.32 n.1, p.24-43, January 2006
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Henry Wolkowicz , Miguel F. Anjos, Semidefinite programming for discrete optimization and matrix completion problems, Discrete Applied Mathematics, v.123 n.1-3, p.513-577, 15 November 2002 | proximal bundle method;eigenvalue optimization;semidefinite programming;convex optimization;large-scale problems |
589295 | Cones of Matrices and Successive Convex Relaxations of Nonconvex Sets. | Let F be a compact subset of the n-dimensional Euclidean space Rn represented by (finitely or infinitely many) quadratic inequalities. We propose two methods, one based on successive semidefinite programming (SDP) relaxations and the other on successive linear programming (LP) relaxations. Each of our methods generates a sequence of compact convex subsets Ck . .) of Rn such that (a) the convex hull of $F \subseteq C_{k+1} \subseteq C_k$ (monotonicity), (b) $\cap_{k=1}^{\infty} C_k = \text{the convex hull of F (asymptotic convergence). Our methods are extensions of the corresponding Lovsz--Schrijver lift-and-project procedures with the use of SDP or LP relaxation applied to general quadratic optimization problems (QOPs) with infinitely many quadratic inequality constraints. Utilizing descriptions of sets based on cones of matrices and their duals, we establish the exact equivalence of the SDP relaxation and the semi-infinite convex QOP relaxation proposed originally by Fujie and Kojima. Using this equivalence, we investigate some fundamental features of the two methods including (a) and (b) above. | Introduction
. Consider a maximization problem with a linear objective function
c T x:
maximize c T x subject to x # F,
where c denotes a constant vector in the n-dimensional Euclidean space R n and
F a subset of R n . We can reduce a more general maximization problem with a
nonlinear objective function f(x) to a maximization problem having a linear objective
function represented by a new variable, x n+1 , if we replace f(x) by x n+1 and then
add the inequality f(x) # x n+1 to the constraint. Thus (1.1) covers such a general
optimization problem. Throughout the paper we assume that F is compact. Then
the problem (1.1) has a global maximizer whenever the feasible region F is nonempty.
For any compact convex set C containing F , the maximization problem
maximize c T x subject to x # C
serves as a convex relaxation problem, which satisfies the properties that
(i) the maximum objective value # of the problem (1.2) gives an upper bound
for the maximum objective value # of the problem (1.1), i.e., # , and
# Received by the editors March 31, 1998; accepted for publication (in revised form) July 19, 1999;
published electronically March 21, 2000.
http://www.siam.org/journals/siopt/10-3/33645.html
Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1
Oh-Okayama, Meguro-ku, Tokyo 152-8552, Japan (kojima@is.titech.ac.jp).
# Department of Combinatorics and Optimization, Faculty of Mathematics, University of Water-
loo, Waterloo, Ontario N2L 3G1, Canada (ltuncel@math.uwaterloo.ca). This work was completed
while this author was visiting Tokyo Institute of Technology, Department of Mathematical and Computing
Sciences, on a sabbatical leave from University of Waterloo. The research of this author was
supported in part by Tokyo Institute of Technology and by a research grant from NSERC of Canada.
(ii) if a maximizer -
lies in F , it is a maximizer of (1.1).
Since the objective function of (1.1) is linear, we know that if we take the convex hull
(defined as the intersection of all the convex sets containing F ) for C in
(1.2), then
(ii) # the set of the maximizers of (1.2) forms a compact convex set whose extreme
points are maximizers of (1.1).
Therefore, if we solve the relaxation problem (1.2) with a convex feasible region C
which closely approximates c.hull(F ), we can expect to get not only a good upper
bound # for the maximum objective value # but also an approximate maximizer of
the problem (1.1). We can further prove that for almost every c # R n (in the sense of
any maximizer x # of (1.2) is an extreme point of c.hull(F ),
which also lies in F ; hence x # is a maximizer of (1.1). This follows from a result due
to Ewald, Larman, and Rogers [5] for consequences of related results; see also [17].
Furthermore, for many representations of various convex sets C, given - x # C, we can
very e#ciently find x # , an extreme point of C, such that c T x # c T - x.
Indeed, the relaxation technique mentioned above has been playing an essential
role in practical computational methods for solving various problems in the fields of
combinatorial optimization and global optimization. It is often used in hybrid schemes
with the branch-and-bound and branch-and-cut techniques in those fields. See, for
instance, [2].
The aim of this paper is to present a basic idea on how we can approximate the
convex hull of F . This is a quite di#cult problem, and also too general. Before making
further discussions, we at least need to provide an appropriate (algebraic) representation
for the compact feasible region F of the problem (1.1) and the compact convex
feasible region C of the relaxation problem (1.2). We employ quadratic inequalities
for this purpose.
denote the set of n - n symmetric matrices and the set of
positive semidefinite matrices, respectively. Given Q
and # R, we write a quadratic function on R n with the quadratic term x T Qx, the
linear and the constant term # as p(-; #, q, Q):
Then the set Q of quadratic functions on R n and the set Q+ of convex quadratic
functions are defined as
and
respectively. We also write p(-) # Q (or Q+ ) instead of p(-; #, q, Q) # Q (or Q+ )
are irrelevant. Throughout the paper, we assume
that the feasible region F of the problem (1.1) is represented by a set of quadratic
inequalities such that
where PF denotes a set of quadratic functions, i.e., PF # Q, and we will derive convex
relaxations, C, represented by convex quadratic inequalities such that
where PC denotes a set of convex quadratic functions, i.e., PC # Q+ . We allow cases
where PF and/or PC involve infinitely many quadratic functions. Thus (1.1) or (1.2)
(or both) can be a semi-infinite quadratic optimization problem (QOP). Here we use
the word "semi-infinite" for optimization problems having a finite number of scalar
variables and possibly an infinite number of inequality constraints.
There are some reasons why we have chosen quadratic inequalities for the representation
of both problems, the maximization problem (1.1) that we want to solve
and its convex relaxation problem (1.2). First, quadratic inequalities form a class
of relatively easily manageable nonlinear inequalities, yet they have enough power
to describe any compact feasible region F in R n . Indeed, if F is closed, then its
complement R n
\F is open so that it can be represented as the union of the open balls
{x
over all x # G for some G # R n
We also know that any single polynomial inequality can be converted into a system
of quadratic inequalities; for example,
can be converted into
Second, we know that we can solve some classes of maximization problems having
linear objective functions and a convex-quadratic-inequality constrained feasible
region C e#ciently. Among others, we can apply interior-point methods [1, 16] to the
problem (1.2) when either PC is finite or PC is infinite, but its feasible region C is
described as the projection of a set characterized by linear matrix inequalities in the
space S n of n - n symmetric matrices onto the n-dimensional Euclidean space R n .
Third, and also most importantly, we can apply the semidefinite programming
(SDP) relaxation, which was originally developed for 0-1 integer programming problems
by Lov-asz and Schrijver [12] and later extended to nonconvex quadratic optimization
problems [6, 18, 19], to the entire class of maximization problems having a linear
objective function and finitely or infinitely many quadratic inequality constraints. See
also [1, 8, 9, 13, 15, 23, 24, 29].
In addition to the reasons above, we should mention that the maximization problem
with a linear objective function and quadratic inequality constraints involves
various optimization problems such as 0-1 integer linear (or quadratic) programming
problems which, in principle, include all combinatorial optimization problems
[1, 9, 18]. Linear complementarity problems [4], bimatrix games, and bilinear matrix
inequalities [14, 20] are also included as special cases.
For some optimization problems, some of the semidefinite programming (SDP)
relaxations we provide may be solved in polynomially many iterations (of an interior-point
method or an ellipsoid algorithm) approximately. Such conclusion requires, in
the case of the ellipsoid method, the existence of a certain polynomial-time separation
oracle for the underlying convex cone constraint (see [9]). In the case of interior-point
algorithms (whose e#ciency in the theory and practice of SDP has been well
established), we need to have an e#ciently computable self-concordant barrier for the
feasible solutions set or at least for the underlying cone constraints (see [16]).
Some of the most exciting activities in combinatorial optimization are currently
centered around the applications of SDP to combinatorial optimization problems (see
[7]). Such activity in theory and practice is fueled by theoretical results establishing
that certain simple SDP relaxations of a combinatorial optimization problem
can be e#ectively utilized in developing polynomial-time approximation algorithms
with worst-case approximation-ratio guarantees much better than those previously
proven using linear programming or other techniques. (See Goemans [7], Goemans
and Williamson [8], Nesterov [15], and Ye [29].) Also outstanding are the results
on the stable set problem establishing the fact that SDP techniques can be used in
optimizing over a relaxation of the stable set polytope which is contained in the polytope
defined by the clique inequalities. (Note that it is NP-hard to optimize over the
latter-mentioned polytope, whereas Gr-otschel, Lov-asz, and Schrijver [9] and Lov-asz,
and Schrijver [12] were able to utilize polynomial-time methods to achieve a better
goal, as far as the proof of approximate optimality of some feasible solutions of the
stable set problem is concerned.)
Given an initial approximation C 0 of F , i.e., a compact convex set C 0 containing
F , both of the methods, proposed in this paper, generate a sequence of compact
convex subsets
(a)
It should be noted that the compactness of each C k and property (b) imply that
(c) if
(detecting infeasibility).
To generate C k+1 at each iteration, the SDP relaxation and the linear programming
relaxation play an essential role, and the entire method may be regarded
as an extension of the Lov-asz-Schrijver lift-and-project procedure for 0-1 integer programming
problems to semi-infinite nonconvex quadratic optimization problems, with
the use of the SDP relaxation in the first method and the LP relaxation in the second
method. The LP relaxation, referred to above, is essentially the same as the
reformulation-linearization technique developed for nonconvex quadratic optimization
problems by Sherali and Alameddine [21]; see also [2, 22]. However, we should
caution the reader that the methods presented here are mostly conceptual in the general
settings, because we need to solve a semi-infinite SDP (or a semi-infinite LP) at
each iteration. For such a task, an e#cient practical algorithm may not be currently
available.
In their paper [6], Fujie and Kojima proposed the semi-infinite convex QOP relaxation
for nonconvex quadratic optimization problems and showed that the semi-infinite
convex QOP relaxation is not stronger than the SDP relaxation in general,
but the two relaxations are essentially equivalent under Slater's constraint qualifica-
tion. We establish the exact equivalence between the two relaxations for semi-infinite
nonconvex quadratic optimization problems without any constraint qualification. Using
this equivalence, we derive some fundamental features of our methods including
(a) and (b) above. One of the common themes in this paper is the usage of cones of
matrices (and duality) in our constructions. This was also one of the themes of [12].
The other themes of this paper are the successive applications of SDP relaxations and
LP relaxations. We call the related procedures the successive SDP relaxation method
and the successive semi-infinite LP relaxation method, respectively.
Section 2 is devoted to preliminaries, where we provide some basic definitions
and properties on quadratic inequality representations for closed subsets of R n , the
homogeneous form of quadratic functions, the SDP relaxation, etc. In section 3, we
present our first method in detail as well as the main results, including the features (a)
and (b). After we present some fundamental characterizations of the SDP relaxation
in section 4, we give proofs of the main results in section 5. In section 6, we apply our
method to 0-1 semi-infinite nonconvex quadratic optimization problems. Incorporating
the basic results on the lift-and-project procedure given by Lov-asz and Schrijver
[12] for 0-1 integer convex optimization problems, we show that our method terminates
in at most (n iterations either to generate the convex hull of the feasible
region or to detect the emptiness of the feasible region, where n denotes the number
of 0-1 variables of the problem. Section 7 contains our second method, which is based
on semi-infinite LP relaxations. We establish the same theoretical properties as we
do for the successive SDP relaxation method. In section 8, we present two numerical
examples showing the worst-case behavior of some of our procedures. In particular,
we know from the second example that the best of our procedures requires infinitely
many iterations to generate the convex hull of F in the worst case.
2. Preliminaries.
2.1. Semi-infinite quadratic inequality representation. In this subsection,
we discuss some representations of a closed subset F of R n in terms of (possibly
infinitely many) quadratic inequalities. If p(-; #, q, Q) # Q, and p(x; #, q, Q) # 0
holds for all x # F , we say that p(x; #, q, Q) # 0 is a quadratic valid inequality for
F and that p(-; #, q, Q) induces a quadratic valid inequality for F . A quadratic valid
inequality p(x; #, q, Q) # 0 for F is
linear
and # R such that # a T x #x # F ,
rank-2 quadratic if
and # R such that a T x # and b T x #x # F ,
spherical if
ellipsoidal if
convex quadratic if Q # S n
respectively. It should be noted that if a quadratic valid inequality p(x; #, q, Q) # 0
for F is rank-2, then the rank of the matrix Q is at most 2 but that the converse is
not necessarily true.
We say that F has a (semi-infinite) quadratic inequality representation P # Q if
holds. To designate the underlying representation P of F , we often write F (P) instead
of F . Whenever F is a closed proper subset of R n , F has infinitely many represen-
tations. We allow the cases where P consists of infinitely many quadratic functions.
Hence can be a semi-infinite system of quadratic inequalities. If
inequality representation of F and if p(-) # c.cone(P), then
is a quadratic valid inequality, where c.cone(P) denotes the closed convex
cone generated by P. Hence if P # P # c.cone(P), then P # is a quadratic inequality
representation of F ; F inequality representation
P of F is finite if it consists of a finite number of quadratic functions, and
infinite otherwise. If F is a compact convex subset of R n , it has a quadratic inequality
representation; in fact, the set of all the linear (rank-2 quadratic or spherical)
valid inequalities for F forms an inequality representation of F . If, in addition, F is
polyhedral, we can take a finite linear inequality representation.
Let C be a compact subset of R n . We use the following symbols:
the set of p(-)'s that induce linear valid inequalities for C,
the set of p(-)'s that induce rank-1 quadratic valid inequalities for C,
the set of p(-)'s that induce rank-2 quadratic valid inequalities for C,
the set of p(-)'s that induce spherical valid inequalities for C,
the set of p(-)'s that induce ellipsoidal valid inequalities for C,
the set of p(-)'s that induce convex quadratic valid inequalities for C,
the set of p(-)'s that induce all quadratic valid inequalities for C.
By definition, we see that
Note that if C is convex, then the equality
holds with each
these, P # (C) is the strongest quadratic inequality representation of C.
2.2. Homogeneous form of quadratic functions-lifting to the space of
symmetric matrices. We introduce a di#erent description of quadratic functions,
which we call the homogeneous form. This form leads us to a lifting of a quadratic
function defined on the Euclidean space to the space of symmetric matrices and to
the SDP relaxation (or to the semi-infinite LP relaxation in section 4.2). For every
quadratic function p(-; #, q, Q) # Q, we connect the variable vector x # R n to the
positive semidefinite matrix
x
and the triplet of the constant # R, q # R n , and Q # S n to the (1
. Then we have the identity
p(x; #, q,
x
Thus, if P # Q is a quadratic inequality representation of F , then
provides an equivalent representation of F ;
Now we have two kinds of description for a quadratic function on R n : the usual
form p(-; #, q, and the homogeneous form introduced above.
The former is used in section 5, where we prove our main results, while the latter is
suitable for the compact description of the SDP relaxation in section 2.3 and the proof
of its equivalence to the semi-infinite convex QOP relaxation in section 4. We will use
both forms in parallel, choosing whichever is convenient to us in a given situation. It
should be noted that the correspondence
is not only one-to-one but also linear. To save notation, we identify the set Q of
quadratic functions with the set S 1+n of (1
any subset of Q with the corresponding subset of S 1+n . Specifically, we write
identify the set of (1
symmetric matrices
with the set Q of quadratic functions from R n to R.
2.3. SDP relaxation. Let P be a semi-infinite quadratic inequality representation
The SDP relaxation -
F (P) of F (P) with the quadratic inequality representation P is
given by
and
and
x
and P . # 1 x T
This implies that x # -
F (P) and F (P) # -
F (P). We also see that -
F (P) is convex.
Hence
F (P). The SDP relaxation was originally proposed for combinatorial
optimization problems and 0-1 integer programming problems [12], and later
extended to quadratic optimization problems. See [1, 6, 8, 9, 15, 19, 18, 23, 24, 29].
3. Main results. Now we are ready to describe our method for approximating
a quadratic-inequality-constrained compact feasible region F of the minimization
problem (1.1). Before running the method, we need to fix a semi-infinite quadratic
inequality representation PF of F , and choose an initial approximation C 0 of the convex
hull of F , i.e., a compact convex set which contains c.hull(F ). Starting from C 0 ,
the method generates a sequence of compact convex sets
we expect to converge to c.hull(F ). At each iteration, we choose a semi-infinite quadratic
inequality representation P k of the kth approximation C k of c.hull(F ). Since
the union (PF # P k ) forms a semi-infinite quadratic inequality representation
of F . We then apply the SDP relaxation to (PF # P k ) to generate the
next iterate C
It should be emphasized that during none of the
iterations do we modify or strengthen the representation PF directly. We only utilize
the semi-infinite quadratic inequality representation of the compact convex set C k
that has been computed in the previous iteration.
Successive SDP Relaxation Method.
Step 0: Let
Step 2: Choose a semi-infinite quadratic inequality representation P k for C k .
Step 3: Let
and
Step 4: Let to Step 1.
We state two convergence theorems below. We choose the spherical inequality
representation at Step 2 of each iteration in the first theorem, while
we choose the rank-2 quadratic inequality representation P 2 (C k ) for C k at Step 2 of
each iteration in the second theorem. Their proofs will be given in section 5.
Theorem 3.1. Assume that PF is a semi-infinite quadratic inequality representation
of a compact subset F of R n , and that C 0 # F is a compact convex subset
of R n . If we choose P of each iteration in the successive SDP
relaxation method, then the monotonicity property (a) and the asymptotic convergence
property (b) stated in the introduction hold.
Theorem 3.2. Under the same assumptions as in Theorem 3.1, if we choose
of each iteration in the successive SDP relaxation method,
then (a) and (b) remain valid.
We know that if P # Q and P # Q are semi-infinite quadratic inequality representations
of C k and if
F (P). Hence, even if we replace
)" in Theorem 3.1 by "P k # P S (C k )" (or "P )" in Theorem
3.2 by "P k # P 2 (C k )"), the properties (a) and (b) remain valid. In particular,
(a) and (b) remain valid when we choose any of P
If we take the linear representation P L (C k ) of C k at every iteration, then we can
prove that
(See Lemma 4.1.) Hence (b) does not follow in general.
In section 8, we will give two numerical examples. The first example shows that
the rank-1 quadratic inequality representation strong enough
to ensure (b). The second example shows that even when we choose the strongest
quadratic inequality representation P # (C k ) of C k for P k at every iteration, not only
does the convergence "C k # c.hull(F )" require infinitely many iterations, but its
speed also becomes extremely slow in the worst case.
4. Fundamental characterization of successive convex relaxation.
4.1. Semi-infinite convex QOP relaxation and its equivalence to SDP
relaxation. The semi-infinite convex QOP relaxation of F (P) with the semi-infinite
quadratic inequality representation P is defined as
We observe that
F (P)
and that the set -
F (P) is a closed convex set. Hence F (P) # c.hull(F
F (P).
The semi-infinite convex QOP relaxation was introduced by Fujie and Kojima
[6]. It was called the relaxation using convex-quadratic valid inequalities for F (P) in
their paper [6]. The following basic properties of the relaxation are essentially due to
them.
Lemma 4.1. Let PF be a semi-infinite quadratic inequality representation of a
closed set F # R n .
Let P be a set of convex quadratic valid inequalities for F , i.e.,
Then
Let P be a set of linear valid inequalities for F , i.e.,
(iii) Let x # c.hull(F ). Suppose that p(x #, q, Q) # 0 for some p(-; #, q, Q) # PF
with a positive definite Q. Then x # -
F (PF ).
Proof. Part (i) follows directly from the definition of the semi-infinite convex
QOP relaxation. Now we show (ii). Let
we see that
Hence it su#ces to show that -
F (PF #P). Let p(-) # c.cone(P F #P)#Q+ .
Then there exist p(-) i # PF positive
m) such that
are linear functions, we see that
F (PF ).
Moreover,
Therefore,
This proves (ii). Finally we will show (iii). Since x # F , there is a p # PF such
that su#ciently small, we obtain that
This implies x # -
F (PF ), and proves (iii).
When P is finite and F (P) satisfies Slater's constraint qualification, Fujie and Kojima
[6] showed that the semi-infinite convex QOP relaxation is essentially equivalent
to the SDP relaxation in the sense that -
F (P) coincides with the closure of -
F (P). The
theorem below shows the exact equivalence between them, without any constraint
qualification, for more general semi-infinite quadratic inequality representation cases.
F (P) is closed, one of the consequences of the next theorem is that -
F (P) is
always closed. Note that we can assume without loss of generality that P is a closed
convex cone, since every closed set F admits such a representation.
Theorem 4.2. Let P be a closed convex cone, giving a semi-infinite quadratic
inequality representation of a closed subset F of R n
its SDP relaxation and its semi-infinite convex QOP relaxation
coincide with each other; -
F (P).
Proof. Using the dual cone
of P, we can express the sets -
F (P) and -
F (P) as follows:
and
# .
For the last identity above, we have used the fact that for any pair of closed convex
cones K 1 and K 2 in R m , we have (K 1 # K 2
First let x # -
F (P). Then there exists an X # S n such that
760 MASAKAZU KOJIMA AND LEVENT TUNC-EL
Consider the identity
-x -X
The first matrix on the right-hand side is in P # and in the second matrix of the
right-hand side, we have X - xx T
since it is the Schur complement of 1 in
the symmetric, positive semidefinite matrix # 1 x T
We have proved x # -
F (P) and
hence -
F (P).
For the converse, let x # -
F (P); that is, there exists some H # S n
such that
The matrix
is positive semidefinite if and only if (H
is. But the latter was
already established. So,
.
Therefore x # -
F (P), and -
F (P) is proved.
4.2. Semi-infinite LP relaxation. In section 7, we will also need an analog of
the above theorem for our successive semi-infinite LP relaxation method. For every
semi-infinite quadratic inequality representation P of a compact subset F of R n , let
us define
and
of Sherali and Alameddine [21]. Here, L denotes the set of linear functions on
The next result can be obtained by following the steps of the proof of Theorem 4.2.
Corollary 4.3. Let P be a closed convex cone, giving a semi-infinite quadratic
inequality representation of a closed subset F of R n
F L (P).
Proof. We observe that
and
Since it is easy to see that #X # S n such that # 1 x T
only if
the proof is complete.
4.3. Invariance under one-to-one a#ne transformation. Let
b be an arbitrary one-to-one a#ne transformation on R n , where A is an n - n non-singular
matrix and b # R n .
Then
of f(F (P)). This means that the semi-infinite SDP and LP relaxations are
invariant under the one-to-one a#ne transformation
We also see that
holds, where U # {L, 1, 2, E, C, #}. Therefore, P L (C),
are invariant under one-to-one a#ne transformations on R n . If in
addition A is a scalar multiple of an orthogonal matrix, then the above identity also
holds for is invariant under such a one-to-one a#ne transformation
on R n .
At each iteration of the successive SDP relaxation method, we observe that
forms a semi-infinite quadratic inequality representation
of f(F ) and P #
inequality representation of f(C k ). Furthermore, if we choose one of the invariant
semi-infinite quadratic inequality representations P L (C k ),
under any one-to-one a#ne transformation for P k , we see
that P U hence the identity above turns out to
be
Here U # {L, 1, 2, E, C, #}. Therefore the successive SDP relaxation method is
invariant under any one-to-one a#ne transformation. The same comment applies to
the successive semi-infinite LP relaxation method, which we will present in section 7.
762 MASAKAZU KOJIMA AND LEVENT TUNC-EL
5. Proofs of Theorems 3.1 and 3.2. We present three lemmas, Lemma 5.1 in
section 5.1, Lemma 5.2 in section 5.2, and Lemma 5.3 in section 5.4. Lemma 5.1 proves
the monotonicity property (a) in Theorems 3.1 and 3.2 simultaneously. Lemma 5.2
is used to prove Theorem 3.1 in section 5.3, and Lemma 5.3 to prove Theorem 3.2 in
section 5.5.
5.1. Monotonicity. We first establish the monotonicity in general.
Lemma 5.1. Let C 0 be a compact convex set containing F . Fix a closed convex
cone S 1+n
# K and
Assume that
# K and
Proof. Since K # S 1+n
contains all symmetric rank-1 matrices of the form
Now, as in the arguments in section 2.3, it follows that c.hull(F
We will show by induction that C k+1 # C k for all
the construction of C 1 and the assumption imposed on C 0 , we first observe that
Now assume that C k # C k-1 for some k # 1. Then P U (C k-1
which implies that PF # P U (C k-1
desired.
5.2. Separating hypersphere. The following lemma easily follows from the
separating hyperplane theorem, and the proof is omitted here.
Lemma 5.2. Let C be a compact convex subset of R n and x # C. Then there
exists a hypersphere S # {x # R n : #x-d#} which strictly separates the point x #
and C such that
where d # R n and # > 0.
5.3. Proof of Theorem 3.1. The monotonicity property (a) follows from Lemma
5.1 by letting K # S 1+n
and U # S. Let C # k=0 C k . We know by (a) that
that all the sets c.hull(F ), C, and C k
are compact sets. To prove (b), we have the following left to show: C # c.hull(F ).
Assume on the contrary that there exists some x # C such that x # c.hull(F ). Then,
by Lemma 5.2, there exists a hypersphere S # {x # R strictly
separates the point x # C from c.hull(F ) such that
there is a quadratic function,
cuts o# x Q). Note that if p 1 (-; #, q, Q) is
such a quadratic function, then so is #p 1 (-; #, q, Q) for any # > 0. Hence we may
assume that the minimum eigenvalue of the matrix Q # S n is at least (-1). Now
consider a quadratic function p 2 (-) defined by
By the definition of # , we see that
This means that the open ball B+ # {x # R with the center d and the
a neighborhood of the compact set C. On
the other hand, the sequence {C k } of compact subsets of R n satisfies
So, we can find a finite positive number # such that the open ball B+ contains C # .
Hence, We also
see that
Thus we have shown that
Therefore, x # C
. This is a contradic-
tion. The theorem is proved.
5.4. A family of inequalities of the convex cone of rank-2 quadratic
valid inequalities for the unit ball. Let B denote the unit ball {x # R
1}. Let Q be an arbitrary n - n symmetric matrix, and let u # R n be an arbitrary
vector on the boundary of B; #u# = 1. We will construct a family of quadratic
valid inequalities, which lie in the convex cone of rank-2 quadratic valid inequalities,
with a parameter # (0, #/8) for the unit ball B satisfying the properties
(i), (ii), and (iii) listed in Lemma 5.3.
We first apply the eigenvalue decomposition to the matrix Q # S n . We may
assume that the first m eigenvalues are nonnegative and the last
are nonpositive for some nonnegative integer m # n. Then we can write the matrix
denote eigenvectors of Q, which are orthogonal to each other, and - j (j = 1, 2, . , m)
and - j (j = m+ 1, . , n) denote the eigenvalues corresponding to them.
For each # (0, #/8), we define
a
a
are nonzero
vectors, and
a
are linear valid inequalities for the unit ball B. For all # (0, #/8), define
In particular, p # (u) # 0 # (0, #/8).
Lemma 5.3.
tends to 0.
(iii) The Hessian matrix of p # (-) coincides with -Q.
Proof. Part (i) was already shown.
(ii) Let j be fixed. It su#ces to show that
converge to zero as # (0, #/8) tends to 0. First, we derive that # j (#) converges to
zero as # (0, #/8) tends to 0. We see from (5.2) that
sin #)
sin #
(b T
sin #
(b T
Since both the numerator and the denominator above converge to zero as # (0, #/8)
tends to 0, we calculate their derivatives at The derivative of the numerator
turns out to be
(b T
which vanishes at On the other hand, the derivative "2 cos #" of the denominator
"2 sin #" in (5.5) does not vanish at converges to 0 as
# (0, #/8) tends to 0. Similarly, we can prove that - # j (#) converges to 0 as # (0, #/8)
tends to 0.
(iii) It follows from the definitions (5.2) and (5.4) that the Hessian matrix of
the quadratic function p # (-)
a
a j (#) T= -
From the lemma above, we see that the cone rich enough to contain
rank-2 quadratic functions with any prescribed Hessian, leading to valid inequalities
that are tight at any given point on the boundary of B.
5.5. Proof of Theorem 3.2. The monotonicity property (a) follows from
Lemma 5.1 by letting K # S 1+n
2. To derive (b), it su#ces to show
that C # k=0 C k # c.hull(F ) as in the proof of Theorem 3.1. Assume on the contrary
that x # c.hull(F ) for some x # C. By Lemma 5.2, there exists a hypersphere
strictly separates the point x # C and c.hull(F )
such that
the successive SDP relaxation method using the
rank-2 quadratic representation for C k at each iteration is invariant under the a#ne
transformation maps d to the origin and the hypersphere
onto the unit hypersphere {x # R
may assume that 1. Thus, we have obtained that
Since u # F , there is a quadratic function p 1 (-; #, q, Q) # PF that cuts o# u;
be the quadratic function introduced
in section 5.4. See (5.2) and (5.4). By Lemma 5.3, we can choose a # (0, #/8)
for which p # (u) # -p 1 (u; #, q, Q)/3 holds. Now we define
766 MASAKAZU KOJIMA AND LEVENT TUNC-EL
By construction, we know that p # k (-) # c.cone(P 2 (C k )). Since both
quadratic functions p # (-) and p # k (-) have the common Hessian matrix -Q,
We will show that
for every su#ciently large k. Then the above two relations imply u # C k+1 for such
a large k. This contradicts the fact
Since the sequence of compact convex subsets
we see that
as k # (j = 2, 3, . , n). By continuity, we see then that for every su#ciently large
Thus we have shown that (5.6) holds for every su#ciently large k. This completes
the proof of Theorem 3.2.
6. Application to 0-1 semi-infinite, nonconvex quadratic optimization
problems. We briefly recall two of the Lov-asz-Schrijver procedures for 0-1 integer
programming problems, and relate them to our successive SDP relaxation method.
Let F be a subset of {0, 1} n whose convex hull is to be approximated. In the Lov-asz-
Schrijver procedures, we assume that a compact convex subset C 0 of R n satisfying
is given in advance. We define
Let K I denote the convex cone spanned by the 0-1 vectors in K
Here the 0th coordinate is special. It is used in homogenizing the sets of interest in
R n . Clearly
The closed convex cone K 0 serves as an initial relaxation of K I . Given the current
relaxation K k of K I , first a convex cone M+ (K k , K k ) in the space of (1
symmetric matrices is defined (the lifting operation). Then a projection of this cone
gives the next relaxation N+ (K k ) of K I .
Now, we define the lifting operation in general. Let K and T be closed convex
cones in R 1+n . A (1 real entries is in
(This condition is equivalent to Y K # T .)
Here, e 0 denotes the unit vector with 0th coordinate 1. Item (ii) above serves
an important role in Lov-asz-Schrijver procedures as well as in some of the SDP
relaxations used by Goemans and Williamson [8], Nesterov [15], and Ye [29]. This
equation is valid simply because for each j for which x j # {0, 1}, the equation x 2
is valid. Indeed, our general framework applies to any compact set in R n , and the
equation Y e not utilized in earlier sections (as it is not valid). In
this section, however, the equation is valid and we utilize it. As will be noted in the
proof of Theorem 6.3, the inclusion of this equation will be guaranteed by our choice
of the initial formulation.
The third condition of Lov-asz-Schrijver procedures is very interesting. They
present a couple of possibilities for the choice of cone T in 0-1 integer programming.
Among them is the cone spanned by all 0-1 vectors with the first component x
This choice, since the cone T # has a very simple set of generators, allows for the
development of polynomial-time algorithms for approximately solving the successive
SDP relaxations as long as the number of iterations of the successive procedure is
O(1). Their result only assumes that a polynomial-time weak separation oracle is
available for K. The key is that since T # has only O(n) extreme rays, it becomes
trivial to check condition (iii) in polynomial time. On the other hand, Lov-asz and
Schrijver [12] note that the choice T # K is also possible and leads to at least as good
relaxations as the former choice for T . (In many cases the successive relaxations for
are significantly tighter than the successive relaxations with the simpler choice
of T .) In the case of the latter choice, the possibility of polynomial-time solvability
of the first few successive relaxations depends on the availability of polynomial-time
algorithms to check Y K # K. Our procedure uses T # K.
Now, we describe the projection step.
We also define the iterated operators N k
use the notation N+ (K), whereas N+ (K, K)
is used in [12].)
Another procedure studied in [12] uses a weaker relaxation by removing the condition
(i) in the lifting procedure. Let M(K,K) and N(K) denote the related sets for
this procedure. We will refer to the first procedure using the lifting M+ (K, K) (and
the projection N+ ) as the N+ procedure. We will call the other (using M(K,K), and
N) the N procedure. Lov-asz and Schrijver prove the following.
Theorem 6.1.
and
768 MASAKAZU KOJIMA AND LEVENT TUNC-EL
Let us see how our successive SDP relaxation method applies to 0-1 nonconvex
quadratic optimization problems. Consider a 0-1 nonconvex quadratic program:
subject to x # F # {x # {0, 1}
We may assume that the set P # contains the quadratic functions x
1, 2, . , n. Then we can replace the 0-1 constraint imposed on the variable x i by the
inequality -x i by adding the quadratic functions -x
1, 2, . , n, to P # , we obtain a quadratic inequality representation PF of the feasible
region F . Let C 0 # [0, 1] n . Note that F #= C 0 #{0, 1} our general setting
here. However, has been assumed for some compact convex subset
C 0 of R n in the Lov-asz-Schrijver procedures discussed above.
Lemma 6.2. Suppose that we take C
and
Proof. Let C #
1 be the semi-infinite convex QOP relaxation of the set F with the
quadratic inequality representation PF #
In view of Theorem 4.2 and Lemma 5.1, we know that
Hence it su#ces to show that
If F contains all the 0-1 vectors, the inclusion relation above obviously holds. Now
assume that x # F is a 0-1 vector. Then there is a quadratic function p 1 (-, #, q, Q) #
PF such that
On the other hand, we know that the quadratic function
with the identity matrix as its Hessian matrix, is a member of c.cone(P 0 ), and that
Hence if # > 0 is su#ciently small, then
This implies that x # C #
As a consequence of the lemma above, we see that the 0-1 nonconvex quadratic
optimization problem (6.1) is equivalent to the 0-1 convex quadratic optimization
problem
subject to x #
Using this observation, we can prove that in the case of 0-1 nonconvex quadratic
optimization problem (6.1), our successive SDP relaxation method converges in (1+n)
iterations.
Theorem 6.3. The successive SDP relaxation method, applied to the 0-1 non-convex
quadratic optimization problem (6.1), using C as the initial approximation
of c.hull(F ) and in each iteration, terminates in at most (1 +n)
iterations with
Proof. We note that by Lemma 6.2, after one iteration of the successive SDP
relaxation method, we obtain the 0-1 convex quadratic optimization problem (6.2)
that can be used with the original Lov-asz-Schrijver procedure. We only have to note
that the successive SDP relaxation method becomes the Lov-asz-Schrijver procedure
after the first iteration. For this purpose, we compare conditions (i), (ii), and (iii) of
the Lov-asz-Schrijver procedure for to the conditions used to construct
in the successive SDP relaxation method. Here
First, we observe that #X # S n such that Y # 1 x T
if and only if # 0,
. Hence (i) is satisfied. For (ii), note that
implies the constraint Y e
implies Y e 0 # Diag(Y ). Finally, for (iii), note that a linear inequality a T x # is
valid for C k if and only if (#, -a T
Therefore, we see that
Step (3.1) of the successive SDP relaxation method implies that
. Thus, we conclude by noting that
Theorem 6.1 implies that n more steps of the procedure is su#cient.
The above discussion and the results show that our successive SDP relaxation
method generalizes the Lov-asz-Schrijver N+ procedure by ignoring condition (ii),
which is no longer valid. Our results in the previous sections already showed that
in this full generality, we still have the asymptotic convergence of the method. It is
therefore interesting to investigate the same questions about the weaker procedure
. What is the generalization of procedure N?
. Does the generalization of procedure N satisfy the same theoretical properties
as the successive SDP relaxation method?
We answer both of these questions in the next section. As is shown in [12], in
some cases the procedure N+ is significantly better than N . Procedure N is weaker,
but the relaxations given by it are always polyhedral sets (so LP techniques can be
employed) and N+ requires more general techniques. Hence, sometimes procedure N
might be more manageable even if the procedure N+ is not.
We should expect that the generalization of procedure N should be only using
condition (iii), Y K # K, in the definition of the lifting. We would also expect that
the generalization should lead to semi-infinite LP (rather than SDP) relaxations. We
show in the next section that the above-mentioned generalization of procedure N
leads to successive semi-infinite LP relaxations and all the analogs of the theoretical
properties established for our successive SDP relaxations can also be established for
the successive semi-infinite LP relaxations.
7. Successive semi-infinite LP relaxation.
Successive Semi-Infinite LP Relaxation Method.
Step 0: Let
Step 2: Choose a quadratic inequality representation P k for C k .
Step 3: Let
#X # S n such that
(The equalities above follow from Corollary 4.3.)
Step 4: Let to Step 1.
Theorem 7.1. Assume that PF is a semi-infinite quadratic inequality representation
of a compact subset F of R n , and that C 0 # F is a compact convex subset of
R n . If we choose P of each iteration in the successive semi-infinite
LP relaxation method, then the monotonicity property (a) and the asymptotic
convergence property (b) stated in the introduction hold.
Proof. We can apply the same proof as the one given for Theorem 3.2 in section 5.5
to the theorem.
Note that we can define another semi-infinite LP relaxation based on the semi-infinite
convex QOP relaxation. Clearly, if Q # S n
So, we can define a semi-infinite LP relaxation based on the above observation:
F L
q O
and
F L
q O
In this case, the equivalence -
F L
F L
is evident. The convergence of the successive
semi-infinite LP relaxation method using -
F L
can be established by following the proofs
of Theorems 3.1 and 3.2. Instead, we note -
F L
F L . Therefore, Theorem 7.1 also
implies that this particular semi-infinite LP relaxation method has the properties (a)
and (b) mentioned in the theorem.
8. Further discussions on successive convex relaxations.
8.1. Conic quadratic inequality representation. The conic quadratic inequality
presented below is a generalization of the linear matrix inequality [3, 28] and
the bilinear matrix inequality [14, 20]. It will be shown that any conic quadratic
inequality can be reduced to a semi-infinite system of standard quadratic inequalities
and vice versa.
Let K and K K} be a closed convex cone in R m
and its dual. Here u - v denotes an inner product of u . For all
lies in K. Now we introduce a conic quadratic
vectors in R m . We may assume
without loss of generality that . The inequality (8.1) turns out to be a system
of m usual quadratic inequalities on R n if we take the nonnegative orthant R m
for the cone K. The inequality (8.1) turns out to be a quadratic matrix inequality,
which is a generalization of linear and bilinear matrix inequalities [3, 28] if we identify
the space of # symmetric matrices with R m and we take the positive semidefinite
of matrices for the cone K, where
We can rewrite the conic quadratic inequality (8.1) as a semi-infinite system of
standard quadratic inequalities in the homogeneous form.
for some P # . This means that we can easily include any conic quadratic
inequality in the semi-infinite quadratic inequality representation of the feasible region
F of the maximization problem (1.1). To see the equivalence between (8.1) and (8.2)
for some P # we observe that (8.1) can be rewritten as
Therefore, if we define
772 MASAKAZU KOJIMA AND LEVENT TUNC-EL
we obtain the desired semi-infinite system (8.2) of standard quadratic inequalities,
which is equivalent to (8.1).
Let F (P) denote the solution set of (8.2) with its quadratic inequality representation
Applying the SDP relaxation to F (P), we obtain
that
and
and
and
The set in the last line corresponds to the SDP relaxation to the solution set of (8.1).
This implies that we can apply the SDP relaxation directly to the conic quadratic
inequality converting it into the semi-infinite system (8.2) of standard
quadratic inequalities.
Conversely, we can reduce any semi-infinite system of standard quadratic inequalities
to a conic quadratic inequality. To show this, consider a semi-infinite system (8.2)
of standard quadratic inequalities in the homogeneous form. We may assume without
loss of generality that P # S 1+n is a closed convex cone. We can rewrite (8.2) as
x
which is a conic quadratic inequality.
Let F denote the solution set of the conic quadratic inequality (8.3) that we have
derived from (8.2) above. Applying the SDP relaxation to F , we obtain that
and
# .
Note that the set in the last line corresponds to the SDP relaxation of the solution
set of the semi-infinite system (8.2) of standard quadratic inequalities.
In view of the discussions above, we know that the conic quadratic inequality
representation is as general as the semi-infinite quadratic inequality representation
and that the SDP relaxations to both representations are equivalent. When we deal
with the semi-infinite convex QOP relaxation, however, the semi-infinite quadratic
inequality representation seems more convenient than the conic quadratic inequality
representation.
8.2. A counterexample to the convergence for the rank-1 quadratic
inequality representation case. The example below shows that the rank-1 quadratic
inequality representation is not strong enough to ensure the convergence of the
successive SDP relaxation method. Let
where denotes the rank-1 quadratic inequality representation of the unit ball,
which consists of all quadratic functions such that (a T x - 1)(a
see that
Theorem 8.1. Suppose that we take P
representation of C k ) in the successive SDP relaxation method applied to the
example above. Then C
Proof. By definition, C which su#ces to establish
the theorem. First observe that C 1 # B. Hence it su#ces to show
equivalently for all p(-) # c.cone(P F ) # Q+ ,
fixed. Then we can choose # i # 0
#) such that
Now assume that # 0 > 0. In this case, we may further
assume without loss of generality that #
It follows from p(-) # Q+ that the Hessian matrix #
I # is positive
semidefinite. Hence if we denote the largest and the smallest eigenvalues of the matrix
We also see that
Hence
8.3. A counterexample to the finite termination for the strongest quadratic
inequality representation case. The example below shows that in the worst
case, even when we take the strongest quadratic inequality representation P # (C k ) for
C k at every iteration,
. the successive SDP relaxation method requires infinitely many iterations, and
. the convergence is extremely slow.
For every
5.
Then
Theorem 8.2. Suppose that we take P strongest quadratic inequality
representation of C k ) in the successive SDP relaxation method applied to the
example above.
(i) C k is symmetric with respect to the x 2 axis:
only if (-x 1 , x 2
(ii) Let
Then
Proof. We will prove (i) and (ii) by induction.
(i) Obviously the assertion is true for Assume that C k is symmetric with
respect to the x 2 axis. Then we know that
This ensures that C k+1 is symmetric with respect to the x 2 axis.
(ii) By definition, we know that # 2. Hence (8.4) holds for Assuming
that (8.4) holds, we prove that (8.5) holds. We first observe that
It su#ces to show that (0, -
# C k+1 or equivalently
Assume on the contrary that
such that
Here we remark that p 1 (-) can be incorporated into p # (-) since p 1 (-) # P k . By the
symmetry with respect to the x 2 axis, we see that
Thus, defining -
we obtain that
It follows from p # and the third inclusion relation of (8.6) that p # (0, -
We may further assume without loss of generality that
redefine all the relations above
remain valid. Since -
we see that Q 11 # 1 and Q 22 # 1. By (8.6) and
hence
776 MASAKAZU KOJIMA AND LEVENT TUNC-EL
Therefore, by the convexity of the quadratic function -
p(x), we obtain that
This contradicts (8.7).
The above example is simple, yet it illustrates great di#culties for the successive
relaxation method. For example, # k+1 /# k # 1. Therefore, the convergence is
slower than linear.
Note that, in any dimension, if we take a pair of ball constraints, one convex
(inclusion), the other nonconvex (exclusion), then both of the successive SDP and
semi-infinite LP relaxation methods stop in one iteration, returning the convex hull
of the intersection. Also, in the above example, if we knew that p 2 (-) a#ects only
the definition of F in the region x 1 # 0 and that p 3 (-) is only e#ective in the region
we could do elementary modifications to the method to speed up convergence
tremendously. This is a good elementary example to illustrate the fact that for such
methods to become more e#cient in practice, hybrid approaches including branch-
and-bound and branch-and-cut seem necessary. We make further remarks in the next
section.
9. Concluding remarks. We propose extensions of two fundamental lift-and-
project procedures N and N+ of Lov-asz and Schrijver [12]. The original procedures
were proposed for 0-1 integer programming problems to compute the convex hull
of feasible (integer) solutions. Our procedure applies to any nonconvex region and
as a result we do not use the key equations, Y e used in N and N+
procedures. Therefore, our relaxations are based either on two conditions: Y is
positive semidefinite and Y K # K (successive SDP relaxation method), or on only
one condition: Y K # K (successive semi-infinite LP relaxation method). In both
cases we established the properties (a) monotonicity and (b) asymptotic convergence.
The weakest version of our procedures satisfying the properties (a) and (b) uses only
rank-2 quadratic valid inequalities. We showed in section 6 that such inequalities
ensure the condition Y K # K. Finally, in section 8 we showed that even the strongest
of such relaxation procedures (using all quadratic valid inequalities) uses infinitely
many iterations to converge. In the above sense, the strongest positive result is given
in section 7 by the successive semi-infinite LP relaxation method based on rank-2
valid inequalities.
On the one hand, theoretically speaking, the best results are given in section 7:
the weakest algorithm achieving the strongest results. Moreover, the successive semi-infinite
LP relaxation method is more likely to be practical for a given general problem.
On the other hand, the relative value of SDP relaxations has been quite impressive
so far on some very special problems (e.g., the stable set problem [12]) and less
impressive on others (e.g., the matching problem [25]). Therefore, one interesting
research direction is to search for interesting classes of nonconvex sets for which the
successive SDP relaxation method is significantly better than the successive semi-infinite
LP relaxation method. For the same reason, (partial) characterizations of
nonconvex sets on which both methods perform comparably are also important.
Our convergence proofs are by contradiction, but the main argument is about
cutting o# a point using valid inequalities induced by the underlying construction.
The strongest convergence result (for the weakest algorithm) uses separating hyper-
spheres. In the other proofs, for the bad points, the separating hyperspheres may have
huge radii and converge to hyperplanes. However, for certain points and shapes, the
advantage of using more general convex quadratic inequalities is clear. This discussion
motivates us to suggest another avenue for research. It would be interesting to find
certain invariants and measures of the input of our procedures that lead to nontriv-
ial, descriptive convergence rates for our methods, perhaps only for some interesting
subclass of problems.
Recently, Kojima and Takeda [11] discussed the computational complexity of the
successive SDP and semi-infinite LP relaxation methods. They gave an upper bound
on the number of iterations which the methods require to attain a convex relaxation of
a quadratically constrained compact set F with a given accuracy # > 0, in terms of #,
the diameter of the initial relaxation C 0 , the diameter of F , and some other quantities
characterizing the Lipschitz continuity and the nonconvexity and nonlinearity of the
quadratic inequality representation PF of F .
The major di#culty in implementing the idea of the successive SDP (or semi-infinite
relaxation method in practice is the solution of a continuum of semi-infinite
SDPs (or semi-infinite LPs) to generate a new approximation C k+1 of the
convex hull of the feasible region F of a nonconvex quadratic program at each itera-
tion. In their succeeding paper [10], the authors propose implementable variants by
introducing two new techniques, a discretization technique for approximating continuum
of semi-infinite SDPs (or semi-infinite LPs) by a finite number of standard SDPs
(or LPs) with a finite number of linear inequality constraints, and a localization technique
for generating a convex relaxation of F that is accurate only in certain directions
in a neighborhood of the objective direction c. They established that, Given any positive
number #, there is an implementable discretized-localized variant of the successive
SDP (or semi-infinite LP) relaxation method which generates an upper bound of the
objective values within # of their maximum in a finite number of iterations. See also
[27] for a practical implementation of this variant and some numerical results.
--R
Interior point methods in semidefinite programming with applications to combinatorial optimization
Linear Matrix Inequalities in System and Control Theory
The Linear Complementarity Problem
The directions of the line segments and of the r-dimensional balls on the boundary of a convex body in Euclidean space
Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming
Discretization and Localization in Successive Convex Relaxation Methods for Nonconvex Quadratic Optimization Problems
Complexity Analysis of Conceptual Successive Convex Relaxation of Nonconvex Sets
2nd rev
A cone programming approach to the bilinear matrix inequality problem and its geometry
On the Generic Properties of Convex Optimization Problems in Conic Form
A recipe for semidefinite relaxation for (0
An Algorithmic Analysis of Multiquadratic and Semidefinite Programming Problems
Control system synthesis via bilinear matrix inequalities
A new reformulation-linearization technique for bilinear programming problems
A reformulation-convexification approach for solving nonconvex quadratic programming problems
Systems Sci.
Dual quadratic estimates in polynomial and boolean programming
On a representation of the matching polytope via semidefinite liftings
Convexity and Optimization in Finite Dimensions I
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Approximating quadratic programming with bound and quadratic constraints
--TR
--CTR
Masakazu Kojima , Levent Tunel, Some Fundamental Properties of Successive Convex Relaxation Methods on LCP and Related Problems, Journal of Global Optimization, v.24 n.3, p.333-348, November 2002
Akiko Takeda , Katsuki Fujisawa , Yusuke Fukaya , Masakazu Kojima, Parallel Implementation of Successive Convex Relaxation Methods for Quadratic Optimization Problems, Journal of Global Optimization, v.24 n.2, p.237-260, October 2002
Mituhiro Fukuda , Masakazu Kojima, Branch-and-Cut Algorithms for the Bilinear Matrix Inequality Eigenvalue Problem, Computational Optimization and Applications, v.19 n.1, p.79-105, April 2001
Henry Wolkowicz , Miguel F. Anjos, Semidefinite programming for discrete optimization and matrix completion problems, Discrete Applied Mathematics, v.123 n.1-3, p.513-577, 15 November 2002 | nonconvex quadratic optimization problem;global optimization;semi-infinite programming;linear matrix inequality;bilinear matrix inequality;semidefinite programming |
589300 | Global Error Bounds for Convex Conic Problems. | This paper aims at deriving and proving some Lipschitzian-type error bounds for convex conic problems in a simple way. First, it is shown that if the recession directions satisfy Slater's condition, then a global Lipschitzian-type error bound holds. Alternatively, if the feasible region is bounded, then the ordinary Slater condition guarantees a global Lipschitzian-type error bound. These can be considered as generalizations of previously known results for inequality systems, which also follow from general results by Bauschke and Borwein in [ SIAM Review, 38 (1996), pp. 367--426] and Bauschke, Borwein, and Li in a 1997 report. However, the proofs in the current paper are considerably simpler. Some of the results are generalized to the intersection of multiple shifted cones (with different shifts). Under Slater's condition alone, a global Lipschitzian-type error bound does not hold. It is shown, however, that such an error bound holds for a specific conic region. For linear systems we establish that the sharp constant involved in Hoffman's error bound is nothing but the condition number for linear programming as used by Vavasis and Ye in [ Math. Programming, 74 (1996), pp. 79--120]. | Introduction
In optimization theory it is often desirable to measure the distance to the solution set from a certain
given point. In general, this distance can be difficult to assess, since one may not have a complete
knowledge about the solution set. However, if the form of the solution set is explicitly given, then
in some cases it is possible to estimate the distance to the solution set by the so-called constraint
violation which is computable. This kind of estimation is termed error bound relation. The first
such result was obtained by Hoffman [7] for systems of linear equalities and inequalities. We shall
discuss Hoffman's error bound in the paper too. A recent extensive survey on various types of error
bound results can be found in Pang [15].
Most papers discussing error bound results assume that the solution set is given by equations and
inequalities, e.g.
For a given point x the amount of constraint violation can be measured as the following quantity
with the notation (y)
A measure for constraint violation is similar to a penalty function in the sense that it takes positive
value for points outside the set, and zero otherwise. Note that a measure for constraint violation
should be easy computable, such as the case for the above defined function v(x). Hoffman's
lemma [7] states that if S 6= ;, and f i and g j are all affine linear functions, then there is a positive
dist (x; S) -v(x) (1.1)
for all x 2 ! n . This means that the distance to S is of the same magnitude as v(x). Such a relation
is known as a Lipschitzian type error bound.
In the case that f i and g j are not linear, the above inequality (1.1) does not hold in general. Early
results concerning nonlinear functions are due to Robinson [17] and Mangasarian [13]. Robinson [17]
showed that for inequality systems if all functions are convex and differentiable, S is bounded and
the Slater condition holds, i.e. there is a - x such that g j (-x) ! 0 for all j, then relation (1.1) holds.
Mangasarian [13] removed the assumption that S is bounded by assuming an additional asymptotic
constraint qualification condition, which however can be difficult to verify in general.
In this paper we consider the following convex conic set:
is a subspace of ! n and K ' ! n is a closed convex cone. Polynomial-time
interior-point algorithms for solving convex optimization problems with convex conic feasible set
were introduced in a systematic manner by Nesterov and Nemirovskii [14]. It turns out that
many important classes of optimization problems, such as linear programming and semidefinite
programming, can be cast in this form. The focus of this paper is to discuss how error bound
type relation can be established for such problems. Throughout this paper we make the following
assumption:
Assumption 1 F 6= ;.
The organization of the paper is as follows. In the next section we prove that with a proper definition
of constraint violation a Lipschitzian type error bound (1.1) can be established for general convex
conic problems, under various conditions on the relations between L and K, including Slater type
conditions. In Section 3 we discuss a link between the constant in Hoffman's error bound and the
so-called condition number for linear programming. Finally, we conclude the paper in Section 4.
We use the following notation in this paper. Matrices are denoted by capital letters, e.g. X. For
indicates the maximum eigenvalue of X, and - min (X) the minimum
eigenvalue of X. We denote n-dimensional Euclidean space by ! n and its nonnegative quadrant
by
. The space of all symmetric n by n matrices is denoted by S n\Thetan and the cone of all
symmetric positive semidefinite n by n matrices by S n\Thetan
. Vector e represents a vector of all ones
with appropriate dimension. For a vector we use the capitalized letter V to denote the
diagonal matrix which takes v as its diagonal elements. For two vectors
as the component-wise Hadamard product. We use the Euclidean norm for vectors
and the spectral norm for matrices. A vector a - 0 means that each component of a is nonnegative,
and X - 0 indicates that X is positive semidefinite.
systems
Consider the convex conic set (1.2). For convenience we further assume that K is a pointed and
solid cone, i.e. K "
The dual of K is
Since K is pointed and solid, K too is a closed, convex, pointed and solid cone.
An immediate next question is: How can we define a constraint violation function for F? For this
purpose we note the following lemma due to Moreau (see Theorem 31.5 in [18]).
Lemma 2.1 For any x 2 ! n there is a unique x p 2 K and x d 2 K such that
In fact, x p is simply the projection of x onto K and kx d k measures the distance from x to K. A
natural definition for the constraint violation for F is now in order:
Definition 2.1 For any x 2 ! n define
dist
as the constraint violation function for F .
It is readily seen that v(x;
It is, however, not immediately clear how the function v(x; F) can be computed. Below we shall
see some examples in which this function is explicitly derived. First we consider the case
, the nonnegative quadrant of ! n . Clearly,
which is exactly the usual definition of the violation for nonnegativity constraints.
Another example is
, the cone of n by n symmetric positive semidefinite matrices.
Consider a given n by n symmetric matrix X. Following Lemma 2.1 we know that there is unique
positive semidefinite matrices X p and X d such that
and X d can be computed as follows. Let an orthonormal matrix and is a
diagonal matrix with eigenvalues of X as its components. Splitting
\Gamma denote the nonnegative and nonpositive parts of respectively, it follows that X
denotes the
minimum eigenvalue of X.
Finally, we consider another popular convex cone: the second order cone K defined as
It can be shown that in this case
2:
In general, Definition 2.1 is only related to the geometry of the object under consideration.
Consider now an arbitrary point z Assume that z 62 F . The following problem yields a
unique point in F with the shortest Euclidean distance to z:
subject to x
Let this optimal solution be -
x. The Karush-Kuhn-Tucker optimality condition for (Proj) is given
as follows:
Hence,
where the first inequality follows from the fact that z p 2 K and - 2 K .
Let the projection of z onto the affine subspace b + L be z l . Then,
dist
Substitute this relation into (2.2) we obtain
(dist (z; dist
In Section 3 we shall discuss how to further bound the errors when K is a polyhedral cone, which is
the situation when the original Hoffman lemma applies. In the rest of this section we assume that
K is a general convex cone. In addition to this we assume that the Slater condition is satisfied, i.e.
Assumption
The following lemma is well-known; see e.g., Duffin [5], Borwein and Wolkowicz [2], Luo, Sturm
and Zhang [11], Nesterov and Nemirovskii [14] and Sturm [21]. For completeness we provide here
a short proof.
Lemma 2.2 Suppose that Assumption 2 holds. Then, for any y 2 must
follow that b T y ? 0.
Proof. Suppose, for the sake of deriving a contradiction, that there is y 6= 0 such that y 2
Consider the hyperplane
For any x while for any x 2 K, since y 2 K we have y T x - 0.
This means that H y separates b +L and K, yielding a contradiction to the fact that b +L intersects
with the interior of K.For fixed - x we consider again the system (KKT) in terms of - and -. After some re-arrangements
this yields 8
which is a closed convex cone as well.
Note that -
case and is omitted in our proof. In many applications, 0 62 L and so
We shall mention another easy case, i.e. - x lies in the interior of K then -
f0g. In this case
x, and therefore
dist (z; F) - dist
due to (2.3). In what remains we shall only concentrate on the situation when - x 62 int K.
Remark that for -
K is known as a face of K .
The condition (2.4) is equivalent to
Lemma 2.3 If Assumption 2 holds then
Proof. Suppose for contradiction that there is y
K . This means that
However, -
which is impossible due to Lemma 2.2. 2
Now we define the minimum angle between L ? and -
K as
Note that both L ? and -
K are closed cones. According to Lemma 2.3, it follows that
for any -
In order to pursue our analysis further, one of the following two mutually exclusive cases will be
considered.
Assumption 3 The set
Assumption
Let us first consider the situation when Assumption 3 holds. In that case we know that there exists
such that for any -
we always have
Now take -
K . Let the projection of 0 onto -
-s
s
s
s
Figure
and the cone -
K .
Let the angle between - and - \Gamma p be '. Clearly, ' -=2. Moreover,
Denote
We are now in a position to prove the following error bound result.
Theorem 2.1 If Assumption 2 and Assumption 3 hold then
dist (z; F) -v(z; F)
for all z
Proof. By (2.5) we have
dist (z; F)= sin ' -dist (z; F):
Using the first equation in (2.4) we also have
')dist (z; -dist (z; F):
Recall relation (2.3). By the above estimations on k-k and k-k, it follows from (2.3) that
(dist (z; F)) 2 -dist (z; F)(kz d k dist
and consequently
dist (z; F) -v(z; F):In the other situation, namely if Assumption 4 holds, then a similar result can be shown.
Theorem 2.2 If Assumption 4 holds, then for any b 2 ! n we must have (b
Moreover, there is a constant - ? 0, independent of b, such that
dist (z; F) -v(z; F)
Proof. First we show that (b Suppose otherwise that there is b with
(b
Then, there will be a hyperplane separating b +L and K, say with such that
Since K is a closed cone, the above separation implies that y T x - 0 for all x 2 K and
Moreover, we also have y T This is in contradiction with the condition
Compared with Lemma 2.3, we have now a stronger relation: f0g. This means that
the proof of Theorem 2.1 can remain exactly the same, except that now ' ? 0 can be taken as the
minimum angle between L ? and K which is independent of b. 2
We remark that both Theorem 2.1 and Theorem 2.2 easily extend to the case when L is a closed
cone.
Theorem 2.3 Suppose that K 1 is a closed convex cone and K 2 is a closed, convex, solid and pointed
cone. Furthermore, suppose that (b compact. Then there
is a constant - ? 0 such that
dist (z; F) -(dist (z; b dist (z; K 2
for all z
Proof. We follow similar lines as in the proof of Theorem 2.1. Consider
subject to x 2 b +K 1
Let the optimal solution be -
x. The Karush-Kuhn-Tucker optimality condition yields:
Let
and
Both -
are closed convex cones.
Now we claim that
Suppose such is not the case. Then, one should be able to find - 6= 0 satisfying
Hence, b T This implies
that contradicting the Slater condition.
are closed convex cones and, moreover, -
2 is a solid pointed cone, we derive
from (2.6) that -
2 can be strictly separated from \Gamma -
1 . Due to compactness of F we may let ' be
a positive lower bound on the minimum angle between this separating hyperplane and -
2 . Then
we have
and consequently
dist (z; b +K 1 )k- dist (z; K 2
dist (z; K 2
The desired result thus follows. 2
Similarly, we have the following result, the proof of which is pretty much the same as that of
Theorems 2.2 and 2.3 and is omitted here.
Theorem 2.4 Suppose that K 1 is a closed convex cone and K 2 is a closed, convex, solid and pointed
cone. Furthermore, suppose that here is a constant - ? 0,
independent of b, such that
dist (z; F) -(dist (z; b dist (z; K 2
for all z
When more than two cones are concerned, a similar result holds under Slater's condition. First we
note the following lemma, see e.g. [11].
Lemma 2.4 Let K be a convex cone and int K 6= ;. Then, x 2 int K if and only if for any
holds that
Theorem 2.5 Let K i be convex cones, m. Suppose that
Then, there is - ? 0 such that
dist
dist (z; K i
for any z 2 ! n .
Proof. Consider
subject to x 2 K
Hence, for the optimal solution -
x the KKT condition yields
with
Let
By Lemma 2.4 there exists
m.
Let
with z ip 2 K i , z id 2 K
ip z due to Lemma 2.1. Moreover, kz id dist (z; K i m.
Therefore,
z
z T
kz id kk- i k:
On the other hand, since
dist (z; K i
and so by letting
it follows that
dist
dist (z; K i ):Theorem 2.1 can be viewed as an analogue to Robinson's result for convex inequality systems. In
the form of convex inequality systems, Theorem 2.2 can be found in Hu and Wang [9] and Deng
and Hu [3]. In particular, Deng and Hu [3] investigated the case when K is the cone of positive
semidefinite matrices. This case is known as linear matrix inequalities (LMIs for short). In its
optimization version it is also called semidefinite programming and has received intensive research
attention recently. Sturm [20] mainly investigated error bounds for LMIs in the absence of Slater's
condition. In fact, in the context of LMIs, both Theorem 2.1 and Theorem 2.2 also follow from the
analysis in [20]. Moreover, an example was given in Sturm [20] showing that Assumption 2 alone
cannot guarantee a global Lipschitzian type error bound even for LMIs. Such an error bound is
only possible when an additional scaling factor is present.
Below we shall discuss how to derive some conditioned error bound relation for the convex conic
problem (1.2) under Assumption 2, without assuming Assumption 3 and Assumption 4.
In this situation the recession cone L " K must be non-empty and it is not contained in the interior
of K.
For a fixed positive angle consider the following cone
the projection of x onto L and the cone L " K has an angle at least -=2
Theorem 2.6 Suppose that Assumption 2 holds. There exists a constant - ? 0 such that
dist (z; F) -v(z; F):
for all z 2 C.
Proof. Observe that if -
x is the projection of z on F , then it must also be the projection of z
on F for any y 2 L ? . This can be seen as follows. The fact that - x is the projection
of z is equivalent to the existence of - 2 L ? and - 2 K such that
(See also (2.1)). Now if z is changed to z y, then we need only to change - to -
satisfy the same set of KKT conditions.
Remark also that to prove the theorem it is sufficient to show that, for any z 2 C, its projection
onto F is contained in a compact set.
Suppose that the theorem is false and that there is a sequence fz :::g, such that the
corresponding projection on F , f-x unbounded. Due to the above remarks
we have made, we need only to consider the projection of z (k) onto the subspace L. Without loss
of generality, assume that z
For sufficiently large k we have
where the first inequality is because -
x (k) must be pointing towards the cone of recession directions
and the last inequality is due to the fact that k-x (k) k ! 1. This contradicts to -
x being
the closest point in F to z (k) .For any given point z C. The
following relation is immediate.
Lemma 2.5 dist (z; F) - dist (z
Proof. Let the projection of z 2 onto F be - x 2 . Then,
dist (z; F) -
where we used the fact that z 1 2 L " K and so -
2.5 and Theorem 2.6 we have
Theorem 2.7 Suppose that Assumption 2 holds. Then
dist (z; F) -v(z
for all z
3 Hoffman's error bound and the condition number
In this section we shall discuss error bounds for the linear system fx j A T x - bg with A 2 ! m\Thetan and
m. This is the setting for which Hoffman's error bound result applies ([7]). Our purpose
is to see how the constant in Hoffman's bound is related to other known quantities for the linear
system. Previous results on the constant of Hoffman's bound can be found, e.g., in [12, 1, 6, 10].
By introducing a slack confine ourselves to the range space of A T , i.e.
Accordingly,
.
For a given z
. Let
which minimizes the distance
to s(z).
Let
ng n K:
Then, for this given s(-x) - 0 we can rewrite (2.1) as
A J -
As (3.1) is a necessary condition for optimality, it is certain that (3.1) is feasible. What remains to
be analyzed is the size of the solution. A key ingredient in our analysis is the following lemma.
Lemma 3.1 Suppose that A has full row rank. Then,
diagonal and D -
Lemma 3.1 was first shown by Dikin [4] and was used in his convergence analysis for affine scaling
methods. Among others, Stewart [19] and Todd [23] rediscovered this result later.
The meaning of Lemma 3.1 can be interpreted as follows. It is well known that
0g and are orthocomplements to each other. Obviously, for
a given positive diagonal matrix D, Null(A) can only intersect with at the origin,
hence there must be a positive angle between them. Lemma 3.1 further states that the minimum
angle between Null(A) and uniformly bounded from below by a positive constant
which is independent of D.
To understand this fact we may consider the following example. Let
simply the line x 1 For a given positive diagonal matrix D, contained
in the first and the third quadrants. The angle between these two subspaces never exceeds -=4.
An important property of the constant -(A) is that it reflects an intrinsic, geometric relationship
of the spaces. Vavasis and Ye [24] used this constant -(A) as a measure of complexity for solving
the related linear programming problem. Their results showed that, in a real-number computation
model, linear program is solvable in polynomial-time, in terms of total number of basic operations,
with respect to the dimension n and the complexity measure log -(A). For problems with integral
input data, this result yields the usual polynomiality complexity result for linear programs in terms
of the input-length.
Holder, Sturm and Zhang [8] showed that -(A) plays an important role in sensitivity analysis
for linear programming. Furthermore, Sturm and Zhang [22] extended some of the results in [8]
to semidefinite programming. It is known however, that Lemma 3.1 cannot extend to general
semidefinite programming for arbitrary invariant scaling of the cone S n\Thetan
Fortunately, in analyzing (3.1) we need only to deal with a polyhedral cone. To see how condition
number -(A) can play a role in error bound analysis, we need to introduce a number of technical
lemmas.
First we note the following equivalent definition of -(A) for arbitrary matrix A due to Vavasis and
Ye [24].
Lemma 3.2 It holds that
kck
positive diagonalg:
For our analysis it is important to know the size of a solution for a linear system. To this end,
we note the following two lemmas. Remark that Renegar [16] studied similar problems in a quite
general framework using a quantity called distance to ill-posedness.
Lemma 3.3 Suppose that A has full row rank. Further assume that fx j
Then, there is a solution -
x in 0g such that
k-xk -(A)kbk:
Proof. Consider a linear program
subject to
and its dual
(D) maximize b T y
subject to A T y
Both (P) and (D) satisfy Slater's condition. Therefore their respective analytic central paths
satisfying the following relation:
x(-)s(-e:
Multiplying the second equation in (3.2) with X(-), the diagonal matrix with x(-) as its diagonal
components, and applying the first equation in (3.2) we obtain
e:
Substituting this into the second equation and finally using the third relation in (3.2) we have
Now we can apply Lemma 3.1 to obtain
The lemma is proven. 2
Next we shall extend this result to the case when Slater's condition is no longer assumed.
Lemma 3.4 Suppose that A has full row rank. Further assume that fx j
Then, there is a solution -
x in 0g such that
k-xk -(A)kbk:
Proof. Let Consider a perturbed set
Clearly, F ffi contains an interior point and therefore Lemma 3.3 can be invoked. Let x
The set fx be a cluster point of x ffi as
and
shall compare the condition number of A and that of its submatrices.
Proof. By Lemma 3.2,
kck
positive diagonalg:
partitioned in accordance with
For fixed c 1 6= 0 and fixed positive diagonal matrix D 1 . Let c positive
diagonal and D 2 ! 0. Clearly, the set of solutions minimizing kD 1=2 converges to the
set of solutions minimizing kD 1=2
)k. For given c and D let y(c; D) be a maximum norm
solution among solutions which minimize kD 1=2 similarly. It follows
that
lim sup
As a consequence,
and so the lemma is proven. 2
Applying Lemmas 3.4 and 3.5 to (3.1) we have
Finally we shall give a bound on the constant in Hoffman's error bound for linear systems.
Theorem 3.1 Suppose that and A has full row rank. It holds that
dist (z; F) -(A)(cond(AA T
for any z 2
Proof. Using (3.1) and (3.3),
By (2.3), on one hand we have
On the other hand,
Combining these two inequalities, the desired result follows. 2
Conclusions
In this paper we discuss error bounds for sets in convex conic form. The notion of constraint
violation is extended to this class of problems. For a number of applications the measure of
constraint violation is easy computable. We show that under Slater's condition, and additionally,
if either the feasible set is bounded or the recession directions satisfy the Slater's condition, then
there is a global Lipschitzian type error bound for general convex conic problems. These results
can be generalized to the intersection of multiple convex cones, or intersection of two shifted convex
cones, one of them being pointed and solid. If only Slater's condition is satisfied without additional
assumptions on the feasible region, then a global error bound is impossible as shown by Sturm [20].
In this case, one may still identify a region in which Lipschitzian type error bound holds. Finally,
we discuss the bounds in Hoffman's lemma for linear systems. It is shown that such a bound is
linked closely with the condition number for linear programming as investigated by Vavasis and
Ye [24].
Acknowledgement
I would like to thank Jos Sturm for pointing out an error in an earlier version
of the paper.
--R
The distance to a polyhedron
Characterizations of optimality for the abstract convex program with finite dimensional range
Computable error bounds for semidefinite programming
Iterative solution of problems of linear and quadratic programming
Linear Inequalities and Related Systems
Approximations to solutions to systems of linear inequalities
On approximate solutions of systems of linear inequalities
sensitivity analysis and parametric programming
On approximate solutions of infinite systems of linear inequalities
The sharp Lipschitz constants for feasible and optimal solutions of a perturbed linear program
Duality results for conic convex programming
A condition number of linear inequalities and equalities
A condition number for differentiable convex inequalities
bounds in mathematical programming
Some perturbation theory for linear programming
An application of error bounds for convex programming in a linear space
Convex Analysis
On scaled projections and pseudoinverses
bounds for linear matrix inequalities
On sensitivity of central solutions in semidefinite programming
A Dantzig-Wolfe-like variant of Karmarkar's interior-point linear programming algorithm
A primal-dual interior point method whose running time depends only on the constraint matrix
--TR | convex conic problems;error bound;condition number |
589314 | Minimizing a Quadratic Over a Sphere. | A new method, the sequential subspace method (SSM), is developed for the problem of minimizing a quadratic over a sphere. In our scheme, the quadratic is minimized over a subspace which is adjusted in successive iterations to ensure convergence to an optimum. When a sequential quadratic programming iterate is included in the subspace, convergence is locally quadratic. Numerical comparisons with other recent methods are given. | Introduction
In this paper we consider the problem of minimizing a quadratic over a sphere:
subject to kxk
where A is a symmetric n n matrix, b 2 R n , T denotes transpose, and k k
is the Euclidean norm. This minimization problem is often called the trust region
subproblem since it must be solved in each step of a trust region algorithm [1, 2, 3, 15,
19]. Problems of this form arise in many other applications including regularization
methods for ill-posed problems [14, 26], and graph partitioning problems [10].
Although the solution to (1) can be expressed in terms of a diagonalization of A,
this representation is practical only when n is small. In this paper, we focus on the
large-scale case. One approach to the large-scale case, developed by Golub and von
Matt in [5] (also see [4]), is to (partially) tridiagonalize A using the Lanczos process
and then solve tridiagonal problems to obtain an approximate solution to (1). For
further developments of this approach, including preconditioning and a Fortran 90
implementation HSL VF05 in the Harwell Subroutine Library, see Gould et al. [7].
For the method developed in this paper, we use an approach in the spirit of the
Golub/von Matt/Gould et al. scheme to obtain a starting guess.
Parametric eigenvalue approaches to the sphere constrained problem (1) are developed
by Sorensen [24] and by Rendl and Wolkowicz [20]. The relationship between
these two approaches is discussed in detail in [20]. Roughly, Sorensen's approach
involves constructing an approximation to the solution of (1) from the solution to
a related eigenvalue problem. Since this approximation may not satisfy the bound
on the norm of the solution, a series of eigenvalue problems are solved, and in the
limit, the bound on the norm of the solution is fullled. In the approach of Rendl
and Wolkowicz, the same eigenvalue problem is solved in each iteration, however, the
bound on the norm of the solution is satised by maximizing a related dual func-
tion. The eigenvalue problems arising in either approach can be solved using Arnoldi
techniques such as those developed in [13]. In the \hard case" (see [16]) where b is orthogonal
to the eigenvectors associated with the smallest eigenvalue of A, Sorensen's
approach needs to be modied. An e-cient algorithm for the hard case is developed
by Rojas in her thesis [21]. She also uses this algorithm to solve some di-cult ill-posed
problems of Hansen [11, 12]. The approach of Rendl and Wolkowicz does not
need modication in the hard case, however, the convergence of algorithms for the
eigenvalue problem may be slower when the computed eigevalue is not simple.
The approach in this paper, which we call the sequential subspace method (SSM),
involves solving (1) with the additional constraint that x is contained in a subspace.
We show that convergence is locally quadratic (locally cubic when if the
subspace contains the iterate generated by one step of the sequential quadratic programming
(SQP) algorithm applied to (1). The convergence is quadratic even when
the original problem is degenerate with multiple solutions, and with a singular Jacobian
for the rst-order optimality system. Descent of the cost at a nonoptimal point
can be ensured by including in the subspace either the cost gradient or an eigenvector
associated with the smallest eigenvalue of A. We observe in numerical experiments
that appropriate small dimensional subspaces are generated by preconditioned
Krylov space and minimum residual techniques. Comparisons with the algorithms of
Sorensen [24], of Rendl and Wolkowicz [20], and of Gould, Lucidi, Roma, and Toint
are given in Section 5.
A solution of the problem
subject to
is any eigenvector associated with the smallest eigenvalue of A. In comparing the SSM
approach to algorithms for solving the eigenproblem, it follows from the discussion
of Sleijpen and Van der Vorst in [22] that an SQP iterate for (2) is closely connected
with the Rayleigh quotient iteration [18, p. 70], which is cubically convergent [18, p.
73]. In [22] approximate solutions to the SQP system are used to build up subspaces
containing the approximation to the eigenvector. In this paper, we solve the SQP
system relatively precisely, and we form a small dimensional subspace containing the
SQP iterate. After computing the new approximation in the subspace, the previous
information is discarded; hence, the computer memory requirements are relatively
small.
Complete diagonalization
If there exists a solution y of (1) with kyk < r, then A is positive semidenite and
y is the global minimizer of the quadratic x T Ax 2b T x. Thus, when a minimizer
of (1) lies in the interior of the constraining sphere, the constraint can be ignored
and the optimization problem can be approached using techniques for unconstrained
optimization. Consequently, we restrict our attention to the following equality constrained
problem:
subject to
The solutions to (3) are characterized by the following result (see [23, Lem. 2.4, Lem.
Lemma 1. The vector x is a solution of (3) if and only if r and there
exists such that A + I is positive semidenite, and
The solution to (3) can be expressed in terms of the eigenpairs of A. Let
T be a diagonalization of A where is a diagonal matrix with diagonal elements
and is the matrix whose columns 1 ,
are
orthonormal eigenvectors of A. Dening
Lemma 2. The vector
is a solution of (3) if and only if c is chosen
in the following way:
(a) Degenerate case: If
then
are arbitrary scalars satisfying the condition
(b) Nondegenerate case: If (a) does not hold, then c
where > 1 is chosen so that
Proof. Simply check that the su-cient optimality conditions of Lemma 1 are
satised. The degenerate case, where the Jacobian of the rst-order optimality system
may be singular, coincides with the \hard case" of More and Sorensen [16] where b
is orthogonal to the eigenspace associated with the smallest eigenvalue of A and the
multiplier is equal to 1 . In the nondegenerate case, the multiplier is chosen so
that A+ I is positive denite and the solution
the constraint x T
In the nondegenerate case, equation (5) leads to upper and lower bounds for the
multiplier . Since i
kbk
r
To obtain a lower bound, observe that
which yields the relation
Utilizing the upper and lower bound u and l and the strict convexity of the left
side of (5) on the interval ( l ; u
it is easy to devise e-cient algorithms to compute
a solution of (5).
3 Incomplete diagonalization, local convergence
At iteration k in the sequential subspace method (SSM) for (3), we impose the additional
constraint that x lies in a subspace S k of R n . Hence, the new iterate x k+1 is a
solution of the problem
subject to
We show that the convergence is locally quadratic, even when the original problem
(3) is degenerate, if we include an SQP iterate associated with x k
in S k
If V is an n l matrix with orthonormal columns that span S k , then (8) is
equivalent to the problem
subject to
After substituting for x, (9) reduces to the following problem in R l :
subject to
l is small, then (10) can be solved by complete
diagonalization as in Section 2. Or if B has a sparse factorization, then (10) can be
solved quickly using the Newton approach developed in [16].
In theory, a tridiagonal B is generated using the Lanczos process [6]. In particular,
if v 1 is a unit vector and v i
is the i-th column of V, then the Lanczos process can be
expressed as follows:
Algorithm 1 (Lanczos)
ksk
Here d is the diagonal and u is the superdiagonal of the tridiagonal matrix B. If
then the Lanczos process is terminated and the column space of
V and AV coincide.
It is well-known that the columns of V generated by this process may deviate
signicantly from orthogonality due to the propagation of rounding errors. And when
this happens, (9) is no longer equivalent to (10). Nonetheless, Gould et al. observe
in [7] that the solution to (10) often provides a good approximation to the solution of
despite the loss of orthogonality. The Lanczos process can be repaired, in order to
restore orthogonality, by using a Householder process to generate the columns of V.
This process, however, requires products between a vector and each of the previously
computed columns of V. Thus the overhead needed to maintain orthogonality grows
like nl 2 in the number of
ops and like nl in storage. This overhead can be signicant
when n or l is large. On the other hand, to compute a high accuracy solution, we
need to maintain orthogonality in order to obtain an equivalent problem (10). This
leads us to focus on approaches that involve subspaces where l is much smaller then
n. In particular, for an implementation (Algorithm 4) of the SSM proposed later, l
is either 4 or 5.
Since sequential quadratic programming (SQP) techniques often converge rapidly,
with a good starting guess, we always include the SQP approximation in the subspace
. If x k
is the current iterate, which we assume satises the constraint
and k is the current approximation to the multiplier associated with the constraint,
then the SQP iterate can be expressed in the following way:
and
z and are solutions of the following linear system:
When the coe-cient matrix in (11){(12) is singular, we let (z; ) be the minimum
residual/minimum norm solution; that is, (z; ) is gotten (in theory) by multiplying
the right side by the pseudoinverse of the coe-cient matrix (see [8]). The SQP method
is equivalent to Newton's method applied to the nonlinear system
A solution x k+1 to the subspace problem (8) is an approximation to the solution of
(3). To obtain an estimate for the multiplier of Lemma 1, we minimize the Euclidean
norm of the residual b Ax k+1 x k+1 over the scalar . This works out to give
(b Ax) T x
We now examine the local convergence of a solution x
of (8) and the multiplier
estimate (14) under the assumption that S k contains
is a solution to (11). Let S denote the set of minimizers of (3), and let be the
multiplier given by Lemma 1. In the nondegenerate setting where A+ I is positive
denite, we show that the iteration is locally, quadratically convergent to the unique
solution of (3). In the degenerate case = 1 where S has more than one element,
we obtain local quadratic convergence to S , where distance is measured in the usual
way:
In the nondegenerate degenerate-case where contains a single
element, we obtain local quadratic convergence for a \safe-guarded" choice of k .
Our convergence result in the special nondegenerate degenerate-case is given later in
Lemma 5, while our local convergence result in either the nondegenerate case or the
degenerate case with multiple solutions is the following:
Theorem 1. Let be the multiplier of Lemma 1 associated with the set of
solutions S of (3), and suppose that either A+ I is positive denite, or
with (4) a strict inequality. Then there exist positive constants and C with the
property that for any (x k ; k ) such that
and for any subspace S k that contains the SQP iterate xSQP associated with (11){
(12), any solution x k+1 of (8) and associated multiplier k+1 given by (14) satisfy the
following estimate:
The eigenvalue problem (2), corresponding to b = 0, is always degenerate (with
multiple solution) and the error has the following special form:
When the multiplier is estimated using (14), it can be shown, when that the
error in the multiplier is bounded by a constant times the error in the solution vector
squared (see the remark at the end of Section 3.1). It follows that for some constant
the same as the convergence result for the Rayleigh quotient iteration.
3.1 Nondegenerate problems
We begin the derivation of Theorem 1 with the nondegenerate case:
Lemma 3. If (3) has a solution x and an associated multiplier with > 1 ,
then there exist a neighborhood N of (x ; ) and a constant C with the property that
for any and for any subspace S k that contains the
SQP iterate xSQP associated with (11){(12), any solution x k+1 of (8) and associated
multiplier k+1 given by (14) satisfy the following estimate:
Proof. Since > 1 , the matrix A+ I is positive denite, and the Jacobian
of the nonlinear system (13) is nonsingular at By the standard convergence
theorem for Newton's method applied to a smooth system of equations, there exist a
neighborhood N of
) and a constant c such that
whenever
Let and be positive scalars chosen so that
for all x 2 R n , let f be the cost function in (3), let L be
the Lagrangian dened by
A Taylor expansion around x yields the following relation:
for any x 2 B r
rg. Combining this with (16) gives
for any x 2 B r
If p is the projection of xSQP onto B r , then
Hence, we have
xSQP for some
, it follows that p 2 S k and f(x k+1 ) f(p). Combining
this inequality with (17) and (18) gives
which implies that
Making this substitution gives
r
Combining (21) with (19), the proof is complete.
Remark. For the eigenvalue problem (2), we have x
In this case, (20) yields
and (21) becomes
3.2 Degenerate problems
Now consider local convergence in the degenerate case where Referring to
Lemma 2, the degenerate case can only happen when i
Any solution to (3) in the degenerate case can be expressed x
and 1 is any linear combination of the vectors i
satisfying the relation
Initially, we suppose that k 1 0, in which case the projection of S on the
eigenspace associated with contains a sphere of radius -. Our convergence result
is the following:
Lemma 4. Suppose that the multiplier of Lemma 1 associated with the set of
solutions S of (3) is given by is the
component of an element of S in the eigenspace associated with E 1 . Then there exist
positive constants and C with the property that for any (x k ; k ) such that
and for any subspace S k that contains the SQP iterate xSQP associated with (11){
(12), any solution x k+1 of (8) and associated multiplier k+1 given by (14) satisfy the
following estimate:
Proof. Initially, let us assume that k is near 1 , but k 6= 1 . In this case,
the linear system (11){(12) is nonsingular, and there exists a unique solution (z; ).
We expand z and x k in terms of the eigenvectors of A writing
and x
. Utilizing (11), we obtain
Substituting this in (12) gives
Let us dene
I)x k and
. For
If x 2 S , then since
we have
Let k be the error at step k dened by
By (26), we have
since
is near - 2 > 0 when x k is near S . From (23), we have
Let x be the closest element of S to x k and dene
. Then we have
By (30) the i
component of in error by O( k ) since i ,
the i
component of x k
, is in error by O( k
by (31).
implies that
. Combining this with (28) and
Hence, for i 2 E+ the i
component of xSQP is in error by O( 2
be the seminorm associated with projection into the eigenspace associated
with
Then we have
for all x ng. Proceeding as we did
earlier, but replacing norms by seminorms,
where p is the projection of xSQP onto the ball B r , and
xSQP for some
by (30) and (32), and z is perpendicular to x k
by (12), we have
This implies that
Consequently, kp x
which combines with (34) to give
By the triangle inequality,
Let k k 1 be the seminorm dened by
and recall that kx k . By the Pythagorean theorem and the fact
that x k+1
has length r, we have
which implies that
The distance from x k+1 to S is given by
where x is any element of S . Relations (35){(38) yield dist(x
Combining these estimates, we have k+1
This analysis was given under the assumption that k 6= 1 . In the special
case k
show how the analysis should be modied. With the change
of variables z =
and the substitution x
, the SQP system
(11){(12) is equivalent, by orthogonal transformation, to
D#
where D is a diagonal matrix with diagonal elements
then the rst s diagonal elements of D and the rst s components of D vanish.
Hence, the rst s equations in (39) imply that The next n s equations give
while the last equation in (39) gives
The minimum norm solution to this last equation is
By (40),
Combining these bounds, we have these relations, all the analysis
from (33) onward can be applied, leading us to the estimate k+1
Lemmas 3 and 4 yield Theorem 1.
3.3 Nondegenerate degenerate-problems
Finally, let us consider the nondegenerate degenerate-case where
and the 1 component of x in the eigenspace associated with the smallest
eigenvalue of A vanishes. Our convergence result is the following:
Lemma 5. If (3) has a solution x is given by (22), then there
exist a neighborhood N of (x ; 1 ) and a constant C with the property that for any
and for any subspace S k that contains the SQP iterate xSQP associated with (11){
(12), the solution x k+1 of (8) and associated multiplier k+1 given by (14) satisfy the
following estimate:
In the case that k
k, C can be chosen so that
Proof. Focusing on the numerator in (24), and substituting
, we
With this substitution for the numerator of in (24), we obtain
the denominator terms in (43) have the following lower bound:
Another lower bound is gotten by neglecting terms corresponding to indices i
where the seminorm k k 1 is dened in (36). Combining (43){(45) yields:
Returning to our previous analysis of the degenerate case, it follows from (29) and
(46) that for
Here we exploit the fact that for . In order to analyze (47), we
consider two separate cases: (i) kx k
x k+ and (ii) kx k
x k+ ,
where is any xed constant satisfying
In case (i),
x k+
x k+
We now derive a similar bound for the left side of (49) in case (ii). In this case, it
follows from (42) that
x k+
for any x 2 R n , where a + subscript on a vector is used to denote its projection on
the eigenspace associated with E+ . After substituting for using (20), we obtain
for any x 2 B r
. Assuming x k
is a unit vector (note that when kx k
only if x
We will establish a uniform bound for the expression (51) when x k
is near x ,
To facilitate this analysis, we rst consider
whether the equation
has a solution of the form
y of this form, the Schwarz inequality gives
jy T
Since the unit vector y is orthogonal to the eigenspace associated with 1 ,
Multiplying (52) by y T and using both (53) and (54) gives
For any x 2 B r , we have
which implies that
since yields the relation:
Referring to (55), we have a contradiction when kx x k+ .
In summary, the equation (52) has no solution over the set Y consisting of those
y that satisfy the following conditions:
lies in the closure of Y, then by (57),
since any solution of (52) satises (55), y cannot be a solution of (52).
Since (52) has no solution over the closure of Y, the following constant - is strictly
positive:
Since
lim
min
(51) is bounded uniformly over all x k near x with
x k+ .
Thus in either case (i) or (ii), the left side of (49) is bounded, and by (47), we have
the same as relation (30) in the degenerate case.
To establish the analogue of (32) for indices i 2 E+ , we need a dierent bound for
the next to last term in (43). From the identity
Hence, we have
Since
implies that
It follows that
This estimate along with the lower bound (44) for the denominator in (43) yields the
relation
The reminder of the analysis is identical to that given for the degenerate case (Lemma
4), starting with (32). Since S it follows from the analysis of Lemma 4 that
In the special case k
Hence, the in (60) can be absorbed in the kx k
completes the proof.
Implementation
In our experimentation with the SSM, we put the following four vectors in S k in each
iteration: xSQP , x k
, and an estimate for an eigenvector of A associated with
the smallest eigenvalue. By including x k in S k , the value of the cost function can only
decrease in consecutive iterations. The multiple b Ax k
of the cost function gradient
ensures descent if the current iterate does not satisfy the rst-order optimality
conditions. The eigenvector associated with the smallest eigenvalue will dislodge the
iterates from a nonoptimal stationary point. We also use this vector in a \safe-guard"
strategy designed to keep A positive denite.
4.1 The SQP system
Now consider the SQP system (11){(12). According to (12), z is orthogonal to the
prior iterate x k . Let P be the matrix that projects a vector into the space perpendicular
to
Multiplying (11) by P yields
according to (12), we have
We haved found that preconditioned Krylov space methods, such as the Gauss-Seidel
scheme in [9], converge very quickly when applied to (61). As a small illustra-
tion, let us consider the second test problem from [24] with
where is a 1000 1000 diagonal matrix with diagonal elements selected randomly
from a uniform distribution on ( :5; :5) and I 2qq T where q is gotten by
rst generating random numbers on ( :5; :5) and then scaling the resulting vector
to have unit length. The vector b is generated in the same way as q. The solid
curve in Figure 1 gives the convergence when a Lanczos type process (Algorithm 1,
with starting vector v is used to generate the matrix V used in (9). The
Lanczos process was modied to ensure orthogonality of the columns of V. For each
value of l in Algorithm 1, we solve the l l tridiagonal problem (10) to obtain an
approximate solution x and associated multiplier for the original problem
(3). In the solid curve of Figure 1, we plot the base 10 logarithm of the norm of the
residual kb (A+I)xk. According to Lemma 1, the residual vanishes at an optimal
solution.
The dashed curve of Figure 1, based on the SSM approach, is gotten in the following
way: Taking Algorithm 1, we generate a V with 40 orthonormal
columns. Solving (10), we obtain starting guess x 0 . In iteration k of the SSM phase,
we start with the vector v we use the Gauss-Seidel/Krylov space
approach of [9] to generate a matrix V, with orthonormal columns, that approximately
contains a solution of (61) in its range. Using the V generated in this way, we
solve (9) to obtain the next iterate x k+1
. The associated multiplier is estimated using
(14). Each kink in the dashed curve of Figure 1 corresponds to the number of iterations
needed to obtain an approximate solution of (61). In this example, roughly 15
multiplications by the elements of the matrix A are used to solve (61). The quadratic
convergence of SSM is re
ected in the rapid decay of the residual norm.
This approach for generating V, using a nonsymmetric Gauss-Seidel matrix,
Krylov spaces, and orthogonalization, can become expensive when n is really large
Matrix-vector products
Figure
1: Convergence of the tridiagonalization approach (solid) and SSM (dashed)
for the second test problem from [24].
since each of the columns of V should be stored in memory. Hence, in the remainder
of this paper, we focus on low-storage symmetric techniques for solving (61), which
we compare to other approaches.
We solve (61) using a preconditioned version of Paige and Saunders' MINRES
algorithm [17]. More precisely, we use Algorithms 3 and 3a in [9] and three dierent
choices for the symmetrizing preconditioner W in that paper: (i) corresponding
to unconditioned iterations, (ii) is the diagonal matrix
whose diagonal matches that of
L is the strictly lower triangular matrix whose lower triangle matches that of C. The
implementations of SSM associated with the latter two preconditioners are denoted
and SSM l
respectively.
Typically, the L matrix associated with
I)P is dense, even when A
is sparse, since P is often dense. Nonetheless, linear systems of the form
can be solved in time proportional to the number of nonzero elements in the lower
triangle of A due to the special structure of C. In terms of the vectors w, q and p
dened by
I)w and
the diagonal d of C can be expressed
while the o-diagonal elements of C are
Exploiting this structure, it can be shown that the solution to (L can
be computed in the following way:
Algorithm
The statement y a i+1:n;i of Algorithm 2 only requires the nonzero
elements in column i of A beneath the diagonal. Hence, the number of
oating point
operations for Algorithm 2 is O(n) plus the number of nonzero elements in the lower
triangle of A.
The analogous procedure for the transposed system is the following:
Algorithm 3
4.2 Positive deniteness
In theory, the MINRES algorithm we use to solve (61) can be applied to any symmetric
matrix. In practice convergence can be extremely slow when C is indenite. For this
reason, we try to choose k so that A+ k I is positive denite. If e is an eigenvector
of the matrix B in (10) associated with the smallest eigenvalue , then the pair (v; ),
approximates an eigenpair of A corresponding to the smallest
eigenvalue. The error in can be estimated in the following way: If is closer to 1
than the other eigenvalues of A, then after substituting
in the residual r = Av v, we have
since
1. Thus j 1 j krk, which implies that
With this insight, we replace the least squares estimate (14) by the following safeguarded
estimate:
When the approximate eigenpair (v; ) is not very accurate, then the safe-guarded
step (62) is a safe, but poor approximation to . Hence, whenever
we apply one iteration of SSM to the quadratic eigenvalue problem (2) in order to
compute a more accurate eigenpair. Due to the third and sixth order estimates in
(15), simply one iteration of SSM for the eigenproblem often yields a highly accurate
eigenpair.
4.3 The algorithm
We now collect our observations and present the algorithm that was used to generate
the numerical results of the next section. To simplify the presentation, we introduce
the following subroutines:
This routine applies Algorithm 1 to the matrix A,
starting from the vector v 1 , to generate a matrix V with columns
l
This routine solves the problem (8) generating a
solution denoted x, and an associated multiplier is a matrix
whose columns are an orthonormal basis for S k , then an estimate (v; ) for the
smallest eigenvalue of A and an associated eigenvector is gotten by computing
the smallest eigenvalue and an associated eigenvector e for
setting
This routine computes a (minimum residual, minimum
solution (z; ) of the following linear system:
Our implementation of the sequential subspace method combines these three routines
and the safe-guarded step (62):
Algorithm 4 (Safe-guarded SSM with Lanczos startup)
while ( == &
it
while ( kb
Algorithm 4
For the computational results reported in the next section, we took
:01ng. The \rand" function appearing at the start of Algorithm 4 generates
a vector with components uniformly distributed on [0; 1].
5 Computational results
In this section we compare the performance of SSM to the performance of the algorithms
in [7], [20], and [24], denoted GLRT, RW, and S respectively, using the three
test problems presented in [24]. The results that we report for S were extracted from
[24], while the results reported for GLRT and RW were obtained using codes provided
by the authors. We thank the authors for providing access to their codes. Each of
these codes used dierent stopping criteria. GLRT stopped when kb (A+I)xk=kbk
was bounded by a given tolerance, while RW stopped when the gap between the value
of the primal and dual problem, and hence the error in the primal cost function, was
smaller than a given tolerance. In order to ensure that each code computed a solution
with the same accuracy, we adjusted the error tolerance parameter of each code until
the value of kb for the computed solution was smaller than a given
tolerance (specied below).
In the rst test problem of [24], is the standard 2-D
discrete Laplacian on the unit square based on a 5-point stencil with equally-spaced
mesh points. Taking a series of 20 problems were
generated where b was a vector with elements uniformly distributed on [0; 1]. Each
of these problems was solved using three dierent tolerances,
In
Table
1 we give the average number of matrix-vector products involving A for each
algorithm. Each iteration of the preconditioned MINRES algorithm with lower trian-
Tolerance S RW GLRT SSM SSM d
SSM l
Table
1: Problem 1, average number of matrix-vector products versus tolerance.
gular preconditioner involves roughly twice as many
ops as an iteration of either the
identity or the diagonal preconditioned schemes. Hence, in doing the bookkeeping,
we charged for two matrix-vector products in each iteration of the triangular preconditioned
scheme. As seen in Table 1, SSM l converges more than twice as fast as the
identity and diagonal preconditioned schemes, and overall, SSM l
uses the smallest
number of matrix-vector products for this test problem. Since the parametric eigenvalue
algorithms S and RW compute an extreme eigenvalue for a series of matrices,
we also list in parentheses in Table 1 the number of these eigenproblems that are
solved. Hence, RW is very economical in terms of the number of these eigenproblems
that are solved.
The second suite of test problems in [24] utilizes the matrix described earlier in
Section 3. In these problems, the radius of the sphere is varied and the number of
matrix-vector products is tabulated. For radii of one or smaller, solutions can be
computed extremely quickly, so we focused on and an error
tolerance of 10 7 . In Table 2 we see that for
had the fewest matrix-
Radius S RW GLRT SSM SSM d SSM l
Table
2: Problem 2, average number of matrix-vector products versus radius.
vector products, while GLRT had the fewest for
The nal problem of [24] again employed the discrete Laplacian matrix, but with
100. The vector b was designed to make the problem degenerate;
rst a random b was generated, then its 1 component was removed. Table 3 gives
the results for the various algorithms.
SSM l
Table
3: Problem 3, average number of matrix-vector products.
We placed an asterisk by the result in Table 3 for GLRT since this routine reduced
the error to 10 4 , not the 10 7 tolerance used by the other routines. Among
the routines that achieved the error tolerance, SSM l
performed the best relative to the
number of matrix-vector products. Note that the number of matrix-vector products
given in Table 3 for S was taken from [24] while Rojas, in her recent thesis [21], developed
a more e-cient implementation of Sorensen's approach for degenerate problems.
In summary, a Lanczos type process seems to be very eective when the problem
is very nondegenerate ( >> 1 ). As the problem becomes more degenerate,
preconditioned schemes such as SSM d or SSM l appear more eective. The number of
times that RW computes an extreme eigenpair is often around 5. For the numerical
experiments reported in this paper, Matlab's eig routine was used to compute this
extreme eigenpair. If this routine for computing an extreme eigenpair could be sped
possibly using the Jacobi type methods of Sleijpen and Van der Vorst [22] or
the truncated RQ iteration of Sorensen and Yang [25], the number of matrix-vector
operations used in the parametric eigenvalue approach would be reduced.
--R
A trust region algorithm for nonlinearly constrained optimization
A trust region strategy for non-linear equality constrained optimization
Least squares with a quadratic constraint
Quadratically constrained least squares and quadratic problems
Matrix Computations
Solving the trust-region subproblem using the Lanczos Method
Applied Numerical Linear Algebra
Iterative methods for nearly singular linear systems
Graph partitioning and continuous quadratic programming
a MATLAB package for analysis and solution of discrete ill-posed problem
Geophysical Data Analysis: Discrete Inverse Theory
Solution of sparse inde
The Symmetric Eigenvalue Problem
A trust region algorithm for equality constrained optimization
A semide
A Large-scale Trust-region Approach to the Regularization of Discrete Ill-posed Problems
A Jacobi-Davidson iteration method for linear eigenvalue problems
Newton's method with a model trust region modi
Minimization of a large-scale quadratic function subject to a spherical constraint
A truncated RQ iteration for large scale eigenvalue calculations
Inverse Problem Theory
--TR
--CTR
Peter A. Graf , Wesley B. Jones, A projection based multiscale optimization method for eigenvalue problems, Journal of Global Optimization, v.39 n.2, p.235-245, October 2007
Stanislav Busygin, A new trust region technique for the maximum weight clique problem, Discrete Applied Mathematics, v.154 n.15, p.2080-2096, 1 October 2006 | symmetric successive overrelaxation;sparse optimization;arnoldi m orthogonalization;gauss-seidel;minimal residual;large-scale optimization;trust region subproblem;quadratic programming;krylov space;preconditioning;quadratic optimization |
589318 | Rescaling and Stepsize Selection in Proximal Methods Using Separable Generalized Distances. | This paper presents a convergence proof technique for a broad class of proximal algorithms in which the perturbation term is separable and may contain barriers enforcing interval constraints. There are two key ingredients in the analysis: a mild regularity condition on the differential behavior of the barrier as one approaches an interval boundary and a lower stepsize limit that takes into account the curvature of the proximal term. We give two applications of our approach. First, we prove subsequential convergence of a very broad class of proximal minimization algorithms for convex optimization, where different stepsizes can be used for each coordinate. Applying these methods to the dual of a convex program, we obtain a wide class of multiplier methods with subsequential convergence of both primal and dual iterates and independent adjustment of the penalty parameter for each constraint. The adjustment rules for the penalty parameters generalize a well-established scheme for the exponential method of multipliers. The results may also be viewed as a generalization of recent work by Ben-Tal and Zibulevsky [SIAM J. Optim, 7 (1997), pp. 347--366] and Auslender, Teboulle, and Ben-Tiba [ Comput. Optim. Appl., 12 (1999), pp. 31--40; Math. Oper. Res., 24 (1999), pp. 645--668] on methods derived from $\varphi$-divergences. The second application established full convergence, under a novel stepsize condition, of Bregman-function-based proximal methods for general monotone operator problems over a box. Prior results in this area required strong restrictive assumptions on the monotone operator. | Introduction
denote the possibly unbounded n-dimensional "box" ([a
This paper considers two closely-related
problems, the minimization problem
min f(x)
(1)
is a closed proper convex function, and the variational inequality
where T is a (possibly set-valued) maximal monotone operator, and NB (x) denotes the cone
of vectors normal to the set B at x. It is well known that, under mild regularity conditions,
(1) is the special case of (2) for which the subgradient mapping of f .
The last decade has seen considerable progress in the theory of proximal point methods
based on generalized distances [11, 13, 19, 5, 21, 31, 14, 2, 3, 17]. Such methods use a scalar-valued
regularization function to derive better-behaved versions of problems (1) and (2). In
this article, we consider separable regularization terms of the form
are scalar functions conforming to very general assumptions (see Assumption
2.1 below). In particular, we assume that as x 2 int B approaches the boundary of
B, kr 1 D(x; y)k !1, where r 1 denotes the gradient with respect to the first vector argu-
ment. The distance-like measure D can be, for example, the squared Euclidean distance, a
Bregman distance [8], or a '-divergence [19] (see Section 2.2 below).
Using these regularization terms, proximal methods for (1) take the form:
x
where ff k is a positive n-dimensional vector whose elements are called stepsizes. Note that
we allow different stepsizes for each coordinate, as suggested by a variety of computational
and theoretical studies [32, 5, 2, 3]. Moreover, since kr 1 D(x; x k )k !1 as x approaches the
boundary of B, the regularization acts not only as a stabilizing proximal term but also as a
kind of barrier function keeping the iterates within int B.
In the case of the variational inequality (2), (3) generalizes to finding x k+1 satisfying the
recursion
RRR
We derive some general results for these types of algorithms in Section 2, assuming that
the stepsizes conform to a special rule that takes into account the curvature of the proximal
term. This rule, although restrictive, appears to cover cases of the greatest practical interest;
as we shall see, it covers the stepsize/penalty selection rules proposed in [32, 5, 2, 3].
Section 3 uses the results of Section 2 to obtain subsequential convergence results for the
generalized proximal minimization algorithm (3).
A critical application of (3), considered in Section 3.2, is when f is minus the dual
function of a convex program such as
s.t.
where are differentiable convex functions. 1 We also assume that this
problem is feasible, i.e., there is - y Choosing B to be
any box containing the nonnegative orthant and f to be the negative of the dual function
of (5), we may implement (3) via a multiplier method in which a sequence of unconstrained
penalized versions of (5) must be solved. This construction leads to a class of multiplier methods
that is extremely broad, subsuming both the classical quadratic augmented Lagrangian
and the exponential method of multipliers [32, 6].
For these multiplier methods, our stepsize choice ensures that for indices i with x k
the corresponding penalty term is augmented so it does not become so "flat" as to permit
infeasibility of primal limit points. Empirically, the technique speeds convergence, and it
also appears in a convergence rate analysis in [32] for the exponential method of multipliers
case. Ben-Tal and Zibulevsky [5] have proved the optimality of the accumulation points
of the exponential method, together with a class of proximal terms closely related to '-
divergences, and their results are extended in [3]. Section 3 places such results in a broader
context that includes Bregman distances.
In Section 4, we restrict our attention to Bregman distances. It has been known for the
better part of a decade that, when D(\Delta; \Delta) is any Bregman distance and the stepsizes do not
vary by coordinate, the recursion (4) converges to a solution of the variational inequality (2)
in various special cases: when the subdifferential of a closed proper convex function
f , or when domT ' int B, meaning that all constraints must already be embedded in the
operator T . In [9], these results were extended to "paramonotone" operators T , a category
which includes as a special case. Unfortunately, many interesting practical cases,
such as the subdifferential maps of saddle functions, are not paramonotone. More recently,
Auslender et al. [2] have obtained strong results for general maximal monotone T , but only
for a specific '-divergence choice of D(\Delta; \Delta). As noted in [4], these results can be extended
to the (generally non-Bregman) case where D(\Delta; \Delta) is obtained by adding a quadratic to any
member of the class \Phi 2 of [3].
1 Actually, the results of Section 3.2 continue to hold [28] if one only supposes that
are closed proper convex and assumes appropriate conditions on the effective domains of the
objective and constraints, as in [24, Chapter 28]. However, this further generality makes the proofs more
convoluted and is dropped for the sake of simplicity in the exposition.
Page 4 RRR 35-99
Section 4 shows convergence, for general maximal monotone T , of the proximal method
(4), where D(\Delta; \Delta) is a Bregman distance, to a solution of (2). We do impose some additional
assumptions, derived from those of Section 2. First, we assume that the Bregman function
used to construct the distance is twice-differentiable, which is not part of the standard
Bregman function setup. Second, in addition to our general stepsize rule, we also require
that the stepsizes do not vary by coordinate, that is, ff
n for all k. The resulting
condition is stronger than the usual requirement that the stepsize is simply bounded away
from zero, but is crucial to the analysis, which blends the techniques of Section 2 with
traditional Fej'er monotonicity arguments. Still, we have managed to substitute conditions
on D(\Delta; \Delta) and ff k , which are parts of the algorithm, for conditions on T , which is part of the
problem to be solved.
Finally, we allow the calculations required for the recursions (3) and (4) to be performed
approximately, as is likely to be necessary in practice. For the rescaling minimization case
of Section 3, we adopt a constructive approximation criterion inspired by [17] and [29].
However, our criterion, which is tailored to the proximal minimization case, appears to be
new. In the variational inequality analysis of Section 4, we use the simple, verifiable criterion
of [14], although extension to the more sophisticated criterion of [29] may well be possible.
In summary, the primary contributions of this paper are:
ffl A novel convergence proof framework for a broad class of proximal algorithms.
ffl Using this framework to establish subsequential convergence of a wide range of proximal
minimization algorithms (3) with differing stepsize parameters for each coordinate; this
result in turn leads to subsequential convergence of a broad class of multiplier methods
with differing penalty parameters for each constraint.
ffl Using the framework to show convergence of "interior" Bregman proximal point algorithms
for maximal monotone operators, with a novel stepsize condition, but without
the usual restrictive assumptions on the operator T .
The new proximal minimization approximation criterion of Section 3 constitutes an additional
contribution.
Fundamental Analysis
This section develops the fundamental analysis necessary for our results. We concentrate
our attention on the variational problem (2), since it subsumes the minimization problem (1)
under mild assumptions.
In order to simplify the notation, we denote, for
d 00
RRR
We are now able to present the necessary assumptions on the functions d
Assumption 2.1 For has the following
properties:
2.1.1. For all y closed and strictly convex, with its minimum at y i .
Moreover, int dom d i (\Delta; y
2.1.2. d i is continuously differentiable over (a
exists and is strictly positive.
2.1.3. For all y essentially smooth [24, Chapter 26].
2.1.4. There exist ae; ffl ? 0 such that if either
The assumption of strict convexity is standard in generalized proximal methods. The
assumption of twice differentiability is also quite common, although many existing results
require only a once-differentiable d i . The essential smoothness assumption makes the distance
act like a barrier function, forcing the iterates defined by the recursion (4), and hence
its approximate version (6) below, to remain in the interior of the box B. In Section 2.2,
we specialize these assumptions to the case of Bregman distances and '-divergences, where
similar comments can be made.
Finally, the fourth part of the assumption is new to the theory of generalized proximal
methods, but is not very restrictive in practice. In particular, we show in Section 2.2 that, for
Bregman distances and '-divergences, this condition can be written in terms of the kernels
used to obtain the regularizations, and that it holds for most of the examples we are aware
of.
In addition, we make the following standard regularity assumption which, in view of the
barrier function properties of d i , is required for any sensible application of (4):
Assumption 2.2 domT " int B 6= ;.
We are now able to present the proximal minimization algorithm:
Rescaling Proximal Method for Variational Inequality (RPMVI)
1. Initialization: Let Choose a scalar c ? 0, and an initial iterate x 0 2 int B.
2. Iteration:
(a) Choose ff k 2 R n
such that ff k
\Phi
(b) Find x k+1 and e k+1 such that
Page 6 RRR 35-99
(c) Let repeat the iteration.
To guarantee the convergence of the RPMVI, we need additional assumptions on the
stepsizes fff k
and the error sequence fe k g; see Assumption 2.3 below.
We define
whence it is clear from (6) that
Assumption 2.3 Let ffi k g be a real sequence converging to zero. The error sequence, fe k g,
the regularization functions d and the stepsizes, fff k
must be chosen
in order to guarantee that:
2.3.1.
ff
2.3.2. If - x is an accumulation point of fx k g, i.e., there is an infinite set K ' N such that
x, then, for each
or there is an infinite set K 0 ' K
such that x
Assumption 2.3.2 may seem artificial at this point, but Sections 3 and 4 will describe
settings where it is easily verifiable.
2.1 Convergence Analysis
We assume throughout this section that Assumptions 2.1 and 2.2 hold, and that sequences
conforming to the recursions of the RPMVI algorithm and Assumption
2.3 exist. In Sections 3 and 4 we will present conditions which, in more specific settings,
guarantee the existence of such sequences.
Lemma 2.4 Let -
x 2 R n be a limit point of fx k g, i.e., x k !K -
x for some infinite set K ' N.
Then for
lim
lim inf
lim sup
Proof. For each i, we consider the three possible cases:
First, suppose i is such that -
For the sake of a contradiction, assume that
using Assumption 2.3.2, there is an infinite set K 0 ' K and a i ? 0 such
RRR
that for all k 2 K 0 , jfl k
x i . Therefore
-ff
ff
[Assumption 2.3.1]
[Choice of ff k
This result contradicts
Next, consider the case - x suppose that lim inf k!K1 fl k
using
Assumption 2.3.2, there must be a i ? 0 and an infinite set K 0 ' K such that for all k 2 K 0 ,
ff
-ff
-cd 00
Let ffl be as in Assumption 2.1.4. If there is an infinite set K 00 ' K 0 such that x
for all k 2 K 00 , we can conclude from the assumption that:
2d 00
aecd 00
since x
for sufficiently large k 2 K 0 .
Page 8 RRR 35-99
As d i (\Delta; x
its minimum at x k\Gamma1
implies that d 0
Hence
ff
for sufficiently large k 2 K 0 , a contradiction with
Finally, the case of -
is analogous to the case -
Lemma 2.5 Let -
x be a limit point of fx k g, i.e., x k !K - x for some infinite set K ' N.
Proof. By Assumption 2.2, there must exist some e
(ex). The
monotonicity of T implies that, for all k - 0,
We will show that unboundedness of ffl k gK would contradict this inequality for some sufficiently
large k.
is unbounded, there must exist an infinite K 0 ' K such that ffl k gK 0 converges in
, with at least one ffl k
implies that for each unbounded
coordinate i, either
or
Therefore, for each unbounded coordinate of ffl k gK 0 , we have
or
On the other hand, for coordinates such that ffl k
also bounded. Thus, for sufficiently large k 2 K must be negative,
contradicting (9). 2
Finally, the main convergence theorem for the RPMVI follows:
RRR
Theorem 2.6 If fx k g is a sequence generated by the RPMVI algorithm with Assumptions
2.1, 2.2, and 2.3 holding, then all the limit points of fx k g are solutions to the variational
inequality problem (2).
Proof. Let -
x be any limit point of fx k g, i.e., x k !K -
x, for some infinite set K ' N. From
Lemma 2.5, we know that the corresponding sequence
must exist some K 0 ' K with
must be outer semicontinuous [27,
12.8(b)], it follows that - fl 2 T (-x). Lemma 2.4 implies that
and these conditions are equivalent to
Incidentally, it is possible to eliminate the requirement of twice-differentiability of d i (\Delta; y i ),
at the cost of some additional complexity in the description of the method. Specifically,
consider replacing Assumption 2.1.4 with the condition that there exist functions
If the stepsizes are now selected so that for some scalar c ? 0, we have for all
and k - 0 that ff k
the conclusions of Theorem 2.6 continue to hold. We
may examine this variation of the analysis in subsequent research. The present approach is
equivalent to taking L i (y choice since d 00 (y
rate of change of d 0 (\Delta; y i ) around y i .
2.2 Some examples of d i
functions
We present some example of d i functions that conform with Assumption 2.1. In particu-
lar, we show that two classes of regularizations widely studied in the literature, Bregman
distances [11, 13] and '-divergences [19], conform to the assumption under very mild restrictions
2.2.1 Bregman distances
Bregman distances were introduced in [8] and have been studied in the context of proximal
methods in [11, 12, 13], as well as many subsequent works. To construct each regularization
one uses an auxiliary convex function h i and defines d i
Nonseparable distances can also be constructed in a similar way, but the
separable case is the most common.
The following properties guarantee that Assumption 2.1 holds for such
Assumption 2.7 For has the following
properties:
2.7.1. h i is closed, int continuously differentiable, with a
strictly positive second derivative throughout (a
2.7.2. h i is essentially smooth.
2.7.3. There exist ae ? 0 and ffl ? 0 such that if either
Note that Assumption 2.7.1 implies that each h i is strictly convex. Assumption 2.7.3 corresponds
to Assumption 2.1.4, since d 00
Fortunately, it is not very restrictive.
Consider the case of finite a i . Since lim x
we know that h 00
must be
unbounded above as x i & a i . To violate the assumption, h 00
would have to oscillate
unboundedly as x i & a i . As far as we are aware, every separable Bregman function proposed
so far conforms not only to Assumption 2.7.3, but to a more stringent, easier-to-verify
condition, as follows:
Lemma 2.8 If there is an ffl ? 0 such that for all x
i is non-increasing,
and for all x 2 (b
i is non-decreasing, then Assumption 2.7.3 holds.
Proof. Suppose that a
Therefore, Assumption 2.7.3 holds with 1. The case b i ! 1 is analogous. 2
Examples of functions h i where all these assumptions hold are:
log x, with a
with with a
Finally, we note that for finite a i we do not yet assume that h i must approach a finite
limit as x i & a i , nor similarly for x i Such an assumption is quite common in
the theory of Bregman distances [11, 13, 9, 29], but, similarly to [21], it is not needed for
the results of Section 3 below. We will use it, however, in the variational inequality analysis
of Section 4.
RRR 35-99 Page 11
2.2.2 '-divergences
The '-divergence regularizations have been studied in the context of proximal methods, for
example, in [19], and more recently in [5, 3]. In these works, the box considered is the
positive orthant, i.e.,
. An auxiliary strictly convex scalar function ' is used to
define the distance d i , but this time by:
The following hypotheses can be used to guarantee Assumption 2.1 when
Assumption 2.9 The function ' : R! (\Gamma1; +1] is such that:
2.9.1. ' is closed and convex, with int
2.9.2. ' is twice differentiable on (0; +1), with ' 00 (t) ? 0 for all t ? 0;
2.9.3.
2.9.4. ' is essentially smooth;
2.9.5. There exists a ae ? 0 such that ae' 0 (t) - ' 00
Slight variations on these assumptions appear, for example, in [5, 3], together with the
following examples:
The next lemma states that Assumption 2.9.5 above implies Assumption 2.1.4:
Lemma 2.10 Let (a be defined as in (10). Then Assumption 2.1.4
is equivalent to the existence of a ae ? 0 such that ae' 0 (t) - ' 00
Proof. First we observe that:
d 00
and so
d 00
Page 12 RRR 35-99
Therefore, Assumption 2.1.4 reduces to
Taking letting y i range over (0; x i ], and setting
Conversely, if (12) is true, (11) holds for an arbitrary choice of ffl ?
We note that in [5], one assumes that the iterations are of the form:
where each ff k
i is greater than c=x k
being a positive constant. In [2, 3], this property is
guaranteed by redefining the distance measure to be
~
~
and assuming stepsizes bounded away from zero. In this case, the iteration is
~
with lim inf k!1 e
rewriting the iteration with
respect to D, instead of ~
D, we recover the rule from [5].
It turns out that these techniques are a special case of our stepsize choice rule, which
gives in the case of a '-divergence that
which is identical if one redefines the constant factor c.
Thus, the reader should note that the class of '-divergences described by Assumption 2.9
encompasses the regularizations studied in [5, 2, 3]. In particular, it includes the classes \Phi 1
and \Phi 2 described in [3].
However, the stepsize rule in the RPMVI is more stringent than the one in [5, 2, 3],
as it also assumes that the stepsize is bounded away from zero. To overcome this slight
restriction, we point out that the assumption ff k
used here only in the first part
of the proof of Lemma 2.4, and it can be replaced by the assumption that d 00
continuous and strictly positive over (a This condition holds for '-divergences, since
d 00
In this sense, the results here can be seen as extensions of those in [5, 2, 3].
RRR
3 Proximal Minimization Methods with Rescaling
This section applies the analysis of the RPMVI method to the minimization problem (1).
We leave Assumption 2.1 as a standing assumption; we also make the following standard
regularity assumption, which in view of the barrier function properties of D, is required for
any sensible application of (3):
Assumption 3.1 dom f " int B 6= ;.
Note that, since int B is open, this assumption implies that ri dom f " int B 6= ;, which
implies that dom @f " int B 6= ;. Then, using [24, Theorem 23.8], one can show that the
minimization problem (1) is equivalent to the variational inequality problem (2) with
Moreover, Assumption 2.2 holds.
Then, we specialize the RPMVI to:
Rescaling Proximal Minimization Method (RPMM)
1. Initialization: Choose c ? 0 and oe 2 [0; 1]. Choose nonnegative scalar sequences fs k g
and fz k g with
2. Iteration:
(a) Choose ff k 2 R n
such that ff k
\Phi
(b) Find x
oe
ae s k+1
oe
with the standing convention that min
\Psi is z k+1 whenever
(c) Let repeat the iteration.
Note that if one chooses s k ; z reduces to the "constructive" criterion
reminiscent of [29].
Page 14 RRR 35-99
3.1 Convergence analysis
We start by showing that the iteration step is well defined if f is bounded below on B:
Lemma 3.2 If f is bounded below on B, then there is a unique point that solves the iteration
step of the RPMM with e a solution to (13)-(14) exists if f is bounded below
on B.
Proof. Let ' be a lower bound of f on B. Given i 2 R, the level set
This last set is a level set of
i ) on B, which must be bounded, since by
Assumption 2.1.1 this function attains its minimum at the unique point x k [24, Corollary
8.7.1]. Therefore, f(\Delta)
attains a minimum on B. The uniqueness of
the minimum follows from the strict convexity of D(\Delta; x k ). 2
To apply the convergence analysis of the previous section to the sequence fx k g computed
by the RPMM, it suffices to show that Assumption 2.3 holds. Verification of Assumption
2.3.1 is straightforward:
Lemma 3.3 With the definition
ae s k
oe
for all k - 1, Assumption 2.3.1 holds for the RPMM.
Proof. From the nonnegativity of fs k g and fz k g, it follows that ffi k g is also nonnegative.
one also has fi k ! 0. Moreover, since oe 2 [0; 1],
oe
ff
for all k, so Assumption 2.3.1 holds. 2
As in (7), we define for all k - 0 and
and let fl k 2 R n be the vector with elements
Lemma 3.4
RRR 35-99 Page 15
Proof. The claim that fl k 2 @f(x k ) follows from the definition of fl k . For the second claim,
we have, using the convexity of d i (\Delta; x
ff
Using (14), it then follows that
ff
ae s k
ae s k
oe fi fi x
ae s k
oe
\Gammas k :Before proving the next result, we state a helpful technical lemma:
Lemma 3.5 [22, Section 2.2] Suppose fa k g, ffl k g ae R are sequences such that fa k g is
bounded below,
exists and is finite, and the recursion a k+1 - a k holds for all k.
is convergent.
It is now possible to establish that Assumption 2.3.2 also holds:
Lemma 3.6 If f is bounded below on B, then ff(x k )g is convergent and
Hence Assumption 2.3.2 holds for the RPMM.
Proof. Using Lemma 3.4,
ns
Then, recalling that fs k g is summable, Lemma 3.5 implies that ff(x k )g is a convergent
sequence. For
Page
Using Lemma 3.4 once again, it follows that
Taking limits, we conclude that fl k
Thus, Theorem 2.6 implies the optimality of all accumulation points of the sequence
g. We strengthen this observation below:
Theorem 3.7 Suppose that Assumptions 2.1 and 3.1 hold, and that f is bounded below on
B. If fx k g has a limit point, then ff(x k )g converges to the infimum of f on B and all limit
points of fx k g will be minimizers of f on B. A condition that guarantees the existence of
limit points of fx k g is the boundedness of the solution set, or any other level set of f .
Proof. As just noted, Lemma 3.6 implies that Assumption 2.3.2 holds, and so Assumption
2.3 holds in its entirety. Assumption 2.1 holds by hypothesis, and, setting
Assumption 3.1 implies Assumption 2.2. Thus, the conclusions of Theorem 2.6 apply. Let
x be a limit point of fx k g, i.e. x k !K - x, for some infinite set K ' N. Theorem 2.6 asserts
that Assumption 3.1, -
x is a minimizer of f on B. Moreover, since
Lemma 2.5 states that ffl k gK is bounded, and since ff(x k )g is convergent by Lemma 3.6,
min
lim
Therefore, lim k!1 f(x k
Finally, the boundedness of any level set of a proper closed convex function implies
boundedness of all level sets [24, Corollary 8.7.1], and Lemma 3.6 states that ff(x k )g is
convergent, consequently it is bounded. So, fx k g is also bounded and has limit points. 2
3.2 Multiplier Methods
We now discuss applying the RPMM to the dual of the convex program (5) to obtain multiplier
methods. The use of proximal methods to derive multiplier methods for constrained
convex optimization is a now-classical subject and may be traced to the seminal paper [26].
In the context of generalized proximal methods, applications can be found, for example,
in [30, 13, 19, 21, 31, 3, 17]. In this section, we consider only the case in which the proximal
step is done exactly, i.e., we will let e as in [30, 13, 19, 17]. Unfortunately,
our approximate-step acceptance rule for the RPMM does not translate directly to an easily
verifiable acceptance criterion for an approximate solution of the penalized problem (17) be-
low. However, partial results in this direction may be obtained under stringent assumptions
on the original problem (5); see Appendix B. A criterion in the spirit of (14) that does not
depend on such assumptions is the subject of ongoing research [15]. We further observe that
the approximation criteria of [17, 29] also do not translate readily to a multiplier method
RRR 35-99 Page 17
setting. On the other hand, under the assumption that the primal objective function g 0
is strongly convex, [26, 21, 3] present some inexact multiplier methods based on a rather
different acceptance rule involving optimizing the augmented Lagrangian function to within
some tolerance ffl of its minimum value.
Consider the convex problem (5), and let ffi C denote the indicator function of a convex
set C. Then we define f to be minus the dual function associated with (5), plus
The dual problem to (5) is then equivalent to the minimization of f . Furthermore, we assume
Assumption 3.83.8.1. The primal problem (5) has a finite optimal value, and it conforms
to the Slater condition.
3.8.2. For all conform to Assumption 2.1 for a i -
3.8.3. There is an - x ? 0 such that - x 2 dom f , where f is as defined in (15).
This assumption has the following consequences: Assumption 3.8.1 implies that the dual
solution set is non-empty and bounded [16] and that there is no duality gap. Assumption
3.8.3 implies that Assumption 3.1 holds for f as defined by (15).
Under Assumption 3.8, if we fix e each iterate x k+1 of the RPMM applied
to the negative dual functional f may be calculated by the following multiplier method
whenever the unconstrained problems (17) have solutions:
\Phi
d \Phi
denotes the monotone conjugate [24, p. 111] with respect to the first argument,
that is, d \Phi
)g. 3 Theorem 3.10 below gives conditions guaranteeing
that a y k+1 satisfying (17) exists.
We relegate the technical aspects of the proof of the equivalence of (16)-(18) to the
RPMM applied to the f defined in (15) to Appendix A, since they are very similar to earlier
2 The case a is of interest because it includes the classical method of multipliers for problems with
inequality constraints [26], along with various extensions described in [13, 20].
3 The classical conjugate / of a function / is defined [24, Chapter 12] via /
for any / : R n ! (1; +1]. The monotone conjugate of / is then the classical conjugate of
, that is,
Page
proofs for various special cases of (17)-(18), for example in [30, 13, 19, 21, 17]. In particular,
Corollary A.4 establishes the equivalence of the two calculations.
Given this equivalence, Theorem 3.7 asserts the subsequential convergence of the sequence
to a dual solution of (5). For the primal sequence, however, it has historically been
harder to prove good behavior. For example, in the case of Bregman distances, a guarantee of
feasibility of primal accumulation points has relied on stringent assumptions like R n
ae int B,
as in [13], or strict complementarity [18].
In the case of the RPMM, with its strong stepsize restrictions, the feasibility, and therefore
optimality, of accumulation points of fy k g is easily demonstrated.
Theorem 3.9 Suppose that Assumption 3.8 holds. Pick a scalar c ? 0, let x 0 2 R n
and
suppose that it is possible to obtain a sequence f(ff k that obeys the recursions (16)-
(18). Then, fx k g is bounded and all its accumulation points are solutions of the dual of (5).
Moreover,
lim sup
lim
and fg 0 (y k )g converges to the optimal value of the primal problem (5). Therefore, any
accumulation point of fy k g solves the primal problem.
Proof. As shown in Corollary A.4, the sequence fx k g is the same as would be computed
by using the RPMM to solve the dual problem, that is, to minimize f . In particular, fx k g
and all its limit points must be nonnegative. Moreover, the Slater condition implies that the
dual function has bounded level sets. Then, the boundedness of fx k g and the optimality of
its limit points follow from Theorem 3.7.
Let us analyze the primal sequence. For each
the same role as
in (7), with e k
Let fx k gK be any convergent subsequence of fx k g, and -
x the respective accumulation
point, x k !K - x. Lemma 2.4 implies that
As fx k g is bounded, the above relations imply that
RRR 35-99 Page 19
Now, suppose for the purposes of contradiction that (20) does not hold. Then, for some
must be an infinite set K ae N and an ffl ? 0 such that
is bounded, there exists a refined subsequence K 0 ' K such that fx k g K 0 is
convergent, with limit -
imply that g i (y k ) !K 0 \Gamma1. Since Lemma 2.5 asserts that fi k
is bounded, we can
conclude that i k
\Gamma1. However, this divergence would imply that x k
i should be 0 for
infinitely many k 2 K 0 ' K, once again a contradiction of (23). Therefore,
lim
and (20) holds.
Finally, we prove that fg 0 (y k )g converges to the optimal value. We may use (17), (18),
and the chain rule to see that y k minimizes the Lagrangian corresponding to the primal
problem with the fixed multiplier x k . Hence,
Let \Gammaf denote the dual optimal value, which is equal to the primal optimal value since
there is no duality gap. Theorem 3.7 states that f(x k Taking limits in (25) and
using (24), it follows that
lim
The feasibility and optimality of the accumulation points of fy k g are then consequences of
the continuity of g i ,
Finally, it is natural to seek conditions under which the penalized subproblems (17) must
have solutions, and the primal sequence fy k g is bounded. The following result addresses
these questions under the standard assumption of a bounded solution set:
Theorem 3.10 Suppose that the primal solution set is bounded. Given any ff k ? 0 and
exist satisfying the recursions (17)-(18). Moreover, the primal
sequence fy k g is bounded.
Proof. For the first assertion, it suffices to show that the penalized problems (17) have
solutions. Given any closed proper convex function /, we define its recession function /1
via dom/ may be chosen arbitrarily [24,
Theorem 8.5]. The boundedness of the primal solution set is equivalent [7, Section 5.3] to:
Page 20 RRR 35-99
Thus, the existence of a solution to (17) is a corollary of Lemma A.5 in the appendix, along
with the sum rule for recession functions [24, Theorem 9.3].
We now prove that fy k g is bounded. Theorem 3.9 shows that the sequences fg i (y k )g,
above. From (27), unboundedness of fy k g would imply that
. But such unboundedness would contradict g 0 (y k )'s
convergence to the optimal value. 2
We remark that the penalty parameter adjustment rule (16), as discussed in Section
2.2.2, essentially subsumes, in a context broader than '-divergences, the corresponding rules
described in [32] for the exponential method of multipliers and in [5, 3, 4] for a general
'-divergence setting.
We end this section giving some examples of d \Phi
functions that may be derived from
separable Bregman distances (see Section 2.2.1). Further examples may be obtained from [21,
28]. For a Bregman-derived distance, we have d i
whence
d \Phi
where h \Phi denotes the standard monotone conjugate of h. Note that when such a d \Phi
used in the minimization operation in (17), the additive terms h i (w are constant
and may be discarded. The following examples may now be easily verified:
may be disregarded; this choice gives the classical quadratic method of multipliers for
inequality constraints.
where the \Gammaw i term may be
disregarded, yielding the exponentional method of multipliers.
4 Bregman Interior Point Proximal Methods for Variational
Inequalities
We now turn our attention to the box-constrained variational inequality problem (2), where
(possibly set-valued) maximal monotone operator. In this section, we
confine ourselves to Bregman distances, as defined in Section 2.2.
We augment Assumption 2.2 as follows:
RRR 35-99 Page 21
Assumption 4.1 T is maximal monotone, the solution set of (2) is non-empty, and there
exists some e
Our goal is to show convergence of an approximate version of the iteration (4), without
further conditions on T . We modify and extend Assumption 2.7 as follows:
Assumption 4.2 For have the same properties
specified in Assumption 2.7, and furthermore, h i is continuous on [a
defining
4.2.1. For all x 2 B and ff 2 R, the level set fy 2 int B j D h (x; y) - ff g is bounded.
4.2.2. If fx k g ae int B converges to x 2 R n , then lim k!1 D h (x; x k
4.2.3. rge h
Note that at finite a i 's and b i 's, the corresponding h i is now required to take a finite
value. The algorithm can now be stated:
Box Interior Proximal Point Algorithm (BIPPA)
1. Initialization: Let
2. Iteration: Choose ff k such that ff k - c maxf1; h 00(x
)g. Find vectors
repeat the iteration.
4.1 Convergence analysis
First, we cite a result showing that the iteration step of BIPPA is well defined:
Lemma 4.3 [13, Theorem 4(i)] Under Assumption 4.2, there is a unique point x k+1 that
solves the iteration step (28) of the BIPPA with e
We note that it is shown in the unpublished dissertation [28] that (28) has a unique exact
solution even if Assumption 4.2.3 does not hold. This result permits one to dispense completely
with Assumption 4.2.3. However, the proof, while essentially a minor modificiation
of that of [1, Theorem A.1], is quite involved, so we do not include it here.
To guarantee the convergence of the BIPPA, we must assume some vanishing behavior
for fe k g; we will use the assumptions of [14]. Although not as general as the criterion
used in RPMM, these conditions are better suited to our analysis, since they will permit
us to use properties associated with Fej'er monotonicity, and are still feasible to enforce
computationally.
Page 22 RRR 35-99
Assumption 4.4 [14] The error sequence fe k g conforms to:X
exists and is finite.
Note that this assumption implies that Assumption 2.3.1 holds
with k1 . We now state some necessary lemmas:
Lemma 4.6 If Assumption 4.4 holds, then the sequence fx k g is bounded and D h
Proof. The result will follow from [14, Lemma 3] once we show that, for z 2 (T +NB
E(z)
exists and is finite. But,X
and Assumption 4.4 implies that the right hand side of this relation is finite. Hence,
exists and is finite. Using Assumption 4.4 once more, we conclude that
E(z) exists and is finite. 2
We also use a key result from Solodov and Svaiter [29]:
Theorem 4.7 [29, Theorem 2.4] Let h i satisfy Assumption 4.2. Given two sequences fx k g ae
B and fy k g ae int B, either one of which is convergent, with lim k!1 D h
the other sequence also converges to the same limit.
This theorem implies that
Bregman function in the classical
sense [8, 10]. Using Theorem 4.7 and Lemma 4.6, we derive:
Corollary 4.8 Under Assumptions 4.1, 4.2, and 4.4, fx k g has at least one limit point.
Moreover, if for some infinite set K ' N, we have x k !K -
x, then x
x. Therefore,
Assumption 2.3.2 holds.
RRR
Before presenting the main convergence theorem for the BIPPA, we present a final technical
lemma that will help us to prove the uniqueness of the accumulations points of fx k g.
Lemma 4.9 Under Assumption 4.4, for all z converges to a
value in [0; +1) which we will denote by d(z).
Proof. Consider any z 2 (T implies that (29) holds. Using
Assumption 4.4 and D h the hypotheses of Lemma 3.5 are satisfied with
a converges, necessarily to
a nonnegative value. 2
Now, the main convergence theorem follows:
Theorem 4.10 Under Assumptions 4.1, 4.2, and 4.4, fx k g converges to a solution of
Proof. Let -
x be an accumulation point of fx k g, i.e. x k !K - x, for some infinite set K ' N.
Such a point exists by Lemma 4.6. From Theorem 2.6,
We now prove the uniqueness of the limit point: from Assumption 4.2.2, we know that
as defined in Lemma 4.9, is zero. Suppose that fx k g has
another accumulation point x k !K 0 x 0 for some infinite set K 0 ' N. We then have that
it follows from Theorem 4.7 that x
Another possible application of our fundamental analysis is to try to generalize the idea
of adding the square of the Euclidean norm and an arbitrary generalized distance to obtain
Fej'er monotonicity to solutions of (2), as in [2, 3] for the special case of '-divergences. The
difficulty here is to generalize the condition that defines the class \Phi 2 in [3]. This topic is the
subject of ongoing research.
--R
An interior-proximal methods for convex linearly constrained problems and its extension to variational problems
A logarithmic-quadratic proximal method for variational inequalities
Interior proximal and multiplier methods based on second order homogeneous kernels.
Modified Lagrangian Methods for Variational Inequality Problems.
Penalty/barrier multiplier methods for convex programming problems.
Nonlinear Programming
Constrained Optimization and Lagrange Multiplier Methods.
The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming.
An interior-point method with Bregman functions for the variational inequality problem with paramonotone operators
An iterative row-action method for interval convex program- ming
The proximal minimization algorithms with D-functions
A convergence analysis of proximal-like minimization algorithms using Bregman functions
Nonlinear proximal point algorithms using Bregman functions
Approximate iterations in Bregman-function-based proximal algorithms
A Practical General Approximation Criterion for Methods of Multipliers Based on Bregman Distances.
A necessary and sufficient condition to have bounded multipliers in nonconvex programming.
Strict convex regularizations
Augmented Lagrangian methods and proximal points methods for convex optimization.
On the twice differentiable cubic augmented Lagrangian.
Proximal minimization methods with generalized Bregman functions.
Introduction to Optimization.
Extension of Fenchel's duality theorem for convex functions.
Convex Analysis.
Conjugate Duality and Optimization.
Augmented Lagrangians and applications of the proximal point algorithm in convex programming.
Variational Analysis Springer-Verlag
T'opicos em M'etodos de Ponto Proximal.
An inexact hybrid generalized proximal point algorithm and some new results on the theory of Bregman functions.
Entropic proximal mappings with applications to nonlinear programming.
Convergence of proximal-like algorithms
On the convergence of the exponential multiplier method for convex programming.
--TR
--CTR
Alfred Auslender , Paulo J. Silva , Marc Teboulle, Nonmonotone projected gradient methods based on barrier and Euclidean distances, Computational Optimization and Applications, v.38 n.3, p.305-327, December 2007
Paulo J. Silva , Jonathan Eckstein, Double-Regularization Proximal Methods, with Complementarity Applications, Computational Optimization and Applications, v.33 n.2-3, p.115-156, March 2006 | convex programming;varphi-divergence;proximal algorithms;variational inequalities;bregman distances |
589323 | On the Complexity of a Practical Interior-Point Method. | The theory of self-concordance in convex optimization has been used to analyze the complexity of interior-point methods based on Newton's method. For large problems, it may be impractical to use Newton's method; here we analyze a truncated-Newton method, in which an approximation to the Newton search direction is used. In addition, practical interior-point methods often include enhancements such as extrapolation that are absent from the theoretical algorithms analyzed previously. We derive theoretical results that apply to such an algorithm, one similar to a sophisticated computer implementation of a barrier method. The results for a single barrier subproblem are a satisfying extension of the results for Newton's method. When extrapolation is used in the overall barrier method, however, our results are more limited. We indicate (by both theoretical arguments and examples) why more elaborate results may be difficult to obtain. | Introduction
. In their 1993 book [16], Nesterov and Nemirovsky derive complexity
results for convex optimization problems. Their basic algorithm is an interior-point
method where each subproblem is solved using a damped Newton method. If a
nonlinear optimization problem is large (and hence complexity is an important issue)
then Newton's method is not normally used because of its computational costs, so
these results might be considered primarily of theoretical interest. Our goal in this
paper is to derive comparable complexity results for algorithms that more closely resemble
practical interior-point algorithms for large-scale optimization (see, e.g., [1, 9,
10, 11, 13, 19, 23]).
The interior-point method we analyze is strongly related to the barrier method
in [13]. The essential features of this algorithm are that each barrier subproblem is
solved approximately using a truncated-Newton method; then the solutions to the
subproblems are extrapolated to obtain an initial guess for a new subproblem. Many
of the enhancements discussed in [13]-such as preconditioning, a specialized matrix-vector
product, and a numerically stable formula for the search direction-fit into the
theoretical framework used here. The major exception is the line search (see below).
We derive a bound on the number of truncated-Newton iterations required to
solve a barrier subproblem to within some tolerance. Each truncated-Newton iteration
involves the approximate solution of the Newton equations via (say) the conjugate-gradient
method, requiring at most O(n 3 ) computations in exact arithmetic, although
typically the number of computations would be O(n) or O(n 2 ), in problems where the
Hessian matrix is sparse. In the algorithm analyzed here, a prescribed step length is
used, so there is no line search. (Here is where the theoretical and practical algorithms
*Received by the editors Month? Date?, 199?; accepted for publication (in revised form) Month?
1997.
yOperations Research and Engineering Department, George Mason University, Fairfax, VA
22030. The work of this author was supported by National Science Foundation grant DMI-9414355.
zOperations Research and Engineering Department, George Mason University, Fairfax, VA
22030. The work of this author was supported by National Science Foundation grant DMI-9414355.
G. NASH AND ARIELA SOFER
differ, since a practical method would likely use an adaptive line search based on
minimizing a one-dimensional approximation to the barrier function.) Ignoring the
computations for evaluating the gradient and Hessian, the algorithm determines the
solution to within a tolerance in a number of operations that is polynomial
in M and the problem dimensions. If polynomial algorithms exist to evaluate the
gradient and Hessian, the overall algorithm for the barrier subproblem is polynomial
in the dimensions of the optimization problem.
The theoretical result that we obtain reduces to the result for Newton's method
if the inner convergence tolerance for the truncated-Newton method is set to zero.
For this reason, we consider this result to be a satisfying extension of the theory for
Newton's method.
In the second major part of the paper we analyze how a simple linear extrapolation
scheme can accelerate the algorithm by providing improved initial guesses for each
subproblem. We show that improved performance can be achieved by an algorithm
based on linear extrapolation, when barrier subproblems are solved exactly. We also
indicate, via an example, that it may be difficult to derive complexity results either
when subproblems are solved inexactly, or when higher-order extrapolation is used.
Our analysis is based on the framework established in Nesterov and Nemirovsky
(1993) and Nemirovsky (1994) as adapted by us in our book [14]. In the rest of the
paper we cite theoretical results for an algorithm based on Newton's method. These
results are due to Nesterov and Nemirovsky, although we frequently cite [14] because
our discussion here more closely follows the organization and notation of that book.
(Related discussions can be found in [3, 8, 22].)
In the barrier method, we assume that the barrier function is "self concordant,"
a property that we define below. Self-concordant barrier functions were introduced in
[16]; recent work on this topic includes [4, 6, 7, 8, 17]. For linear programs and convex
quadratic programs, the ordinary logarithmic barrier function can be used. Related
barrier functions can be used for semi-definite programming [18, 20, 21, 24]. It is
possible to prove that, for any convex feasible region with the properties we specify
below, there exists an appropriate self-concordant barrier function, although in general
it may not be practical for computation. Thus the results we describe here provide a
general theoretical approach for solving convex programming problems.
Two major results are required to prove that the overall algorithm is a polynomial
algorithm. The first states that if the truncated-Newton method is applied to a single
barrier subproblem, and the initial guess is "close" to the solution, then the number
of iterations required to find an approximate solution of this subproblem is bounded.
This is the topic of Section 3. The second states that the linear extrapolation can
improve the initial guess for the next subproblem. This topic is addressed in Section
4.
These two results are less obvious than they might at first seem. If any constraint
in the convex programming problem is binding at the solution, then the solution will
be on the boundary of the feasible region where the barrier function has a singular-
ity. Since standard convergence results for Newton-type methods assume that the
Hessian at the solution has a bounded condition number, a traditional analysis is not
appropriate.
To analyze the behavior of Newton-type methods in this case, we must in some
manner take this singularity into account. To do this, we define a norm k\Deltak x in terms of
the Hessian of the barrier function evaluated at a point x. We will measure "closeness"
in terms of this norm.
This norm depends on the Hessian, and changes as the variables change. For this
COMPLEXITY OF A PRACTICAL INTERIOR-POINT METHOD 3
norm to be useful, the rate of change of the Hessian matrix must not be "too great."
This reasoning leads to the imposition of a bound on the third derivatives of the barrier
function in terms of the Hessian (see Section 2). This bound is all that is required to
prove the first major result corresponding to the behavior of the truncated-Newton
method on a single barrier subproblem.
To prove that the approximate solution of one subproblem will not be too far
from the solution of the next subproblem, it is necessary that the values of the barrier
functions not change "too quickly" as the barrier parameter changes. To guarantee
this, we impose a bound on the first derivatives of the barrier functions in terms of
the Hessian (see Section 4). By measuring all quantities in terms of the Hessian, we
are able to circumvent the difficulties associated with the singularity of the barrier
function at the solution.
If the barrier function has these properties, then an interior-point method can be
designed so that the optimal solution of a convex programming problem can be found
(to within some tolerance) using a polynomial number of truncated-Newton iterations.
Practical experience suggests that an improved barrier method can be obtained if
the approximate solutions to the subproblems are extrapolated to produce an initial
guess for the next subproblem. We analyze this idea in Section 4. We are able to
derive some theory to support this idea in the case when linear extrapolation is used,
and when the subproblems are solved exactly. Through an example, we suggest that
it may be difficult to obtain a comparable result for either a more elaborate algorithm
(with higher-order extrapolation) or a more realistic algorithm (with the subproblems
solved inexactly).
2. Basics. In this section, we define self concordance and establish some basic
lemmas. An extensive discussion of the theory of self concordance is given in the book
by Nesterov and Nemirovski [16]. The presentation here parallels our book [14].
Let S be a bounded, closed, convex subset of ! n with nonempty interior int S.
(The assumption that S is bounded is not that important, since we could modify the
optimization problem by adding artificial, very large bounds on the variables.) Let F
be a convex function defined on the set S, and assume that F has three continuous
derivatives. Then F is self concordant on S if:
(i) (barrier property) F along every sequence f x i g ae int S converging
to a boundary point of S.
(ii) (differential inequality) F satisfies
for all x 2 int S and all
In this definition,
that is, it is a third-order directional derivative of F .
As an example, the logarithmic barrier function
log(a T
is self concordant on the set
\Psi .
4 STEPHEN G. NASH AND ARIELA SOFER
The constant 2 in the definition is arbitrary. If instead
for some constant C, then the scaled function -
concordant. The
number 2 is used in the definition so that the function F log x is self concordant
without any scaling.
We make several assumptions to simplify our discussion. They are not essential;
in fact, almost identical results can be proved without these assumptions. We assume
that r 2 F (x) is nonsingular for all x 2 int S. (See Theorem 2.1.1 in [16] for an approach
that avoids this assumption.) This allows us to define a norm as follows:
We also assume that F has a minimizer x 2 int S. Because F is convex, these
assumptions guarantee that x is the unique minimizer of F in S.
The following lemmas indicate some basic properties of self-concordant functions.
The first shows that the third-order directional derivative can be bounded using this
norm.
Lemma 1. If F is self-concordant on S then
for all x 2 int S and for all
Proof. See [8] or [16].
The next lemma bounds how rapidly a self-concordant function F (x) and its
Hessian can change if a step is taken whose norm is less than one. The first result is
an analog of a Taylor series expansion for a self-concordant function. The second is a
bound on how rapidly the norm can change when x changes.
Lemma 2. Let F be self concordant on S. Let x 2 int S and suppose that khk x !
1. Then x
where
The lower bound in (1) is satisfied even if khk x - 1. Furthermore, for any g 2 ! n ,
Proof. See [16].
Our convergence results for a barrier subproblem are phrased in terms of a quantity
called the "Newton decrement." It is defined below. The Newton decrement
measures the norm of the Newton direction, but indirectly it can be interpreted as a
"proximity measure" for the distance to the barrier trajectory. We use the Newton
decrement in place of more traditional measures of convergence, such as
COMPLEXITY OF A PRACTICAL INTERIOR-POINT METHOD 5
If x 2 int S and pN is the Newton direction for F at x, then the Newton decrement
of F at x is
Consider the Taylor series approximation to F
The Newton direction pN minimizes this approximation and is the solution to
The optimal value of the Taylor series approximation is
indicating why ffi (F; x) is called the Newton decrement.
We have the following lemma.
Lemma 3. The Newton decrement satisfies
Proof. See [16].
We will obtain bounds on F
in terms of the Newton
decrement, and we will also measure the progress at each iteration of the truncated-
Newton method in terms of the Newton decrement. Thus, statements about the
convergence of the method in terms of the Newton decrement will indirectly provide
us with information about convergence as measured in the more traditional ways.
3. Convergence of a Truncated-Newton Method. We now study the consequences
of using a truncated-Newton method, rather than Newton's method, to
minimize a self-concordant function:
minimize
where S and F are as in the previous section.
In the truncated-Newton method, a search direction p will be computed that
satisfies the acceptance criterion
where x is the current estimate of the solution to the barrier subproblem, pN is the
Newton direction, and ffl is some tolerance. For simplicity, we assume that the tolerance
ffl is fixed, although similar results could be derived in the case where ffl varied from
iteration to iteration.
It is not that important how the search direction is computed, as long as the number
of arithmetic operations is polynomial in the dimensions of the problem. Practical
truncated-Newton methods often use the conjugate-gradient method which, in exact
arithmetic, is guaranteed to converge to the Newton direction in a finite number of
operations. Thus, in exact arithmetic, this would be an appropriate procedure.
6 STEPHEN G. NASH AND ARIELA SOFER
The acceptance criterion (2) is impractical since it involves the Newton direction
pN . It is, however, closely related to practical rules for terminating the inner iteration
of a truncated-Newton method. If we define
to be the value of the quadratic model for F (x) at p, then
x
x
In [12] it is recommended that the inner iteration of a truncated-Newton method be
terminated based on the value of the quadratic model, and the barrier method in [13]
uses a related rule.
Other practical acceptance rules are based on the value of the relative residual,
and have the form
for some tolerance j [2]. It is straightforward to derive thatp cond 2 (r 2 F (x))
where is the condition number of r 2 F (x) in the 2-norm. These inequalities
provide a further demonstration of the relationship of (2) with practical
termination rules for the inner iteration of the truncated-Newton method.
Some useful consequences of the acceptance criterion (2) are stated in the following
lemma.
Lemma 4. Suppose that the acceptance criterion (2) is satisfied at x. then
and
Proof. The first result is a straightforward consequence of (2). The second is
obtained by squaring the acceptance criterion:
COMPLEXITY OF A PRACTICAL INTERIOR-POINT METHOD 7
The function F will be minimized using a "damped" truncated-Newton method,
that is, a step is taken along the truncated-Newton direction but with a specified step
length that is less than one. If we denote the search direction at x by p, then the
method is defined by
p:
The reason for including this step length is that the resulting displacement will always
have norm less than one, so that Lemma 2 applies. It also guarantees that the damped
truncated-Newton step is well-defined, in the sense that the iterates remain in int S.
As the method converges and the truncated-Newton direction approaches zero,
the step length approaches one, so that (asymptotically) rapid rates of convergence
can be attained. The rest of this section develops the properties of the damped
truncated-Newton method.
The next lemma gives a lower bound on how much the function F will be decreased
by a step of the damped truncated-Newton method.
Lemma 5. If x+ is the result of the damped truncated-Newton iteration, then
Proof. Let be the step length. Using Lemma 2 and (4) we
obtain
x
The desired result is just a re-arrangement of this last inequality.
The lemma provides a lower bound for F The lower bound is zero
when kpk and is positive and strictly increasing for kpk x ? 0. (The derivative of
the right-hand side with respect to kpk x is positive.) The result gives a lower bound on
how much progress is made at each truncated-Newton iteration. Figure 1 illustrates
this lower bound for various values of ffl.
If kpk x remains large, then the truncated-Newton method must decrease the value
of F (x) by a nontrivial amount. Since the function is bounded below on S, this cannot
go on indefinitely, and kpk x must ultimately become small.
The next theorem analyzes the convergence of the method in the case where
are related by (3), these two results provide a
bound on the number of truncated-Newton iterations required to solve the optimization
problem to within some tolerance. (This argument is made precise in Theorem
7.) The theorem also determines a bound on kx \Gamma x k x
in terms of ffi (F; x), and thus
shows that if ffi (F; x) is small, then the norm of the error is small as well.
Theorem 6. If x 2 int S then
Let x be the minimizer of F in S, and assume that ffi
8 STEPHEN G. NASH AND ARIELA SOFER
Figure
Bound on F various values of ffl.
bound
bound
bound
bound
Proof. The proof is in two parts, proving each of the results in turn.
Part 1: We will derive the bound on ffi (F; x+ ). Let p be the truncated-Newton
direction at x, pN be the Newton direction at x, and
ffp. For any h 2 ! n we define
This function is twice continuously differentiable for
By Lemma 1 and Lemma 2 we have
x+tp
x
Note that Z ffh Z
x d-
x
Then
x d-
dt
x
x khk x
COMPLEXITY OF A PRACTICAL INTERIOR-POINT METHOD 9
x khk x
x
x khk x
x
Since this is true for all h we have
This completes Part 1 of the proof.
Part 2: The proof of
can be found in [14] or
[15].
We conclude with a summary theorem. It provides a bound on the number of
truncated-Newton iterations required to minimize F to within a tolerance.
Theorem 7. Let S be a bounded, closed, convex subset of ! n with non-empty in-
terior, and let F (x) be a convex function that is self concordant on S. Given an initial
guess x0 2 int S, the damped truncated-Newton method is defined by the recurrence
where p is an approximation to the Newton direction pN at
for some tolerance 0
F
At some iteration i we must have
and then for every j - i we have
G. NASH AND ARIELA SOFER
The number of truncated-Newton steps required to find a point x
bounded by
for some constant C that depends on ffl and that decreases as ffl ! 0. If then the
number of steps is bounded by
for some constant -
C.
Proof. The bound on F derived in [16]. The remaining conclusions
are consequences of the earler results. The formulas for the constants C and -
C can
be derived as follows. Let be the first iteration for which
Such an index i must exist, since F is bounded below on S. If
then by (3)
It follows from Lemma 5 that
Thus the number of initial iterations i is at most
The progress of the later iterations is described by Theorem 6. If j - i, then
Hence the number of later iterations is at most
Summing these two bounds determines the constant C.
then at the later iterations
and a similar analysis determines -
C.
and the truncated-Newton method computes the Newton direction, then
the theorem is the same as that reported for Newton's method in [14, 15]. It establishes
the polynomiality of the truncated-Newton method applied to a barrier subproblem
(see Section 1 for details).
4. Extrapolation. In Section 3 we analyzed the behavior of a truncated-Newton
method when applied to a single barrier subproblem. We now consider the overall
interior-point method based on solving a sequence of subproblems. Our main concern
is with the effects of extrapolating the (approximate) solutions of several subproblems
to obtain an improved initial guess for the next subproblem.
COMPLEXITY OF A PRACTICAL INTERIOR-POINT METHOD 11
The complexity results for the overall method depend on an additional assump-
tion, that is, a bound on the first-derivative of the barrier function. Although we do
not make much direct use of this assumption in this paper, it underlies many of our
comments, and so we state it here.
Let S be a set with the same properties as in Section 3. Following [16], a self-
concordant function F on S is a self-concordant barrier function for S if, for some
for all x 2 int S and all h 2 ! n . We may assume, without loss of generality, that
- 1.
The function F
barrier function with
the function
log(a T
is a self-concordant barrier function with for the set
\Psi .
A self-concordant barrier function exists for any closed convex set S [16]; evaluating
such a barrier function may not be computationally practical, however.
We also assume that the convex program is written in the following standard
subject to x 2 S
where c 6= 0. An optimization problem with a general nonlinear objective function
can be converted to this form by adding an additional variable and constraint.
The problem (P) will be solved using a path-following method of the following
form. For ae ? 0 we define
where F is a self-concordant barrier function for the set S with parameter - 1, and
where the Hessian of F is nonsingular for all x 2 int S. (The nonsingularity assumption
is not essential; see [16].) Let x (ae) be the minimizer of F ae (x) for x 2 int S. Our
method will generate x
A complexity result for a particular algorithm based on this approach and Theorem
7 can be proved as in [14, 15]. In this algorithm, x i 2 int S is accepted as an
approximate minimizer of F ae i if
for some
2 , and then x i is used as an initial guess for minimizing F ae i+1 .
Theorem 8. Suppose that we solve the problem (P) on a bounded, closed, convex
domain S using the path-following method described above, where F is a self-
concordant barrier function with parameter - 1. Let 1be the parameter in
the acceptance test, and assume that the penalty parameters are updated via
G. NASH AND ARIELA SOFER
Assume that the method is initialized with ae 0 and x0 2 int S where x0
satisfies the acceptance test for F ae 0
. Then
ae 0
If a truncated-Newton method based on (2) is used to minimize F ae i to within proximity
-, then the number of truncated-Newton iterations required to find an approximate
solution to subproblem i does not exceed a constant N -;';ffl depending only on -, ', and
ffl. In particular, the total number of truncated-Newton iterations required to find an x
satisfying c T bounded above by
log
with constant C -;';ffl depending only on -, ', and ffl.
It is possible that a better initial guess for a subproblem (and hence a better
algorithm) can be obtained by extrapolation of previous solutions. The technique
of extrapolation-initially proposed by Fiacco and McCormick [5]-approximates the
barrier trajectory x(ae) by a polynomial of degree q. The coefficients of the polynomial
are computed from the solutions of q subproblems, and are then used to predict
the solution of the barrier subproblem for the new value of ae. The results in [5] indicate
that extrapolation is a powerful computational tool. Our own computational
experiments [13] also indicate that better initial guesses (and better overall perfor-
mance) can be obtained by extrapolating the solutions of a sequence of subproblems.
An example illustrating the usefulness of extrapolation is summarized in Table 1; for
details, see [13]. In this case, the use of cubic extrapolation reduces the number of
truncated-Newton iterations by a factor of 2.1, and the number of gradient evaluations
by a factor of 2.7.
(Complexity results for a predictor-corrector method can be found in [15]; this
predictor-corrector method is a form of extrapolation, but appears to be less practical
for large nonlinear programs than the technique used here. It is not clear that
this predictor-corrector approach can be extended effectively to a truncated-Newton
method.)
The lemma below gives theoretical support to these computational results. The
technique of extrapolation may have limitations, however, as we shall discuss in the
latter part of this section.
We first examine linear extrapolation and assume that x
linear extrapolation predicts that
where
More generally, \Delta i can be considered as a "search direction" along the barrier tra-
jectory, as in the lemma below. The result shows that linear extrapolation can be
used to produce initial guesses that are at least as good as when extrapolation is not
used. For simplicity, we choose fl i j fl for all i, although it would be easy to extend
the result to the case where the fl i 's are not constant. (The lemma uses our general
COMPLEXITY OF A PRACTICAL INTERIOR-POINT METHOD 13
Table
Effect of Cubic Extrapolation (iter = number of truncated-Newton iterations,
ng = number of gradient evaluations). Based on [13] with elaborate
barrier algorithm, test problem 51, 1000 variables.
Extrapolation No Extrapolation
ae iter ng iter ng
28 277
Totals 51 367 107 982
assumption that r 2 F is positive definite. If r 2 F is only positive semi-definite, then
the lemma is still true under the assumption that c T \Delta i 6= 0.)
Lemma 9. Let ae assume that x
Define the linear extrapolation direction
If
then
that is, \Delta i is a descent direction for F i+1 at x i . We define
with a more specific upper bound provided by (1).
Proof. We first prove that c T x series expansion gives
Because x i 6= x positive definite, the last term is positive. Since
14 STEPHEN G. NASH AND ARIELA SOFER
which implies that
F
Similarly, by switching the roles of x i and x i\Gamma1 , we obtain
If we multiply the first of these inequalities by ae i , the second by ae i\Gamma1 , and rearrange,
then
Combining this with the above results
We now use this result to prove that \Delta i is a descent direction. Since x
Using these results, we obtain
This completes the first part of the proof.
The second part of the proof relies on (1). Using the formulas above, the upper
bound in (1) becomes
Note that by the assumptions in the lemma.
If we solve OE 0 we obtain the solution
It is straightforward to verify that OE 00 (ff so that this is a local minimizer of OE.
we obtain that
and so x
Since the upper bound on F i+1 is decreasing at
is a local minimizer of the upper bound, this completes the proof.
COMPLEXITY OF A PRACTICAL INTERIOR-POINT METHOD 15
This result could be extended to the case where the subproblems are not solved
exactly, as long as the magnitudes of
were sufficiently small so as not
to interfere with the inequalities in the lemma. It appears to be difficult to generalize
the above result greatly, however, as the discussion below indicates.
Lemma 9 shows that an appropriate step along the extrapolation direction can
produce a decrease in the objective value of the barrier function for the updated
barrier parameter. However taking the "full" extrapolation step (fl i =fl
guaranteed to be beneficial, no matter how slight the change in the barrier parameter.
This is true even when the subproblems are solved exactly. To see this, let
and consider the extrapolated point
The effect of extrapolation is described by
Now
d
d
0:
Thus, for a small change in the barrier parameter (i.e., as to first order
there is no improvement in the objective value of the new barrier function at the
extrapolated point.
Somewhat more insight can be obtained by analyzing / 00 (0):
The second term is positive by convexity, but the first term is less than or equal to
zero (see Lemma 9). If we define
then we can write
From (5) and our comments above we obtain
G. NASH AND ARIELA SOFER
If kvk x i
- then the lower bound on / 00 (0) is positive, and hence no improvement
in the objective value is obtained. Thus the full extrapolation step is not guaranteed
to be useful.
So far, we have only considered linear extrapolation. We now indicate via an
example that higher-order extrapolation may lead to predicted solutions that are far
worse than those obtained without extrapolation. In this example, we consider both
exact and approximate solutions to subproblems. As above, an approximate solution
x i to a barrier subproblem will be accepted if x i 2 int S and if
for some - 0.
Consider the one-variable problem
subject to x - 0:
This problem is already in standard form, and log(x) can be used as a barrier function
with
It is easy to analyze this example. The solution to the optimization problem is
The solution to the barrier subproblem is x 1=ae. The norm has the
value jh=xj. The Newton decrement is xaej. The solution to
a subproblem is accepted if
The penalty parameter is updated via ae
Here we use four values of 0:5. The choice
corresponds to solving the subproblems exactly. We initialize the penalty
parameter with ae In each case, the approximate solution to the subproblem
is chosen at the upper bound of the acceptable range. This choice is consistent with
the class of theoretical algorithms that we study in this paper, and is plausible for a
practical algorithm.
In
Figure
2 we show the results of applying quadratic extrapolation on this prob-
lem. The exact solutions of the first seven barrier subproblems are marked with \Theta.
The vertical bars indicate the range of x values that satisfy the acceptance criterion for
a subproblem. The dotted lines show the path of extrapolated approximate solutions,
with a * used to indicate the extrapolated initial guess.
In each of the four cases, the extrapolated values are exceedingly poor initial
guesses for the next subproblem. In fact, the extrapolation paths move away from,
rather than toward, the solution of the next subproblem.
In
Figure
3 we show the results of applying linear extrapolation. In this case,
the approximate solutions are chosen as the lower bound of acceptable values for the
first subproblem, and the upper bound for the second subproblem. This is allowed
by the theory, but we think it unlikely that a practical algorithm could produce such
approximate solutions.
In this case, even with linear extrapolation, the extrapolated points can (when
- is sufficiently large) point away from the solution of the next subproblem. Hence
extrapolation is worse than doing nothing. As mentioned, we do not expect these
circumstances to arise for a practical algorithm. Nevertheless, any complexity theory
COMPLEXITY OF A PRACTICAL INTERIOR-POINT METHOD 17
Figure
Quadratic Extrapolation of Approximate Solutions.
rho
x
Quadratic: kappa=0
rho
x
Quadratic: kappa=.1
rho
x
Quadratic: kappa=.25
rho
x
Quadratic: kappa=.5
Figure
Linear Extrapolation of Approximate Solutions.
rho
x
rho
x
rho
x
rho
x
for such an algorithm would have to rule out this possibility, and hence would have to
be based on a more elaborate theoretical framework than that used here.
This example suggests that an algorithm that uses extrapolation must monitor
the effectiveness of the extrapolation scheme, and must not use it blindly. It also
suggests that the algorithm would have to be carefully designed so that inaccuracies
in the solutions of barrier subproblems did not interfere with the performance of the
extrapolation scheme.
We have not been able to derive complexity results for a more elaborate algorithm
of this type. This example leads us to think that this would be a difficult enterprise.
G. NASH AND ARIELA SOFER
--R
Computational experience with penalty/barrier methods for nonlinear programming
Interior Point Approach to Linear
A sufficient condition for self-concord- ance
Sequential Unconstrained Minimization Techniques
Osman G- uler
Two interior-point algorithms for a class of convex programming problems
Interior point methods via self-concordance or relative Lipschitz condition
A practical interior-point method for convex pro- gramming
An unconstrained optimization technique for large-scale linearly constrained minimization problems
Assessing a search direction within a truncated-Newton method
A barrier method for large-scale constrained optimiza- tion
Linear and
Interior point polynomial time methods in convex programming
Homogeneous interior-point algorithms for semidefinite program- ming
An infinitely summable series implementation of interior-point methods
On the long step path-following method for semidefinite programming
Complexity Issues
Interior methods for constrained optimization
Extending primal-dual interior point algorithms from linear programming to semidefinite programming
--TR | convex programming;interior-point method;truncated-Newton method;self-concordance;large-scale optimization;complexity |
589351 | An Unsymmetrized Multifrontal LU Factorization. | A well-known approach to computing the LU factorization of a general unsymmetric matrix A is to build the elimination tree associated with the pattern of the symmetric matrix A + AT and use it as a computational graph to drive the numerical factorization. This approach, although very efficient on a large range of unsymmetric matrices, does not capture the unsymmetric structure of the matrices. We introduce a new algorithm which detects and exploits the structural asymmetry of the submatrices involved during the processing of the elimination tree. We show that with the new algorithm, significant gains, both in memory and in time, to perform the factorization can be obtained. | Introduction
We consider the direct solution of sparse linear equations based on a multifrontal approach.
The systems are of the form A is an n n unsymmetric sparse matrix. The
multifrontal method has been developed by Du and Reid [11, 12] for computing the solution
of indenite sparse symmetric linear equations using Gaussian elimination and then has been
extended to solve more general unsymmetric matrices by Du and Reid [13].
The multifrontal method belongs to the class of methods that separate the factorization into
an analysis phase and a numerical factorization. The analysis phase involves a reordering
step, which will reduce the ll-in during numerical factorization and a symbolic phase that
This work was supported by the Director, Oce of Science, Division of Mathematical, Information, and
Computational Sciences of the U.S. Department of Energy under contract number DE-AC03-76SF00098.
y amestoy@enseeiht.fr, ENSEEIHT-IRIT, 2 rue Camichel 31071 Toulouse (France) and PRamestoy@lbl.gov,
NERSC, Lawrence Berkeley National Laboratory, 1 Cyclotron Rd. Berkeley CA 94720
z NERSC, Lawrence Berkeley National Laboratory, 1 Cyclotron Rd., Berkeley CA 94720
builds the computational tree, so called elimination tree [9, 18, 20], whose structure gives the
dependency graph of the multifrontal approach. The analysis phase is generally not concerned
with numerical values and is only based on the sparsity pattern of the matrix.
As far as the analysis phase is concerned, the approaches introduced by Du and Reid
for both symmetric and unsymmetric matrices are almost identical. When the matrix is
unsymmetric, the structurally symmetric matrix the summation is
performed symbolically, is used in place of the original matrix A. The elimination tree of
the unsymmetric LU factorization is thus identical to that of the Cholesky factorization of the
symmetrized matrix M.
To control the growth of the factors during LU factorization, partial threshold pivoting is
used during the numerical factorization phase. The pivot order, used during the analysis to
build the elimination tree might not be respected. Numerical pivoting can then result in an
increase in the estimated size of the factors and in the number of operations. To improve the
numerical behaviour of the multifrontal approach it is common to involve a step of preprocessing
based on the numerical values. In fact if the matrix is not well-scaled, which means that the
entries in the original matrix do not have the same order of magnitude, a good prescaling
of the matrix can have a signicant impact on the accuracy and performance of the sparse
solver. In some cases it is also very benecial to precede the ordering by performing an
unsymmetric permutation to place large entries on the diagonal. Du and Koster [10] have
designed algorithms to permute large entries onto the diagonal and have shown that it can very
signicantly improve the behaviour of multifrontal solvers.
The multifrontal approach by Du and Reid [13] is used in the Harwell Subroutine Library code
3] and in the distributed memory code MUMPS developed in the context of the PARASOL
project (EU ESPRIT IV LTR project 20160) [4, 5]. Another way to represent the symbolic LU
factorization of a structurally unsymmetric matrix is to use directed acyclic graphs (see for
example [14, 15]). These structures more costly and complicated to handle than a tree, capture
better the asymmetry of the matrix. Davis and Du [6] implicitly use this structure to drive
their unsymmetric-pattern multifrontal approach.
We explain, in this article, how to use the simple elimination tree structure of the symmetric
matrix M to detect, during the numerical factorization phase, structural asymmetry in the
factors. We show that, with the new factorization phase, we very signicantly reduce the
computational time, the size of the LU factors and the total memory requirement with respect
to the standard multifrontal approach [13]. In Section 2, we rst recall the main properties of
the elimination tree and describe the standard multifrontal factorization algorithm. We then
introduce the new algorithm and use a simple example to show the benets that can be expected
from the new approach. In Section 3, our set of test matrices is introduced. We analyse the
performance gains (in terms of size of the factors, memory requirement and factorization time)
of the new approach with respect to the standard multifrontal code on our set of test matrices.
We add some concluding remarks in Section 4
2 Description of the multifrontal factorization algorithms
Let A be an unsymmetric matrix and let M denote the structurally symmetric matrix A+A T .
The elimination tree is dened using the structure of the Cholesky factors of M. If the matrix
M is reducible then the tree will be a forest. Liu [18] denes the elimination tree as the
transitive reduction of the directed graph of the Cholesky factors of M. The characterization
of the elimination tree and the description of its properties are beyond the scope of this article.
In our context, we are interested in the elimination tree only as the computational graph for
the multifrontal factorization. For a complete description of the elimination tree the reader can
In the multifrontal approaches, we actually use an amalgamated elimination tree, referred to as
the assembly tree [12] which can be obtained from the classical elimination tree. Each node
of the assembly tree corresponds to Gaussian elimination operations on a full submatrix, called
a frontal matrix. The frontal matrix can be partitioned as shown in Figure 1.
fully summed rows -
partly summed rows -
fully summed columns
partly summed columns
Figure
1: Partitioning of a frontal matrix.
Each frontal matrix factorization involves the computation of a block of columns of L,
termed fully summed columns of the frontal matrix, a block of rows of U, termed fully
summed rows, and the computation of a Schur complement matrix F 22 F 21 F 1
called a
contribution block. The rows (columns) of the F 22 block are referred to as partly summed
rows (columns).
original nonzero
new entry in M
Figure
2: Example of matrix A and
The unsymmetric matrix A, on the left-hand side of Figure 2, will be used to illustrate the
main properties of the assembly tree and to introduce the new algorithm. In Figures 2 and 3
an \X" denotes a nonzero position from the original matrix A and a \ i
" corresponds to a
new entry introduced during symmetrization. In Figure 3, we indicate the structure of the lled
matrix is the matrix of the Cholesky factor of M. Entries with an \F"
corresponds to ll-in entries in the L factor.
F
F
F
original nonzero
Fill-in (w.r.t. M)
new entry in M
Figure
3: Structure of the Cholesky factors of the matrix M.
The matrix M F is used to dene the assembly tree (see Figure 4) associated with the multifrontal
LU factorization of the matrix A. From the fact that the factorization is based on the assembly
tree associated with the Cholesky factorization of M F , it results that
where Struct() denotes the matrix pattern. Let us denote by structural zero a numerical
zero that does not result from numerical cancellation. Typically, due to the symmetrization,
the matrix M might contain many structural zeros that will propagate during the numerical
factorization phase. What has motivated our work is the following question. Is it possible,
during the processing of the assembly tree to eciently detect and remove structural zeros that
appear in matrix M F and that are direct or indirect consequence of the symmetrization of
Although it is not so clear from the structure of the matrix M F , we will show that
blocks of structural zeros can be identied during the processing of the assembly tree.
In the following, we rst describe how the assembly tree is exploited during the standard
multifrontal algorithm. We then report and analyse the sparsity structure of the frontal matrices
involved in the processing of the assembly tree associated with our example matrix. Based on
these observations, we will introduce the new factorization algorithm.
The assembly tree is rooted (a node of the tree called the root is chosen to give an orientation
to the tree) and is processed from the leaf nodes to the root node. If two nodes are adjacent
in the tree, then the one nearer the root is the parent node, and the other is termed its
child. Each edge of the assembly tree indicates a data dependency between parent and child.
It involves sending a contribution block from the child to the parent. A parent node process
will start when the processes associated with all of its children are completed.
U
U
U
U
original nonzero
arrowhead of var. 3
contribution block
Root
(2)
(1)
Figure
4: Assembly tree associated with our test matrix.
For example, in Figure 4, node (3) must wait for the completion of nodes (1) and (2) before
starting its computations. The subset of variables which can be used as pivots (boldface
variables in Figure 4) are the fully summed variables of node (k). The contribution blocks
of the children and the entries from the original matrix corresponding to the fully summed
variables of node (k) are used to build the frontal matrix of the node. This will be referred
to as the assembly process. During the assembly process of a frontal matrix, we need for
each fully summed variable j, to access the nonzero elements in the original matrix that are in
rows/columns of indices greater than j. A way to eciently access the original matrix is to store
it in arrowheads according to the reordered matrix. For example during the assembly process
of node (3) the arrowheads of variables 3 and 4 from matrix A together with the contribution
blocks of nodes (1) and (2) are used to assemble the frontal matrix of node (3). One should
note that, by construction, the list of indices in the partly summed rows is identical to that
of the partly summed columns (row and column indices of block F 22 in Figure 1). Therefore,
during the assembly process, only the list of row indices of the partly summed rows is built. This
list is obtained by merging all the row and column indices of the arrowheads of the matrix A
with the row indices of the contribution blocks of all the sons. Once the structure of the frontal
matrix is built, the numerical values from both the arrowheads and the contribution blocks can
be assembled at the right place in the frontal matrix. The
oating point operations involved
during the assembly process will be referred to as assembly operations (only additions)
whereas
oating-point operations involved during the factorization of the frontal matrices will
be referred to as elimination operations.
Partial threshold pivoting is used to control the element growth in the factors. Note that
pivots can be chosen only from within the block F 11 of the frontal matrix. The LU factors
corresponding to the fully summed variables are computed and a new contribution block is
produced. When a fully summed variable of node (k) cannot be eliminated during the node
process because of numerical considerations, then the corresponding arrowhead in the frontal
matrix is added to the contribution block and the fully summed variable will be included in the
fully summed variables at the parent of node (k). This process creates additional ll-in in the
LU factors.
In a multifrontal algorithm, we have to provide space for the frontal matrices and the
contribution blocks, and to reserve space for storing the factors. We need working space to
store both real and integer information. This will be referred to as the total working space
of the factorization phase. The same integer array can be used to describe a frontal matrix, its
corresponding LU factors and its contribution block. The management of the integer working
array can thus be done in a simple and ecient way. In a uniprocessor environment, it is possible
to determine the order in which the assembly tree will be processed. Furthermore, if we process
the assembly tree with a depth rst search order, we can use a stack to manage the storage
of the factors and the contribution blocks. This mechanism is ecient both in terms of total
memory requirement and amount of data movement (see [12]). A stack mechanism, starting
from the beginning of the real working array, is used to store the LU factors. Another stack
mechanism starting from the end of the real working array is used to store the contribution
blocks. After the assembly phase of a node the working space used by the contribution blocks
of its children can be freed and, because the assembly tree is processed with a depth rst search
order, the contribution blocks will always be at the top of the stack. In the remainder of this
paper, the maximum stack size of the contribution blocks will be referred to as the maximum
stack size.
The standard and new algorithms for multifrontal factorization
During a multifrontal factorization, each frontal matrix can be viewed as the minimum structure
to perform the elimination of the fully summed variables and to carry the contribution blocks
from all of its sons. In Figure 5, we have a closer look at the frontal matrices involved in the
processing of the assembly tree of Figure 4 to identify the structural zeros.
We report, beside each node, the structure of the factorized frontal matrix assuming that the
pivots are chosen down the diagonal of the fully summed block and in order (i.e. no numerical
pivoting is required). An \X" corresponds to a nonzero entry and a \O" corresponds to a
structural zero.
One can see that, for our example, the frontal matrices have many structural zeros. There
are two kinds of structural zeros: those forming a complete zero column (or row), and more
isolated zero entries in a nonzero column or row (for example entries (4,3) and (4,7) in the
e
standard algorithm.
frontal matrix of node (3)). If one knows how to detect a partly summed row (or column)
with only structural zeros then the corresponding row (or column) can be suppressed from the
frontal matrix because this row (or column) will not add any contribution to the father node.
Structural zero rows (or columns) can be detected during the assembly process of a frontal
matrix because of the following property: if a row (or column) index does not appear in the
row (or column) indices both of the arrowheads of the original matrix and of the contribution
blocks of the sons, then this index will correspond to a row (or column) with only structural
zeros. This property is used to deduce the assembly process of the new algorithm. Note that
if the matrix is not structurally decient then each fully summed row (or column) must have
at least one nonzero entry. Therefore, we can restrict our search for zero rows (columns) to the
partly summed rows (columns).
In the new assembly algorithm, the list of indices of the partly summed rows of a frontal matrix
is dened as the merge of the row indices in the arrowheads of the fully summed variables of the
node with the row indices of the contribution blocks of its sons. The column indices are dened
similarly. As it is illustrated in Figure 6, the new assembly process can result in signicant
modications in the processing of the assembly tree. For example, on node (1), row 3 and
column 5 are suppressed from the frontal matrix; on node (2), all the partly summed rows are
suppressed; on node (3), row 7 and column 5 are suppressed. As it can be noticed in Figure 6,
frontal matrices naturally become unsymmetric in structure.
We nally indicate in Figure 7 the structure of the LU factors obtained with the new algorithm.
Figure
Processing the assembly tree associated with the matrix A in Figure 2 using the new
algorithm.
This should be compared to the matrix M F in Figure 3 showing the structure of the factors
obtained with the standard algorithm. It can be seen that nonzero entries corresponding to
original nonzero
Fill-in (w.r.t. M)
new entry in M
Figure
7: Structure of the LU factors obtained with the new algorithm.
ll-in (for example (7,4) in M F ) or introduced during the symmetrization of M (for example
in M F ) might be suppressed by the new algorithm. On the other hand, the new algorithm
will never suppress structural zeros in a block of fully summed variables (for example (4,3) in
node (3) of Figure 5). On our small example, the total number of entries in the factors reduces
from 31 to 23.
Comparing Figures 5 and 6, one can notice that the new algorithm might also lead to a signicant
reduction in both the number of operations involved during the assembly process and the
maximum stack size. The latest combined with a reduction in the size of the factors will result
in a reduction in the total working space. On our example the number of assembly operations
drops from 30 to 20 (18 entries from A plus 2 from the contribution blocks). The maximum
stack size reduces from 8 to 1 (obtained in both cases after stacking the contribution blocks of
nodes (1) and (2)).
3 Results and performance analysis
We describe in Table 1 the set of test matrices (order, number of nonzero entries, structural
symmetry and origin). We dene the structural symmetry as the percentage of the number
of nonzeros matched by nonzeros in symmetric locations over the total number of entries. A
symmetric matrix has a value of 100. Although, our performance analysis will focus on matrices
with a relatively small structural symmetry, all classes of unsymmetric matrices are represented
in this set. The selected matrices come from the forthcoming Rutherford-Boeing Sparse Matrix
Collection [8] 1 , Tim Davis collection 2 , and SPARSEKIT2 3 .
The Harwell Subroutine Library [16] code ma41 has been used to obtain the results for the
standard multifrontal method. The factorization phase of ma41 has then been modied with the
new algorithm. The ma41 code has a set of parameters to control its eciency. We have used the
default values for our target computer. Approximate minimum degree ordering [1] has been used
to reorder the matrix. As we have mentioned in the Introduction, it is often quite benecial for
very unsymmetric matrices to precede the ordering by performing an unsymmetric permutation
to place large entries on the diagonal and then scaling the matrix so that the diagonal entries
are all of modulus one and the o-diagonals have modulus less than or equal to one. We use
the Harwell Subroutine Library code mc64 [10] to perform this preordering and scaling on all
matrices of structural symmetry smaller than 55. When mc64 is not used, our matrices are
always row and column scaled (each row/column is divided by its maximum value). All results
presented in this section, have been obtained on one processor (R10000 MIMPS RISC 64-bit
processor) of the SGI Cray Origin 2000 from Parallab (University of Bergen, Norway). The
processor runs at a frequency of 195 Mhertz and has a peak performance of 400 M
ops per
second.
Web page http://www.cse.clrc.ac.uk/Activity/SparseMatrices/
Web page http://www.cise.ufl.edu/ davis/sparse/
3 Web page http://iftp.cs.umn.edu/pub/sparse/
Matrix name Order No. entries StrSym Origin (Discipline)
av4408 4408 95752 0 Vavasis (Partial di. eqn.) [21]
bbmat 38744 1771722 54 Rutherford-Boeing (CFD)
cavity26 4562 138187 95 SPARSEKIT2 (CFD)
ex11 16614 1096948 100 SPARSEKIT2 (CFD)
goodwin 7320 324784 64 Davis (CFD)
lhr14c 14270 307858 1 Davis (Chemical engineering)
lhr17c 17576 381975 0 Davis (Chemical engineering)
lhr34c 35152 764014 0 Davis (Chemical engineering)
lhr71c 70304 1528092 0 Davis (Chemical engineering)
lns 3937 3937 25407 87 Rutherford-Boeing (CFD)
Rutherford-Boeing (Economics)
Rutherford-Boeing (Demography)
raefsky6 3402 137845 2 Davis (Structural engineering)
Rutherford-Boeing (Chemical engineering)
rim 22560 1014951
sherman5 3312 20793 78 Rutherford-Boeing (Oil reservoir simul.)
shyy161 76480 329762 77 Davis (CFD)
twotone 120750 1224224 28 Davis (Circuit simulation)
wang4 26068 177196 100 Rutherford-Boeing (Semiconductor)
Table
1: Test matrices. StrSym denotes the structural symmetry.
In the following graphs, we report the performance ratios of the new factorization algorithm
over the standard algorithm. Matrices are sorted by increasing structural symmetry of the
matrix to be factored, i.e. after application of the column permutation when mc64 is used. We
use the same matrix order in the graphs and in the complete set of results provided in Tables 2
and 3. In this way, one can easily nd, given a point in the graph, its corresponding entry in
the tables.
On the complete set of test matrices, we rst analyse in Figure 8 what is probably of main
concern for the user of a sparse solver, i.e. the time to factor the matrix and the total working
space (as dened in the previous section). In Figure 8, we divide the matrices into three
categories: matrices of structural symmetry smaller than 50 for which the time reduction is
between 20% and 80%, matrices whose structural symmetry is between 50 and 80 for which the
time reduction is between 3% and 20% and nearly structurally symmetric matrices for which
there is almost no dierence between the standard and new version. It is interesting to notice
that even on symmetric matrices the added work to detect asymmetry does not aect much the
performance of the factorization. In the remainder of this paper, we will not report results on
almost structurally symmetric matrices (symmetry greater than 80).
43 43 48
new
over
standard
version
total working space
Figure
8: Study of the factorization time and the total working space. sy (in abscissa)
corresponds to structurally symmetric matrices.
In
Figure
9, we relate the gain in the factorization time with the reduction in the number of
elimination operations and in the number of assembly operations. Although the number of
operations due to the assembly is always much smaller that the number of operations involved
during factorization (see Tables 2 and 3), the assembly process can still represent an important
part of the time spent in the factorization phase (see for example [2]). This is illustrated in
Figure
9 where we see that the high reduction in the number of assembly operations (more
than 50%) signicantly contributes to reducing the factorization time. Note that on a relatively
large matrix (twotone) of symmetry 57 still signicant gains in time and in number of assembly
operations (more than 40%) can be obtained. In Figure 10, we relate the total working space
reduction to the size of the factors and to the maximum stack size. Although a reduction in the
maximum size of the stack might not always introduce a reduction in the total working space,
we see that in practice it is often the case. An extreme example of this reduction is matrix
orani678 of symmetry 9 (see Table 2) for which maximum stack size is reduced by more than
one order of magnitude (5482454 to 457312). Finally, we notice that a large reduction in the
maximum stack size (Figure 10) will generally correspond to a large reduction in the number
of assembly operations (Figure 9).
Structural Symmetry
new
over
standard
version
oper for factors
oper for assem.
Figure
9: Impact of the reduction in the number of
oating point operations on the time.
43 43 48
new
over
standard
version
maximum size of stack
space for LU factors
total working space
Figure
10: Correlation between the factor size, the maximum stack size, and the total working
space.
Matrix Total Operations in Facto.
(Str.Sym.) Version LU stack space Elimin. Assemb. Time
raefsky6 Stnd 1509016 606458 2017106 4.795E+08 4.348E+06 2.17
(2) New 998064 145575 1119135 2.313E+08 1.049E+06 1.14
raefsky5 Stnd 1757680 378792 2095082 3.746E+08 3.874E+06 1.83
av4408 Stnd 551354 227787 677254 6.872E+07 1.451E+06 0.46
lhr14c Stnd 2167304 415090 2439502 2.092E+08 7.944E+06 1.77
lhr34c Stnd 5613656 755710 6249187 6.282E+08 2.064E+07 5.24
lhr17c Stnd 2813418 639172 3204801 3.089E+08 1.085E+07 2.65
lhr71c Stnd 11615170 729857 12711920 1.402E+09 4.304E+07 13.16
twotone Stnd 22086166 15899616 34489449 2.933E+10 2.171E+08 183.84
onetone1 Stnd 4713485 3348215 6212037 2.282E+09 2.675E+07 14.54
rdist1 Stnd 258096 53767 279999 8.150E+06 5.054E+05 0.13
Table
2: Comparison of the standard (Stnd) and the new algorithms on matrices of structural
symmetry < 50.
Matrix Total Operations in Facto.
(Str.Sym.) Version LU stack space Elimin. Assemb. Time
bbmat Stnd 44111480 8351266 48035816 3.676E+10 2.283E+08 185.75
utm3060 Stnd 324640 78679 806970 2.683E+07 6.973E+05 0.22
utm5940 Stnd 701496 131839 2799224 6.640E+07 1.529E+06 0.51
onetone2 Stnd 2253553 898540 385816 5.085E+08 7.628E+06 3.88
goodwin Stnd 1264140 307777 1385604 1.612E+08 2.841E+06 1.05
rim Stnd 4127204 833290 4371615 5.648E+08 9.194E+06 3.37
shyy161 Stnd 7437816 377535 290589 9.945E+08 1.178E+07 6.56
shyy41 Stnd 251336 28523 8137032 1.036E+07 6.337E+05 0.14
sherman5 Stnd 167412 61227 217769 1.284E+07 4.414E+05 0.13
lns 3937 Stnd 285517 89578 335285 1.920E+07 5.482E+05 0.20
cavity15 Stnd 202629 33004 230111 1.033E+07 3.453E+05 0.10
cavity26 Stnd 394164 58589 447293 2.433E+07 6.877E+05 0.22
ex11 Stnd 11981558 3960507 13614400 6.678E+09 3.836E+07 27.76
fidapm11 Stnd 15997220 4863371 19069150 9.599E+09 4.705E+07 39.78
olaf1 Stnd 5880174 1506068 6794919 1.965E+09 1.685E+07 8.91
wang4 Stnd 11561486 5063375 15900237 1.048E+10 4.087E+07 46.19
Table
3: Comparison of the standard (Stnd) and the new algorithms on matrices of structural
symmetry >= 50
Concluding remarks
We have described a modication of the standard multifrontal LUfactorization algorithm that
can lead to a signicant reduction in both the computational time and the total working space.
The standard multifrontal algorithm [13] for unsymmetric matrices is based on the assembly
tree of a symmetrized matrix and involves frontal matrices symmetric in structure. Therefore, it
produces LU factors such that the matrix F = L+U is symmetric in structure. This approach
is currently used in the context of two publically available packages (ma41 [2, 3] and MUMPS [5, 4])
and has the advantage, with respect to other unsymmetric factorization algorithms [6, 7, 17],
of having the LU factorization based on the processing of an assembly tree, while the other
approaches explicitly or implicitly use a more complex to handle graph structure.
We have demonstrated that, based on the same assembly tree, one can derive a new multifrontal
algorithm that will introduce asymmetry in the frontal matrices and in the matrix of the factors
F. The detection of the asymmetry is only based on structural information and is not costly to
compute as it has been illustrated on structurally symmetric matrices, for which both algorithms
behave similarly. On a set of unsymmetric matrices, we have shown that the new algorithm will
reduce both the factor size and the number of operations by a signicant factor. We have also
observed that the reduction in the number of indirect memory access operations involved during
the assembly process is generally much higher than the reduction in the number of elimination
operations. Finally, we have noticed that the gain in the maximum stack size is also relatively
high and is comparable to the gain in the number of assembly operations.
Space for Operations
LU Stack Total Elim Assemb. Time
mean 0.79 0.35 0.69 0.67 0.41 0.59
median 0.80 0.35 0.76 0.68 0.40 0.61
50 Structural symmetry < 80
mean 0.93 0.85 0.93 0.89 0.82 0.86
median 0.95 0.91 0.95 0.93 0.89 0.90
Table
4: Performance ratios of the new algorithm over the standard algorithm.
To conclude, we report in Table 4 a summary of the results (mean and median) obtained on
the test matrices with structural symmetry smaller than 80. For very unsymmetric matrices
(structural symmetry smaller that 50), we obtain an average reduction of 31% in the total
working space and of 41% in the factorization time. The maximum stack size and the number
of assembly operations are reduced by respectively 65% and 59%. Finally, it is interesting to
observe that, even on fairly symmetric matrices (50 structural symmetry < 80), it can still
be worth trying to identify and exploit asymmetry during the processing of the assembly tree.
Acknowledgements
. We want to thank Horst Simon and Esmond Ng who gave us the
opportunity to work at NERSC (LBNL) for one year. We are grateful to Sherry Li for helpful
comments on an early version of this paper.
--R
Exploiting structural symmetry in unsymmetric sparse symbolic factorization.
Elimination structures for unsymmetric sparse lu factors.
A scalable sparse direct solver using static pivoting.
The role of elimination trees in sparse factorization.
The multifrontal method for sparse matrix solution: Theory and Practice.
A new implementation of sparse Gaussian elimination.
--TR
--CTR
Gianmarco Manzini , Mario Putti, Mesh locking effects in the finite volume solution of 2-D anisotropic diffusion equations, Journal of Computational Physics, v.220 n.2, p.751-771, January, 2007
Abdou Guermouche , Jean-Yves L'excellent, Constructing memory-minimizing schedules for multifrontal methods, ACM Transactions on Mathematical Software (TOMS), v.32 n.1, p.17-32, March 2006
Xiaoye S. Li , James W. Demmel, SuperLU_DIST: A scalable distributed-memory sparse direct solver for unsymmetric linear systems, ACM Transactions on Mathematical Software (TOMS), v.29 n.2, p.110-140, June
Timothy A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method, ACM Transactions on Mathematical Software (TOMS), v.30 n.2, p.165-195, June 2004
Wook Ryol Hwang , Martien A. Hulsen , Han E. H. Meijer, Direct simulation of particle suspensions in sliding bi-periodic frames, Journal of Computational Physics, v.194 n.2, p.742-772, March 2004
See Jo Kim , Wook Ryol Hwang, Direct numerical simulations of droplet emulsions in sliding bi-periodic frames using the level-set method, Journal of Computational Physics, v.225 n.1, p.615-634, July, 2007 | elimination tree;gaussian elimination;unsymmetric matrices;multifrontal methods;sparse linear equations |
589352 | Improved Symbolic and Numerical Factorization Algorithms for Unsymmetric Sparse Matrices. | We present algorithms for the symbolic and numerical factorization phases in the direct solution of sparse unsymmetric systems of linear equations. We have modified a classical symbolic factorization algorithm for unsymmetric matrices to inexpensively compute minimal elimination structures. We give an efficient algorithm to compute a near-minimal data-dependency graph for unsymmetric multifrontal factorization that is valid irrespective of the amount of dynamic pivoting performed during factorization. Finally, we describe an unsymmetric-pattern multifrontal algorithm for Gaussian elimination with partial pivoting that uses the task- and data-dependency graphs computed during the symbolic phase. These algorithms have been implemented in WSMP---an industrial strength sparse solver package---and have enabled WSMP to significantly outperform other similar solvers. We present experimental results to demonstrate the merits of the new algorithms. | Introduction
. Typical
dire ct
solve rs for
ge ne ral
sparse syste ms of
line ar
e quations of
the form
have four distinct
phase s: analysis, comprising orde
ring for fill-in
re duction and symbolic factorization;
nume rical factorization of
the sparse coe #cie nt matrix A into triangular factors L and U using
Gaussiane limina-
tion with partial pivoting; forward and
backwarde limination to
solve for x using
the triangular factors L and U and
the right-hand-side ve ctor b; and
ite rative re fine me nt
of
the compute d solution. In this
pape r,
we de scribe some of
the algorithms that
are use d in
the unsymme tric symbolic and
nume rical factorization
phase s of
the Watson
Sparse Matrix
Package (WSMP)-a
high-pe rformance and robust
software for solving
ge ne ral
sparse line ar
syste ms.
The se algorithms
are crucial to WSMP's
pe rformance ,
which has
be e n shown to
be significantly
be tte r than that of
othe r similar
solve rs [18].
An important contribution of this
pape r is to show that, contrary to
conve ntional
wisdom, it is
possible to symbolically
de te rmine a static communication
patte rn for
paralle l
unsymme tric
sparse LU
ve n in
the pre se nce of partial pivoting.
The proce ss of factoring a
sparse matrix can
be e xpre sse d by a
dire cte d acyclic
task-de pe nde ncy graph (task-DAG).
The ve rtice s of this DAG
corre spond to
the tasks
of factoring rows or columns, or groups of rows and columns, of
the sparse matrix,
and
the e dge s
corre spond to
the de pe nde ncie s
be twe e n
the tasks. A task is
re ady
fore xe cution if and only if all tasks with
incominge dge s to it
have comple te d. In
addition to a task-DAG,
the re is a
data-de pe nde ncy graph (data-DAG)
associate d
with
sparse matrix factorization.
The ve rte x
se t of
the data-DAG is
the same as that
of
the task-DAG for a
give n
sparse matrix.
Ane dge from a
ve rte x i to a
ve rte x j in
the data-DAG
de note s that at
le ast
some of
the output data of task i is
re quire d as
input by task j. In this
pape r,
we de fine task i as
the task of computing column i of
L and row i of U .
Once the tasks
are de fine d,
the task-DAG is
unique to a
sparse # Received by the editors October 3, 2001; accepted for publication (in revised form) by E. G. Ng
October 18, 2002; published electronically December 3, 2002.
http://www.siam.org/journals/simax/24-2/39604.html
T.J.W atson Research Center, P.O. Box 218, Yorktown Heights, NY 10598 (anshul@watson.
ibm.com).
matrix for a
give n
pe rmutation of rows and columns;
howe ve r,
the data-DAG is a
function of
the sparse factorization algorithm. Multifrontal algorithms [9, 14, 23] for
sparse factorization can work with a minimal data-DAG
(i.e ., a data-DAG with
the smalle st
possible
dge s) for a
give n matrix.
In
the case of
symme tric
sparse matrice s,
the minimal task- and data-DAGs for
the factorization
proce ss
are a
tre e calle d
the e limination
tre e [22].
Howe ve r, for un-
symme tric
sparse matrice s,
the task- and data-DAGs
are ge ne ral DAGs.
More ove r,
the e dge -se t of
the minimal data-DAG for
unsymme tric
sparse factorization can
be a
rse t of
the e dge -se t of a task-DAG.
Gilbe rt and Liu [16]
de scribe e limination
structure s for
unsymme tric
sparse LU factors and
give an algorithm for
sparse unsym-
me tric symbolic factorization.
The se e limination
structure s
are two DAGs that
are transitive re ductions of
the graphs of
the factor
matrice s L and U ,
re spe ctive ly, and
can
be use d to
de rive a task-DAG for
sparse LU factorization.
Some re se arche rs
have argue d that computing
ane xact
transitive re duction can
be tooe xpe nsive [9, 15] and
have propose d using subminimal DAGs with
more e dge s than
ne ce ssary.
Howe ve r,
trave rsing
unne ce ssary
dge s during
nume rical factorization can
be a
source of
ove rhe ad.
More ove r, in a
paralle l
imple me ntation,e xtra
dge s can
be pote ntial
source s of
unne ce ssary synchronization or communication.
In this
pape r,
we show how a
re lative ly straightforward modification to
Gilbe rt
and Liu's symbolic factorization
algorithme nable s
ane #cie nt computation of
the minimale limination DAGs.
We also
de fine a
se t
dge s that must
be adde d to
the task-DAG in
orde r to
ge ne rate a minimal data-DAG that is valid as long as
partial pivoting with dynamic row and
columne xchange s is not
pe rforme d during
factorization. Finally,
we de scribe how
supple me nting this data-DAG
furthe r with a
small
se t
ofe xtrae dge s can
yie ld a
ne ar-minimal data-DAG that is
su#cie nt to
handle an arbitrary
numbe r of pivot
failure s and
the re sulting row and
columne xchange s
during
nume rical factorization. A pivot
failure occurs
the pivot
pre dicte d
by
the analysis
phase must
be alte re d during
nume rical factorization
be cause the nume rical
value of
the pivot is too small. By
me ans
ofe xpe rime nts on a
suite of
unsymme tric
sparse matrice s from
re al applications,
we show that computing
the final data-DAG
xtre me ly fast.
Furthe rmore , for
the matrice s in our
te st
suite , this
data-DAG has only a slightly
highe r
ofe dge s than
the task-DAG
constructe d
using
comple te transitive re duction.
The multifrontal
me thod [9, 14, 23] for
sparse matrix factorization usually
o#e rs
a significant
pe rformance advantage ove r
conve ntional factorization
sche me s by
pe r-
mittinge #cie nt utilization of
paralle lism and
me mory
hie rarchy. Du# and
Re id [14]
de scribe d a
symme tric-patte rn multifrontal algorithm for
unsymme tric
matrice s that
ge ne rate s
ane limination
tre e base d on
the symme tric
structure of
the union of
the structure s of A and
the transpose of A to
guide the nume rical factorization. This
algorithm works on
square frontal
matrice s
(se e se ction 4.1) and can incur a substantia
ove rhe ad for
ve ry
unsymme tric
matrice s
due to
unne ce ssary data
de pe nde ncie s
in
the e limination
tre e and
due toe xtra
ze ros in
the artificially
symme trize d frontal
matrice s. Davis and Du# [9] and
Hadfie ld [20]
introduce d an
unsymme tric-patte rn
multifrontal algorithm that
ove rcome s
the de ficie ncie s of a
symme tric-patte rn algo-
rithm. Our
powe rful symbolic
phase e nable s us to
use a much
more simplifie d and
e #cie nt
ve rsion of
the unsymme tric-patte rn multifrontal algorithm with partial pivoting
We de scribe the unsymme tric-patte rn multifrontal algorithm that is
use d in
WSMP
ande xpe rime ntally
compare it with
othe r
state -of-the -art
sparse unsymme tric
factorization
code s.
UNSYMMETRIC SPARSE MATRIX FACTORIZATION 531
Table
Test
ith their order (N), number of nonzeros (NNZ), and the application area of origin.
Number Matrix N NNZ Application
Finite element analysis
3 bayer01 57735 277774 Chemistry
4 bbmat 38744 1771722 Fluid dynamics
programming
6 e40r0000 17281 553956 Fluid dynamics
7 e40r5000 17281 553956 Fluid dynamics
simulation
9 epb3 84617 463625 Thermodynamics
engineering
14 mixtank 29957 1995041 Fluid dynamics
engineering
simulation
simulation
pre2 659033 5959282 Circuit simulation
19 raefsky3 21200 1488768 Fluid dynamics
22 tib 18510 145149 Circuit simulation
twotone 120750 1224224 Circuit simulation
wang3old 26064 177168 Circuit simulation
simulation
In
Table
1.1,
we introduce the suite of randomly
chose n
te st
matrice s that
we will
use ine xpe rime nts throughout this
pape r.
The table shows
the orde r
ofe ach matrix,
the numbe r of
nonze ros in it, and
the application
are a of
the origin of
the matrix. All
matrice s in our
te st
suite arise in
re al-life proble ms and
are in
the public domain.
The e xpe rime nts
re porte d in this
pape r
we re conducte d on an IBM RS6000 WH-2 with a
Gbyte s of RAM, 8
Mbyte s of
le ve l-2
cache , and 64
Kbyte s
of
le ve l-1
cache .
The organization of this
pape r is as follows.
ction 2
introduce s
the te rms,
conve ntions, and notations
use d in
the pape r. A symbolic factorization algorithm that
compute s
the structure of
the triangular factors and
minimale limination
structure s
is
de scribe d in
se ction 3. In
se ction 4,
we de scribe how to
compute ne ar-minimal
data-DAGs for
unsymme tric multifrontal factorization.
The nume rical factorization
algorithm is
discusse d in
de tail in
se ction 5.
We finish with concluding
re marks in
se ction 6.
The last
subse ction
ofe ach major
se ction
containse xpe rime ntal
re sults
pe rtaining to
the algorithms in that
se ction.
2. Terminology and conventions.
We assume that
the original n - n
sparse unsymme tric
coe #cie nt matrix is
irre ducible and cannot
be pe rmute d into a block-triangular
form. This is not a
se rious
re striction,
be cause a
ge ne ral matrix can first
be re duce d to a block-triangular form and
the n only
the irre ducible diagonal blocks
ne e d
to
be factore d [12].
We assume that
the coe #cie nt matrix A is
factore d into a
lowe r
triangular matrix L and an
uppe r triangular matrix U .
Multiple row and column
pe rmutations may
be applie d to A during various
stage s of
the solution
proce ss.
Howe ve r, for
the sake of clarity,
we will always
de note the coe #cie nt matrix by A and
the factors by L and U .
The state of
pe rmutation of A, L, and U will usually
be cle ar
from
the conte xt.
We de note the dire cte d graph
corre sponding to an n - n matrix M by
re graph may not always
be associate d
with
ane xplicitly
de fine d matrix.
Howe ve r,
the n
ane dge i#j # EM if
and only if m ij is a structural
nonze roe ntry in
the sparse matrix M .
The transpose of a matrix M is
re pre se nte d by M # . If i#j # EM ,
the n j#i # EM # , and
vice -ve rsa.
the se t of
indice s of
the columns in M that
have a structural
nonze roe ntry in row i. This is also
the se t of all
ve rtice s to which i has an outbound
e dge inG M . Similarly, Struct(M #,i ) is
the se t of
indice s of
the rows in M that
have a structural
nonze roe ntry in column i and is also
the se t of all
ve rtice s from which i
has an
inbounde dge inG M . A
dire cte d path from
node i to
node j in
the dire cte d
graphG M is
de note d by i#j.
The transitive re ductionG M O(VM , EM O ) of a graph
the graph with
the smalle st
dge s that has a
dire cte d path
i#j if and only
ifG M has a
dire cte d path i#j.
Since we are primarily
de aling with
the nonze ro
structure of
matrice s
rathe r than
the actual
value s,
we may also
loose ly
re fe r to M O as
the transitive re duction of M
O is a
transitive re duction
ofG M .
The le ading i - i submatrix of M is
de note d by M i and
the corre sponding graph and
its
transitive re duction
andG M O
re spe ctive ly.
The e dge s and paths in
some of
the graphs
use d in this
pape r
are labe le d. An
e dge in a
labe le d graph can
have one of
the thre e labe ls-L, U, or LU.
De pe nding on
its
labe l,
ane dge can
be an
L-e dge , a
U-e dge , or an
LU-e dge . L-, U-, and
LU-e dge s
from
ve rte x i to j
are de note d by i L
#j,
iU #j, and i
LU #j,
re spe ctive ly. An L-path from i
to j,
de note d by i L
#j, is a
dire cte d path containing only L- and
LU-e dge s. Similarly,
a U-path from i to j,
de note d by
iU #j, is a
dire cte d path containing only U- and
LU-e dge s. If an
L-e dge i L
#je xists in
the graph,
the
L-pare nt(i). Similarly, if
iU #je xists,
the
U-pare nt(i), and if i
LU #je xists,
the
LU-pare nt(i).
We de fine 1 a
supe rnode [q : r] as a maximal
se t of
conse cutive indice s {q, q
1, . , r} such that for all
and
matrice s L and U ,
we de fine
supe rnodal
matrice s L and U such
thate ach
supe rnode [q : r] in L
and U is
re pre se nte d by a
single row and column in L and U .
re m # n is
the total
numbe r of
supe rnode s.
Furthe rmore , if
and r < s,
the n g < h; that is,
the column and row
indice s in L and U maintain
the re lative orde r of
supe rnode s in L and U .
3. Computing a task-DAG and the structures of L and U .
Gilbe rt and
pre se nt an
unsymme tric symbolic factorization algorithm to
compute the structure s of
the factors L and U and
the ir
transitive re ductions L O and U O . Figuresummarize s
Gilbe rt and Liu's algorithm.
The algorithm
compute s
the structure
of L, U , and L O row by row and
compute s
the structure of U O by columns.
The total
time that
the algorithm shown in
Figure
spe nds in
ste p 1 is
bounde d
by flops(LU O ) [16], which is
the numbe r of
ope rations
re quire d to multiply
the sparse matrice s L and U O . Similarly,
the time spe nt in
ste p 3 is
bounde d by flops(UL O ).
The total computational cost of
ste ps 2 and 4 is
O| )). This is
be cause transitive re duction is
pe rforme d on n rows of U and columns of L, and
the ith
ste p
could
pote ntially
trave rse alle dge s
inG L O
andG U O
Ste ps 2 and 4 of
Gilbe rt and
Liu's algorithm
are much
more costly than
ste ps 1 and 3.
The cost of
the se ste ps
Other definitions of supernodes in the context of unsymmetric sparse factorization have been
used in the literature [11].
UNSYMMETRIC SPARSE MATRIX FACTORIZATION 533
to n do
1.
Compute
trave rsingG U O
and using
the fact that # j < i, j # Struct(L i,# ) if and only if # k # j such that k #
the re is a path k#j in U O
i-1 .
2.
Transitive ly
re duce Struct(L i,# )
usingG L O
ande xte nd it
toG L O
3.
Compute
4.
Transitive ly
re duce Struct(U #,i )
usingG U #O
ande xte nd it
toG U #O
end for
Fig . 3.1. Gilbert and Liu's unsymmetric symbolic factorization algorithm [16].
to n do
1.
Transitive ly
re duce Struct(L i,# )
usingG L O
ande xte nd it
toG L O
2.
Compute
3.
Transitive ly
re duce Struct(U #,i )
usingG U #O
ande xte nd it
toG U #O
4.
Compute
end for
Fig . 3.2. A modified symbolic factorization algorithm.
has
prompte d
re se arche rs to
se e k
alte rnative s, such as computing fast but
incomple te transitive re duction [9, 15].
The use of such
alte rnative s
toG L O
andG U O with
more e dge s
thanG L O
andG U O ,
re spe ctive ly, can
incre ase the cost of
ste ps 1 and 3, as
we ll
as that of
nume rical factorization.
3.1. A modification to Gilbert and Liu's algorithm.
We now
de scribe a
re lative ly
simple modification to
the algorithm shown in
Figure
3.1.
We start by
splitting
the original
coe #cie nt matrix into a
lowe r triangular part
store d by columns
and an
uppe r triangular part
store d by rows. In our
modifie d symbolic factorization
algorithm,
we compute the structure of L by
the columns
(i.e ., L # by rows) and that
of U by
the rows. This is
achie ve d by simply
re formulating
the algorithm shown in
Figure
3.1 to
pe rform only
ste ps 2 and 3, but
twice fore ach i on two
se ts of
ide ntical
data
structure s-one corre sponding to L # and
the othe r
corre sponding to U .
The modifie d algorithm is shown in
Figure
3.2.
Note that in
the algorithm of
Figure
3.2,
ste ps 3 and 4
are ide ntical to
ste ps
1 and 2,
re spe ctive ly.
The first two
ste ps
compute the ith rows of L O and U and
the last two
ste ps
compute the ith columns of U O and L. An actual
code of this
algorithm can
use the same pair of
routine s with
di#e re nt
argume nts to
imple me nt
all four
ste ps.
The re duction in
the size of
the code by half,
howe ve r, is a
se condary
be ne fit of
the modifie d algorithm.
The primary
advantage of this
sche me is that it
allows
imme diate de te ction of
supe rnode s during symbolic factorization. This, as
we shalle xplain in
se ction 3.2, allows us to avoid computing and
storingG L O
andG U #O
e xplicitly.
Inste ad,
we can work only with
the ir
supe rnodal
counte rpartsG L O and
G U #O .
3.2. Use of supernodes to speed up transitive reduction. Most
mode rn
sparse factorization
code s
re ly
he avily on
toe #cie ntly
utilize me mory
hie rarchie s and
paralle lism in
the hardware .
are so crucial to high
pe r-
formance in
sparse matrix factorization that
the crite rion for
the inclusion of rows
and columns in
the same supe rnode is
ofte n
re laxe d [7] to
incre ase the size of
the supe rnode s.
Conse cutive rows and columns with
ne arly
the same but not
ide ntical
structure s
are ofte n
include d in
the same supe rnode , and artificial
nonze roe ntrie s with
a
nume rical
value of 0
are adde d to maintain
ide ntical row and column
structure s for
all
me mbe rs of a
supe rnode .
The rationale is that
the slight
incre ase in
the numbe r
of
nonze ros and floating-point
ope rations
involve d in
the factorization is
more than
compe nsate d for by a
highe r factorization
spe e d.
WSMP's LU factorization algorithm also works on
the re laxe d
ge ne rate
d by its symbolic factorization. In
the symbolic factorization algorithm, as soon as
are compute d in
the ith
ite ration of
the oute r loop,
the y
can
be compare d with Struct(L #,i-1 ) and Struct(U i-1,# ) to
de te rmine if
the y
be long
to
the curre nt
supe rnode . A
ne w row-column pair is
adde d to
the curre nt
supe rnode if its
structure
ide ntical or
ne arly
ide ntical to
the pre vious row-column pair.
If
the ith row-column pair fails to
me e t
the crite rion for
me mbe rship into
the curre nt
supe rnode ,
the n a
ne w
supe rnode is
starte d at i.
The use of
supe rnode s allows us to significantly
re duce the cost of computing
the transitive re ductions. In
ste p 1 of
the algorithm shown in
Figure
3.2,
inste ad of
transitive ly
re ducing
the e ntire Struct(L i,# ),
we re duce only
the se t {h
re
Ste p 3 is
tre ate d similarly. As a
re sult of working only
with
supe rnode s,
the uppe r bound on
the cost of computing
the transitive re duction
de cre ase s from
O| )) to
O| )). This is
be cause only
the supe rnodal
DAGsG L O
andG U O
are se arche d
duringe ach of
the n
transitive re duction
ste ps. Strict
supe rnodal
andG U O would
have at
le ast
fe we re dge s
thanG L O
andG U O ,
re m is
the numbe r of
supe rnode s.
The re ason
is that U O and L #O do not contain
anye dge s i#j,
re
supe rnode .
The use of
re laxe d
re duce s
the numbe r
ofe dge s
e ve n
furthe r
be cause some pote ntiale dge s of
the form i#j,
re
be e liminate d from
the task-DAG
node s i and
are artificially
me rge d.
3.3. Task-DAGs for LU factorization. In this
pape r,
we will
re fe r to two
type s of task-DAGs: a
conve ntional DAG
de note d by
and a
supe rnodal DAG
de note d by T S . Each
ve rte x of
the conve ntional task-DAG
re fe rs to
the task of
computing a
single row of U and
the corre sponding column of L. On
the othe r hand,
a
ve rte x of
the supe rnodal task-DAG
corre sponds to a
se t of row-column pairs that
constitute a
supe rnode . Although, in a practical
imple me ntation,
we always work
with
supe rnodal DAGs,
we will
ofte n
use conve ntional task- and data-DAGs in
the re mainde r of
the pape r to
the e xposition
simple . All
re sults and
de scriptions
pre se nte d in
te rms of
the conve ntional DAGs map naturally to
the supe rnodal
case .
We first show how to
compute TC in
te rms of
the conve ntional
structure s L #O
and U O .
The transpose matrix L # is
use d to
indicate that for all i#j
Theorem 3.1.
is a task-DAG for LU factorization if its
ve rte x
se
{1, 2, . , n} and
itse dge -se
Proof. To
prove that
is a task-DAG,
we show that E T C is
su#cie nt to
re p-
re se nt a
prope r
orde ring of
the ne limination tasks
de note d by V T C . Struct(L #,i ) can
contribute to Struct(L #,j ) only if i # Struct(U #,j ), and if this is
the case ,
the n
the symbolic factorization algorithm of
Figure
3.2e nsure s that U O
containse ithe r i#j
UNSYMMETRIC SPARSE MATRIX FACTORIZATION 535
Table
Comparison of conventional symbolic factorization (due to Gilbert and Liu
[16])w ith supernodal
| is the size of the largest diagonal block in the matrix
onw hich symbolic
factorization is performed; Nsup is the number of supernodes; t C and t S are the times in seconds
of the
tw are the number of edges in the
task-DAGs produced by the
tw algorithms.
Matrix |V | Nsup
Conventional
Supernodal
|t Ct S
bbmat 38744 4877 10. 41260 1.7 6077 7.9 6.8 5.9
e40r5000 17281 2755 .60 19891 .16 3182 6.3 6.3 3.8
fidap011 16614 1262 2.3 16613 .42 1261 13. 13. 5.5
mil053 530238 166155 15. 530237 4.5 166154 3.2 3.2 3.3
mixtank 29957 2984 7.8 30949 1.2 3203 10. 9.7 6.5
nasasrb 54870 3808 4.9 54869 .97 3807 14. 14. 5.1
pre2 629628 243693 30. 765210 6.4 317216 2.6 2.4 4.7
raefsky3 21200 1282 2.1 21199 .41 1281 17. 17. 5.1
raefsky4 19779 1359 2.9 19778 .50 1358 15. 15. 5.8
tib 17583 7823 .11 22904 .07 10060 2.2 2.3 1.6
twotone 105740 34304 2.6 126656 .91 44856 3.1 2.8 2.9
wang3old 26064 8451 3.1 26063 .54 8450 3.1 3.1 5.7
wang4 26068 8254 3.0 26067 .53 8253 3.2 3.2 5.7
or i#j.
The same is
true for Struct(U i,# ), Struct(U j,# ), and L #O .
The re fore ,e ve ry
row-column pair i that
update s row and column j must
be e liminate d
be fore j.
The ore m 3.1 can
be e asilye xte nde d to
the supe rnodal
case .
The supe rnodal
task-DAG T S is
de fine d by a
ve rte x
se
ane dge se
re m is
the numbe r of
supe rnode s.
3.4. Experimental results. In
Table
3.1,
we compare Gilbe rt and Liu's symbolic
factorization algorithm [16] with
the supe rnodal symbolic factorization algorithm
de scribe d in
se ction 3.2.
We re port
the ir CPU
time s
tC and t S ,
re spe ctive ly, and
the numbe re dge s in task DAGs
and T S
ge ne rate d by
the m.
The last column of
Table
3.1 shows
the factor by which
the supe rnodal symbolic
factorization is
faste r than
the conve ntional algorithm.
The table also shows
ave rage supe rnode size (n/m) and
the ratio
dge s in
and T S
fore ach matrix.
The se two
ratios
are close ly
re late d.
The ratio of
tC and t S
be ars
some corre lation to
the ratio
dge s in
and T S , but
the actual ratio is matrix
de pe nde nt.
Note that only
the time of
transitive re duction
ste ps 1 and 3 of
the algorithm in
Figure
3.2 is
re duce d by
the use of
supe rnode s;
the time of computing
the structure s of L and U in
ste ps 2 andre mains mostly
unchange d
(othe r than
some re duction in
the numbe r of
structure s
me rge d
due to
supe rnode re laxation).
The re fore ,
the actual
re duction
achie ve d in
the symbolic factorization
time de pe nds on
the re lative amounts of
time spe nt in
536 ANSHUL
transitive re duction and computing L and U
structure s.
More ove r,
Table
re ports
the numbe r
dge s in
the task-DAGs, not
the numbe r
dge s in
the actual
lowe r
and
uppe r triangular
transitive ly
re duce d graphs that
are trave rse d during symbolic
factorization.
Re call that
the e dge -se t of a task-DAG is
the union of
the e dge -se ts
of
the corre sponding
lowe r and
uppe r triangular
transitive ly
re duce d graphs.
The amount of structural
symme try in
the matrix
a#e cts
the numbe r of
commone dge s
be twe e n
the uppe r and
lowe r
transitive ly
re duce d graphs, which in turn
de te rmine s
the actual
dge s in
the task-DAG.
Eise nstat and Liu [15]
pre se nt an
alte rnative to
comple te transitive re duction
to
re duce the cost of this
ste p in
sparse unsymme tric symbolic factorization.
The y
propose e xploiting structural
symme try in
the matrix to
compute partial
transitive re ductions. Although
the y
pre se nte xpe rime ntal
re sults on a
di#e re nt
se t of much
smalle r
matrice s, it
appe ars that
the use of
supe rnode s as
propose d in
se ction 3.2
can
achie ve much
highe r
spe e dups in symbolic factorization
while computinge xact
transitive re ductions than
the partial
transitive re duction
sche me propose d in [15].
Howe ve r,
Eise nstat and Liu's algorithm too can
be spe d up by
the use of
supe rnode s.
A
supe rnodal
ve rsion of this algorithm has
be e n
imple me nte d in
the Supe rLU dist [21]
sparse solve r
package .
We compare d our symbolic factorization
time with that of
Supe rLU dist and found
the latte r to
be slowe r by about 25%
ove rall on our
te st
suite . This could
be partly
due to
imple me ntation
di#e re nce s and partly
due to
the fact that
while Eise nstat and Liu's algorithm
save s
time in
the transitive re duction
computation, it
spe ndse xtra
time in
me rging
structure s
due to
re dundante dge s in
the DAG. It
appe ars that
the use of
supe rnode s in
Gilbe rt and Liu's algorithm can
spe e d
up its
transitive re ductione nough for it to match or
outpe rforme ve n a
supe rnodal
ve rsion of
Eise nstat and Liu's algorithm
ine xe cution
time .
4. Data-DAGs for unsymmetric multifrontal LU factorization.
The original
multifrontal algorithm [14, 23] was
de scribe d in
the conte xt of a
symme tric-
patte rn
coe #cie nt matrix but has
be e n
applie d to
matrice s with
unsymme tric patte
rns by introducing
ze ro-value de ntrie s at
appropriate locations to
conve rt
the original
matrix into
one with
the patte rn of A+A # [14, 2, 4]. This can
cause a substantial
ove rhe ad for
ve ry
unsymme tric
matrice s
due to
the e xtra computation
pe rforme d on
the introduce de ntrie s and
the re sulting fill-in. Davis and Du# [9] and
Hadfie ld [20]
introduce d an
unsymme tric-patte rn multifrontal algorithm to
ove rcome this short-
coming. In this
se ction,
we de ve lop
ne ar-minimal data-DAGs for
the unsymme tric
multifrontal algorithm-an
aspe ct of
unsymme tric multifrontal factorization that has
not
be e n
we ll
inve stigate d in
pre vious works. As
we shall show in
se ction 5,
the availability of a
ne ar-minimal data-DAG aids in
the e #cie nt
imple me ntation of
the nume rical factorization
phase . It would also
he lp
minimize the synchronization and
communication
ove rhe ads in a
paralle l
imple me ntation.
4.1. Outline of the symmetric multifrontal algorithm.
The symme tric-
patte rn multifrontal algorithm is
guide d by an
asse mbly
ore limination
tre e [22, 23,
19], which
se rve s as both
the task- and
data-de pe nde ncy graphs for
the factorization
proce ss.
The data
associate d
withe ach
supe rnode of
the e limination
tre e is a
square frontal matrix. A frontal matrix F g
associate d with a
supe rnode
de nse matrix
whose dime nsions
are e qual
)| or| Struct(U q,#
)| .
The contiguous
local row and column
indice s in
the de nse frontal matrix
corre spond to noncontiguous
global
indice s of
the matrix L
Eache ntry in a frontal matrix
corre sponds to
a structural
nonze roe ntry in
the global matrix.
Afte r a frontal matrix F g is fully
asse mble d or
populate d,
the le ading r - q columns
corre sponding to
UNSYMMETRIC SPARSE MATRIX FACTORIZATION 537
the supe rnode (also known as
the pivot block)
are factore d and
be come parts of
the factors U and L,
re spe ctive ly.
The re maining trailing part of
the frontal matrix is now
calle d
the update or
the contribution matrix,
de note d by C g .
The contribution matrix
corre sponding to a
supe rnode is
asse mble d
comple te ly into
the frontal matrix of its
only
pare nt
supe rnode and is
ne ve r
acce sse d again. This is
be cause if
the pare nt of
supe rnode
the e limination
tre e ,
the n Struct(L #,r
The same is
true for columns of U
due to
symme try.
In a
re cursive formulation of
the symme tric-patte rn multifrontal algorithm,
the task
corre sponding to a
supe rnode first
comple te s
ide ntical subtasks
fore ach of its
childre n in
the e limination
tre e ,
the n
asse mble s
the ir contribution
matrice s into its
frontal matrix, and finally
pe rforms
the partial factorization on
the frontal matrix.
Calling a
re cursive proce dure to
pe rform
the task
de scribe d
above on
the root su-
pe rnode of
the e limination
tre e comple te s
the factorization of a
sparse matrix with a
symme tric
structure .
4.2. Outline of the unsymmetric multifrontal algorithm.
The ove rall
structure of an
unsymme tric-patte rn multifrontal algorithm is similar to its
symme t-
ric
counte rpart and can
be e xpre sse d in
the form of a
re cursive proce dure starting at
the root
(the supe rnode with no
outgoinge dge s) of
the task-DAG.
Howe ve r,
the re are two major
di#e re nce s.
The first
di#e re nce is in
the control-flow. In
the unsymme tric
multifrontal algorithm,
be fore starting a subtask for a child,
the task
corre sponding
to
the pare nt
supe rnode must
che ck to
se e if
the child
supe rnode has
alre ady
be e n
proce sse d by
anothe r
pare nt. Only
the first
pare nt to
re ach a child actually
pe rforms
the re cursive computation starting at that child.
The se cond
di#e re nce is in
the data-
flow, or
the way contribution
matrice s
are asse mble d into frontal
matrice s. This is
e xplaine d
be low in
gre ate r
de tail.
Re call that
the e dge -se t E T C of
the task-DAG
is
the union of
the e dge -se ts
O of
the transitive re ductions of L # and U ,
re spe ctive ly.
We now assign
labe ls to
the e dge s in
.
The e dge s
contribute d to E T C
sole ly by E L #O
are labe le d as
L-e dge s.
dge s
contribute d to E T C
sole ly by E U O
are labe le d as
U-e dge s.
The third
type of
labe l,
the LU-labe l, is
assigne d to
the e dge s that
be long to
the inte rse ction E L #O and E U O . Finally, an
L-e dge i L
#j is
conve rte d to an
LU-e dge i
LU #j
if
the re is a U-path
iU #j in
, and a
U-e dge iU #j is
conve rte d to i
LU #j if
the re is an
L-path i L
#j in
.
The e dge s of
the supe rnodal task-DAG T S
are de fine d similarly.
Unlike the symme tric multifrontal algorithm,
the frontal and contribution matrice
s in
the unsymme tric multifrontal algorithm
are , in
ge ne ral,
re ctangular
rathe r
than
square .
Furthe rmore , a contribution matrix in
the unsymme tric multifrontal
algorithm can
pote ntially
be asse mble d into
more than
one frontal matrix
be cause a
supe rnode in
the data-DAG can
have more than
one pare nt. As
de scribe d in [20],
the asse mbly of contribution
matrice s into
the pare nt frontal
matrice s in
the unsymme tric
multifrontal algorithm
proce e ds as follows.
#h
be an
L-e dge in
the data-DAG,
re
have an
inde x i in common,
the n
alle le me nts of row i
of U in C g can
pote ntially
be asse mble d into F h . Similarly, if
gU #h is a
U-e dge and
have an
inde x i in common,
the n
alle le me nts of column
i of L in C g can
pote ntially
be asse mble d into F h . Finally, if g
LU #h is an
LU-e dge ,
the n
the e ntire trailing submatrix of C g with global row and column
indice s
gre ate r
than
ore qual to s can
be asse mble d into F h .
Ce rtaine ntrie s of C g may
have pote ntial
de stinations in
the frontal
matrice s of
Multifrontal factorization guided by the task-DAG
Matrix
LU
U
LU
LU
LU U
intended destination
task-DAG
U
Fig . 4.1. An example of the inability of a task-DAG to guide complete assembly of all contribution
matrices in the unsymmetric multifrontal algorithm. An 'X' denotes a nonzero in the coe#cient
matrix and a '+' denotes a nonzero created due to fill-in.
more than
one pare nt of
ge ve n if
the data-DAG contains no
unne ce ssarye dge s. This
is
be cause C g can
have common rows (columns) with
the frontal
matrice s of
more than
one among g's LU- and
L-pare nts
(U-pare nts).
The unsymme tric multifrontal
algorithm
muste nsure that
anye ntry of a contribution matrix is not
use d to
update more than
one frontal matrix. Additionally, a
corre ct data-DAG must
have su#cie nt
outgoinge dge s from all
supe rnode s so
thate ache ntry of a contribution matrix has a
pote ntial
de stination in at
le ast
one frontal matrix.
4.3. Inadequacy of task-DAG for unsymmetric multifrontal algorithm.
By
me ans of a
smalle xample in
Figure
4.1,
we show that if
the task-DAG
de fine d
in
se ction 3.3 is
use d as a data-DAG,
the n all contribution
matrice s may not
be fully
absorbe d into
the ir
pare nt frontal
matrice s.
The figure shows a
sparse matrix
with factorization fill-in,
the transitive ly
re duce d DAGs L #O and U O , and
the task-
DAG with
itse dge s
labe le d as
de scribe d in
se ction 4.2. For
the sake of
clarity,e ach
supe rnode is
chose n to
be of
size 1.
The figure shows all frontal and contribution
(shade d portions)
matrice s and
the flow of data from
the contribution to frontal
matrice s along
the e dge s of
the task-DAG.
Note that
alle dge s may not
le ad to a data
LU #5. It
asily
se e n that
the U-e dge 1U #4, which is
abse nt from
the task-DAG
(be cause it is
re move d
while transitive ly
re ducing U to U O ), is
ne ce ssary
for
the comple te asse mbly of C 1 .
4.4. A data-DAG for a predefined pivot sequence. Having shown that
the minimal task-DAG cannot
se rve as a data-DAG for
unsymme tric multifrontal
we now
de fine a data-DAG that is
su#cie nt for
the prope r
asse mbly
of all contribution
matrice s, as long as rows and columns
are note xchange d among
di#e re nt
supe rnode s for pivoting.
We will
use D N to
de note such a DAG,
whe re the supe rscript N stands for "no pivoting." A data-DAG D P that can
accommodate
UNSYMMETRIC SPARSE MATRIX FACTORIZATION 539
pivoting will
be de scribe d in
se ction 4.5.
Theorem 4.1. If a column
s all of
the following
conditions,
the n a
U-e dge i U
#j is
ne ce ssary for C i to
be comple te ly
asse mble d into its
pare nts' frontal
matrice s:
1.
The LU-pare nt of i, if
ite xists, is
gre ate r than j.
2.
None of i's
U-pare nts
are in Struct(U #,j ).
3.
The re e xists a k # Struct(L #,i ) such that k > j.
The transpose of this
the ore m can
be state d similarly.
Proof.
The contribution matrix C i has a column that
contribute s to L #,j ,
be cause ,
at
the le ast,
the re is
ane le me nt
corre sponding to L k,j in C i . At
the same time ,
none of i's
U-pare nts' frontal
matrice s
have column j, so
the y cannot absorb L #,j from C i .
Since the LU-pare nt of i is
gre ate r than j, it too cannot absorb L #,j from C i .
The addition of
iU #j
make s it
possible for C i to
contribute L #,j to F j .
The transpose case can
be prove n similarly.
The ore m 4.1
capture s
the situation
illustrate d in
Figure
4.1 for 4, and
pre scribe s
the addition of
toe nsure comple te asse mbly of C 1 .
Theorem 4.2. If D N is a DAG
forme d by adding all
possible e dge s to
according to
The ore m 4.1,
provide d that
the se e dge s don't
alre adye xist,
the n D N is a
data-de pe nde ncy DAG for
the unsymme tric multifrontal algorithm without pivoting.
Proof. To show that D N is a data-DAG,
we must show that
itse dge -se t is
su#cie nt for
the comple te absorption of all contribution
matrice s into
the ir
pare nt
frontal
matrice s.
We prove this by contradiction.
Without loss of
ge ne rality,
assume that
ane le me nt
corre sponding to L k,j in C i is
not
asse mble d.
Note that i < j < k. If L k,j is in C i ,
the
ithe r
the re is a U-path
iU #j in
.
If
the n
alle ntrie s with row
indice s
gre ate r than
ore qual to j in column
of C i will
be absorbe d by F j , and
the se e ntrie s
include the one corre sponding to
L k,j . If
the n a U-path
iU #je xists in
the re are two
possibilitie s:
e ithe r
LU-pare
LU-pare nt(i) > j.
LU-pare nt(i). If l # j,
the n
the e ntire trailing submatrix of C i with row and column
indice s
gre ate r than l, including
be asse mble d into F l . If l > j,
the n
conside r two
furthe r
possibilitie s:e ithe r
one of i's
U-pare nts is in Struct(U #,j ) or is not. If
one is,
the n its frontal matrix
will absorb column j from C i . If
none of i's
U-pare nts is in Struct(U #,j ),
the n all
conditions for
the applicability of
The ore m 4.1
are satisfie d.
The re fore ,
iU #j would
have be e n
adde d to D N and would
have cause d
the e ntry
corre sponding to L k,j in
C i to
be absorbe d into F j . Thus, it is not
possible for
the e ntry
corre sponding to
L k,j to
be le ft
unasse mble d in any C i . Similarly, it can
be shown that
the e ntry
corre sponding to any U j,k cannot
be le ft
unasse mble d in any C i .
Having shown that
the e dge -se t of D N is
su#cie nt for
unsymme tric multifrontal
factorization without pivoting,
we now show that not
alle dge s that D N
inhe rits from
may
be ne ce ssary if pivoting is not
pe rforme d during factorization.
Theorem 4.3. For LU factorization without pivoting,
ane dge i U
#j (i L
in
is
re dundant if
the maximum
inde x in Struct(L #,i ) (Struct(U i,# )) is
smalle r
than j.
Proof.
Re call that Struct(L #,j
the maximum
inde x in Struct(L #,i ) is
smalle r than j,
the n
doe s not
contribute to Struct(L #,j ).
The proof for
L #O and Struct(U i,# ) is similar.
Note that
The ore m 4.3 is valid only if row and
columne xchange s
are not
pe r-
forme d during LU factorization.
Othe rwise , additional fill-in
cause d by pivoting could
cre ate an
inde x
gre ate r than
ore qual to j in Struct(L #,i ) or Struct(U i,#
),e ve n if it is
not
pre dicte d by
the symbolic factorization on
the original
pe rmutation of
the matrix.
The re fore ,
alle dge s in
could
pote ntially
be use d.
Supe rnodal
ve rsions of
The ore ms 4.1-4.3 for T S can
be prove n similarly. To
summarize the re sults of this
subse ction,
we have shown how to construct a data-
DAG for
unsymme tric multifrontal factorization without pivoting from a task-DAG
and
we have shown that although
the task-DAG is
de rive d from
the strict
transitive re ductions of L # and U (or L # and U), it may still pass
one dge s to
the data-DAG
that
are re dundant if pivoting is not
pe rforme d during factorization.
The re fore ,
the data-DAG is not minimal.
Howe ve r, if pivoting is
pe rforme d,
the n
pote ntially all
the e dge s could
ge t
use d.
4.5. Supplementing the data-DAG for dynamic pivoting.
We will now
show that
the e dge -se t of data-DAG D N
constructe d in
se ction 4.4 may not
be su#-
cie nt if pivoting is
pe rforme d during factorization.
We also discuss how to
supple me nt
to
ge ne rate a data-DAG D P
whose e dge -se t is
su#cie nt to
handle any amount of
pivoting.
We start with an
ove rvie w of
the pivoting
me thodology in
the unsymme tric
multifrontal algorithm, which has
be e n
de scribe d in
de tail in [20].
If a
diagonale le me nt A i,i (q # i # r) in a
supe rnode [q : r] fails to
me e t
the pivoting
crite rion,
the n first an
atte mpt is
made toe xchange row and column i with
a row j and a column k such that i <
the pivoting
crite rion. Such
intrasupe rnode pivoting has
noe #e ct on
the structure of
the factors
and factorization can
continue as usual.
Howe ve r, it may not always
be possible to
find a
suitable row-column pair within a
supe rnode 's pivot block to satisfy
the pivoting
crite rion. In this situation,
inte rsupe rnode pivoting is
ne ce ssary. If
the LU-pare nt of
the data-DAG and a
suitable ith pivot cannot
be found
within
the pivot block of F g ,
the n all row-column pairs from i to r
are symme trically
pe rmute d to
ne w locations from s - (r
Thus,e #e ctive ly,
shrinks to [q
the supe rnode [s :
t]e xpands to [s - (r
As a
side e #e ct of this pivoting,
the re is additional fill-in in all
the ance stors of g in
the data-DAG that
are smalle r than h. In particular,
the columns of L of all of g's
U-ance stors
smalle r than h
ge te xtra row
indice
the rows of U of all of g's
L-ance stors
smalle r than h
ge te xtra column
indice
failure in
supe rnode h is
handle d similarly in a
re cursive manne r.
In D N ,
whose construction is
de scribe d in
se ction 4.4, all
supe rnode s may not
have an
LU-pare nt to support
the symme tric pivoting
me thod
de scribe d
above .
The re fore ,
as
the first
ste p towards
de riving D P from D N ,
we alte r
the e dge -se t of
the latte r
as follows.
Fore ach g from 1 to m
re m is
the total
numbe r of
supe rnode s),
the smalle st
supe rnode h to which both g L
#h and
gU #he xist is
de signate d as
the LU-pare nt of g; that is, if
ane dge g#h
doe s
note xist,
the n an
LU-e dge g
LU #h is
adde d to
the data-DAG, or if an L- or a
U-e dge g#he xists,
the n it is
conve rte d to an
LU-e dge .
The n,
alle dge s g#k such that k > h
are de le te d. If
the original matrix is
not
re ducible to a block-triangular form,
the n
afte r this
modification,e ach
supe rnode othe r than
the root
supe rnode has an
LU-pare nt to
accommodate row-column pairs
that fail to satisfy
the pivoting
crite rion in
the ir original locations [20]. It
asily
se e n that this modification has
noe #e ct on
The ore ms 4.1-4.3
be cause g L
#h
(gU #h) is
in
the modifie d D N only if g L
#h
(gU #h) is in
the original D N as
de fine d in
se ction 4.4.
UNSYMMETRIC SPARSE MATRIX FACTORIZATION 541
old
indices
Original matrix
Multifrontal factorization without any pivot failure
FLU
Factorization with failure of pivot 1
Handling failure of pivot 1
new
unassembled intended destination
data-DAG13LU
U
LU
LU
Fig . 4.2. An example factorization to
show how the failure of pivot 1 is handled by a symmetric
permutation of
row and column 1 to merge
themw ith their LU-parent supernode, 4. An 'X' denotes
a nonzero in the coe#cient matrix and a '+' denotes a fill-in. The circled `X' and '+' are created
due to pivoting. A '0' denotes a fill-in predicted by the original symbolic factorization that has a
value of zero due to pivoting-related movement of
row s and columns. The figure also
show s that the
absence of 2 L
#4 leaves the entry U 1,5 unassembled from C 2 .
Figure
4.2 shows how
the failure of pivot row and column 1 is
atte mpte d in
the unsymme tric multifrontal factorization of a small 5 -
5e xample matrix. Row-column
1 is
symme trically
pe rmute d to a
ne w location
adjace nt to 1's
LU-pare nt 4 in
the data-
DAG. This
re sults in an addition of row
inde x 1 to 1's
U-pare nt 2 and an addition of
column
inde x 1 to 1's
L-pare nt 3. Additionally,
afte r moving to
the ir
ne w locations,
row 1 in U and column 1 in L
ge t fill-in in column and row positions
re row 4 in
U and column 4 in L
have nonze ros
(i.e ., U 1,5 , L 4,1 , and L 5,1 ).
Figure
4.2 also shows
that
afte r pivoting,
the ne w row 1 of C 2 cannot
be fully
asse mble d in
the abse nce of an
L-e dge 2 L
#4.
arly, in addition to adding
dge s as
de scribe de arlie r, D N
re quire s
furthe r modifications in
orde r to
se rve as a data-DAG for
unsymme tric multifrontal
algorithm with dynamic pivoting.
Figure
give s
anothe re xample of a factorization
whe re D N is
unable to
guide a
comple te asse mbly in
the e ve nt of a pivot
failure .
Note that
thise xample satisfie s
the first two conditions of
The ore m 4.1.
Howe ve r,
since it
doe s not satisfy condition 3, no
dge s
are adde d and
the abse nce of
pre clude s a
comple te asse mbly of C 2 into its
pare nts' frontal
matrice s
We now
state and
prove a
the ore m that
pre scribe s a modification of D N to
pre ve nt
the situation
illustrate d in
Figure
4.3.
Theorem 4.4. If a column
s all of
the following
conditions,
the n a
U-e dge i U
#j is
ne ce ssary for C i to
be comple te ly
asse mble d into its
pare nts' frontal
matrice s in
the e ve nt of
failure of pivot k.
1.
The LU-pare nt of i is
gre ate r than j.
2.
None of i's
U-pare nts
are in Struct(U #,j ).
3. A
xists such that
the re is a U-path k U
#i in
LU-pare nt(k) > j.
The transpose of this
the ore m can
be state d similarly.
Proof.
Note that
The ore m 4.4 is
ve ry similar to
The ore m 4.1.
The only
di#e re nce is condition 3. If pivot k fails,
the n it will add a row in Struct(L #,i ) that
corre sponds
to
LU-pare nt(k) -1, which is
the ne w location of k and is
gre ate r than j - 1,
the ne w
inde x for j. Thus,
the failure of pivot k transforms condition 3 of
The ore m
4.4 into condition 3 for
the applicability of
The ore m 4.1, which has
alre ady
be e n
prove d.
The ore m 4.4
state s
thate ve n if Struct(L #,i )
doe s not
have any
inde x
gre ate r than
j but all
othe r conditions for
the applicability of
The ore m 4.1
are satisfie d and
iU #j
is not
pre se nt in
the DAG,
the n pivoting may
re sult in
incomple te asse mbly
unle ss
thise dge is
adde d. This is
be cause pivoting can
cre ate a
nonze roe ntry L k,i such that
j. This is what
happe ns in
the e xample shown in
Figure
4.3 for
the original
indice s. Pivoting
change s i, j, and k to 1, 7, and 8,
re spe ctive ly.
In light of
The ore m 4.4,
we introduce anothe r modification to D N .
Inste ad of using
The ore m 4.1 strictly to
de rive D N from
we omit
che cking for condition 3 and
de rive D N by adding all
those e dge s to
practice ) that satisfy conditions 1
and 2.
Now, by
me ans of
The ore m 4.5,
we will show that
the data-DAG D N
,e ve n
afte r
the modifications
de scribe d
above , is not
su#cie nt
toe nsure comple te asse mbly of
all contribution
matrice s in
the e ve nt of
inte rsupe rnode pivoting.
The re ade r can
ve rify that
Figure
the transpose case of
The ore m 4.5 for
5. Finally,
The ore m 4.6 will show that
supple me nting
the data-
DAG with
additionale dge s
pre scribe d by
The ore m 4.5
make s it
su#cie nt to
handle all contribution
matrice s in
the face of
inte rsupe rnode pivoting. As
we dide arlie r in
this
pape r, for
the sake of clarity and simplicity,
we will
state and
prove The ore ms 4.5
and 4.6 in
the conte xt of
conve ntional DAGs with
single -node supe rnode s.
The re sults
naturallye xte nd to
supe rnodal DAGs.
Theorem 4.5. If h is
the LU-pare nt of j and all of
the following conditions hold,
the n a
U-e dge i U
#h is
ne ce ssary for C i to
be comple te ly
asse mble d into its
pare nts'
frontal
matrice s in
the e ve nt that j fails to
me e t
the pivot
crite rion in its original
location.
1.
The re e xists an L-path j L
#i such that i < h and
LU-pare nt(i) > h.
2.
None of i's
U-pare nts
are in Struct(L #,j ).
3.
Eithe r # k # Struct(L #,i ) such that k > h, or
the re is a U-path k U
LU-pare
The transpose case can
be state d similarly.
UNSYMMETRIC SPARSE MATRIX FACTORIZATION 543
indices
indices
new
F 9
F 486423748
9 LU
UU
U
U
U
U
U
Original matrix data-DAG9 X
unassembled5
Handling failure of pivot 1
intended destination
LU
LU
LU
LU
LU
LU
Fig . 4.3. An example factorization to
show that the edges in D N are not su#cient to assemble
its parents' frontal matrices in the event of the failure of pivot 1. The convention for
representing di#erent types of structural nonzeros is the same as in Figure 4.2.
Proof. If pivot j fails,
the n, along with
the othe r
faile d
LU-childre n of h, it
occupie s a
ne w position just
be fore h.
Since the re is an L-path j L
#i, column j is
adde d to C i
afte r
the failure of pivot j; that is, in
the ne w matrix
afte r pivoting, j #
We know that
the LU-pare nt of i is
gre ate r than
the ne w j,
be cause LU-pare nt(i) > h.
Since none of i's
U-pare nts
we re in
the old Struct(L #,j ),
the y
are not in
the ne w Struct(U #,j
)e ithe r. Thus
the first two conditions for
the applicability
of
The ore ms 4.1 and 4.4
are satisfie d. Condition 3 of
The ore m 4.5
quivale nt to
condition 3 of
The ore ms 4.1 and 4.4.
The re fore , a
U-e dge iU #j is
ne e de d for
prope r
multifrontal factorization of
the ne w matrix
afte r
pe rmuting j to its
ne w location.
Since , in its
ne w location, j is
me rge d with h into a common
supe rnode , a
U-e dge iU #h in
the original matrix would
have su#ce d.
The proof of
the transpose case is
similar.
Theorem 4.6. If D P is a DAG
forme d by adding all
possible e dge s according to
The ore m 4.5 to D N ,
the n D P is an
ade quate data-DAG for
unsymme tric multifrontal
factorization with
pote ntially
unlimite d
inte rsupe rnode pivoting.
Proof.
We prove this by showing that with D P , it is not
possible for
anye le me nt of
a contribution matrix C i to
re main
unasse mble d. Without loss of
ge ne rality,
conside r
ane le me nt
corre sponding to L k,j in C i . If L k,j is in C i ,
the ne ithe r k # Struct(L #,i )
the original L and U
pre dicte d by symbolic factorization, or
row k or column j or both
we re adde d to C i
due to pivoting. If row k and column j
are parts of
the original
structure of C i ,
the n
The ore m 4.2 has
alre ady shown that
the e dge -se t of D N , which is a
subse t of
the e dge -se t of D P , is
su#cie nt to
asse mble L k,j .
We now show that L k,j will
be absorbe d from C i by
one of i's
pare nts in D P
adde d to C i
due to pivoting,
irre spe ctive of
whe the r row k
be longe d to
the original Struct(L #,i ) or if it too was
adde d to C i
due to pivoting.
LU-pare nt(i) and
LU-pare nt(j).
We conside r two
case s: (1) g # h
and (2) g > h. If g # h, F g will
have both row k and column j and will absorb
the e le me nt
corre sponding to L k,j from C i . If g > h,
the n
the first condition for
the applicability of
The ore m 4.5 has
be e n
d. Now
we conside r two
furthe r
sce narios: (2a) At
le ast
one of i's
U-pare nts is in
the original Struct(L #,j ), or (2b)
none of i's
U-pare nts is in
the original Struct(L #,j ). In
case of (2a),
afte r pivoting,
at
le ast
one of i's
U-pare nts is in
the ne w Struct(U #,j ) and
the frontal matrix of this
U-pare nt will absorb column j from C i , including
the e ntry
corre sponding to L k,j . In
case (2b),
the se cond condition for
the applicability of
The ore m 4.5 has
be e n
d.
Finally,
whe the r row k was in
the original Struct(L #,i ) or was
adde d to C i
due to
the failure of a
U-de sce nde nt k, in its final location, k must
be gre ate r than h.
The re ason
is that if j # k # h
(i.e ., k's
ne w location is in
the e xte nde d
supe rnode h),
the n h
must
be an
LU-ance stor of i
be cause
implie s that
the re are both i L
#h and
iU #h in
the data-DAG. But that is not
possible be cause we are alre ady working
unde r
the assumption that
the LU-pare nt g of i is
gre ate r than h.
The re fore , k > h and
the third condition of
The ore m 4.5 has also
be e n
d. As a
re sult,
The ore m 4.5
would
have e nsure d that a
U-e dge iU #h is
pre se nt in D P to
asse mble column j from
Similarly,
we can
prove that
noe ntry
corre sponding to any U j,k will
be le ft
unasse mble d in C i .
4.6. Experimental results. In
se ctions 4.4 and 4.5,
we showe d how to supple
me nt
the e dge -se t of
the task-DAG to construct a data-DAG for
the unsym-
me tric multifrontal algorithm.
Table
showse xpe rime ntal
re sults of WSMP's
UNSYMMETRIC SPARSE MATRIX FACTORIZATION 545
Table
Time required for
and the number of edges in each DAG.
Matrix Symbolic Supplement-1 Supplement-2 Total Time
|ED P |
bbmat 1.7 6077 .06 6142 .00 6181 1.76 1.02
mil053 4.5 166154 .51 166154 .07 166154 5.08 1.00
mixtank 1.2 3203 .05 3203 .00 3203 1.25 1.00
pre2 6.4 317216 .84 320063 .16 320942 7.40 1.01
twotone .91 44856 .12 45866 .01 45918 1.04 1.02
wang3old .54 8450 .03 8450 .00 8450 0.57 1.00
imple me ntation of
the proce dure s to
ge ne rate the various DAGs.
Thre e DAGs
are conside re d in
Table
4.1:
the supe rnodal task-DAG T S ,
the supe rnodal data-DAG
D N for
unsymme tric multifrontal factorization without pivoting, and
the supe rnodal
data-DAG D P for
unsymme tric multifrontal factorization with pivoting.
The table shows
the time to
compute e ach of
the DAGs and
the numbe r
dge s in
the m for
the
matrice s in our
te st
suite .
T S is
compute d by
the basic symbolic factorization algorithm
de scribe d in
se ction
the re fore , t S is
the basic symbolic factorization
time .
We re fe r to
the proce ss of
computing D N from T S as
Supple me nt-1.
Supple me nt-1
che cks for
the first two
conditions of
The ore m 4.1 to find
the e dge s to
be adde d to E T S and
the n adds
outgoing
dge s from
supe rnode s without
LU-pare nts to
ld E D N .
Supple me nt-2
is
the proce ss that
addse dge s
base d on
the first two conditions of
The ore m 4.5 to
to
ld
The e xe cution
time of
Supple me nt-1 and
Supple me nt-2 is
de note d
by t 1 and t 2 ,
re spe ctive ly.
Note that not all
the e dge s in D N and D P
are ne ce ssary. For
the sake of com-
putational
spe e d,
Supple me nt-1 and
Supple me nt-2 in WSMP do not
che ck for all
the conditions of
The ore ms 4.1, 4.4, and 4.5
while addinge dge s.
The last conditions of all
thre e the ore ms
are skippe d.
Eve n if all conditions of
the se the ore ms
we re che cke d, not
all
the e dge s in
the re sulting data-DAGs may
be ne ce ssary.
The re fore , D N and D P
are not minimal data-DAGs for
unsymme tric multifrontal factorization.
Howe ve r, as
Table
4.1 shows,
the se DAGs do not
have many
more e dge s than T S for most
re al-life matrice s.
The ave rage fore xce sse dge s in
supe rnodal D P
ove r T S is only about 4%
for our
te st
suite .
We have shown that
the e dge s in
the task-DAGs
or T S
are
insu#cie nt to
dire ct
the data flow in
unsymme tric multifrontal factorization. On
the othe r hand,
the e dge s in D P
are su#cie nt,e ve n with pivoting.
The re fore ,
the numbe r
dge s in a truly minimal
supe rnodal data-DAG is
some whe re be twe e n
the numbe r
dge s in T S and in
the supe rnodal D P .
The e xpe rime ntal
re sults in
Table
4.1 show
that
the se two
rs
are usually fairly
close .
The table also shows that
the time re quire d to construct D N and D P is also small
compare d to
the basic symbolic
torization
time . Thus,
the me thodology
de scribe d in this
se ction for
the construction
of data-DAGs for
unsymme tric multifrontal factorization
nt in both
time and
the numbe r of
DAGe dge s. A comparison of
the
Table
4.1 with
WSMP's LU factorization
time give n in
Table
5.1 shows that
the total symbolic
time is usually significantly
le ss than
the nume rical factorization
time .
5. Implementation details of unsymmetric factorization. A
brie f
outline of
the unsymme tric multifrontal algorithm
base d on
the work of
Hadfie ld [20] and
Davis and Du# [9] is found in
se ction 4.2.
We now add
more de tails to it and
pre se nt
a
comple te algorithm that is
imple me nte d in WSMP. WSMP is
ge are d towards multiple
factorizations of
matrice s with
the same sparsity
patte rn but
di#e re nt
nonze ro
value s.
The re fore , symbolic
phase is
pe rforme d only
once and its output is
use d in
all
subse que nt
nume rical
ve n with
di#e re nt pivot
se que nce s
re sulting
from
di#e re nt
nume rical
value s.
A
fundame ntal
data-structure in our
unsymme tric multifrontal algorithm is
the frontal matrix. A frontal matrix is
associate d
withe ach
supe rnode .
Figure
5.1 shows
the organization of a typical frontal matrix for a
supe rnode
The core of this frontal matrix is
a| Struct(L #,q
)| -| Struct(U q,#
)| portion,
re
are pre dicte d by
the symbolic factorization. In
the abse nce of pivoting,
the first r - q columns of this matrix would
be factore d and would
be save d as parts of U and L,
re spe ctive ly.
The re maining trailing
submatrix would
constitute the contribution matrix
whose conte nts would
be absorbe d
into
the frontal
matrice s of
the pare nts of g in D P .
Extra rows
F [q:r]
s [q:r]
Extra columns
r
pivot block
pivot block
Fig . 5.1. Organization of a typical frontal matrix for a supernode r]). The p failed
pivots from the LU-children of the supernode are appended at the beginning of the frontal matrix
and the extra
row s and columns inherited from U- and L-descendents, respectively, are appended at
the end.
UNSYMMETRIC SPARSE MATRIX FACTORIZATION 547
In
the pre se nce of
nume rical
pivoting,e xtra pivots as
we ll as
othe r rows and
columns may
be adde d to
the frontal matrix
de pe nding on
the labe ls and pivot
failure s
of
the childre n of g in D P . Extra pivots (row-column pairs with
the same indice s)
are adde d to F g if
some of
the pivots of g's
LU-childre n fail to satisfy
the pivoting
crite rion.
The LU-childre n of g
the mse lve s may
have inhe rite d
some or all of
the se faile d pivots from
the ir own
LU-childre n.
The re fore ,
faile d pivots from any of
the LU-de sce nde nts of g
cane nd up in its frontal matrix. If p such pivots
are adde d,
the n
the size of
the pivot block
incre ase s from r - q
The frontal matrix F g can similarly
inhe rite xtra rows
corre sponding to
faile d
pivots in its
U-de sce nde nts
whose LU-pare nts
are gre ate r than g
ande xtra columns
corre sponding to
faile d pivots in its
L-de sce nde nts
whose LU-pare nts
are gre ate r than
g.
Irre spe ctive of
the ir
ne w
indice s,
the se e xtra rows and columns
are always ap-
pe nde d at
the e nd of
the original rows and columns of F g and a
d list of
the ir
indice s is
maintaine d
ate ach
supe rnode .
Eve ntually,
the se are asse mble d into
the e xtra pivots of
the frontal
matrice s of
the LU-pare nts of
the supe rnode s
whe re the se pivots
faile d.
The row and column
structure s
pre dicte d by symbolic factorization
are intact for
future factorizations of
matrice s with
the same nonze ro
patte rn.
The additions to
the se structure s
due to pivoting, which
de pe nd on
the nonze ro
value s in
the matrix
be ing
factore d,
are maintaine d
se parate ly and
are discarde d
be fore e ach
ne w factorization.
The availability of a static data-DAG D P that is
su#cie nt for handling an arbitrary
amount of dynamic pivoting is critical to our
imple me ntation of
the unsymme tric
multifrontal algorithm.
Figure
give s a
high-le ve l
pse udocode of our factorization
algorithm.
The algorithm starts with
the root
supe rnode of task- and data-DAGs. At
any
supe rnode , first, it
re cursive ly factors all
the unfactore d
childre n of that
supe r-
node .
The n it looks at
the faile d pivots (if any) of its
childre n to
figure out
the numbe r and
indice s of
the e xtra rows, columns, and pivots, if any, and accordingly
allocate s a frontal matrix of
the appropriate size . In
the ne xt
ste p,
the contribution
from
the original
coe #cie nt matrix and
the contribution
matrice s of
the curre nt
supe rnode 's
childre n
are accumulate d in
the appropriate locations
inside the frontal
matrix. Finally,
the algorithm
proce e ds to factor
the pivot block of
the frontal matrix
and
update s
the re mainde r of
the frontal matrix.
The le ading
succe ssfully
factore d
rows and columns
are save d as portions of U and L for
use during triangular
solve s.
The re maining contribution matrix
ve ntually
asse mble d into
the frontal
matrice s
of its
pare nts and is
re le ase d by
the last
pare nt to pick up its contribution.
The frontal matrix of
the LU-pare nt of a
supe rnode picks up all its
faile d pivot
row-column pairs as
we ll as
the e ntire trailing submatrix of its contribution matrix
with row and column
indice s
gre ate r than
ore qual to
the first
inde x of
the pare nt
supe rnode .
The re maining rows and columns of a
supe rnode 's contribution matrix
are asse mble d into
the frontal
matrice s of its L- and
U-pare nts in D P . It is
possible for
more than
one L- or
U-pare nts' frontal
matrice s to
have the same row or column
indice s in common with
the child's contribution matrix.
Howe ve r,e ache le me nt of
a contribution matrix must
be adde d
intoe xactly
one frontal matrix.
Some simple bookke e ping to
track of rows and columns that
have be e n
asse mble d
su#ce s to
e nsure this condition for
the re lative ly
fe w rows and columns that
have the pote ntial
to
be copie d into
the frontal
matrice s of
multiple L- and
U-pare nts,
re spe ctive ly.
Figure
5.2 and
the de scription in this
se ction show that WSMP's
unsymme tric
multifrontal algorithm is fairly straightforward to
imple me nt.
The static task- and
data-DAGs
compute d during
the symbolic
phase and
the use of
re cursion
make the
548 ANSHUL
function uns mf (root) {
/* 1.
Re cursive calls to root's
childre n */
fore ach child k of root in T S do
if not
alre ady
proce sse d k then
Call uns mf (k);
Flag
supe rnode k as
alre ady
proce sse d;
end for
/* 2.
Colle ct pivoting info to
de te rmine size of F root */
fore ach child k of root in D P do
if k is an L-child then
if k has
faile d pivots then
Add
the m to
the sorte d list of F root
'se xtra columns;
hase xtra columns then
Add
those whose LU-pare nt is
gre ate r than root to
the sorte d list of F root
'se xtra columns
while che cking for
duplicate s;
else if k is a U-child then
if k has
faile d pivots then
Add
the m to
the sorte d list of F root
'se xtra rows;
hase xtra rows then
Add
those whose LU-pare nt is
gre ate r than root to
the sorte d list of F root
'se xtra rows
while che cking for
duplicate s;
else if k is an LU-child then
if k has
faile d pivots then
Add
the m to
the sorte d list of F root
'se xtra pivots;
hase xtra columns then
Add
those whose LU-pare nt is
gre ate r than root to
the sorte d list of F root
'se xtra columns
while che cking for
duplicate s;
hase xtra rows then
Add
those whose LU-pare nt is
gre ate r than root to
the sorte d list of F root
'se xtra rows
while che cking for
duplicate s;
end for
/* 3.
Initialize root's frontal matrix */
Allocate F root of
appropriate size and fill it with
ze ros;
Populate F root
withe ntrie s from A
corre sponding to
supe rnode root;
/* 4.
Asse mbly from
childre n's contribution
matrice s into F root */
fore ach child k of root in D P do
Copy
appropriate contribution from C k into F root ;
if root is
the last
pare nt of k to pick up C k 's contribution then
Fre e the space occupie d by C
end for
5.
Nume rical factorization */
Factor
the pivot block of F root and
update the trailing part to
ld C root ;
function uns mf.
Fig . 5.2. A simple and e#cient unsymmetric multifrontal algorithm.
UNSYMMETRIC SPARSE MATRIX FACTORIZATION 549
nume rical factorization algorithm much
simple r to
de scribe and
imple me nt than
the e arlie r
de scriptions of
the unsymme tric
patte rn multifrontal algorithm in [20] and [9].
Othe r than UMFPACK [8], WSMP is
the only
sparse solve r
available that is
base d on
an
unsymme tric
patte rn multifrontal algorithm. It is also
the first such
paralle l
solve r
available for
ge ne ral
use . Although
Hadfie ld [20]
provide de xpe rime ntal
re sults from a
paralle l
imple me ntation on
the nCUBE, a practical
paralle l
solve r did not
re sult from
thate #ort.
The algorithm of
Figure
5.2 is not only
re lative ly
simple in
de scription but is
also computationally
le an
be cause it
the none sse ntial non-floating-point
ope rations and can
handle pivot
failure s
fairlye #cie ntly. It is also
note worthy that
for structurally
symme tric
matrice s,
the algorithm in
Figure
5.2 naturally
re duce s
to a
symme tric-patte rn multifrontal algorithm
guide d by
ane limination
tre e , which
re place s both T S and D P .
Othe r than a
state me nts
fore ach
supe rnode ,
the re is no
ove rhe ad in using this algorithm for structurally
symme tric
matrice s.
5.1. Experimental results.
We now
compare the unsymme tric LU factoriza-
tion
time of WSMP with that of
thre e state -of-the -art multifrontal
sparse solve rs,
name ly, MUMPS
ve rsion 4.1.6 [4, 5], MA41 [2, 3], and UMFPACK
ve rsion 3.2 [8]. A
de taile d
comparative study that
include s
more solve rs can
be found in [18].
The software
compare d in this
se ctione mploy
di#e re nt variants of
the multifrontal
me thod.
MUMPS contains a
symme tric-patte rn multifrontal factorization
code base d on
the classical multifrontal algorithm [14]. MA41, in
some se nse , is a hybrid
be twe e n sym-
me tric and
unsymme tric
patte rn multifrontal
solve rs. It
use s
ane limination
tre e to
guide factorization, but
the frontal
matrice s
are prune d of
all-ze ro rows and columns.
UMFPACK 3.2 contains a variation of
the unsymme tric-patte rn multifrontal algorithm
[9] that
use s
ane limination
tre e de rive d from
the structure of A # A.
Apart from
the factorization algorithm,
the re are othe r significant
di#e re nce s
among
the four
software package s that
a#e ct
the ir
pe rformance . First,
the y
use diffe
re nt
sche me s for
fill-re ducing
ring. By
de fault, WSMP
use s a
symme tric
pe rmu-
tation
base d on a
ne ste d-disse ction
ring [17]
compute d on
the structure of A+A # .
MUMPS and MA41
use a
symme tric
pe rmutation
base d on
the approximate mini-
mum
de gre e (AMD) algorithm [1]
applie d to
the structure of A+A # . UMFPACK
use s
a column AMD algorithm [10] to
pre pe rmute only
the columns of A and
compute s a
row
pe rmutation
base d on
nume rical and sparsity
crite ria during factorization.
The se cond
di#e re nce is
the use of a maximal matching algorithm [13] to
pe rmute the rows
of
the coe #cie nt matrix to
maximize the product of
the magnitude s of its diagonal
e ntrie s. As shown in [6, 18], this can
a#e ct factorization
time s
be cause it
change s
the amount of structural
symme try and
the amount of
nume rical pivoting during factor-
ization. WSMP
use s this
pre proce ssing on all
matrice s, MUMPS and MA41
use it
only if
the structural
symme try in
the original matrix is
le ss than 50%, and UMF-
doe s not
use it at all.
The third
di#e re nce is that WSMP
re duce s
the coe #cie nt
matrix into a block-triangular form,
while MUMPS, MA41, and UMFPACK 3.2 do
not.
Table
5.1 shows
nume rical factorization
time s and
ope ration counts of MUMPS,
MA41, UMFPACK, and WSMP run with
the options in MUMPS, MA41, and WSMP
change d to
minimize the di#e re nce s
be twe e n
the code s
othe r than
the factorization
algorithm.
We switche d o#
the pe rmutation to a
he avy-diagonal form and
the associate
d scaling in MUMPS, MA41, and WSMP. For WSMP,
inste ad of its
de fault
ne ste d-disse ction
ring,
we use d an
approximate minimum fill
orde ring, which is
ve ry similar to AMD.
Eve n with
the se change s,
di#e re nce s
re main
be twe e n
the four
Table
LU factorization times and operation counts of MUMPS, MA41, UMFPACK 3.2, and WSMP
w ith similar permutation options. The best time is in boldface and the second best time is underlined.
3.2W SMP
Matrix time ops time ops time ops time ops
af23560 3.89 2.56 3.58 2.54 8.59 3.46 4.06 3.22
bbmat 54.3 41.6 56.3 41.1 78.7 39.1 27.6 21.5
e40r0000 4.93 2.53 3.63 1.58 6.23 2.17 0.80 .419
ecl32 64.2 64.6 67.1 64.4 191. 112. 139. 77.6
fidap011 8.58 7.01 8.79 6.96 17.0 8.51 6.51 5.74
mil053 43.5 31.8 40.0 31.8 107. 46.2 28.2 20.8
mixtank 151. 141. 152. 64.1 363. 243. 76.3 64.6
nasasrb 12.8 9.45 11.9 9.43 55.9 28.2 10.4 8.78
pre2 fail fail fail fail fail fail 346. 301.
raefsky3 4.44 2.90 3.88 2.90 16.0 7.87 4.88 4.17
raefsky4 107. 74.4 92.9 44.7 25.0 12.9 43.4 22.5
twotone 56.5 38.3 37.6 31.8 30.1 10.8 2.87 1.49
wang3old 72.9 57.8 57.7 51.0 40.6 24.2 45.8 32.3
wang4 11.8 10.5 12.2 10.5 53.4 30.7 8.84 7.94
code s. For
instance , MUMPS, MA41, and WSMP first
pe rmute the matrix such
that it has a diagonal of structural
nonze ros. This initial
pe rmutation is
the same for
MUMPS and MA41
be cause both
use the same code to
compute it.
Howe ve r, it can
be di#e re nt for WSMP.
The pivoting
strate gy of UMFPACK
base d on row
is
inhe re ntly
di#e re nt from
the symme tric
inte rsupe rnode pivoting
strate gy
use d in
MUMPS, MA41, and WSMP. WSMP's algorithms work only with a
pe rmutation to
the block-triangular form, which is not
imple me nte d in MUMPS, MA41, and UMF-
PACK.
Howe ve r, with
the e xce ption of comp2c,
the e #e ct of block-triangularization
on
the ope ration count for factorization is insignificant, if any. As a
re sult of
the se di#e re nce s and
due to
the fact that MUMPS may
pe rform
more ope rations than
ne c-
e ssary on structurally
unsymme tric
matrice s,
the factorization
ope ration counts for
the four
code s in
Table
are di#e re nte ve n with a similar
ring algorithm for
fill-re duction.
In
Table
5.1,
the faste st factorization
time fore ach matrix is in
boldface and
the se cond
faste st
time is
unde rline d. Although
di#e re nce s
othe r than
the factorization
algorithm
itse lf
a#e ct
the pe rformance of
the se code s, it
ise asy to
se e the broad
picture thate me rge s from
Table
5.1. Most of
the boldface e ntrie s
are in
the WSMP column
and most of
the unde rline de ntrie s
are in
the MA41 column. For many
matrice s,
the e #e ct of
the algorithmic
choice s of
the software ise vide nt in
the factorization
statistics in
Table
5.1. MUMPS usually
re quire s
more floating-point
ope rations for
factorization than MA41 and WSMP
be cause it
use s artificially
symme trize d frontal
matrice s
padde d with
ze ros. For
the same re ason, UMFPACK is
faste r than MUMPS
UNSYMMETRIC SPARSE MATRIX FACTORIZATION 551
for
ve ry
unsymme tric
matrice s (such as
baye r01,
one tone 2, and
twotone
howe ve r,
it is
slowe r for
matrice s with
more structural
symme try (such as fidap011, mil053,
and wang4) partly
be cause it
use s a
fill-re ducing
pe rmutation on
the columns of
the coe #cie nt matrix
be fore starting LU factorization. MA41
o#e rs a significant
improve me nt
ove r MUMPS for
matrice s with a
ve ry
unsymme tric
structure , such as
comp2c,
one tone 1, and
twotone . It
se e ms that MA41's
me chanism for finding all-
ze ro rows and columns incurs a slight
ove rhe ad that it cannot
o#se t for
matrice s
with a
ne arly
symme tric
structure (such
ase cl32, fidap011, and wang4), for which
it is
some what
slowe r than MUMPS. It is
cle ar from
Table
5.1 that WSMP has
the smalle st
ove rall factorization
time se ve n
its
de fault options
are modifie d to
compare it with
the othe r
solve rs. With its
de fault options, WSMP's factorization
time s
are usually much
smalle r [18] than
those shown in
Table
5.1.
6. Concluding remarks. This
pape r
de scribe s
sparse unsymme tric symbolic
and
nume rical factorization algorithms that
improve pre vious similar algorithms. Our
phase , in particular, is
more powe rful than
othe rs
de scribe d in
the lite rature . It
ine xpe nsive ly
compute s
minimale limination
structure s that
are transitive re ductions of
the uppe r and
lowe r triangular factors of
the original
coe f-
ficie nt matrix. In addition, it
compute s
ne ar-minimal
data-de pe nde ncy DAGs for
unsymme tric multifrontal factorization with and without pivoting. A data-DAG that
has only a slightly
highe r
ofe dge s than a minimal task-DAG and that is
capable ofe xpre ssing all
possible data-de pe nde ncie s in
the face of dynamic pivoting
is a
fe ature of our symbolic
phase .
We show how this data-DAG aids in a
high-pe rformance imple me ntation of
the unsymme tric multifrontal LU factorization
algorithm. This factorization algorithm is not only
faste r than
othe r
sparse LU factorization
algorithms but is also
simple r than
the unsymme tric multifrontal algorithms
de scribe d
pre viously in
the lite rature .
Our algorithms do not
introduce additional
ove rhe ads
while factoring
matrice s
with a
symme tric
nonze ro
patte rn.
Whe n
pre se nte d with a
sparse matrix with a
symme tric
structure , both
the symbolic and
the nume rical factorization algorithms
and
the data-structure s
ge ne rate d by
the m
grace fully transform into
the ir
symme tric
counte rparts without
re quiring any significant amount
ofe xtra
proce ssing or
storage .
In a
distribute d-me mory
paralle l
imple me ntation of
unsymme tric
sparse LU
the e dge s of
the data-DAG
conne cting tasks
mappe d onto
di#e re nt
proce sse s
de te rmine the inte rproce ss communication
patte rn.
The static and
ne ar-
minimal
nature of
the data-DAG
use d in our algorithms would
be e xtre me ly
use ful for
pote ntial
paralle l
imple me ntations of
unsymme tric multifrontal factorization,
re changing
the data-DAG dynamically could
be cumbe rsome and
ine #cie nt and
the unne ce ssary
dge s could
incre ase synchronization and communication
ove rhe ads.
Acknowledgments
.
The author
wishe s to thank
Andre w Conn,
Fre d Gustavson
Jose ph Liu, Sivan
Tole do, and
the anonymous
re fe re e s
fore xtre me ly
use ful
comme nts
one arlie r drafts of this
pape r.
--R
An approximate minimum degree ordering algorithm
Vectorization of a multiprocessor multifrontal code
Memory management issues in sparse multifrontal methods on multiprocessors
A fully asynchronous multifrontal solver using distributed dynamic scheduling
Multifrontal parallel distributed symmetric and unsymmetric solvers
Analysis and comparison of tw sparse solvers for distributed memory computers
The influence of relaxed supernode partitions on the multifrontal method
An unsymmetric-pattern multifrontal method for sparse LU factorization
A Column Approximate Minimum Degree Ordering Algorithm
A supernodal approach to sparse partial pivoting
Direct Methods for Sparse Matrices
On algorithms for permuting large entries to the diagonal of a sparse matrix
The multifrontal solution of unsymmetric sets of linear equations
Exploiting structural symmetry in unsymmetric sparse symbolic factorization
Elimination structures for unsymmetric sparse LU factors
Fast and e
Recent advances in direct methods for solving unsymmetric sparse systems of linear equations
Highly scalable parallel algorithms for sparse matrix factorization
On the LU Factorization of Sequences of Identically Structured Sparse Ma- tricesw ithin a Distributed Memory Environment
A scalable sparse direct solver using static pivoting
The role of elimination trees in sparse factorization
The multifrontal method for sparse matrix solution: Theory and practice
--TR
--CTR
Timothy A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method, ACM Transactions on Mathematical Software (TOMS), v.30 n.2, p.165-195, June 2004 | sparse matrix factorization;parallel sparse solvers;multifrontal methods |
589360 | Numerical Approximation of an SQP-Type Method for Parameter | This paper deals with the numerical approximation of the Levenberg--Marquardt SQP (LMSQP) method for parameter identification problems, which has been presented and analyzed in [M. Burger and W. Mhlhuber, Inverse Problems, pp. 943--969]. It is shown that a Galerkin-type discretization leads to a convergent approximation and that the indefinite system arising from the Karush--Kuhn--Tucker (KKT) system is well-posed.In addition, we present a multilevel version of the Levenberg--Marquardt method and discuss the simultaneous solution of the discretized KKT system by preconditioned iteration methods for indefinite problems. From a discussion of the numerical effort we conclude that these approaches may lead to a considerable speed-up with respect to standard iterative regularization methods that eliminate the underlying state equation. The numerical efficiency of the LMSQP method is confirmed by numerical examples. | Introduction
Parameter identication denotes the procedure of determining unknown parameters appearing
in an underlying state equation (usually a partial dierential equation), from indirect
measurements related to the solution of this equation. Such problems frequently appear in
many applications, where mathematical models of physical, chemical, biological or economical
processes are used (cf. e.g. [1, 12, 16] and the references there).
Since such problems are ill-posed in general, i.e., the parameter to be reconstructed does
not depend on the observation in a stable way, regularization methods have to be used in
order to compute a stable approximation of the parameter in presence of data noise. Due to
the ill-posedness of the identication problem, the numerical approximation of such problems
is not a simple task. The standard approach that can be found in literature is based on an
a-priori elimination of the state equation, and an application of a discretized regularization
method to the resulting operator equation involving the parameter-to-output map, which is
the operator mapping the parameter to the corresponding observation. The main part in the
This work has been supported by the Austrian National Science Foundation FWF under project grants F
13/08 and F 13/09.
INTRODUCTION
evaluation of this map is the solution of the underlying state equation for given parameter,
which is numerically realized by standard discretizations such as nite elements.
This approach, in particular combined with iterative regularization methods (cf. [17] for an
overview), has been applied with success even to rather complicated parameter identication
problems (cf. e.g. [9, 24, 25]). However, since this methods need a high number of direct solves
(i.e., solutions of the state equation), ne discretizations of the parameter yield a considerable
computational eort, which results in high CPU-times or even in the impossibility to use ne
discretizations. Another drawback of this approach is that the discretizations of state and
parameter are rather independent, which makes the numerical analysis extremely di-cult.
Therefore, fundamentally dierent methods for the solution of parameter identication
problems have been investigated recently, whose common idea is to avoid the a-priori elimination
of the state equation (cf. [10, 20, 26]). The aim of this paper is to discuss the numerical
approximation of an iterative regularization method based on the idea of sequential quadratic
programming (cf. [10]). We investigate Galerkin-type discretizations in the product space
for parameter, state variable and a corresponding Lagrangian variables, which leads to a sequence
of well-posed indenite systems. With this approach we are able to show convergence
of the numerical approximation both for the quadratic programming problem arising in each
iteration step and for the overall minimization procedure.
The general setup in this paper is as follows: we assume that we are given a noisy measurement
z - satisfying
where the exact data satisfy
z := E^u; (1.2)
Our aim is to identify the parameter q 2 Q ad Q (where
Q ad is a closed subset of Q with non empty interior) in the underlying equation
is a continuous nonlinear operator with
In this setup X, X , Q and Z are Hilbert spaces, and X can be identied with the dual of
X. Finally, we assume that e is continuously Frechet-dierentiable on X Q and that the
partial derivative e u 2 L(X; X ) is self-adjoint and satises the coercivity condition
he u (u; q)v; vi e kvk 2
for some e
The above setup is typical for a partial dierential equation of elliptic type, which is
also the main type of application we have in mind. We want to mention that the innite-
dimensional analysis carried out in the preceding paper [10] was not restricted to elliptic
problems, but only assumed well-posedness of the state equation for given parameter. How-
ever, since the numerical approximation techniques for elliptic problems dier from the ones
for parabolic or hyperbolic problems (cf. e.g. [32] for an overview), one cannot expect a successful
unied approach to corresponding parameter identication problems. For this reason
we start with an investigation of the elliptic case in this paper, but we want to mention that
the numerical identication of parameters in transient equations or even mixed systems of
equations is an important and challenging problem for future research.
In [10], it has been mentioned that the parameter identication problem in the above setup
is an ill-posed inverse problem and we have proposed the following iterative regularization
method based on the idea of sequential quadratic programming:
Programming Method).
Let
be a given initial value and let ( k ) k2N be a bounded sequence of positive real numbers.
The Levenberg-Marquardt sequential quadratic programming (LMSQP) method consists of
the iteration procedure
where is the minimizer of the quadratic programming problem2
subject to the linear constraint
The iteration procedure is stopped as soon as
with appropriately chosen > 1.
Due to the results of [10], the LMSQP-method is a convergent regularization method,
in particular the quadratic programming problems of the form (1.7), (1.8), which have to
be solved in each iteration step, are well-posed. Our aim in this paper is to investigate
the numerical approximation of the LMSQP-method by a Galerkin-type approach. We shall
show below that this leads to an indenite system in each iteration step, whose solution is an
approximation of optimal order to the solution of (1.7), (1.8). Moreover, we show that the
reconstructions obtained with the discretized LMSQP-method converge to a solution of the
parameter identication problem as the noise level and the discretization size tend to zero, if
an appropriate stopping rule is used, which relates the residual to the noise level and some
measures for the discretization.
Moreover, we shall discuss the solution of the discretized Karush-Kuhn Tucker system,
which is an indenite linear system to be solved for the discretized equivalents of state, parameter
and Lagrangian variable. The standard approaches to the solution of such discretized
problems arising from partial dierential equations are reduced SQP-methods, where state and
Lagrangian variable are eliminated a-priorily. We recall the basic properties of the reduced
SQP-approach, but we mainly focus on the iterative solution of the whole system with appropriate
preconditioning. This promising approach has been employed recently for parameter
identication (cf. [20, 26]) and optimal control problems (cf. [2, 3, 4, 5]) with good numerical
results, in particular with respect to e-ciency.
The paper is organized as follows: in Section 2 we investigate the numerical approximation
of the LMSQP-method by a Galerkin-type approach and discuss the well-posedness,
stability and approximation properties of the discretized Karush-Kuhn-Tucker (KKT) system;
the convergence of the discretized solutions is shown in Section 3. Some further numerical
methods and the implementation of the outer iteration, i.e., the SQP-iteration under the
assumption that we are able to solve the quadratic optimization problems arising in each
step of the LMSQP method, are examined in Section 4. We discuss the correct scaling of
variables, globalization strategies as well as a multi-level approach, which leads to a further
speed-up of the method. Section 5 is devoted to the inner iteration, i.e., the numerical solution
of the discretized KKT-system. Some basic properties of this symmetric indenite
problem are studied, as well as its iterative solution with appropriate preconditioning. As a
rst application we investigate the identication of a potential in an elliptic boundary-value
problem, where we can give quantitative error estimates in terms of the discretization sizes.
Some numerical experiments related to this identication problems are presented in Section
7, before we nally conclude and give an outlook to further interesting problems related to
this topic in Section 8.
Discretization Techniques
In the following we investigate the discretization of the LMSQP-method by a Galerkin ap-
proach. First of all, we assume that we have discretized data z -; 2 Z Z of the form
z
is the orthogonal projector onto the nite-dimensional subspace Z . Note
that we can give an error estimate for z -; using (1.1) and kR
Now let X h X, Q h Q be nite-dimensional subspaces of X and Q, with the corresponding
orthogonal projectors
. Then we can discretize the
LMSQP-Method as follows:
be as above and let
be a given initial value. Moreover, let ( k ) k2N be a bounded sequence of positive real numbers.
The Galerkin Levenberg-Marquardt sequential quadratic programming (GLMSQP) method
consists of the iteration procedure
where is the minimizer of the quadratic programming problem2 kR (Eu z - )k 2
subject to the linear constraint
2.1 The Discretized Karush-Kuhn-Tucker System 5
Note that the constraint (2.5) can be rewritten in operator form as
~
to be solved for (u; with the notation
and P
is the adjoint of P h . Under the assumption (1.5), we obtain that
for all v 2 X h , i.e., the discrete bilinear form associated with the operator P
coercive
on X h . This implies by the Lax-Milgram theorem, that (2.6) is uniquely solvable with respect
to u for given q 2 Q h . Consequently, in an analogous way to the proof of Proposition 2.1 in
[10] we may show the following result on the well-posedness of the quadratic programming
problem that has to be solved in each step of Method 2:
Proposition 2.1. Let e be continuously Frechet-dierentiable, let (1.5) hold and let k > 0.
Then the quadratic programming problem (2.4), (2.5) has a unique solution
which is also the only local minimum.
2.1 The Discretized Karush-Kuhn-Tucker System
In [10], the Karush-Kuhn-Tucker system for the innite-dimensional version of the LMSQP-
method has been derived and analyzed in the framework of linear saddle point problems.
Now we will discuss the discretized analogue of this system, namely the rst-order optimality
conditions for the quadratic programming problem (2.4), (2.5).
The Lagrangian of (2.4), (2.5) is given by
for
are equal to the identity on X h and Q h ,
respectively, we can rewrite the Lagrangian as
~
with the operators K k and L k dened by (2.7), (2.8). The KKT-system can now be deduced
by computing the partial derivatives of the Lagrangian with respect to u, q and , i.e.,
solves the linear saddle-point problem@ P
~
~
~
~
As in [10], we dene the symmetric bilinear form a
a
and the bilinear form b
Moreover, we use the right-hand sides
Then the KKT-system (2.12) can be interpreted as the Galerkin approximation of an indenite
variational problem, i.e., (u; q; is the solution of
a
In an analogous way to the proof of Theorem 2.3 in [10] we can show that the bilinear
form a satises the kernel-ellipticity condition on X h Q h , i.e., there exists a constant a > 0
such that
a
and that b satises the LBB-condition
sup
for some b > 0. This implies the following well-posedness result (cf. [7, 8]) for the discretized
problem (2.17), (2.18):
Theorem 2.2. Let e be continuously Frechet-dierentiable, let (1.5) hold and let k > 0.
Then the indenite system (2.17), (2.18) has a unique solution (u; q;
which depends continuously on the right-hand sides f k and g
k .
Since the constants a and b are the same as in the corresponding innite-dimensional
conditions in X Q, they are in particular independent of the discrete subspaces X h and
This allows to deduce an approximation result for the solutions of (2.17), (2.18) to the
solution of the innite-dimensional KKT-System, given in variational
form as
a k (u; q;
with a k given by
as above and g k dened by
2.1 The Discretized Karush-Kuhn-Tucker System 7
Theorem 2.3. Suppose that the assumptions of Theorem 2.2 are satised and let
denote the unique solution of (2.17), (2.18). Then there exists a constant c > 0 independent
of X h and Q h such that
where (u; q; ) denotes the unique solution of (2.19), (2.20) and
Proof. First, let (~u h ; ~
the solution of (2.17), (2.18) with a
k replaced by a k ,
k . Then Theorem 2.1 in [8] implies the existence of a constant c 1 > 0 (independent of X h
and Q h ) such that
Moreover, the stable dependence of the solutions of (2.17), (2.18) on the right-hand side
implies the existence of c 2 > 0 with
sup
hg
ja
sup
(R
(R
and with the triangle inequality we may conclude (2.23).
Theorem 2.3 provides an error estimate for the solutions of the discretized saddle-point
problem (2.17), (2.18), consisting of two parts corresponding to the numerical approximation
in the image space Z and in the pre-image spaces X and Q. An obvious estimate for the rst
term is
;h inf
~
which possibly does not lead to a quantitative estimate, since there is no additional information
on the smoothness of the noisy data. An alternative estimate is
~
The inmum of can usually be estimated more easily, since the exact data z are
smoother due to the fact that ^ u is the solution of the state equation for some parameter ^
q.
E.g., if the state equation is of elliptic type with solution ^
is the embedding operator, and R results from a standard nite element discretization on a
grid with neness , then we have at least
Another important observation is that the last term vanishes if the discrete spaces Z and
are equal, which can be achieved in some applications.
The second term in (2.23) shows that the Galerkin approximation of the KKT-system is
of optimal order in X h Q h X h ; it can be estimated by standard methods for nite element
discretizations; quantitative estimates can be obtained using the regularity of the iterates.
This part depends of course strongly on the specic application.
3 Convergence Analysis
In this section we will analyze the Galerkin LMSQP method with respect to convergence, i.e.,
the convergence of the reconstruction obtained with an appropriate stopping rule as the noise
level and the measure for the discretization neness tend to zero. With
identify the innite-dimensional case, i.e., We assume that
the discrete subspaces satisfy
If we denote by e k and f k the error terms
we can rewrite the Karush-Kuhn-Tucker system (2.12) as@ P
~
~
~
~
~
~
r kA (3.2)
where the r k denotes the remainder
As in [10], we require a condition on the nonlinearity, which is summarized in the following:
Assumption 1. Let (1.5) be satised and dene the remainder r(u; q) by
Then we assume that there exists a constant
kEe
and that there exists a solution (^u;
of the parameter identication
problem.
If we dene the discretization measures h , h by
where
kEe
and by
Then for all (u;
holds, where
Remark 1. If X h , Z and Q h are standard nite-element spaces on some triangulations, then
h , and h can be estimated by the approximation error of these elements. In particular,
if the discretization parameter (i.e., the maximal size of a triangle) tends to zero and if the
triangulation is regular, one can guarantee that h , and h tend to zero (cf. [32] for further
details).
For the choice of the stopping index we use a numerical version of (1.9), which involves
the discretization measures dened above:
For an appropriate choice of , this allows us to prove the following monotonicity property of
the iterates:
Lemma 3.1. Let Assumption 1 be fullled, let the noise be bounded by (1.1), and assume
that
In addition, k is chosen such that k k 1 for all k 2 N and that
s
and the stopping index k is chosen according to the generalized discrepancy principle (3.11)
with
then and the estimates
and
hold for all k < k .
Proof. Assume that q
we deduce the identity
The noise bound (1.1) implies that
and using the Cauchy-Schwarz inequality together with (3.9) we obtain the estimate
(3.16) follows from dividing (3.15) by k and the fact that
s
By induction we can now show that q k 2 B (q 0 ) for k < k and satisfying (3.14).
In an analogous way to the proof of Lemma 3.2 in [10] we can prove the following statement
on the niteness of the stopping index k if - > 0:
Lemma 3.2. Under the assumptions of Lemma 3.1, the discrepancy principle (3.11) yields
a nite stopping index k if
and is chosen according to (3.14).
One observes that in the above estimates, the term - ;h now plays the same role as the noise
level - in the innite-dimensional setup. Therefore it is also possible to prove convergence
as in the same way as convergence in the innite-dimensional case for - ! 0 (cf.
[10, Theorem 3.5]). Consequently, we do not give the detailed convergence proof, but refer to
[10] for further details on the technique of the proof. We only recall the basic assumptions
on e and give the nal convergence result, where we use the notation (u -;;h
k ) for the
iteration according to (2.12) with initial value (P h
discretization
parameter h and .
Assumption 2. In addition to Assumption 1, assume that e is of the form
with continuously Frechet-dierentiable (nonlinear) operators A
, such that
Moreover, we assume that A and N satisfy the nonlinearity conditions
kEe
and
kEe
for some positive constants
2 and
3 .
Theorem 3.3 (Convergence). Let Assumption 2 and (3.12) be fullled with , suciently
small, and let the noise be bounded by (1.1). Moreover, let k be chosen such that
and that (3.13) is satised. If the perturbed iteration is stopped with
according to the generalized discrepancy principle (3.11) with
(uniformly bounded in h and ) satisfying (3.14), then
(q -;;h
where (u; q) is a solution of (1.3) with
Proof. Analogous to the proof of Theorem 3.5 in [10].
4 Numerical Realization of the SQP-Iteration
In the following we want to discuss some numerical methods and variants for the 'outer
iteration', i.e., the Galerkin LMSQP algorithm under the assumption that we are able to
solve the discretized KKT-system numerically. The 'inner iteration', namely the numerical
solution of the indenite system (2.12) will be investigated in Section 5.
4.1 Scaling of State Variable, Parameter and Lagrangian Variable
The performance of an iteration algorithm often depends crucially on the way the problem
is formulated. Scaling is a well-known technique for reformulating an optimization problem
whose main objective is twofold: on the one hand all the variables should be of similar
magnitude, on the other hand also the value of the derivatives should all be of similar size. In
unconstrained optimization, a problem should be rescaled in such a way, that changes of the
iterate in one direction do not result in by far larger changes of the value of the objective than
changes in another direction. In constrained optimization the above statements are also true
for each constraint. Additionally the set of constraints should be well balanced with respect
to each other such that each constraint has equal weight. Furthermore the set of constraints
should be balanced with respect to the objective. As scaling is of high practical importance
for any optimization problem, many aspects can be found in monographs on optimization (cf.
e.g. [19, 30]).
We want to consider only the last aspect in this context, i.e., the scaling of the state
constraint with respect to the objective which is also of high importance for achieving fast
convergence of the outer iteration. For the inner iteration, the aspect of scaling can be
included in the construction of a good preconditioner. The outer iteration of an SQP method
tries to attain two goals at the same time: feasibility of the iterate with respect to the state
constraint and optimality of the iterate with respect to the objective. One aspect dominating
the other results usually in bad convergence properties: If the feasibility aspect dominates,
only very small changes of the iterate are possible in order to ensure "almost" feasibility. If
the optimality aspect dominates, any violation of the state constraint is reduced too slowly.
For the LMSQP method in the form of (2.4),(2.5) it turned out that in many situations
the feasibility aspect is strongly dominating. Using line search methods for globalization (see
also Subsection 4.2) this results usually in step lengths much smaller than one. Replacing the
state constraint by a preconditioned state constraint leads to a better balanced formulation
and to much faster convergence. Furthermore a step length parameter equal to one is accepted
in almost all steps. Another aspect of this kind of rescaling is treated in Subsection 6.2.
4.2 Globalization Strategies
The LMSQP method is a variant of Newton's method and therefore only locally convergent
(see also the analysis in Section 3). For this reason, globalization strategies, such as trust
region methods or line search strategies (which are the two most popular classes of globalization
techniques in optimization), are needed.
The basic idea of trust region methods is to add an additional constraint on the maximal
increment to the quadratic optimization problem for the correction step of the current iterate,
i.e. instead of (2.4), (2.5) one would solve (2.4), (2.5) and k(u u k ; q q k )k k with k chosen
appropriately. Trust region methods have been successfully applied to PDE constrained
optimization problems (see e.g. [14, 38]), often using a reduced SQP approach. We want to
mention that a similar eect as with trust-region methods could be reached in principle by
controlling the penalty parameter k , which also restricts the step size and produced good
numerical results (see Example 7.1). For a comprehensive overview of trust region methods
we refer to Conn et. al. [13].
In the code used for two-dimensional problems (cf. [10] and Example 7.2), we use a line
search algorithm for globalization. In contrast to trust region methods, the calculation of the
increment is split into two phases: rst of all, a search direction is determined, and secondly
the estimation of a step length parameter indicating how far into the search direction one
should go. For the computation of the search direction we solve the optimization problem
and (2.5). In order to determine the step length we cannot use the objective itself as a
criterion (as in unconstrained optimization), but have to use a merit function which balances
the minimization of the objective with the feasibility with respect to the state constraint.
Applied to a discretized optimization problem of the form
~
subject to an equation constraint of the form
possible choices are the l 1 -merit function
4.3 Nested Multi-Level Optimization Techniques 13
and its variants, and the augmented Lagrangian
where is an estimate of the Lagrangian variable corresponding to the discretized equation
constraint. Both merit functions are exact in the sense that for su-ciently large, minimizers
of the original constrained optimization problem also minimize the merit function.
A crucial property in the design of a merit function is that it should accept step length
one close to a solution in order to preserve the quadratic convergence of the SQP method.
The augmented Lagrangian works well, as long as the estimate for the Lagrangian multiplier
is accurate enough, whereas the l 1 -merit function sometimes suers from the so-called
"Marathos-eect", i.e. it does not accept unit step length and therefore causes a slow-down
of the convergence. A strategy to overcome this di-culty using a second order correction
can be found in [30]; nevertheless, it performed very well in our numerical experiments (see
Example 7.2).
4.3 Nested Multi-Level Optimization Techniques
Important tools for the e-cient numerical approximation of innite-dimensional optimization
problems are multi-level optimization methods. In the nested multi-level setup, one starts the
optimization procedure at a coarse level X
, where the iteration procedure can be
carried out e-ciently. If an appropriate stopping rule is satised, one interpolates the state
and parameter obtained in this way to a ner level X h 2
(for serving now
as a starting value on this level. This procedure is repeated until the nest level is reached.
Usually, nested space are used in this approach, i.e., X h 1
(for
which leads to simple interpolation operators. Since one cannot choose the discretization of
the data arbitrarily in general , we consider only the case of xed here, but a multi-level
approach in can be realized in an analogous way, if necessary.
Nested multi-level methods outperform standard discretization techniques in many cases
(cf. e.g. [21, 22, 29]); usually a considerable number of iterations is needed on the coarse level
only , where the numerical eort per iteration is very low. On the nest levels, the stopping
rule is often satised already after one iteration step and so the overall eort is less than for a
direct discretization on the nest level. For the Galerkin LMSQP method, we can formulate
a multi-level algorithm as follows:
Algorithm 4.1 (Nested Multi-Level Galerkin LMSQP). Given a decreasing sequence
'=1;:::;L with nested spaces X h ' X h '+1 , Q h ' Q h '+1 (e.g. h non-increasing
sequence ' satisfying (3.14), the nested multi-level Galerkin LMSQP method consists
of the following iterative procedure:
1.
2. Perform the Galerkin LMSQP method until the stopping criterion (3.11) is satised
with stopping index k (').
3. If stop the iteration, else prolongate the iterate (u '
k ) to the ner level
which results in a new starting value (u '+1
and go to step 2.
The analysis in Section 3 shows that for '
, the estimate
holds, where ' is the error corresponding to the interpolation of the iterates from level ' 1
to level ', i.e.,
kR Ee '
Z
kf '
This monotonicity estimate corresponds very well to the intuition that only few iterations are
needed on the ne levels, in particular if '
k is decreasing, which leads to
kR Ee '
Z '
For a ne level with small , we can expect that
and the second term '
can be expected to be negligible. I.e., the stopping
rule at level ' is probably satised with k
Under typical conditions, where X h ' and Q h ' correspond to standard nite-element spaces
on dierent renement levels of an initial triangulation of a
domain
one can show that at
least consequently
ch
ch 1
for some constant c 2 R+ , where
Together with the above estimate one can show with a standard proof technique that the
converges to a solution (u; q) of the parameter identication problem for
5 Numerical Solution of the KKT-System
In the following we will discuss the numerical solution of the discretized KKT-system (2.12)
for xed iteration number k. We have seen above that the Galerkin-type approximation (2.12)
of the original KKT-system is stable and convergent, now we discuss some of its structural
properties, which are important for the application of iterative solution methods and for the
construction of preconditioners.
5.1 The System Matrix M 15
Choosing bases
of the nite-dimensional subspaces X h and Q h , we may represent
via
with coordinate vectors V; In order to transform (2.12) into a linear
system for V , S and , we dene the matrices
and the vectors
This allows us to rewrite the discretized KKT-system (with penalty parameter
respectively as
where M is the matrix in (5.6) and
The structural properties of M and its sub-matrices will be examined in the following section.
5.1 The System Matrix M
Due to the well-posedness result on the discretized KKT-system (2.12) (cf. Theorem 2.2), we
may conclude that the system matrix M is regular. In order to obtain further insight into
the structure of M , we investigate the properties of the sub-matrices G, H, K and L:
Proposition 5.1. The matrices K 2 R mm and H 2 R nn are symmetric positive denite,
and the matrix G 2 R mm is symmetric positive semi-denite. If, in addition, the operator
is injective on X h , then G is regular, too.
Proof. Let u and q be as in (5.2), then there exist constants c 1 (h) and c 2 (h) such that
where j:j denotes the euclidean norm in R n and R m , respectively. Thus, we have
and
Moreover, the identity
implies that G is positive semi-denite and regular under the assumption that E is injective
on X h . The symmetry of the matrices G, H and K can be veried in a similar way, using
the symmetry of scalar products and the self-adjointness of the operator K k .
The matrix L 2 R mn is di-cult to analyze, it is neither symmetric nor regular in general
(in particular if n 6= m). However, some fundamental properties of M (such as its regularity)
rely rather on G, H and K than on L. Moreover, the classical splitting of a symmetric
saddle-point problem as@ G 0 K T
I
I
where H := H and C is the Schur-complement
is possible if we only know that G and H are regular. In particular, we may conclude that
M has n +m positive and m negative eigenvalues.
5.2 Reduced SQP Approaches
The basic idea of reduced SQP-methods is the a-priori elimination of the equality constraint,
which can be written in matrix form as
which is equivalent to an elimination of V and in (5.6).
Due to Proposition 5.1, K is a regular, symmetric matrix and thus, we may compute
which yields after some calculations the n n-system
with
The reduced SQP-approach seems of particular interest if n m, which is a frequently
used discretization strategy for parameter identication and optimal control problems (cf. e.g.
[35, 36, 37]). The original matrix M is an indenite matrix of size (2m+n) (2m+n), while
5.3 Simultaneous Solution of the KKT-System 17
the reduced system matrix M r in (5.12) is of size n n. However, M r is not a sparse matrix
even if all the sub-matrices of M are sparse, since it involves the inverse of K. Moreover,
the evaluation of M r is more expensive than the evaluation of the original system matrix M ,
since it involves the solution of two systems of the form
with dierent right-hand sides g, while for the evaluation of M only direct evaluations of K are
needed, which are very cheap for typical nite element discretization of the state constraint.
In practice, one usually tries to compensate this disadvantage of reduced SQP-methods by
using a Broyden-type update for the reduced system matrix instead of the exact matrix M r ,
which leads to e-cient optimization algorithms for small n.
5.3 Simultaneous Solution of the KKT-System
Recently, the simultaneous solution of KKT-systems by iterative methods has been investi-
gated, in particular in connection with optimal control problems (cf. [2, 4, 5, 20]). Compared
to the reduced SQP-approach, a simultaneous solution strategy has the obvious advantage
that the allocation and evaluation of the system matrix M is much cheaper than of M r . The
pay-o is that M is indenite and larger than M r , which might cause additional eort. How-
ever, the main eort in the reduced SQP-approach is related to the evaluation or assembly of
the system matrix M r , respectively, and therefore a simultaneous solution of the KKT-system
can result in a tremendous speed-up of the SQP-method, in particular for ne discretizations.
At a rst glance, it seems rather straight-forward to solve (5.7) by a standard iterative
method for indenite systems such as inexact Uzawa methods (cf. [6, 15]) or Krylov-subspace
methods such as GMRES (cf. [34]), MINRES (cf. [31]) and QMR (cf. [18]). However, in
the case of large-scale problems, we have to expect a large condition number (note that
is usually small and that M is singular for and a complicated eigenvalue pattern of
the matrix M , which might cause iterative methods to diverge or to need a high number of
iterations. Therefore, an appropriate preconditioning technique seems necessary for any of
the methods. We do not go into details here, but refer to the forthcoming paper [11] for a
discussion of preconditioners.
In the following we distinguish two types of solvers that seem appropriate for the solution
of the indenite system (5.7) and discuss their basic properties with respect to the special
structure of M .
Inexact Uzawa Iterations
Inexact Uzawa methods and similar iteration procedures have been developed for the solution
of the classical Stokes system and similar problems (cf. [32] for an overview). The classical
Uzawa method is just a gradient method for the dual of the corresponding Lagrange functional,
the inexact Uzawa method can be interpreted as a preconditioned version (cf. [32]). Following
the exposition by Zulehner [39], we can write an inexact Uzawa method for a system of the
form (5.6) as
A
followed by
A is a preconditioner for the diagonal matrix
A :=
C is a preconditioner for the Schur-complement C dened by (5.8). In terms of (5.7) we
can write the inexact Uzawa iteration as
M is a preconditioner for the system matrix, given by
.
A convergence analysis of this method is available only in the case when A is a regular
matrix (cf. [6, 39]), which means that we have to assume that G is regular. The latter
is true e.g. if the data z represent distributed data for the state, i.e., E is an embedding
operator. In this case, the structure of A is rather simple and it is not a di-cult task to
construct a preconditioner, even exact preconditioning seems possible (note that G is just a
mass matrix for a typical nite element discretization). Since the matrices G and H do not
change during the SQP-iteration we may even compute decompositions in a preprocessing
step. The construction of a preconditioner for the Schur-complement C is more di-cult and
must take into account the specic nature of the underlying state equation.
Krylov-Subspace Methods
The Krylov-subspace methods GMRES and QMR are variants of the CG-algorithm that are
applicable to indenite problems, too. The basic idea of such methods is a defect minimization
in the Krylov-subspace
generated by X 1 , in the k-th iteration step. Since preconditioned CG-methods are probably
the most successful class of iteration methods for positive denite systems, such methods
seem very attractive also in the indenite case, although additional di-culties may arise (cf.
e.g. [34]).
The convergence analysis in [34] and [18] shows that the error bounds obtained for both
methods are essentially the same, and mainly dependent on the eigenvalue distribution and
the condition number of the system matrix M . Therefore, appropriate preconditioning is
again of high importance, in this case also with the possibility that G is singular. We refer
to [11] for a detailed discussion of this problem.
As a rst application we investigate the identication of the potential q in the elliptic boundary
value problem
in
@
from a state observation in L
2 which is a well-studied problem in literature (cf. e.g. [33]).
In [10], it has been shown that in the setup (d denotes the space dimension)
the operators
satisfy all assumptions needed for the convergence analysis of the LMSQP-method. Now we
shall study a concrete nite-element discretization of the KKT-system and the derivation of
estimates for the numerical errors , h and h .
6.1 Error Estimates for the Discretized KKT-System
In this case we can write the whole KKT-system in classical form as
in
in
in
again with homogenous Dirichlet boundary conditions upon v and on @ where L d is a
dimension-dependent dierential operator of order 2d corresponding the norm in H
7 e.g.,
we have
supplemented by homogenous boundary conditions up to order d 1. If f 2 L
and
3 a standard elliptic regularity argument shows that ^
0(for all k 2 N. In the same way we can show that k 2 H
and s k 2 H
7 This
additional regularity can be employed to derive standard error estimates for nite-element
discretizations of the KKT-system (2.12).
If we use piecewise linear nite elements on regular triangulations T and T h for the
discretization spaces Z and X h , where and h represent the neness of the grids, then a
classical approximation result for nite elements (cf. [32, p.96]) implies that
Of course, one could also use piecewise constant elements on T , which would yield
However, in practical applications a higher-order approximation in is often desirable, since
can be signicantly larger than a reasonable choice of h. A canonical approximation of the
parameter q is a nite element space of order greater or equal d on a regular triangulation T ~ h
assumptions on the exact solution ^
q one can obtain quantitative estimates
for h in terms ~
h. At a rst glance it seems surprising that one needs a-priori assumptions
on the parameter, but not on the state in order to derive error estimates. However, due to
the ill-posedness of the identication problem with respect to the parameter q, such a-priori
knowledge seems to be necessary. The approximation of the state corresponds rather to the
approximation of the underlying elliptic state equation, which is well-posed with respect to
u and yields further regularity. We nally want to mention that according to the theory
developed above, one could choose T ~ h independent of T h , but this would cause unnecessary
complications in the implementation of the method.
We note that alternatively one can use the space
for d 3, which yields
An appropriate discretization strategy is e.g. to choose Q h as the space of piecewise constant
elements on an underlying grid T ~ h . The advantage of this approach is that elements of order
greater than one, which are necessary for
(d 2), can be avoided.
6.2 Structure of the System Matrix
For the potential identication problem, some parts of the system matrix M in (5.6) are well-
understood. First of all, G is an L 2 -mass matrix and it is positive denite if the triangulations
T and T h coincide, which we assume in the following. The eigenvalues of G are then all of
order h d . The matrix H is the stiness matrix for the dierential operator L d , with minimal
eigenvalue of order h d and maximal eigenvalue of order h d .
The matrix K is the sum of a stiness matrix for the Laplacian and a weighted mass
matrix (with weight q k in the L 2 -scalar product), where one can expect the rst part in
this sum to be dominating. Thus, the stiness matrix ^
K for the Laplacian will be a good
preconditioner for K. The maximal and minimal eigenvalues of K and
K are of order h d 2
and h d , respectively. The remaining part in the system matrix, namely the matrix L, is
di-cult to understand, since its elements are weighted L 2 -scalar products of basis functions
of dierent nite element spaces. However, the spectral norm of L can be estimated, it is of
order ~ h d .
The construction of preconditioners for G and H is well-investigated, even exact preconditioning
seems to be applicable. For K it seems reasonable to use a preconditioner ^
K for the
Laplacian, e.g. a multi-grid preconditioner. With preconditioning for K, the system matrix
can be transformed to
~
with the corresponding Schur-complement
~
K is an appropriate preconditioner for K, then we can estimate the minimal eigenvalue by
min ( ~
C) min
and the maximal eigenvalue by
O
Hence, the condition number of ~
C is independent of h, but only depends on and ~ h
h . One
observes that the condition number is decreasing as ~ h tends to h from above (note that
usually ~ h h). For the Uzawa iteration, one can choose the preconditioner ^
C in this case as
a multiple of ^
K 1 or even of G 1 . If ~ h h, the Uzawa iteration seems not to
be optimal, in this case one can apply either a reduced SQP-approach or use Krylov-subspace
methods with dierent preconditioning strategies. For the details on the latter we refer to
[11].
7 Numerical Experiments
In order to test our theoretical results, we numerically solve some model problems, which have
already been investigated with respect to the convergence behavior of the LMSQP-method in
[10].
Example 7.1. Our rst example is the identication of the potential q in (6.1), (6.2) from a
state observation u 2 L
2(with
The exact potential is given by
which is an element of
3 This problem was implemented in the software-system
MATLAB.
The data are generated by solving the state equation on a ne grid and subsequent interpolation
to a coarser grid; the noise is an additive high-frequency perturbation. We used
uniform grids with m nodes for the discretization of the state u and the Lagrange-parameter
and n nodes for the parameter q, i.e., . The parameters
k are chosen according to which lead to convergence of the
method even for starting value q 0.
The KKT-system (5.6) is solved by the QMR method, using an Uzawa-type preconditioner
as described in Section 6.2, with ^
K a preconditioner for the Laplacian and ^
. The convergence results for the overall LMSQP-method have been shown in [10] and
compared to a Levenberg-Marquardt method following the feasible path. It turned out that
both methods lead to almost the same iteration sequence q k . In particular, the number of
iterations needed until the stopping rule is satised, is the same for both methods. Now
we compare the numerical e-ciency of the LMSQP-method with feasible path approaches,
namely the Levenberg-Marquardt method (LM) on the feasible path (with the same Galerkin
discretization as for LMSQP and solution of the Gauss-Newton system by a preconditioned
CG-method) and a Broyden-type variant of the Levenberg-Marquardt method (cf. [23] for
further details).
22 7 NUMERICAL EXPERIMENTS
Table
1: CPU-time (in seconds) needed for the LMSQP-method, the LM-method and a
Broyden-type variant of the LM-method.
For this sake we choose dierent discretization levels (xed during the iteration) and
measure the CPU-time needed for the LMSQP-method, until the stopping rule is satised (for
xed noise level -). From the results shown in Table 1 one observes that the LMSQP-method
with simultaneous solution of the KKT-system outperforms the feasible-path approaches for
all dierent discretizations. Since the LMSQP and the LM-method need the same number of
outer iterations, the dierence in the numerical eort is caused by the fact that the eort for
the evaluation of the system matrix in the LM-method is signicantly higher than evaluation
and preconditioning of the system matrix in the simultaneous LMSQP-method. Obviously,
the gain in the numerical eort for the evaluation of the system matrix increases with the
number of discretization points, which explains the extremely large CPU-time for the LM-
method at the nest discretization level
is much faster than the LM-method, which is again caused by the fact that the evaluation of
the system matrix can be carried out e-ciently. However, the number of iterations needed
for the Broyden-type variant is much larger than for the other two methods, which use the
full information about the derivatives.
Finally, we investigate the spectral condition of the system matrix M as well as of the
matrix ~
M dened by (6.13), where we use a preconditioner for the Laplacian as ^
K. >From the
left picture in Figure 1, which shows the condition number as a function of the discretization
size h (in logarithmic scale) for xed one observes that the condition number of M
grows quadratically with h 1 , while the condition number of ~
M is much smaller and almost
independent of h. The second part of Figure 1 shows a plot of the condition numbers vs.
the parameter in doubly logarithmic scale, from which it seems that the growth of the
condition number as ! 0 is slower for ~
M than for the original matrix M . In both cases, the
condition number seems to be a convex function of , which has a unique minimum at some
. However, this value is rather large and values of that are signicantly larger than
are not of interest for our purpose, since they would cause a tremendous slow-down of the
outer iteration. Therefore we can focus our attention to the case < , where the condition
number increases in a monotonically with 1 .
Example 7.2. Our second numerical example is the identication of the conductivity q 2
log(cond(M))
Condition Number vs. Discretization Size
Original
Preconditioned State
log b
log(cond(M))
Condition Number vs. b
Original
Preconditioned State
Figure
1: Plot of the spectral condition of the matrix M vs. the discretization size h (in
logarithmic scale, left) and vs. the parameter (in doubly logarithmic scale, right). The
solid line shows the condition number of the original matrix M , the dashed line of the matrix
~
M with preconditioned state equation.
in
in
on
@
from a state observation u 2 L
2 The
domain
is a ball in R 2 with missing rst quadrant,
i.e., in radial
coordinates
The exact parameter to be reconstructed is ^
q 1, the right-hand side in (7.1) is given by
r
with
The corresponding solution of the state equation is ^
3r). The data are generated
using the exact solution ^ u perturbed by uniformly distributed random noise. For the discretization
we used triangular nite elements with piecewise quadratic shape functions for
the state u and the Lagrange parameter and piecewise constant shape functions for the
parameter q. The results were calculated using the nite element code FEPP [27], developed
at the Department for Analysis and Computational Mathematics of the University of Linz.
We want to mention that this identication problem is quite challenging not only due to
the complicated geometry, but also due to the fact that q is not identiable along a level
line in the interior, where u attains an extremum. This does not destroy the theoretical
identiability results, because it is a set of Lebesgue-measure zero, but it can be expected to
create numerical di-culties.
Results for exact data can be found in Table 2. The good performance of the method
with respect to both, CPU time and number of outer iterations can be observed clearly.
Especially for problems with ne discretizations of the parameter q, this method can still be
realized e-ciently, while classical approaches do not yield results in reasonable time. A plot
Level dim q dim u avg QMR it SQP it time
Table
2: CPU-time and number of inner (QMR) and outer (SQP) iterations for exact data
Figure
2: Parameter distribution for exact data at level 4
of the parameter q can be found in Figure 2, from which one observes that the parameter is
reconstructed very well except in a neighborhood of the level curve 0g.
Additional speed-up can be gained using a multi-level approach as described in Subsection
4.3. We used nested spaces for q and u by subdividing each triangular element into four smaller
elements, when rening the mesh. Table 3 presents results for this approach. It can be seen
that on ne discretization levels one SQP step is su-cient for fullling the stopping criterion,
which corresponds very well to the theoretical predictions made in Section 4.3. A comparison
of the results to the ones in Table 2 shows that for xed discretization level, the solution of
the identication problem on level 5 is only slightly faster than the identication of q on level
6 (with about the fourfold number of parameters) using a multi-level approach. A plot of
Level dim q dim u avg QMR it SQP it time acc. time
Table
3: CPU-time per level, accumulated time and number of inner (QMR) and outer (SQP)
iterations for exact data using a nested multi-level approach
Figure
3: Parameter distribution for exact data at level 4 using a nested multi-level approach
the parameter can be found in Figure 3. Here the approximation of the parameter in the
area where it can not identied is by far better than in the classical approach using only one
discretization level (compare Figure 2).
8 Conclusions and Outlook
We have developed a framework for Galerkin-type approximations of the LMSQP-method
for parameter identication problems in elliptic partial dierential equations and we have
discussed the implementation of the Galerkin LMSQP-method with iterative solution of the
KKT-system. The numerical results show that the resulting iteration method clearly outperforms
state-of-the-art methods for iterative regularization and provides a tool for the e-cient
solution of identication problems with ne discretizations. Moreover, we have developed a
multi-level version of the Galerkin-LMSQP method, which yields a further speed-up.
The crucial point for the possibility to obtain an e-cient implementation of the LMSQP-
method is the preconditioning of the KKT-system, which is then solved iteratively as an
indenite problem in the product space for state, parameter and Lagrangian variable. The
construction of such preconditioners is not a simple task and has not been discussed in detail in
the present paper, but will be investigated in [11], where dierent preconditioning techniques
will be compared.
Other numerical aspects to be investigated in future research are adaptive discretization
strategies and fast parallel solvers based on domain-decomposition techniques. The adaptive
discretization of optimal control problems, which is a closely related subject, has been
discussed by Becker et al. [3]; possibly the ideas of this work can be carried over to identi-
cation problems, too. The parallel solution of optimal control problems has been investigated
by Lions and Pironneau [28] in the case of quadratic problems; recently Biros and Ghattas
[4, 5] performed a numerical study of a parallel solver with an SQP-method for the outer and
preconditioned Krylov-subspace methods for the inner iteration. Many of their ideas seem
to be applicable also for parameter identication problems that are solved with the LMSQP-
method, which rises the hope that e-cient parallel versions of the LMSQP-method can be
designed also for large-scale identication problems such as impedance tomography.
Finally, we want to recall that the framework of this problem does not apply to transient
problems of parabolic or hyperbolic type. Since numerical methods for dierent types of
partial dierential equations have many type-specic features in general, it is not surprising
that also the numerical treatment of parameter identication problems should depend on
the type of the underlying state equation. However, it seems possible to construct e-cient
and convergent discretized methods at least in the case of parabolic equations, which is an
important task for future research.
Acknowledgments
The authors thank Dr. Walter Zulehner (University of Linz) and Dr. Joachim Schoberl (cur-
rently Texas A & M University) for useful and stimulating discussions on the preconditioning
of the indenite system (5.6).
--R
Estimation Techniques for Distributed Parameter Systems
Preconditioners for Karush-Kuhn-Tucker matrices arising in the optimal control of distributed systems
Parallel Lagrange-Newton-Krylov-Schur methods for PDE-constrai- ned optimization problems
Parallel Lagrange-Newton-Krylov-Schur methods for PDE-constrai- ned optimization problems
Analysis of the inexact Uzawa algorithm for saddle point problems
On the existence
Mixed and Hybrid Finite Element Methods
Iterative regularization of a parameter identi
Inverse Problems in Partial Di
Inexact and preconditioned Uzawa algorithms for saddle point problems
Inverse Problems in Di
Convergence rate results for iterative methods for solving non-linear ill-posed problems
QMR: a quasi-minimal residual method for non-Hermitian linear systems
Preconditioned all-at-once methods for large
The numerical solution of a control problem governed by a phase
On Broyden's method for the regularization of nonlinear ill-posed prob- lems
A projection-regularized Newton method for nonlinear ill-posed problems with application to parameter identi cation problems with nite element discretization
Sur le controle parallele des systemes distribues
Solution of sparse inde
Numerical approximation of partial di
Determination of a source term in the linear di
GMRES: a generalized minimal residual algorithm for solving non-symmetric linear systems
Control applications of reduced SQP methods
Partially reduced SQP methods for large-scale nonlinear optimization problems
Solving discretized optimization problems by partially reduced SQP methods
Global convergence of trust-region interior-point algorithms for in nite-dimensional nonconvex minimization subject to pointwise bounds
Analysis of iterative methods for saddle point problems: a uni
--TR | parameter identification;iterative regularization;galerkin methods;sequential quadratic programming;indefinite systems |
589719 | Mappings for conflict-free access of paths in bidimensional arrays, circular lists, and complete trees. | Since the divergence between the processor speed and the memory access rate is progressively increasing, an efficient partition of the main memory into multibanks is useful to improve the overall system performance. The effectiveness of the multibank partition can be degraded by memory conflicts, that occur when there are many references to the same memory bank while accessing the same memory pattern. Therefore, mapping schemes are needed to distribute data in such a way that data can be retrieved via regular patterns without conflicts. In this paper, the problem of conflict-free access of arbitrary paths in bidimensional arrays, circular lists and complete trees is considered for the first time and reduced to variants of graph-coloring problems. Balanced and fast mappings are proposed which require an optimal number of colors (i.e., memory banks). The solution for bidimensional arrays is based on a particular Latin Square. The functions that map an array node or a circular list node to a memory bank can be calculated in constant time. As for complete trees, the mapping of a tree node to a memory bank takes time that grows logarithmically with the number of nodes of the tree. The problem solved here has further application in minimizing the number of frequencies assigned to the stations of a wireless network so as to avoid interference. | Introduction
In recent years, the traditional divergence between the processor speed and the memory access rate is
progressively increasing. Thus, an efficient organization of the main memory is important to achieve
high-speed computations. For this purpose, the main memory can be equipped with cache memories
which have about the same cycle time as the processors - or can be partitioned into multibanks.
Since the cost of the cache memory is high and its size is limited, the multibank partition has mostly
been adopted, especially in shared-memory multiprocessors [3]. However, the effectiveness of such a
memory partition can be limited by memory conflicts, that occur when there are many references to
the same memory bank while accessing the same memory pattern. To exploit to the fullest extent the
performance of the multibank partition, mapping schemes can be employed that avoid or minimize the
memory conflicts [15]. Since it is hard to find universal mappings - mappings that minimize conflicts
for arbitrary memory access patterns - several specialized mappings, designed for accessing regular
patterns in specific data structures, have been proposed in the literature (see [12, 2] for a complete list
of references).
In particular, for bidimensional arrays, Budnik and Kuck [7], Balakrishnan et al. [4], Kim and
Prasanna [12], and Das and Sarkar [8] studied mappings that provide conflict-free access to rows,
columns, positive and negative diagonals, subarrays, and distributed subarrays. The techniques used
range from Latin squares to Perfect Latin squares, from linear mappings to quasi-groups [11]. Subse-
quently, mappings for other data structures like complete trees and binomial trees have been devised.
In particular, mappings that provide conflict-free access to complete subtrees, root-to-leaves paths, sub-
levels, and composite patterns obtained by their combination, have been investigated in [8, 9, 1, 10, 14].
The mapping schemes proposed in those papers are optimal, i.e., they use as few memory modules as
possible; balanced, i.e., the nodes of data structures are distributed as evenly as possible among the
banks; fast, i.e., the bank address to which a node is assigned is computed quickly with no knowledge
of the entire structure mapping; and flexible, i.e., they can be used for templates of different size.
In the present paper, optimal, balanced and fast mappings are designed for conflict-free access of
paths in bidimensional arrays, circular lists, and complete trees. With respect to the above mentioned
papers, paths in bidimensional arrays and circular lists are dealt with for the first time. Moreover, access
to any (not only to root-to-leaves) paths in complete trees is provided. The remainder of this paper
is organized as follows. In Section 2, the conflict-free access problem is formally stated. In Section 3,
the problem of accessing paths in bidimensional arrays is solved. The proposed solution is a variant
of a graph-coloring, which requires an optimal number of colors and is achieved using a combinatorial
object similar to a Latin Square. As a byproduct, the memory bank to which an array node is assigned
is computed in constant time. In Section 4, the problem of accessing paths in circular lists is optimally
solved and the function that maps a circular list node to a memory bank can be calculated in constant
time. In Section 5, the same problem on complete trees is also optimally solved via a variant of a graph-coloring
problem. The time needed to assign a tree node to a memory bank grows logarithmically with
the number of nodes of the tree. Conclusions are offered in Section 6.
Conflict-Free Access
When storing a data structure D, represented in general by a graph, on a memory system consisting
of N memory banks, a desirable issue is to map any subset of N arbitrary nodes of D to all the N
different banks. This problem can be viewed as a coloring problem where the distribution of nodes of
D among the banks is done by coloring the nodes with a color from the set f0;
it is hard to solve the problem in general, access of regular patterns, called templates, in special data
structures - like bidimensional arrays, circular lists, and complete trees - are considered hereafter.
A template T is a connected subgraph of D. The occurrences fT of T in D are the
template instances. For example, if D is a complete binary tree, then a path of length k can be a
template, and all the paths of length k in D are the template instances.
After coloring D, a conflict occurs if two nodes of a template instance are assigned to the same
memory bank, i.e., they get the same color. An access to a template instance T i results in c conflicts if
belong to the same memory bank.
Given a memory system with N banks and a template T , the goal is to find a memory mapping
that colors the nodes of D in such a way that the number of conflicts for accessing any
instance of T is minimal. In fact, the cost for T i colored according to U , CostU (D; defined as
the number of conflicts for accessing T i . The template instance of T with the highest cost determines
the overall cost of the mapping U . That is,
A mapping U is conflict-free for T if
Among desirable properties for a conflict-free mapping, a mapping should be balanced, fast, and
optimal. A mapping U is termed balanced if it evenly distributes the nodes of the data structure among
the N memory banks. For a balanced mapping, the memory load is almost the same in all the banks.
A mapping U will be called fast if the color of each node can be computed quickly (possibly in constant
time) without knowledge of the coloring of the entire data structure. Among all possible conflict-free
mappings for a given template of a data structure, the more interesting ones are those that use the
minimum possible number of memory banks. These mappings are called optimal. It is worth to note
that not only the template size but also the overlapping of template instances in the data structure
determine a lower bound on the number of memory banks necessary to guarantee a conflict-free access
scheme. This fact will be more convincing by the argument below for accessing paths in D.
E) be the graph representing the data structure D. The template P k is a path of
length k in D. The template instance P k [x; y] is the path of length k between two vertices x and y in
V , that is, the sequence of vertices such that (v h ; v h+1
The conflicts can be eliminated on P k [x; y] if are assigned to all different memory
banks. The conflict-free access to P k can be reduced to a classical coloring problem on the associated
graph GDP k
obtained as follows. The vertex set of GDP k
is the same as the vertex set of GD , while
the edge (r; s) belongs to the edge set of GDP k
iff the distance d rs between the vertices r and s in GD
satisfies d rs - k, where the distance is the length of the shortest path between r and s. Now, colors
must be assigned to the vertices of GDP k
so that every pair of vertices connected by an edge is assigned
a couple of different colors and the minimum number of colors is used. Hence, the role of maximum
clique in GDP k
is apparent for deriving lower bounds on the conflict-free access on paths. A clique K
for GDP k
is a subset of the vertices of GDP k
such that for each pair of vertices in K there is an edge.
By well-known graph theoretical results, a clique of size n in the associated graph GDP k
implies that at
least n different colors are needed to color GDP k
. In other words, the size of the largest clique in GDP k
is a lower bound for the number of memory banks required to access paths of length k in D without
conflicts.
On the other hand, the conflict-free access to P k on GD is equivalent to color the nodes of GD in
such a way that any two nodes which are at distance k or less apart have assigned different colors.
Unfortunately, this latter coloring problem is NP-complete [13] for general graphs. This justifies the
investigation either for good heuristics for general graphs or optimal algorithms for special classes of
graphs. In the next three sections, optimal mappings for bidimensional arrays, circular lists and complete
binary trees will be derived for conflict-free accessing P k .
Accessing Paths in Bidimensional Arrays
Let a bidimensional array A be the data structure D to be mapped into the multibank memory system.
An array r \Theta c has r rows and c columns, indexed respectively from 0 to r \Gamma 1 (from top to bottom)
and from 0 to c \Gamma 1 (from left to right), with r and c both greater than 1.
The graph E) representing A is a mesh, whose vertices correspond to the elements of
A and whose arcs correspond to any pair of adjacent elements of A on the same row or on the same
column. For the sake of simplicity, A will be used instead of GA since there is no ambiguity. Thus, a
generic node x of A will be denoted by its row index and j is its column index.
least
l (k+1) 2m
memory banks are required for conflict-free accessing P k in A.
Proof Consider a generic node of A, and its opposite node at distance k on the same
column, i.e., All the nodes of A at distance k or less from both x and y are mutually at
distance k or less, as shown in Figure 1. Therefore, in the associated graph GAP k , they form a clique,
and they must be assigned to different colors. In details, such a clique, denoted as KA (x; k), is defined
as follows:
KA
ni
l km
l km
l km
l km
l kmo
Summing up over t, the size of the clique results to be
''- k-
Hence, at least
l (k+1) 2m
colors are required. 2
(b)
x
x
y
y
Figure
1: A subset KA (x; of nodes of A that forms a clique in GAP k
Below, a conflict-free mapping is given to color all the nodes of an array A using as few colors as in
Lemma 1. Therefore, the mapping is optimal. From now on, the color assigned to node x is denoted
by fl(x).
Algorithm Array-Coloring (A; k);
l (k+1) 2m
and
even
if k is odd
ffl Assign to each node x = (i; A the color
Intuitively, the above algorithm first covers A with a tessellation of basic sub-arrays of size M \Theta M .
Each basic sub-array S is colored in a Latin Square fashion as follows:
ffl the colors in the first row of S appear from left-to-right in the sequence 0;
ffl the color sequence for a generic row is obtained from the sequence at the previous row by a \Delta
left-cyclic shift.
For the coloring of A, decomposed into 6 basic sub-arrays of size M \Theta M , is illustrated in
Figure
2.
Theorem 1 The Array-Coloring mapping is optimal, fast, and balanced.
Proof To prove optimality, it must be shown that the mapping is conflict-free and that the
minimum number of colors is used.
Figure
2: An array A of size 16 \Theta 24 with a tessellation of 6 sub-arrays of size 8 \Theta 8 colored by the
Array-Coloring algorithm to conflict-free access P 3 .
Consider a generic node x = (g; f) of A and the associated clique KA (x; k), defined in Lemma 1. In
order to prove that the mapping is conflict-free, one only needs to show that all the nodes of KA (x; k),
which are mutually at distance no more than k, are assigned by the Array-Coloring algorithm to different
colors. Formally, consider an arbitrary pair of nodes belonging to KA (x; k),
such that 0, the roles of w and z could be swapped). Then the mapping is
conflict-free if the Array-Coloring algorithm guarantees that the colors fl(w) and fl(z) are different.
Moreover, let oe(w; z) be the difference between
the two colors assigned to w and z. Then, the mapping is conflict-free if the following two conditions
simultaneously hold:
(1)
In order to show that the conditions in (1) hold for any pair of nodes of KA (x; k), the two cases k
even and k odd must be distinguished.
When k is even, one has that
l (k+1) 2m
that oe(w; z)
the congruence oe(w; z) 6j 0 mod M is equivalent to oe(w; z) 6= 0 and oe(w; z) 6= M .
Clearly, oe(w; which is verified only if either z = w or j' \Gamma jj is a
multiple of k + 1. But, since j' \Gamma jj - k implies oe(w; z) 6= 0, no two distinct nodes of KA (x; can have
the same color.
Thus, it remains to prove that oe(w; z) 6= M . Assume by contradiction that oe(w;
Therefore, three cases may occur:
l M
In case (i), oe(w;
which contradicts the fact that jj \Gamma 'j - k.
In case (ii), oe(w; z) can be equal to M if and only if
is,
. Thus, in case (ii), for any pair of nodes z and w of
KA (x; which do not satisfy the first condition in (1), it results that is equal to a positive integer
and precisely,
But this violates the second condition in (1) because (i
Finally, in case (iii), oe(w; if and only if jj \Gamma
1). That is, for any pair
of nodes z and w of KA (x; not satisfying the first condition in (1), it yields precisely,
But again this violates the second condition in (1) because the distance between w and z is (i
In conclusion, for k even, any two nodes whose colors differ exactly by M are k
relative positions are depicted in Figure 3(a).
When k is odd, it follows that
l (k+1) 2m
that oe(w; z)
equivalent to oe(w; z) 6= 0 and oe(w; z) 6= M .
Clearly, oe(w; which is verified only if either
or is a multiple of k. Hence, two distinct nodes of KA (x; which have the same
color are at distance (i
It remains to prove that oe(w; z) 6= M . As before, three cases may occur:
l M
Note that
2 and
l M
Repeating the same reasoning done for k even, one can show again that any two nodes whose colors
differ by M are k apart. Their relative positions are illustrated in Figure 3(b).
So, the Array-Coloring Algorithm is conflict-free. Moreover, since it uses the minimum number of
colors, the proposed mapping is optimal.
(b)
(a)
Figure
3: Relative positions in A of two nodes which are assigned to the same color: (a) k even, (b) k
odd.
It is easy to see that the time required to color all the nodes of an array is O(n). Moreover,
to color only a single node x = (i; j) of the tree requires only O(1) time, since
and hence the mapping is fast.
In order to prove that the mapping is balanced, observe that each color appears once in each sub-row
of size M . Hence, the number m of nodes with the same color verifies rb c
e: 2
Observe that the Array-Coloring Algorithm guarantees conflict-free access to some paths longer than
k. Specifically, it is possible to access without conflicts any horizontal path of length M and any vertical
path of length
g.c.d.(M;\Delta) because L is the minimum integer such that Finally,
since the distance between two consecutive nodes on the same diagonal of A is 2, any b k
consecutive
elements on a diagonal can be accessed with no conflicts.
Accessing Paths in Circular Lists
Let a circular list C be the data structure D to be mapped into the multibank memory system. A
circular list of n nodes, indexed consecutively from 0 to n \Gamma 1, is a sequence of n nodes such that node
i is connected to both nodes (i \Gamma 1) mod n and (i
The graph E) representing C is a ring, whose vertices correspond to the elements of C
and whose arcs correspond to any pair of adjacent elements of C. For the sake of simplicity, C will be
used instead of GC since there is no ambiguity.
At least M memory banks are required for conflict-free accessing P k in C.
Proof For conflict-free accessing P k in C two nodes with the same color must be at distance at
least k + 1. When all the nodes are mutually at distance less than k and must all be colored
with different colors. When each color may appear at most
times. Therefore,
Figure
4: Conflict-free access to P 4 in a circular list C of 13 nodes colored by the Circular-List-Coloring
algorithm with 7.
at least
\Upsilon colors are needed. Observed that
follows that at
least
memory banks are required. 2
Below, an optimal conflict-free mapping is provided to color all the nodes of a circular list C using
as few colors as in Lemma 2. As before, the color assigned to node x is denoted by fl(x).
Algorithm Circular-List-Coloring (C; k);
ffl Assign to node x 2 C, the color
Note that a linear (that is, non circular) list L can be optimally colored to conflict-free access P k with
which matches the trivial lower bound given by the number of nodes in P k . In fact,
L can be optimally colored by a naive algorithm which assigns to node x the color
Such a naive algorithm does not work for circular lists. For example, consider the circular list C of
nodes, shown in Figure 4, to be colored to access P 4 . Applying the naive algorithm with M
only the first 10 nodes can be feasibly colored with 5 colors, but 3 additional colors are then required
for feasibly coloring the last 3 nodes, for a total of 8 colors. In contrast, the optimal Circular-List-
(b)
(a)
Figure
5: A circular list C of 17 nodes colored to conflict-free access P 3 according to: (a) the Circular-
Coloring algorithm requires 7 colors only. Moreover, it is worth to point out that the naive algorithm
does not always work for circular lists even when applied with M
. For
instance, for 5. Applying the naive algorithm with M
to this instance, 15 nodes can be colored using 5 colors, but 2 additional colors are needed for feasibly
coloring the last 2 nodes for a total of 7 colors (as shown in Figure 5(b)). Instead, the optimal coloring
provided by the Circular-List-Coloring algorithm uses only 5 colors, as shown in Figure 5(a). Indeed,
the naive algorithm always produces a feasible (although not necessarily optimal) coloring if applied
using
Theorem 2 The Circular-List-Coloring mapping is optimal, fast, and balanced.
Proof To prove optimality, two cases may be distinguished. If
and the Circular-List-Coloring algorithm reuses the same color at distance M . Hence, no
conflict arises. If n 6j 2. Two nodes get the same color only if they
are at distances M or M \Gamma 1, which are both greater than or equal to k+ 1. Hence, as before, no conflict
arises. Since the algorithm uses as few colors as possible, the mapping is optimal. It is also fast since
each node is colored in constant time. Finally, each color is assigned to exactly n
nodes when n is a
multiple of M , and no more than
l min(n;')
l max(n\Gamma';0)
nodes are colored with the same color in
all the other cases. 2
It is interesting to note at this point that, given a circular list of n nodes, the minimum number
of colors required to conflict-free access P k satisfies the following properties (see Figure 6):
ffl Up to results, i.e. all the nodes must have different colors. Indeed,
all of them are mutually at distance no more than k and, therefore, they form a clique on the
graph
depends on both n and k, and, for a fixed k, is not a monotone
M(n,
Figure
The number of colors M(n; 6) required to conflict-free access P 6 when n ranges between 1
and 58.
function of n. In contrast, for arrays and trees (as will be proved in the next section), M depends
only on k and is monotone.
Accessing Paths in Complete Trees
Let a rooted complete binary tree B be the data structure to be mapped into the multibank memory
system. The level of node x 2 B is defined as the number of edges on the path from x to the root,
which is at level 0. The maximum level of the nodes of B is the height of B. Let LevB (i) be the set of
all nodes of B at level i - 0.
A complete binary tree of height H is a rooted tree B in which all the leaves are at the same level
and each internal node has exactly 2 children. Thus, LevB (i) contains 2 i nodes. The h-th ancestor of
the node (i; j) is the node (i
its children are the nodes (i
in the left-to-right order.
From now on, the generic node x, which is the j-th node of LevB (i), with counting from left
to right, will be denoted by Therefore, the generic path instance P k [x; y] will be denoted by
Lemma 3 At least
memory banks are required to conflict-free access P k in B.
Proof Consider a generic node x = (i; j). All the 2 b k
nodes in the subtree S of height b k
rooted at the b k
c-th ancestor of x are mutually at distance not greater than k.
In addition, consider the d k
, ancestors of x, on the path I of length d k
2 e from
the b k
c-th ancestor of x up to the k-th ancestor of x. All these nodes are at distance not greater than
k from node x, and together with the nodes of S they are at mutual distance not greater than k.
x
(a)
(b)
Figure
7: A subset KB (k) of nodes of B that forms a clique in
Moreover, for
nodes in the complete subtree of height
rooted at the - j 's child which does not belong to I. Such nodes are at distance
not greater than k from x. Furthermore, these nodes, along with the nodes of S and I, are all together
at mutual distance not greater than k.
Hence, in the associated graph GDP k
there is at least a clique of size
From that, the claim easily follows. Figure 7 shows a subset KB (k) of nodes of B which are at pairwise
distance not greater than k, for 4, and hence forms a clique in the associated graph GBP kAn optimal conflict-free mapping to color a complete binary tree B acts as follows.
A basic subtree KB (k) defined as in the proof of Lemma 3 is identified and colored. Such a tree is
then overlaid to B in such a way that the uppermost
levels of B coincide with the lowermost
levels of KB (k). Then, the complete coloring of B is produced level by level by assigning to each node
the same color as an already colored node.
Formally, for a given k, define the binary tree KB (k) as follows:
ffl KB (k) has a leftmost path of k nodes.
ffl the root of KB (k) has only the left child;
ffl a complete subtree of height is rooted at the right child of the node at level i on the leftmost
path of KB (k).
(a) (b)6 7 8 934
Figure
8: Coloring of B for conflict-free accessing: (a) P 3 , (b) P 4 . (Both KB (3) and KB (4) are depicted
by dash splines.)
The
nodes of KB (k) must be colored with 2 b k
different colors. Thus,
the uppermost
levels of B are already colored.
For the sake of simplicity, to color the remaining part of B, the levels are counted starting from
the root of KB (k). That is, the level of the root of B will be renumbered as level
1. Now, fixed
the algorithm to color B acts as follows.
Algorithm Binary-Tree-Coloring (B; k);
ffl Color KB (k) with M colors;
ffl Visit the tree B in breadth first search, and for each node x of B, with
mod 2;
- Assign to x the same color as that of the node y
and
Examples of colorings to conflict-free access P 3 and P 4 are illustrated in Figure 8.
x
y
Figure
9: For inherits the same color as node y
Theorem 3 The Binary-Tree-Coloring mapping is optimal, fast and balanced.
Proof. To prove that the mapping is optimal, it must be shown that it is conflict-free and it uses
as few colors as those given by Lemma 3. First, observe that the 2 b k
c leaves of a subtree of height
are at mutual distance not greater than k, and therefore they must be colored with all different
colors. Thus, let each level of B be partitioned (starting from the leftmost node) into consecutive
blocks of size 2 b kc . The block b(i; w), with w - 0, at level i of B consists of the 2 b kc consecutive
nodes (i; w2 b k
which must all be assigned to a different color.
Consider the node x = (i; j) to be colored. The node x = (i; belongs to the block b
and it appears in the (- position inside the block. Consider the leftmost node z of b x , where
. Then, a generalization KB (z; of KB (k) can be defined depending on z.
KB (z; includes the following nodes of B:
ffl the nodes on the path \Gamma of length k from the father of z up to the (k 1)-th ancestor of z;
ffl for
the nodes of the complete binary tree S q of height k \Gamma q rooted at the child,
which does not belong to \Gamma, of the q-th ancestor of z;
ffl the nodes of the complete binary tree S of height
rooted at the
-th ancestor of z.
It is crucial to note that all the following nodes are at distance k + 1 from all the nodes in b x :
(i) the root of KB (z; k),
(ii) the leaves of S q , with
(iii) the leaves of S, which are not parents of any node in b x .
The nodes of b
are colored from left to right copying the same colors used in the
nodes of KB (z; specified in (i), (ii), and (iii) above, and considered by increasing level and from left
to right, as illustrated in Figures 10 and 11 for k even and odd, respectively. In particular,
is assigned to the same color as the root of KB (z; k), which is the
ancestor of x;
2, the 2 k\Gammaq nodes of b x , (i; b j
are assigned to the same colors as the leaves of the tree S q .
Observe that the number of nodes colored with the two steps above is 1
When k is odd, this is enough to color the entire block since 2 d k
c . In fact, the
set of nodes of KB (z; specified in (iii) above is empty for k odd. In contrast, when k is even, only
the first half of the block has been colored since 2 d k
to color the second half of
the block, one further step is required, which uses the colors of the nodes of KB (z; specified in (iii)
above:
ffl The rightmost 2 b k
nodes of b x are assigned to the same colors as the rightmost (resp., leftmost)
leaves of the complete binary tree rooted at (
1)-th ancestor of z, depending on the
fact that the
-th ancestor of z is a left (resp., right) child of its father.
In order to prove that the mapping is conflict-free, an inductive reasoning on the level i of the tree
is followed. The basis for the induction is when the tree coincides with KB (k) and it is colored,
by definition, with all different colors. For i ? k, consider a generic node x = (i; j), its block b x and
its leftmost node z. By inductive hypothesis, all the nodes in the tree up to level are colored in a
conflict-free manner, but with color repetitions. In particular, the subtree KB (z; k) is conflict-free and
since its nodes are mutually at distance at most k they must have been assigned to all different colors.
The algorithm colors b x copying the colors of some nodes in KB (z; k), specified in (i), (ii), and (iii),
which are exactly at distance k + 1 from the nodes of b x . Therefore, there are no color repetitions in b x
and no conflict can arise. Note that nodes in different blocks at level i may inherit the same color, but
since any two nodes in different blocks are at distance at least k conflict can arise. Therefore, all
the nodes in the tree up to level i are colored in a conflict-free manner.
Finally, since the tree is colored with the colors of KB (k), whose number equals the lower bound of
Lemma 3, the tree-coloring mapping is optimal.
It is easy to see that the time required to color all the n nodes of a tree is O(n). However, to color
only a single node x of the tree requires only O(log n) time since, in the worst case, all the nodes along
a path from x up to the root must have been colored.
One can readily see that, if the height H of the tree B is a multiple of k, then the nodes of B can be
partitioned into
subsets, each of which induces a copy of KB (k). Therefore, each color
is used m times, and the mapping is balanced. 2
z
G
Figure
10: The generalization KB (z; 6) of KB (6) for the node z. The root of KB (z; 6), the leaves of the
and the rightmost leaves of S are used to color the nodes in the block b z .
G
z
Figure
11: The generalization KB (z; 5) of KB (5) for the node z. The root of KB (z; 5) and the leaves of
the subtrees S 4 and S 5 are used to color the nodes in the block b z .
The results shown for binary trees can be extended to a q-ary tree Q, with q - 2.
Corollary 1 At least
memory modules are required to conflict-free access P k in a q-ary tree Q. 2
Similarly to the binary case, for a given k, define a q-ary tree K q
ffl K q
Q (k) has a leftmost path of k
ffl the root of K q
Q has only the leftmost child;
ffl a complete subtree of height is rooted at the q \Gamma 1 rightmost children of the node at level i
on the leftmost path of K q
Such a K q
Q (k) is then overlaid to Q in such a way that the uppermost
levels of Q coincide with
the lowermost
levels of K q
Then, the complete coloring of Q is produced level by level by
assigning to each node the same color as an already colored node.
For the sake of simplicity, to color the remaining part of Q, the levels are again counted starting
from the root of K q
That is, the level of the root of Q will be renumbered as level
1. Now,
the algorithm to color Q is the following:
Algorithm q-ary-Tree-Coloring (Q; k);
ffl Color K q
ffl Visit the tree Q in breadth first search, and for each node x of Q, with
do:
mod q;
- Assign to x the same color as that of the node y
and
By a reasoning similar to that employed for complete binary trees, the optimality of the q-ary-Tree-
Coloring Algorithm easily follows.
6 Conclusions
In this paper, the problem of conflict-free accessing arbitrary paths P k in particular data structures,
such as bidimensional arrays, circular lists and complete trees, has been considered for the first time
and reduced to variants of graph-coloring problems. Optimal, fast and balanced mappings have been
proposed. Indeed, the memory bank to which a node is assigned is computed in constant time for arrays
and circular lists, while it is computed in logarithmic time for complete trees. However, it remains as
an open question whether a tree node can be assigned to a memory bank in constant time.
On the other hand, the conflict-free access to P k on an arbitrary data structure D is NP-complete
[13], and this justifies the investigation of good heuristics. This problem is equivalent to the classical
node coloring problem in the associated graph GDP k
. Therefore, it can be solved by the most effective
coloring heuristic known so far, that is, the saturation-degree heuristic [6], which works as follows. Let
N(x) be the neighborhood of node x in the associated graph GDP k
. At each iteration, the saturation-
degree heuristic selects the node x to be colored as one with the largest number of different colors
already assigned in N(x). Ties between nodes are broken by preferring the node x with the largest
number of colored nodes in N(x). Once selected, node x is assigned the lowest color not yet assigned
in N(x).
As experimentally proved in [5], the saturation-degree heuristic is especially effective when the
minimum number of colors is given by the size of the largest clique K of GDP k
. Therefore, it should
work efficiently also for the conflict-free access problem, and, in particular, for d-dimensional arrays
as well as for generic, i.e. not necessarily complete, trees. Indeed, it is expected in such cases that
the minimum number of required memory banks be equal to the lower bound given by the size of the
largest clique K of GDP k
, as happened for bidimensional arrays and complete trees. Unfortunately,
the resulting coloring is not guaranteed to be optimal, fast or balanced. Moreover, it is still an open
question to determine whether the problem of conflict-free accessing paths on d-dimensional arrays and
generic trees is NP-complete.
Finally, in a more practical perspective, the number of memory banks available could be fixed to a
constant -, depending on the memory configuration. Then, if the number of memory modules M(k)
required for a given P k is larger than -, no conflict-free access is possible. However, assume that P k 0 is
the longest path that can be accessed without conflicts using - memory banks, i.e. M(k 0 ) -. Then,
accessing P k , no more than d k
conflicts may arise. Hence, the proposed mappings are scalable.
Acknowledgement
The authors are grateful to Richard Tan for his helpful comments, and to Thomas McCormick for having provided the
reference [13].
--R
"Toward a Universal Mapping Algorithm for Accessing Trees in Parallel Memory Systems"
"Multiple Template Access of Trees in Parallel Memory Systems"
"Accounting for Memory Bank Contention and Delay in High-Bandwidth Multiprocessors"
"On Array Storage for Conflict-Free Memory Access for Parallel Processors"
"Assigning Codes in Wireless Networks: Bounds and Scaling Properties"
"The Organization and Use of Parallel Memories"
"Conflict-Free Data Access of Arrays and Trees in Parallel Memory Systems"
"Parallel Priority Queues in Distributed Memory Hypercubes"
"Load Balanced Mapping of Data Structures in Parallel Memory Modules for Fast and Conflict-Free Templates Access"
New York
"Optimal Approximation of Sparse Hessians and its Equivalence to a Graph Coloring Problem"
"Conflict-Free Template Access in k-ary and Binomial Trees"
"Theoretical Limitations on the Efficient Use of Parallel Memories"
--TR
Conflict-free template access in <italic>k</italic>-ary and binomial trees
Accounting for Memory Bank Contention and Delay in High-Bandwidth Multiprocessors
Multiple templates access of trees in parallel memory systems
Assigning codes in wireless networks
New methods to color the vertices of a graph
Latin Squares for Parallel Array Access
Optimal and Load Balanced Mapping of Parallel Priority Queues in Hypercubes
Load Balanced Mapping of Data Structures in Parallel Memory Modules for Fast and Conflict-Free Templates Access
Toward a Universal Mapping Algorithm for Accessing Trees in Parallel Memory Systems
--CTR
Alan A. Bertossi , Cristina M. Pinotti , Richard B. Tan, Channel Assignment with Separation for Interference Avoidance in Wireless Networks, IEEE Transactions on Parallel and Distributed Systems, v.14 n.3, p.222-235, March
Sajal K. Das , Irene Finocchi , Rossella Petreschi, Conflict-free star-access in parallel memory systems, Journal of Parallel and Distributed Computing, v.66 n.11, p.1431-1441, November 2006 | bidimensional array;conflict-free access;complete tree;path template;multibank memory system;mapping scheme;frequency assignment;circular list |
589754 | Compiler-optimized simulation of large-scale applications on high performance architectures. | In this paper, we propose and evaluate practical, automatic techniques that exploit compiler analysis to facilitate simulation of very large message-passing systems. We use compiler techniques and a compiler-synthesized static task graph model to identify the subset of the computations whose values have no significant effect on the performance of the program, and to generate symbolic estimates of the execution times of these computations. For programs with regular computation and communication patterns, this information allows us to avoid executing or simulating large portions of the computational code during the simulation. It also allows us to avoid performing some of the message data transfers, while still simulating the message performance in detail. We have used these techniques to integrate the MPI-Sim parallel simulator at UCLA with the Rice dHPF compiler infrastructure. We evaluate the accuracy and benefits of these techniques for three standard message-passing benchmarks on a wide range of problem and system sizes. The optimized simulator has errors of less than 16% compared with direct program measurement in all the cases we studied, and typically much smaller errors. Furthermore, it requires factors of 5 to 2000 less memory and up to a factor of 10 less time to execute than the original simulator. These dramatic savings allow us to simulate regular message-passing programs on systems and problem sizes 10 to 100 times larger than is possible with the original simulator, or other current state-of-the-art simulators. | Introduction
Predicting parallel application performance is an essential step in developing large applications on highly scalable
parallel architectures, in sizing the system configurations necessary for large problem sizes, or in analyzing
alternative architectures for such systems. Considerable research is being done on both analytical and simulation
models for performance prediction of complex, scalable systems. Analytical methods typically require custom
solutions for each problem and may not be tractable for complex interconnection networks or detailed modeling
scenarios; simulation models are likely to be the primary choices for general-purpose performance prediction. As
is well known, however, detailed simulations of large systems can be very computation-intensive and their long
execution times can be a significant deterrent to their widespread use.
The current generation of parallel program simulators use two techniques to reduce model execution times: direct
execution and parallel simulation. In direct execution, the simulator uses the available system resources to
directly execute portions of the program. Parallel simulation distributes the computational workload among
multiple processors, while using appropriate synchronization algorithms to ensure that execution of the model
produces the same result as if all events in the model were executed in their causal order. However, the current
state of the art is such that even using direct execution and parallel simulations, the simulation of large
applications designed for architectures with thousands of processors can run many orders of magnitude slower
than their physical counterparts.
In this paper, we propose, implement, and evaluate practical, automatic optimizations that exploit compiler
support to enable efficient simulation of very large message-passing parallel programs. Our goal is to enable the
simulation of target systems with thousands of processors, and realistic problem sizes expected on such large
platforms. The key idea underlying our work is to use compiler analysis to isolate fragments of local
computations and message data whose values do not affect the performance of the program. For example,
computations that determine loop bounds, control flow, or message patterns and volumes all have an effect on
performance, whereas computations of many array values have no significant effect on performance. These
computations can be abstracted away while simulating the rest of the program in detail to predict the performance
characteristics of the application. Similarly, it is also possible to avoid performing data transfers for many
messages whose values do not affect performance, while simulating the performance of the messages in detail.
There are two major aspects to the compiler analysis required to accomplish this optimization: identifying the
values within the program that could affect program performance, and isolating the computations and
communications that determine these values. To perform the first step, we use a compiler-synthesized static task
graph model [4, 5], an abstract representation that identifies the sequential computations (tasks), the parallel
structure of the program (task scheduling, precedences, and explicit communication), and the control-flow that
determines the parallel structure. The symbolic expressions in the task graph for control flow conditions,
communication patterns and volumes, and scaling expressions for sequential task execution times directly capture
all the values (i.e., the references within those expressions) that impact program performance. The second step
uses a compiler technique called program slicing [21] to identify the portions of the computation that determine
these values. The compiler can then emit simplified MPI code that contains exactly the computations that must be
actually executed during the simulation (in addition to the communication), while the remaining code fragments
are abstracted away. The compiler also needs to estimate the execution time of the abstracted code by using
parameterized by direct measurement. In addition to reducing simulation times, these
optimizations can dramatically reduce the memory requirements for the simulation (if major program arrays are
only referenced in redundant computations, they do not have to be allocated at all during the simulation). The
memory savings can potentially allow much larger problem sizes and architectures to be studied than would
otherwise be feasible.
In order to demonstrate the impact of these optimizations, we have combined the MPI-Sim parallel simulator [6,
25-27] with the dHPF compiler infrastructure [2], to develop a program simulation framework that incorporates
the new techniques described above. The original MPI-Sim simulator used both direct execution and parallel
simulation to achieve substantial reductions in the simulation time of parallel programs. dHPF, in normal usage,
compiles an HPF program to MPI (or to a variety of shared memory systems), and provides extensive parallel
program analysis capabilities. The integrated tool can allow us to evaluate the impact of the preceding
optimizations with existing MPI and HPF programs without requiring any changes to the source code. In
previous work, we modified the dHPF compiler to automatically synthesize the static task graph model and
symbolic task time estimates for MPI programs compiled from HPF source programs. 1 In this work, we use the
static task graph plus program slicing to perform the simulation optimizations described above. We have also
extended MPI-Sim to exploit the information from the compiler, and avoid executing significant portions of the
computational code. The hypothesis is that this will significantly reduce the memory and time requirements of the
simulation and therefore enable us to simulate much larger systems and problem sizes than were previously
possible.
We use a number of widely used benchmarks to evaluate the utility of the integrated framework: Sweep3D [1], a
benchmark; SP from the NPB benchmark suite [8] and Tomcatv, a SPEC92 benchmark. The
simulation models of each application were validated against measurements over a range of problem sizes and
numbers of processors. The errors in the predicted execution times, compared with direct measurement, were at
1 In the future, we plan to synthesize this information for existing MPI codes as well. The dHPF infrastructure supports very general
computation partitioning, communication analysis, and symbolic analysis capabilities that make this feasible for a wide class of MPI
programs.
most 16% in all cases we studied, and often were substantially less. The validation has been done for the
distributed memory IBM SP architecture, as well as the shared memory SGI Origin 2000 (note, that MPI-Sim
simulates the MPI communication, not the communications via shared memory). The optimizations had a
significant impact on the performance of the simulators: the total memory usage of the simulator using the
compiler synthesized model was a factor of 5 to 2000 less than the original simulator, and the simulation time was
typically lower by a factor of 5-10. These dramatic savings allow us to simulate systems or problem sizes that are
10-100 times larger than is possible with the original simulator, without significant reductions in the accuracy of
the simulator. For example, we were successful in simulating the execution of a configuration of Sweep3D for a
target system with 10,000 processors! In many cases, the simulation time was faster than the original program.
The remainder of the paper proceeds as follows. Section 2 first describes the state of the art of parallel program
simulation, to set the stage for our work. Section 3 provides a brief overview of MPI-Sim and the static task
graph model. Section 4 describes the optimization strategy and the compiler and simulator extensions required to
implement the strategy. Section 5 describes our experimental results, and Section 6 presents our main
conclusions.
Related Work
Because analytical performance prediction can be intractable for complex applications, program simulations are
commonly used for such studies. It is well known that simulations of large systems tend to be slow. To improve
the simulators, direct-execution has been used [20, 26, 28]. Direct execution simulators make use of available
system resources to directly execute portions of the application code and simulate architectural features that are of
specific interest, or are unavailable. For example, simulators can be used to study various architectural
components such as the memory subsystem or the interconnection network. Specifically, if one is interested in
determining if a faster communication fabric for a network of workstations is of value for a given set of
applications, one can run the application on the currently available machines and only simulate the projected
network's behavior. The benefits of this direct-execution simulation are obvious: first, one can estimate the value
of the new hardware without the expense of purchasing it; second, one can do the simulation fast-there is no
need to simulate the workstation's behavior (for example down to the level of memory references) since that part
of the hardware is readily available.
Many of the early simulators were designed for sequential execution [9, 13, 14]. However, even with the use of
abstract models and direct execution, sequential program simulators tended to be slow with slowdown factors
ranging from 2 to 35 for each process in the simulated program [9]. Several recent efforts have been exploring
the use of parallel execution [10, 16, 17, 23, 24, 27, 28] to reduce the model execution times, with varying degrees
of success. In order to have multiple simulation processes and maintain accuracy, simulations use protocols to
synchronize the processes. One of the widely used protocols is the Quantum protocol, which lets the processes
compute for a given quantum before synchronizing them. In general, synchronous simulators that use the
quantum protocol must trade-off simulation accuracy with speed-frequent synchronizations slowdown the
simulation, but synchronizing less frequently introduces errors, by possibly executing statements out-of-order.
Both LAPSE [16, 17] and Parallel Proteus use some form of program analysis to increase the simulation window
beyond a fixed quantum. MPI-Sim uses parallel discrete event simulation with the conservative protocol [24, 27].
Supported protocols include the Null Message Protocol (NMP) [11], the Conditional Event Protocol (CEP) [12],
and a new protocol, which is a combination of the two [22]. As discussed in the next section, MPI-Sim exploits
the determinism present in the communication pattern of the application to reduce, and in many cases, completely
eliminate synchronization overheads.
Although simulation protocol optimizations have reduced simulation times, the resulting improvements are still
inadequate to simulate the very large problems that are of interest to high-end users. For instance, Sweep3D is a
kernel application of the ASCI benchmark suite released by the US Department of Energy. In its largest
configuration, it requires computations on a grid with one billion elements. The memory requirements and
execution time of such a configuration makes it impractical to simulate, even when running the simulations on
high performance computers with hundreds of processors.
To overcome this computational intractability, researchers have used abstract simulations, which avoid execution
of the computational code entirely [18, 19]. However, this leads to major limitations that make the approach
inapplicable to many real world applications. The main problem with abstracting away all of the code is that the
model is essentially independent of program control flow, even though the control flow may affect both the
communication pattern as well as the sequential task times. Also, the preceding solution requires significant user
modifications to the source program (in the form of a special input language) in order to express required
information about abstracted sequential tasks and communication patterns. This makes it difficult to apply such a
tool to existing programs written with widely used standards such as Message Passing Interface (MPI) or High
Performance Fortran (HPF).
3 Background and Goals
3.1 MPI-SIM: Parallel Simulation of MPI programs using Direct Execution
The starting point for our work is MPI-Sim, a direct-execution parallel simulator for performance prediction of
MPI programs. MPI-Sim simulates an MPI application running on a parallel system (referred to as the target
program and system respectively). The machine on which the simulator is executed (the host machine) may be
either a sequential or a parallel machine. In general, the number of processors in the host machine will be less than
the number of processors in the target architecture being simulated, so the simulator must support multi-threading.
The simulation kernel on each processor schedules the threads and ensures that events on host processors are
executed in their correct timestamp order. A target thread is simulated as follows. The local code is simulated by
directly executing it on the host processor. Communication commands are trapped by the simulator, which uses
an appropriate model to predict the execution time for the corresponding communication activity on the target
architecture.
supports most of the commonly used MPI communication routines, such as point-to-point and collective
communications. In the simulator, all collective communication functions are implemented in terms of point-to-
point communication functions, and all point-to-point communication functions are implemented using a set of
core non-blocking MPI functions.
In general, the host architecture will have fewer processors than the target machine (for sequential simulation, the
host machine has only one processor); this requires that the simulator provide the capability for multithreaded
execution. Since MPI programs execute as a collection of single threaded processes, it was necessary to provide a
capability for multithreaded execution of MPI programs in MPI-Sim. Further, memory and execution time
constraints of sequential simulation led to the development of parallel implementations of MPI-Sim. MPI-Sim
has been ported to multiple parallel architectures including a distributed memory IBM SP2 as well as a shared-memory
SGI Origin 2000.
The simulation kernel provides support for sequential and parallel execution of the simulator. Parallel execution is
supported via a set of conservative parallel simulation protocols [26], which typically work as follows: Each
2 In the future, we plan to synthesize this information for existing MPI codes as well. The dHPF infrastructure supports very general
computation partitioning, communication analysis, and symbolic analysis capabilities that make this feasible for a wide class of MPI
programs.
application process in the simulation is modeled by a Logical Processes (LP) 3 . Each LP can execute
independently, without synchronizing with other LPs, until it executes a wait operation (such as an MPI-Recv,
MPI-Barrier etc.); a synchronization protocol is used to decide when such an LP can proceed. We briefly
describe the default protocol used by MPI-SIM. Each LP in the model computes local quantities called Earliest
Output Time (EOT) and Earliest Input Time (EIT) [7]. The EOT represents the earliest future time at which the
LP will send a message to any other LP in the model; similarly the EIT represents a lower bound on the receive
timestamp of future messages that the LP may receive. Upon executing a wait statement, an LP can safely select
a matching message (if any) from its input buffer, that has a receive timestamp less than its EIT. Different
asynchronous protocols differ only in their method for computing EIT. Our implementation supports a variety of
such protocols as mentioned previously. The primary overhead in implementing parallel conservative protocols is
due to the communications to compute EIT and the blocking suffered by an LP that has not been able to advance
its EIT. We have suggested and implemented a number of optimizations to significantly reduce the frequency and
strength of synchronization in the parallel simulator thus reducing unnecessary blocking in its execution [26, 27].
Our optimizations were geared towards exploiting determinism in applications. For instance, consider an LP that
is blocked at a receive statement and its input buffer contains a single message. In general, the LP cannot proceed
by removing that message from the buffer as it might be possible that another message destined for this LP is in
transit, and that message has a lower timestamp. However, if the receive statement is known by the process to be
deterministic, it follows that there must exist a unique message that matches the receive statement. As soon as the
LP receives this message, it can proceed without the need for any synchronizations with other LPs in the model.
In the best case, if every receive statement in the model is known to be deterministic, no synchronization
messages will be generated in the model and the parallel simulation can be extremely efficient.
The preceding optimizations have two limitations: first, it works only with communications statements that are a
priori known to be deterministic. Second, the use of direct execution in the simulator implies that the memory and
computation requirements of the simulator are at least as large as that of the target application, which restricts the
target systems and application problem sizes that can be studied even using parallel host machines. The compiler-directed
optimizations discussed in the next section are primarily aimed at alleviating these restrictions.
3.2 The Static Task Graph Representation
As will be seen in the next section, the compiler analysis to be performed can be greatly facilitated by exploiting
an appropriate abstract representation for the parallel behavior of the program. As part of the POEMS project [3,
15], we have developed an abstract program representation called the static task graph (STG) that captures
extensive static information about a parallel program [5]. The STG is designed to be computed automatically by a
parallelizing compiler. It is a compact, symbolic representation of the parallel structure of a program,
independent of specific program input values or the number of processors. Each node of the STG represents a set
of possible parallel tasks, typically one per process, identified by a symbolic set of integer process identifiers. To
illustrate, the STG for the example MPI program is shown in Figure 1. The compute node for the loop nest
represents a set of tasks, one per process, denoted by the symbolic set of process ids }0
p . Each
node also includes markers describing the corresponding region of source code of the original program (for now,
each node must represent a contiguous region of code). Each edge of the graph represents a set of edges
connecting pairs of parallel tasks described by a symbolic integer mapping. For example, the communication edge
in the figure is labeled with a mapping indicating that each process p ( 1
sends to process
nodes fall into one of three categories: control-flow, computation and communication. Each computational
node includes a symbolic scaling function that captures how the number of loop iterations (if any) in the task
3 In general, an LP can be used to simulate multiple application processes.
scales as a function of arbitrary program variables. Each communication node includes additional symbolic
information describing the pattern and volume of communication.
Overall, the STG serves as a general, language- and architecture-independent representation of message-passing
programs. In previous work, we extended the dHPF compiler to synthesize static (and dynamic) task graphs for
MPI programs generated by the dHPF compiler from HPF source programs [4]. In the future, we will extract task
graphs directly from existing MPI codes. This compiler support is extremely valuable because it enables the
techniques developed in this paper to be applied fully automatically, i.e., without user intervention, for efficient
simulation of parallel programs.
Compiler-Supported Techniques for Efficient Large-Scale Simulation
This section begins by motivating the overall strategy we use to address the key restriction on simulation
scalability identified above, namely, the time and cost required for simulating the detailed computations of the
target program. We then describe more specifically how this strategy is accomplished.
4.1 Optimization Strategy and Challenges
Parallel program simulators used for performance evaluation execute or simulate the actual computations of the
target program for two purposes: (a) to determine the execution time of the computations, and (b) to determine the
impact of computational results on the performance of the program, due to artifacts like communication patterns,
loop bounds, and control-flow. For many parallel programs, however, a sophisticated compiler can extract
extensive information from the target program statically. In particular, we identify two types of relevant
information often available at compile-time:
1. The parallel structure of the program, including the sequential portions of the computation (tasks), the
mapping of tasks to threads, and the communication and synchronization patterns between threads.
2. Symbolic estimates for the execution time of isolated sequential portions of the computation.
If this information can be provided to the simulator directly, it may be possible to avoid executing substantial
portions of the computational code during simulation, and therefore reduce the execution time and memory
requirements of the simulation.
To illustrate this goal, consider the simple example MPI code fragment in Figure 1. The code performs a "shift"
communication operation on the array D, where every processor sends its boundary values to its left neighbor, and
then the code executes a simple computational loop nest. In this simple example, the communication pattern and
the number of iterations of the loop nest depend on the values of the block size per processor (b), the array size
(N), the number of processors (P), and the local processor identifier (myid). Therefore, the computation of these
values must be executed or simulated during the simulation. However, the communication pattern and loop
iteration counts do not depend on the values stored in the arrays A and D, which are computed and used in the
computational loop nest (or earlier). We refer to these latter values as redundant computations (from the point of
view of performance estimation). If we can estimate the performance of the computational loop nest analytically,
we could avoid simulating the code of this loop nest, while still simulating the communication behavior in detail.
We could achieve this optimization by using the compiler to generate the simplified code shown on the right in
the figure. In this code, we have replaced the loop nest with a call to a special simulator-provided delay function.
We have extended MPI-Sim to provide such a function, which simply forwards the simulation clock on the
double precision A(NMAX, 1
double precision
call mpi_comm_size(MPI_COMM_WORLD, P, ierr)
call mpi_comm_rank(MPI_COMM_WORLD, myid, ierr)
read(*, N)
if (myid .gt.
<SEND D(2:N-1, myid*b+1) to processor myid-1>
endif
if (myid .lt. P) then
<RECV D(2:N-1, (myid+1)*b+1) from processor myid+1>
endif
do
do
endif
integer, allocatable :: dummy_buf
call mpi_comm_size(MPI_COMM_WORLD, P, ierr)
call mpi_comm_rank(MPI_COMM_WORLD, myid, ierr)
call read_and_broadcast(w_1)
read(*, N)
allocate dummy_buf((N-2)*2)
if (myid .gt.
<SEND dummy_buf(:) to processor myid-1>
endif
if (myid .lt. P) then
<RECV dummy_buf(:) from processor myid+1>
endif
call delay((N-2) * (min(N,myid*b+b) -
Figure
1: Example to illustrate (a) a simple MPI program, (b) task-graph for MPI program, and
(c) simplified MPI program for efficient simulation.
(a) Original MPI Code (c) Simplified MPI Code
(b) Task Graph for
Original MPI Code
Task Pairs: {[p] - [q]:
DO
Compute
DO I
Control-flow edge
C Communication edge
Compute
Tasks:
simulation thread by a specified amount. The compiler estimates the cost of the loop nest in the form of a simple
scaling function shown as the argument to the delay call. This function describes how the computational cost
varies with the retained variables (b, N, P and myid), plus a parameter w 1 representing the cost of a single loop
iteration. We currently obtain the value of w 1 by direct measurement for one or a few selected problem sizes and
number of processors, and use the scaling function to compute the required delay value for other problem sizes
and number of processors. Note in the example that the compiler has avoided allocating the arrays A and D,
which significantly reduces the memory required to simulate the program.
As an additional optimization, if the compiler can prove that the data transferred in the message is also
"redundant", the simulator can also avoid performing an actual data transfer, although it will simulate the message
operation in detail. It can also avoid allocating any memory for the message buffer. This message optimization
can lead to further savings in simulation time and memory usage.
This paper develops automatic compiler-based techniques to perform the optimizations described above, and
evaluates the potential benefits of these techniques. In particular, our goal is to use the compiler-generated static
task graph (plus additional compiler analysis) to avoid simulating or executing substantial portions of the
computational code of the target program and sending unnecessary data. We use the task graph to identify the
computational tasks that are candidates for elimination, to compute the scaling expressions for those delay
functions, and (most importantly) to identify which values computed in the program have an impact on
performance. We use additional compiler analysis to distinguish the computations that compute such values, i.e.,
those that are not redundant as defined above.
More specifically, there are four major challenges we must address in achieving the above goals, of which the first
three have not been addressed in any previous system known to us:
a) We must transform the original parallel program into a simplified but legal MPI program that can be
simulated by MPI-Sim. The simplified program must include only the computation and communication code
that needs to be executed by the simulator. It must yield the same performance estimates as the original
program for total execution time (for each individual process), total communication and computation times, as
well as more detailed metrics of the communication behavior.
b) We must be able to abstract away as much of the local computation within each task as feasible and eliminate
as many data structures of the original program as possible, by isolating the redundant computations in the
program.
c) We must identify the messages whose contents do not directly affect the computation at the receiver, and
exploit this information to reduce simulation time and memory usage.
d) We must estimate the execution times of the abstracted computational tasks for a given program size and
number of processors. Accurate performance prediction for sequential code is a challenging problem that has
been widely studied in the literature. We use a fairly straightforward approach described in Section 4.5.
Refining this approach is part of our ongoing work in the POEMS project.
The following subsections describe the techniques we use to address these challenges, and their implementation in
dHPF and MPI-Sim. We first describe the basic process of using the task graph to generate the simplified MPI
program, then describe the compiler analysis needed to identify redundant computations, and finally discuss the
approach we use to estimate the performance of the eliminated code.
4.2 Translating the static task graph into a simplified MPI program
The STG directly identifies the local (sequential) computational tasks, control flow, and communication tasks and
patterns of the parallel program. By using the compiler-generated STG as the basis for our analysis, we can avoid
having to perform a complex, ad hoc analysis to identify these components.
Given this information, the first step is to identify contiguous regions of computational tasks and/or control-flow
in the STG that can be collapsed into a single condensed (or collapsed) task, such as the loop nest of Figure 1.
Note that this is simply a transformation of the STG for simplifying further analysis and does not directly imply
any changes to the parallel program itself. We refer to the task graph resulting from this transformation as the
condensed task graph. In later analysis, we can consider only a single computational task or a single collapsed
task at a time for deciding how to simplify the code (we refer to either as a single sequential task).
The criteria for collapsing tasks depend on the goals of the performance study. First, as a general rule, a collapsed
region must not include any branches that exit the region, i.e., there should be only a single exit at the end of the
region. Second, for the current work, a collapsed region must contain no communication tasks because we aim to
simulate communication precisely. Finally, deciding whether to collapse conditional branches involves a difficult
tradeoff: it is important to eliminate control-flow that references large arrays in order to achieve the savings in
memory and time we desire, but it is difficult to estimate the performance of code containing such control-flow.
We have found, however, that there are typically few branches that involve large arrays that do have a significant
impact on program performance. For example, one minor conditional branch in a loop nest of Sweep3D depends
on intermediate values of large 3D arrays. The impact of this branch on execution time is relatively negligible,
but detecting this fact, in general, can be difficult within the compiler because it may depend on expected problem
sizes and computation times. Therefore, there are two possible approaches we can take. The more precise
approach is to allow the user to specify through directives that specific branches can be eliminated and treated
analytically for program simulation. A simpler but more approximate approach is to eliminate any conditional
branches inside a collapsible loop nest, and rely on the statistical average execution time of each iteration to
provide a good basis for estimating total execution time of the loop nest. With either approach, we can use
profiling to estimate the branching probabilities of eliminated branches We have currently taken the second
approach, but the first one is not difficult to implement and could provide more precise performance estimates.
While condensing the task graph, we also compute a scaling expression for each collapsed task that describes how
the number of computational operations scales as a function of program variables. We introduce time variables
that represent the execution time of a sequence of statements in a single loop iteration (denoted w i for task i). The
approach we use to estimate the overall execution time of each sequential task is described in Section 4.5.
Based on the condensed task graph (and assuming for now that the compiler analysis of Section 4.3 is not
needed), we generate the simplified MPI program as follows. We retain any control-flow (loops and branches) of
the original MPI code that is retained in the condensed task graph, i.e., the control-flow that is not collapsed.
Second, we retain the communication code of the original program, in particular only the calls to the underlying
message-passing library. If a program array that is otherwise unused is referenced in any communication call, we
replace that array reference with a reference to a single dummy buffer used for all the communication. (Note that
without the message optimization described later in this section, the simulator must still perform the actual data
transfer between processes when simulating the message. The message optimization attempts to eliminate this
data transfer itself.) We use a buffer size that is the maximum of the message sizes of all communication calls in
the program and allocate the buffer statically or dynamically (potentially multiple times), depending on when the
required message sizes are known. Third, we replace the code sequence for each sequential task of the task graph
by a call to the MPI-Sim delay function, and pass in an argument describing the estimated execution time of the
task. We insert a sequence of calls to a runtime function, one per w i parameter, at the start of the program to read
in the value of the parameter from a file and broadcast it to all processors. Finally, we eliminate all the data
variables not referenced in the simplified program.
4.3 Program slicing for identifying redundant computations and data
The major challenge in performing the transformations mentioned earlier correctly and effectively is to identify
the redundant computations, i.e., the ones that can be safely eliminated. The solution we propose is to use
program slicing to retain those parts of the computational code (and the associated data structures) that affect the
program execution time. Given a variable referenced in some statement, program slicing finds and isolates a
subset of the program computation and data that can affect the value of that variable [21]. The subset has to be
conservative, limited by the precision of static program analysis, and therefore may not be minimal.
The key requirement in applying program slicing is to identify the variable values that affect the execution time of
the program. Once again, the compiler-generated static task graph captures this information directly and
precisely, allowing us to avoid a complicated and ad hoc analysis of the entire source code. In particular, the
values that affect performance are exactly the variable references that appear in the retained control-flow of the
condensed graph, in the scaling functions of the sequential tasks and communication events, and in the source and
destination expressions of the communication descriptors (or the communication calls themselves).
Once these values are identified, program slicing can be used to isolate the computations and data that affect those
variable values. Program slicing is essentially a reachability analysis on the dependence graph of the program,
including both data and control dependences. In particular, given a particular target reference, we use a
reachability analysis to identify the statements in the program that can affect the value of that reference through
some chain of dependences (i.e., through some feasible path in the dependence graph). Because this is a well-known
compiler technique, we omit the details here. A state-of-the-art algorithm for program slicing is described
in [21] and was used as the basis for our implementation. Applying this technique, however, requires that the
target reference be part of the program so that it appears in the program dependence graph computed by the
compiler. Some of the expressions of the static task graph are not directly derived from corresponding
expressions in the program, and therefore cannot be used as starting points for program slicing. For such
expressions, we introduce dummy procedure call statements at appropriate points in the target program, passing
those expressions as arguments, and then rebuild the program dependence graph. Now, these expressions can be
used as starting points for slicing. The dummy procedure calls can later be eliminated.
Obtaining the memory and time savings we desire requires full interprocedural program slicing, so that we
completely eliminate the uses of as many large arrays as possible. General interprocedural slicing is a challenging
but feasible compiler technique that is not currently available in the dHPF infrastructure. For now, we take limited
interprocedural side effects into account, in order to correctly handle calls to runtime library routines (including
communication calls and runtime routines of the dHPF compiler's runtime library). In particular, we assume that
these routines can modify any arguments passed by reference but cannot modify any global (i.e., common block)
variables of the MPI program. This is necessary and sufficient to support single-procedure benchmarks. We
expect to incorporate full interprocedural slicing in the future, to support continuing work in POEMS.
The final output of the slicing analysis is the set of computations that must be retained in the simplified MPI code,
while the remaining computations of the program (except for I/O statements and communication calls) can be
considered redundant. The code generation for the simplified MPI program (described in the previous section) is
modified slightly to use this information. For each sequential task, the non-redundant computations are retained
in the generated program, while the rest of the task is replaced with a single call to the simulator delay function.
For precise performance prediction, the simulator delay calls should not include the time for the retained
computations since those will be simulated (and their time accounted for) explicitly. The execution time
estimates computed above, however, apply to the entire task. In practice, we have found that the amount of non-redundant
code is very small for most tasks and therefore we do not adjust the execution time estimates to account
for this retained code.
4.4 Message optimization for simulating redundant messages
As noted in the previous Section, the data transferred in some of the messages may also be "redundant" from the
point of view of performance. If such cases can be identified, we can avoid performing the data transfers during
the simulation, potentially leading to additional time and memory savings. Although this is conceptually similar
to redundant computations, we discuss this "message optimization" separately because the mechanism for
achieving this optimization is somewhat different, as explained below.
First, the compiler can identify redundant messages as a direct result of the program slicing analysis described
above. In particular, the technique described above to account for interprocedural side-effects during slicing
directly identifies those message receive calls that receive redundant values. The corresponding message send
calls are already known to the compiler. The compiler provides this information to the simulator by flagging the
MPI calls that are redundant. The buffers used by these messages are not allocated in the resulting simplified MPI
program.
The actual message optimization is as follows. If a call is not flagged, MPI-Sim simulates the call in detail (by
sending the necessary protocol messages and predicting the end-to-end latency for the messages) and sends the
data to the receiving simulation thread, so that the actual data is available to the simulated application. However,
if the call is flagged by the compiler as "redundant", then MPI-Sim still simulates the call in the detail with
respect to the MPI communication protocol, but sends only an empty message to the receiving simulation thread.
Since "redundant" receives are also flagged, the receiver does not copy the data in the buffer. The messages need
to be present in the simulated application because they provide information about the synchronization in the
program. Although this optimization does not reduce the number of messages sent, the size of the messages is
reduced, and the memory used by the messages does not need to be allocated. This results in lower latencies
incurred by the messages that are sent between processors as well as smaller communication overheads due to
copying the data enclosed in the messages into/from the communication buffers. It also results in lower memory
usage by the simulator.
4.5 Estimating task execution times
The main approximation in our approach is to estimate sequential task execution times without direct execution.
Analytical prediction of sequential execution times is an extremely challenging problem, particularly with modern
superscalar processors and cache hierarchies. There are a variety of possible approaches with different tradeoffs
between cost, complexity, and accuracy.
The simplest approach, and the one we use in this paper, is to measure task times (specifically, the w i ) for one or a
few selected problem sizes and number of processors, and then use the symbolic scaling functions derived by the
compiler to estimate the delay values for other problem sizes and number of processors. Our current scaling
functions are symbolic functions of the number of loop iterations, and do not incorporate any dependence of
cache working sets on problem sizes. We believe extensions to the scaling function approach that capture the non-linear
behavior caused by the memory hierarchy are possible.
Performance
Estimates
Measured
task times
Simplified
MPI code
MPI code
with timers
Parallel
Program dHPF
Parallel
System
Figure
2: Compilation, parameter measurement and simulation for a parallel program.
Two alternatives to direct measurement of the task time parameters are (a) to use compiler support for estimating
sequential task execution times analytically, and (b) to use separate offline simulation of sequential task execution
times [15]. In both cases, the need for scaling functions remains, including the issues mentioned above, because it
is important to amortize the cost estimating these parameters over many prediction experiments.
The scaling functions for the tasks can depend on intermediate computational results, in addition to program
inputs. Even if this is not the case, they may appear to do so to the compiler. For example, in the NAS benchmark
SP, the grid sizes for each processor are computed and stored in an array, which is then used in most loop bounds.
The use of an array makes forward propagation of the symbolic expressions infeasible, and therefore completely
obscures the relationship between the loop bounds and program input variables. We simply retain the executable
scaling expressions, including references to such arrays, in the simplified code and evaluate them at
execution time.
We have been able to automate fully the modeling process for a given HPF application compiled to MPI. The
modified dHPF compiler automatically generates two versions of the MPI program. One is the simplified MPI
code with delays calls described previously. The second is the full MPI code with timer calls inserted to perform
the measurements of the w parameters. The output of the timer version can be directly provided as input to the
delay version of the code. This complete process is illustrated in Figure 2.
We performed a detailed experimental evaluation of the compiler-based simulation approach. We studied three
issues in these experiments:
1. The accuracy of the optimized simulator that uses the compiler-generated information, compared with both
the original simulator and direct measurements of the target program.
2. The reduction in memory usage achieved by the optimized simulator compared with the original and the
resulting improvements in the overall scalability of the simulator in terms of system sizes and problem sizes
that can be simulated.
3. The performance of the optimized simulator compared with the original, in terms of both absolute simulation
times and in terms of relative speedup as compared to sequential model execution, when simulating a large
number of target processors.
Results in each of the above categories are presented for both types of the optimizations considered in this paper:
elimination of local computations and elimination of data contents from large messages.
We begin with a description of our experimental methodology and then describe the results for each of these
issues in turn.
5.1 Experimental Methodology
We used three real-world benchmarks (Tomcatv, Sweep3D and NAS SP) and one synthetic communication
kernel (SAMPLE) in this study. Tomcatv is a SPEC92 floating-point benchmark, and we studied an HPF version
of this benchmark compiled to MPI by the dHPF compiler. Sweep3D, a Department of Energy ASCI benchmark
[1], and SP, a NAS Parallel Benchmark from the NPB2.3b2 benchmark suite [8], are MPI benchmarks written in
Fortran 77. Finally, we designed the synthetic kernel benchmark, SAMPLE, to evaluate the impact of the
compiler-directed optimizations on programs with varying computation granularity and message communication
patterns that are commonly used in parallel applications.
For Tomcatv, the dHPF compiler automatically generates three versions of the output MPI code: (a) the normal
MPI code generated by dHPF for this benchmark, where the key arrays of the HPF code are distributed across the
processors in contiguous blocks in the second dimension (i.e., using the HPF distribution (*,BLOCK)); (b) the
simplified MPI code with the calls to the MPI-Sim delay function, making full use of the techniques described in
Section 4; and (c) the normal MPI code with timer calls inserted to measure the task time parameters, as described
in Section 4.5. Since dHPF only parses and emits Fortran and MPI-Sim only supports C, we use f2C to translate
each version of the generated code to C and run it on MPI-Sim. For the other two benchmarks, Sweep3D and
NAS SP, we manually modified the existing MPI code to generate the simplified MPI and the MPI code with
timers for each case (since the task graph synthesis for MPI codes is not implemented yet). These codes serve to
show that the compiler techniques we developed can be applied to a large range of codes with good results.
For each application, we measured the task times (values of w i ) on 16 processors. These measured values were
then used in experiments with the same problem size on different numbers of processors. The only exception was
NAS SP, where we measured the task only for a single problem size (on 16 processors), and used the same task
times for other problem sizes as well. Recall that the scaling functions we use currently do not account for cache
working sets and cache performance. Changing either the problem size or the number of processors affects the
working set size per process and, therefore, the cache performance of the application. Nevertheless, the above
measurement approach provided very accurate predictions from the optimized simulator, as shown in the next
subsection.
All benchmarks, except SAMPLE, were evaluated for the distributed memory IBM SP (with up to 128
processors); the SAMPLE experiments were conducted on the shared memory SGI Origin 2000 (with up to 8
processors).
5.2 Validation
The original MPI-Sim was successfully validated on a number of benchmarks and architectures [6, 26, 27]. The
new techniques described in Section 4, however, introduce additional approximations in the modeling process.
The key new approximation is in estimating the sequential execution times of portions of the computational code
(tasks) that have been abstracted away. Our aim in this section is to evaluate the accuracy of MPI-Sim when
applying these techniques.
For each application, the optimized simulator (henceforth denoted as MPI-SIM-TG) was validated against direct
measurements of the application execution time and also compared with the predictions from the original
simulator. We studied multiple configurations (problem size and number of processors) for each application. In all
cases MPI-SIM-TG is validated against the measured system. 4
We begin with Tomcatv, which is handled fully automatically through the steps of compilation, task
measurements, and simulation shown in Figure 2. The size of Tomcatv used for the validation was 2048-2048.
Figure
3 shows the results from 4 to 64 processors. Even though MPI-Sim with the analytical model (MPI-SIM-
TG) is not as accurate as MPI-Sim with direct execution (MPI-SIM-DE), the error in the performance predicted
by MPI-SIM-TG was below 16% with an average error of 11.3% against the measured system.
4 The message optimizations further introduced do not modify the underlying communication model and thus do
not affect validation.
Validation of MPI-SIM for Tomcatv2060100140
number of processors
runtime
(in
sec)
measured
Figure
3: Validation of MPI-Sim for (2048-2048) Tomcatv (on the IBM SP).
Figure
4 shows the execution time of the model for Sweep3D with a total problem size of 150-150-150 grid cells
as predicted using MPI-SIM-TG, MPI-SIM-DE, as well as the measured values, all for up to 64 processors. The
predicted and measured values are again very close and differ by at most 9.8%. On average, MPI_SIM_DE
differed from the measured value by 3.7% and MPI-SIM-TG by 7.2%.
number of processors
runtime
(in
sec)
measured
Figure
4: Validation of Sweep3D on the IBM SP, Fixed total Problem Size.
Finally, we validated MPI-SIM-TG on the NAS SP benchmark. The task times were obtained from the 16
processor run of the class A, the smallest of the three built-in sizes (A, B and C) of the benchmark, and used for
experiments with all problem sizes. Figures 5 and 6 show the validation for class A and the largest size, class C.
The validation for class A is good (the errors are less than 7%). The validation for class C is also good with an
average error of 4%, even though the task times were obtained from class A. This result is particularly interesting
because, for programs of the same size, class C on average runs 16.6 times longer than class A. This demonstrates
that the compiler-optimized simulator is capable of accurate projections across a wide range of scaling factors.
Furthermore, cache effects do not appear to play a great role in this code or the other two applications we have
examined. This is illustrated by the fact that the errors do not increase noticeably when the task times obtained on
a small number of processors were used for a larger number of processors.
Validation for SP Class A100300500700
number of processors
runtime
(in
sec)
measured
Figure
5: Validation for NAS SP, class A on the IBM SP.
Validation for SP class C5001500250016 36 64 100
number of processors
runtime
in
seconds measured
Figure
Validation for NAS SP, class C on the IBM SP.
Figure
7 summarizes the errors that MPI-SIM-TG incurred when simulating the three applications. All the errors
are within 16%. The figure emphasizes that the compiler-supported approach combining analytical model and
simulation is very accurate for a range of benchmarks, system sizes, and problem sizes.
It is hard to explore these errors further without detailed analysis of each application. Therefore, to better quantify
what errors can be expected from the optimized simulator, we used our SAMPLE benchmark, which allows us to
vary the computation to communication ratio as well as the communication patterns.
%Error between MPI-SIM-TG Predictions and the
Measured
number of processors
Tomcatv
Sweep3D(150cubed)
Figure
7: Percent Error Incurred by MPI-SIM-TG when Predicting Application Performance.
was validated on the Origin 2000. Two common communication patterns were selected: wavefront and
nearest neighbor. For each pattern, the communication to computation ratio was varied from 1 to 100 to a ratio of
1 to 1.
Figure
8 plots the total execution time for the program and MPI-SIM-TG prediction. In order to
demonstrate better the impact of computation granularity on the validation, Figure 8 plots the percentage variation
in the predicted time as compared with the measured values. As can be seen from the figure, the predictions are
very accurate when the ratio of computation to communication is large, which is typical of many real-world
applications. As the amount of computation granularity in the program decreases, the simulator incurs larger
errors. This can expected because both measurement errors and task time estimation errors can become relatively
more significant. Nevertheless, the graph shows that the predicted values differ by at most 15% from the
measured values, even for small communication to computation ratios.
Validation of SAMPLE Measured vs.
Predicted with Optimization on Origin 2K50015000.01 0.0125 0.0167 0.025
Communication to Computation Ratio
Time
in
seconds Wvfrnt-Measured
NN-Measured
Figure
8: Validation of SAMPLE on the Origin 2000.
Percent variation of measured time
from predicted time5150.01 0.03 0.10 0.30 0.50 0.70 0.90
Communication to Computation ratio
Difference wvfrnt nn
Figure
9: Effect of Communication to Computation Ratio on Predictions.
The accuracy of MPI-SIM-TG for large computation to communication ratio (below 5% error) indicates that the
slightly higher errors we observed for Tomcatv, Sweep3D and NAS SP must be due to the presence of small
computation to communication ratios.
5.3 Expanding the simulator to larger systems and problem sizes.
The main benefit of using the compiler-generated code is that we can decrease the memory requirements of the
simplified application code. Since the simulator uses at least as much memory as the application, decreasing the
amount of memory for the application decreases the simulator's memory requirements, thus allowing us to
simulate large problem sizes and systems.
Number of
processors
total memory use
total
memory use
Memory Reduction
Factor
Sweep 3D, 4-4-255
per Proc. Problem Size 4900 2884MB 30MB 96
Sweep 3D, 6-6-1000
per Proc. Problem Size 6400 215GB 122MB 1762
Tomcatv, 2048-2048 4 236MB 118.4KB 1993
Table
1: Memory Usage in MPI-SIM-DE and MPI-SIM-TG for the benchmarks.
Table
1 shows the total amount of memory needed by MPI-Sim when using the analytical (MPI-SIM-TG) and
direct execution (MPI-SIM-DE) models. For Sweep3D, with 4900 target processors, the analytical models reduce
memory requirements by two orders of magnitude for the 4-4-255 per processor problem size. Similarly, for the
6-6-1000 problem size, the memory requirements for the target configuration with 6400 processors are reduced
by three orders of magnitude! Three orders of magnitude reduction is also achieved for Tomcatv, while smaller
reductions are achieved for SP. This dramatic reduction in the memory requirements of the model allows us to (a)
simulate much larger target architectures, and (b) show significant improvements in execution time of the
simulator.
To illustrate the improved scalability achieved in the simulator with the compiler-derived analytical models, we
consider Sweep3D. In this paper, we study a small subset of problems that are of interest to application
developers. They are represented by the 20 million cell total problem size, which can be divided into 4-4-255,
7-7-255, and 28-28-255 per processor problem sizes which need to run on 4,900, 1,600 and 100 processors,
respectively.
The scalability of the simulator for the 4-4-255 problem size can be seen in Figure 10. The memory requirements
of the direct execution model restricted the largest target architecture that could be simulated to 2500 processors.
With the analytical model, it was possible to simulate a target architecture with 10,000 processors. Since the
application's predicted runtime for 10,000 processors is 11.0955 seconds and the runtime of the simulator for that
configuration is 148.118 seconds, the simulator's slowdown is only 13.35! Note that instead of scaling the system
size, we could scale the problem size instead (for the same increase in memory requirements per process), in order
to simulate much larger problems.
Validation and Scalability of Sweep3D (4x4x255/proc)26101 10 100 1000 10000
number of processors
runtime
(in
sec)
Measured
Figure
10: Scalability of Sweep3D for the 4-4-255 per Processor Size (IBM SP).
5.4 Performance of MPI-Sim
The benefits of compiler-optimized simulation are not only evident in memory reduction but also in improved
performance. We characterize the performance of the simulator in four ways:
1. performance gains when using the message optimization (MPI-SIM-TGMO) and MPI-SIM-TG as compared
to MPI-SIM-DE,
2. absolute performance (i.e., total simulation time) of MPI-SIM-TG vs. MPI-SIM-DE and vs. the application,
3. parallel performance of MPI-SIM-TG, in terms of both absolute and relative speedups, and
4. performance of MPI-SIM-TG when simulating large systems on a given parallel host system.
Effect of Optimizations on Simulator's Performance
To illustrate the performance improvements between MPI-SIM-DE, MPI-SIM-TG, which takes advantage of only
the local optimizations and MPI-SIM-TGMO, which additionally optimizes the messages being sent, we
conducted experiments on the three benchmarks. In case of Sweep3D we compared the performance of the three
versions of the simulator when each had a given number of host processors available. The problem size per
processor was fixed, and the number of target processors in the experiment was increased. This study
demonstrates the ability of each simulator to efficiently simulate large problem sizes.
For NAS SP, since the problem size of the application is given (here class C), we fixed the number of target
processors and varied the number of host processors available to the simulator. This study illustrates not only the
relative performance of the simulators, but also their ability to use computational resources.
Figures
11, 12 and 13 show the performance of MPI-SIM-TGMO, MPI-SIM-TG and MPI-SIM-DE when
simulating Sweep3D for three sizes per processor sizes: 7-7-255, 14-14-255 and 28-28-255. All simulators use
host processors to simulate up to 4,900 target processors. The improvements in performance between MPI-
SIM-DE and MPI-SIM-TG for the above sizes are on the average 39.7%, 67.28% and 88.07% respectively. As
the problem size per processor grows larger, the amount of computation per processor increases thus the amount
of computation abstracted away increases resulting in runtime savings.
7x7x255 Per Processor Size, 64 Host Processors20060010 100 1000 10000
target processors
runtime
in
sec. MPI-SIM-TGMO
Figure
11: Sweep3D, 7x7x255 Per Processor Size, (MPI-SIM-TGMO is MPI-SIM-TG+ the message
optimization).
14x14x255 Per Processor Size, 64 Hosts2006001000
target processors
runtime
in
sec
MPI-SIM-TGMO
Figure
12: Sweep3D, 14-14-255 Per Processor Size.
28x28x255 Per Processor Size, 64 host procs200600100010 100 1000 10000
target procs
runtime
in
sec
MPI-SIM-TGMO
Figure
13: Sweep3D, 28-28-255 Per Processor Size.
Although the biggest performance gain is in the computation optimization, reducing the size of the messages sent,
where possible, is beneficial. The simulation, MPI-SIM-TGMO, runs faster than the simulation, which just
optimizes the computation (MPI-SIM-TG). The improvements for the sizes 7-7-255, 14-14-255 and 28-28-255
are 28.04%, 31.23% and 13.9% respectively. The benefits of the message optimizations are limited for the
Sweep3D application, because it uses a large number of barrier synchronizations as well as collective operations
such as (MPI_Allreduce). These operations either take no data or only single data items.
We also observed great performance improvements for the NAS SP benchmark, class C, the largest size available
in the suite. Figures 14 and 15 show the performance of MPI-SIM-TG and MPI-SIM-TGMO for two target
processor configurations: 16 and 64. The simulations were run on a variety of host processors from 1 to 64. First,
both MPI-SIM-TG and MPI-SIM-TGMO ran faster than the actual application. The measured runtime of the
application executing on 16 processors is 2623.38 seconds, whereas running on 64 processors it is 790.67
seconds.
Additionally, Figures 14 and 15 illustrate that the simulation can run an order of magnitude faster than MPI-SIM-
when the message optimization is used. In Figure 14, the jump in runtime for MPI-SIM-TG (from 1 to 2 host
processors) is due to the large communication costs. The size of the messages sent between processors is 605,161
doubles. Therefore the cost of sending these messages increases considerably when more than one processor is
used. When only 2 host processors are used this increased cost is not compensated by the increased computational
power. However, as the number of host processors increases, better performance is achieved. Since the size of
these large messages can be reduced to 0 in the MPI-SIM-TGMO simulation, this communication overhead is
significantly reduced and the simulator performs substantially better than MPI-SIM-TG. As the number of target
processors increases (to 64 in Figure 15), the size of the messages in the simulation is reduced (to 370,441 for the
target processor code.) Still, using the message optimization results in an order of magnitude decrease in the
simulator's runtime.
Absolute Performance, Local Code Optimization Only
To compare the absolute performance of MPI-Sim, we gave the simulator as many processors as were available to
the application (#host processors = # target processors).
class C2006001000
host processors
runtime
in
sec.
MP I-S IM-TG
MP I-S IM-TGMO
Figure
14: A 16 Target Processor Simulation of NAS SP, Class C Running on Various Number of Host
Processors.
Processors , NAS SP C lass C100300500700
host Processors
runtime
in
seconds MP I-S IM-TG
MP I-S IM-TGMO
Figure
15: A 64 Target Processor Simulation of NAS SP, Class C Running on Various Number of Host
Processors.
Figure
shows the absolute performance for Sweep3D with a total problem size of 150 3 . MPI-SIM-DE is on the
average 2.8 times slower than the actual application (Measured in the Figure). However, MPI-SIM-TG is initially
faster then the measured application starting at 13 times faster when running on 4 processors, gradually becoming
only 2.2 times faster for processors and finally being twice as slow as the application running on 64 processors.
Message optimizations present in MPI-SIM-TGMO further decrease the simulators' runtime by on the average
18% as compared to MPI-SIM-TG. Both MPI-SIM-TG and MPI-SIM-TGMO are always faster (on the average
and 18.5 times faster respectively) than MPI-SIM-DE, showing the clear benefits of compiler optimizations.
However, as the number of processors increases the amount of communication relative to the computation
increases thus exposing the overhead of simulating the communications and making MPI-SIM-TG and MPI-SIM-
TGMO slower than the application.
cubed Sweep3D, Total Problem Size1010000
number of processors
Runtime
in
seconds
Measured
MPI-SIM-TGMO
Figure
Absolute Performance of MPI-Sim for Fixed Total Problem Size Sweep3D. (Vertical Scale is
Logarithmic)
Figure
17 shows the runtime of the application and the measured runtime of the two versions of the simulator
running NAS SP class A. We observe that MPI-SIM-DE is running about twice slower than the application it is
predicting. However, MPI-SIM-TG is able to run much faster than the application, even though detailed
simulation of the communication is still performed. In the best case (for 36 processors), it runs 2.5 times faster.
For 100 processors, it runs 1.5 times faster. The relative performance of MPI-SIM-TG decreases as the number of
processors increases because the amount of computation in the application decreases with increased number of
processors and thus the savings from abstracting the computation are decreased.
Absolute Performance of MPI-Sim for NAS SP206010014030 50 70 90
Number of processors
Runtime
in
Seconds Measured
Figure
17: Absolute Performance of MPI-Sim for the NAS SP Benchmark, class A.
Even more dramatic results were obtained with Tomcatv, where the runtime of MPI-SIM-TG does not exceed 2
seconds for all processor configurations as compared to the runtime of the application which ranges from 130 to
seconds (Figure 18). This is due to the ability of the compiler to abstract away most of the computation. All
that the simulator needs to directly execute is the skeleton code that controls the flow of the computation and
communication patterns.
Absolute Performance of MPI-Sim for Tomcatv20601001400
number of processors
runtime
(in
seconds) application
Figure
Absolute Performance of MPI-Sim for Tomcatv (2048x2048).
Parallel Performance
To evaluate the parallel performance of the simulator, we study how well can it take advantage of increasing
system resources (her processors) to solve a given problem (fixed total problem size). Figures 14 and 15
indirectly demonstrate the performance of the simulator; to illustrate the performance better, the speedup achieved
for the 16 target configuration is depicted in Figure 19. Although MPI-SIM-TGMO, has a smaller runtime than
MPI-SIM-TG, it scales well for only up to 8 host processors. This is because, as the number of host processors
increases, the communication overhead between the host begins to dominate the runtime. On the other hand, MPI-
SIM-TG, which had to send large messages, suffers most when more than one host is used, but then is able to
distribute that overhead among more processors.
16Target NAS SP, Class C0.51.52.53.5
number of host processors
MPI-SIM-TGMO
Figure
19: Speedup of MPI-Sim for NAS SP.
Clearly, the performance of the simulator is better when larger systems are simulated. For the 64-target processor
case (
Figure
15), the runtime decreases steadily as the number of processors is increased. However, using more
than host processors actually increases the simulator's runtime. (64 Target Class C could not be run on a single
processor due to memory constraints, so direct speedup comparisons are not possible.)
Better scalability is seen for the Sweep3D application. Figure 20 shows the performance of MPI-SIM-TG and
MPI-SIM-DE simulating the 150 3 Sweep3D running on 64 target processors when the number of host processors
is varied from 1 to 64. The data for the single processor MPI-SIM-DE simulation is not available because the
simulation exceeds the available memory. Clearly, both MPI-SIM-DE and MPI-SIM-TG scale well. The speedup
of MPI-SIM-TG is also shown in Figure 21. The steep slope of the curve for up to 8 processors indicates good
parallel efficiency. For more than 8 processors the speedup is not as impressive, reaching about 15 for 64
processors. This is due to the decreased computation to communication ratio in the application. Still, the runtime
of MPI-SIM-TG is on the average 5.4 times faster than that of MPI-SIM-DE.
Runtime of S imu lator Vs. Application (150x150x150 Sweep3d ,
64Target proc)100300500700
number host processors
runtime
(in
sec)
MP I-SIM -DE
MP I-SIM -TG
Measured
Figure
20: Parallel Performance of MPI-Sim.
Speedup of MPI-SIM-TG (150cubed Sweep3D, 64
Target Processors)515
berofprocessors
speedup MPI-SIM-TG
Figure
21: Speedup of MPI-SIM-TG for Sweep3D.
Performance for Large Systems
To quantify further the performance improvement for MPI-SIM-TG, we have compared the running time of the
simulators when predicting the performance of a large system; in this case we want to simulate a billion-cell
problem for Sweep3D. This application's developers envision this problem to utilize 20,000 processors, which
corresponds to a 6-6-1000 per processor problem size. Figure 22 shows the running time of the simulators as a
function of the number of target processors, when 64 host processors are used. The problem size is fixed per
processor, so the problem size increases with the increased number of processors. The figure clearly shows the
benefits of the optimizations. In the best case, when the performance of 1,600 processors is simulated
(corresponding to the 57.6 million problem size) the runtime of the optimized simulator is nearly half the runtime
of the original simulator. However, even with the optimizations, the memory requirements are still too large to be
able to simulate the desired target system.
MPI-SIM runtime for the 6x6x1000 per processor size
host processors)20060010000 500 1000 1500 2000 2500 3000
number of target host processors
runtime
in
seconds
Figure
22: Performance of MPI-SIM when Simulating Sweep3D on Large Systems.
6 Conclusions
This work has developed a scalable approach to detailed performance evaluation of communication behavior in
Message Passing Interface (MPI) and High Performance Fortran (HPF) programs. Our approach is based on
using compiler analysis to identify portions of the computation whose results do not have a significant impact on
program performance, and therefore do not have to be simulated in detail. The compiler builds an intermediate
static task graph representation of the program which enables it to identify program values that have an impact on
performance, and also enables it to derive scaling functions for computational tasks. The compiler then uses
program slicing to determine what portions of the computations are not needed in determining performance.
Finally, the compiler abstracts away those parts of the computational code (and corresponding data structures),
replacing them with simple, analytical performance estimates. It also flags messages for which the data transfer
does not have to be performed within the simulation. All of the communication code is retained by the compiler,
and is simulated in detail by MPI-Sim.
Our experimental evaluation shows that this approach introduces relatively small errors into the prediction of
program execution times. The benefit we achieve is significantly reduced simulation times (typically more than a
factor of 2) and greatly reduced memory usage (by two to three orders of magnitude). This gives us the ability to
accurately simulate detailed performance behavior of systems and problem sizes that are 10-100 times larger than
is possible with current state-of-the-art simulation techniques.
In our current work, we are also exploring a number of alternative combinations of modeling techniques. For
example, we can use detailed simulation for the sequential tasks, instead of analytical modeling and measurement.
This will not only allow to get accurate estimates of task execution times, but also enable us to study the
application's performance on a processor and memory architecture different from the currently available
platforms. Within POEMS, we aim to support any combination of analytical modeling, simulation modeling and
measurement for the sequential tasks and the communication code. The static task graph provides a convenient
program representation to support such a flexible modeling environment [5].
One potential limitation of our work is that the benefits would not be as large for applications where the
parallelism and communication patterns depend extensively on intermediate results of the computations. In
particular, so-called irregular applications may have this property. Evaluating the benefits for such applications
requires further research, and perhaps a refinement of the techniques developed here.
Another interesting direction is whether the techniques described here can be extended to other types of
distributed applications (i.e., non-scientific applications) that use network communication intensively. If very fast
simulation techniques could be developed for such applications, they could prove extremely valuable in
controlling runtime optimization decisions such as object migration, load balancing, or adaptation for quality-of-
service requirements, which are critical decisions for many distributed applications.
Acknowledgements
This work was supported by DARPA/ITO under Contract N66001-97-C-8533, "End-to-End Performance
Modeling of Large Heterogeneous Adaptive Parallel/Distributed Computer/Communication Systems,"
(http://www.cs.utexas.edu/users/poems/). The work was also supported in part by the ASCI ASAP program
under DOE/LLNL Subcontract B347884, and by DARPA and Rome Laboratory, Air Force Materiel Command,
USAF, under agreement number F30602-96-1-0159. We wish to thank all the members of the POEMS project
for their valuable contributions. We would also like to thank the Lawrence Livermore National Laboratory for the
use of their IBM SP. This work was performed while Adve and Sakellariou were with the Computer Science
Department at Rice University.
--R
"The ASCI Sweep3D Benchmark Code,"
"Using integer sets for data-parallel program analysis and optimization.,"
"POEMS: End-to-end Performance Design of Large Parallel Adaptive Computational Systems,"
"Compiler Synthesis of Task Graphs for a Parallel System Performance Modeling Environment.,"
"Application Representations for a Multi-Paradigm Performance Modeling Environment for Parallel Systems,"
"Performance Prediction of Large Parallel Applications using Parallel Simulations,"
"Parsec: a parallel simulation environment for complex systems,"
"The NAS Parallel Benchmarks 2.0,"
"PROTEUS: a high-performance parallel-architecture simulator,"
"Optimistic simulation of parallel architectures using program executables,"
"Distributed simulation: a case study in design and verification of distributed programs,"
"The Conditional Event Approach to Distributed Simulation,"
"The Rice parallel processing testbed,"
"Multiprocessor Simulation and Tracing using Tango.,"
"POEMS: End-to-end Performance Design of Large Parallel Adaptive Computational Systems.,"
"A Distributed Memory LAPSE: Parallel Simulation of Message-Passing Programs,"
"Parallelized direct execution simulation of message-passing parallel programs,"
"FAST: a functional algorithm simulation testbed,"
"Functional Algorithm Simulation of the Fast Multipole Method: Architectural Implications,"
"Improving the Accuracy vs. Speed Tradeoff for Simulating Shared-Memory Multiprocessors with ILP Processors,"
"Interprocedural slicing using dependence graphs,"
"Transparent implementation of conservative algorithms in parallel simulation languages,"
"Reducing Synchronization Overhead in Parallel Simulation,"
"An adaptive synchronization method for unpredictable communication patterns in dataparallel programs,"
"Parallel Simulation of Data Parallel Programs,"
"MPI-SIM: using parallel simulation to evaluate MPI programs,"
"Asynchronous Parallel Simulation of Parallel Programs,"
"The Wisconsin Wind Tunnel: VIrtual Prototyping of Parallel Computers,"
--TR
The rice parallel processing testbed
Interprocedural slicing using dependence graphs
PROTEUS: a high-performance parallel-architecture simulator
The Wisconsin Wind Tunnel
A distributed memory LAPSE
Reducing synchronization overhead in parallel simulation
Optimistic simulation of parallel architectures using program executables
Parallelized Direct Execution Simulation of Message-Passing Parallel Programs
Transparent implementation of conservative algorithms in parallel simulation languages
Using integer sets for data-parallel program analysis and optimization
Poems
MPI-SIM
Performance prediction of large parallel applications using parallel simulations
Asynchronous Parallel Simulation of Parallel Programs
Improving lookahead in parallel discrete event simulations of large-scale applications using compiler analysis
Parsec
POEMS
An adaptive synchronization method for unpredictable communication patterns in dataparallel programs
Compiler Synthesis of Task Graphs for Parallel Program Performance Prediction
Parallel Simulation of Data Parallel Programs
FAST
Improving the Accuracy vs. Speed Tradeoff for Simulating Shared-Memory Multiprocessors with ILP Processors
--CTR
Yasuharu Mizutani , Fumihiko Ino , Kenichi Hagihara, Fast performance prediction of master-slave programs by partial task execution, Proceedings of the 4th WSEAS International Conference on Software Engineering, Parallel & Distributed Systems, p.1-7, February 13-15, 2005, Salzburg, Austria | performance modeling;parallel simulation;parallelizing compilers |
589795 | Local behavior of the Newton method on two equivalent systems from linear programming. | Newton's method is a fundamental technique underlying many numerical methods for solving systems of nonlinear equations and optimization problems. However, it is often not fully appreciated that Newton's method can produce significantly different behavior when applied to equivalent systems, i.e., problems with the same solution but different mathematical formulations. In this paper, we investigate differences in the local behavior of Newton's method when applied to two different but equivalent systems from linear programming: the optimality conditions of the logarithmic barrier function formulation and the equations in the so-called perturbed optimality conditions. Through theoretical analysis and numerical results, we provide an explanation of why Newton's method performs more effectively on the latter system. | Introduction
Newton's method is generally accepted as an effective tool for solving a system of nonlinear
It is a locally and quadratically convergent method
under reasonable assumptions (see e.g. Dennis and Schnabel (Ref. 1)). It is often not fully
appreciated, however, that Newton's method can exhibit significantly different local and global
behavior on two equivalent systems. By equivalent systems, we refer to two systems of nonlinear
equations that can be derived from one another and essentially share the same set of solutions
(though some auxiliary variables/equations may be present in one but not in another). In this
paper, we compare the behavior of Newton's method applied to two well-known equivalent
systems of nonlinear equations associated with linear programming.
The first of these equivalent systems consists of the first-order optimality conditions of the
log-barrier formulation of the linear program. The second system consists of equations in the
perturbed first-order optimality conditions for the linear program. Though the two nonlinear
systems have essentially the same set of solutions, El-Bakry, Tapia, Tsuchiya, and Zhang (Ref. 2)
show that Newton's method necessarily generates different iterates for the two systems. In
this paper, we show that Newton's method applied to the perturbed optimality conditions for
the linear program has a larger sphere of convergence than Newton's method applied to the
optimality conditions of the log-barrier formulation of the linear program.
Of these two equivalent systems, the perturbed first-order optimality conditions are widely
used in interior-point methods for linear programming. However, the reasons for favoring this
system have not been fully analyzed. In this paper, we provide an explanation on why the
system associated with the perturbed optimality conditions is the system of choice.
The paper is organized as follows. In Section 2, we present the two equivalent nonlinear
systems under consideration. In Section 3, we introduce the notion of the sphere of convergence
of Newton's method and provide theoretical results on the radius of the sphere of convergence
of Newton's method applied to the two equivalent systems. In Section 4 we present numerical
results supporting the theory we developed in the previous section. Finally, we make some
concluding remarks in Section 5.
2. Two Equivalent Formulations
In this section, we introduce the linear programming problem and the two equivalent nonlinear
systems under consideration. We consider the linear programming problem in the standard form
m. The Lagrangian function
associated with problem (1) is
where y
are, respectively, the vectors of Lagrange multipliers associated with
the equality and the inequality constraints. The first-order optimality conditions for problem
(1) are
2.1. Two Equivalent Systems
We derive one of the equivalent systems by formulating problem (1) in the logarithmic barrier
framework. This framework, which was first introduced by Frisch (Ref. 3), consists of solving
a sequence of equality constrained minimization problems with decreasing values of the barrier
parameter - ? 0. For problem (1) and a given value of - ? 0, the log-barrier subproblem has
the following form
log x i
Assume that the feasible set fx : for every value of - ? 0,
there exists a unique solution x
- of the log-barrier subproblem. Under mild assumptions (see
e.g. Fiacco and McCormick (Ref. 4)), as - ! 0 the sequence of iterates fx
converges to a
solution x of problem (1), i.e. lim -!0 x
The optimality conditions for the log-barrier subproblem are derived by differentiating the
Lagrangian function,
log
where y is the vector of Lagrange multipliers associated with the equality constraints,
and setting the gradient of the Lagrangian equal to zero. Then the optimality conditions are
Observe that the Jacobian of FB is given by
\Gamma-X \Gamma2 A T
If
In applications of Newton's
method near the solution, the Jacobian necessarily becomes ill-conditioned as - approaches zero
(see, (Ref. 5, 6)).
Now we derive a nonlinear system equivalent to system (3). Consider the introduction of an
auxiliary variable, z 2 R n , and define
which is written equivalently as Substituting z into system (3) and adding the
equation that relates x; z and - yields the system
The Jacobian of F P is given by
The Jacobian is nonsingular if solution where x and z contain
zero components, the Jacobian may or may not be nonsingular, depending on the degeneracy
of the solution.
Kojima, Mizuno and Yoshise (Ref. 7) first proposed to use system (5) to solve the linear
program in a primal-dual interior-point method. El-Bakry, Tapia, Tsuchiya, and Zhang (Ref. 2)
show that although systems (3) and (5) are equivalent, Newton's method necessarily generates
different iterates for the two systems.
Two things are worth noting in comparing the two systems (3) and (5). First, while FB and
are undefined for
are defined. Second, while F 0
B is dependent on -, F 0
P is
not. These differences will greatly affect the behavior of Newton's method when applied to the
two systems as - approaches zero.
We remark that although the Jacobian F 0
does not depend on -, we will nevertheless
use the present notation for the Jacobian to stress its association with system (5) for a
given value of - 0.
System (5) can also be obtained by considering the first-order optimality conditions (2) of
the linear program and perturbing the complementarity equation,
2.2. Central Paths
Assume that the strictly feasible set f(x; nonempty.
Let
the solution to system (3) for a particular value of - ? 0, and similarly let
the solution to system (5). Then by the central path for system (3), we mean
the set
The set of points in CB forms a continuous path such that lim -!0 (x
(Ref. 4, 8)).
We remark that systems (3) and (5) are equivalent, in the sense that for - ? 0, and (x
for z
-(X
system (5), we have lim -!0 (x
system (5) the central path is defined as the set
2.3. Assumption and Notation
Throughout the paper, we make use of the following assumption and notation.
Nondegeneracy Assumption. Let the matrix A be of full rank m, and let (x ; y ; z ) be a
primal and dual nondegenerate solution of system (2). Without loss of generality, we assume
that the first m components of x are positive and the remaining (n \Gamma m) components are zero.
The nondegeneracy assumption guarantees that (x ; y ; z ) is an isolated solution point in the
primal-dual space. It is also well known that the pair (x ; z ) satisfies strict complementarity:
x
0g. Then by the nondegeneracy assumption,
ng. The matrix A will be partitioned into
where A B denotes the matrix consisting of the columns of A indexed by B and similarly for
AN . Note that If u is a vector, then its uppercase counterpart U will denote the
diagonal matrix whose diagonal consists of the elements of u. For a vector u 2 R n , u B is the
vector of the first m components of u and uN is the vector of the remaining (n \Gamma m) components
of u. The quantity u 2 represents the vector u whose components are individually squared. All
norms k \Delta k are assumed to be the Euclidean norm unless otherwise noted.
3. Sphere of Convergence: Analysis
Standard local theory of Newton's method applied to a nonlinear system (see e.g. (Ref. 1))
provides the existence of a neighborhood about a solution in which Newton's method is well-
defined. More importantly, starting from any point in the neighborhood, Newton's method
guarantees convergence to the solution. For systems (3) and (5), such a neighborhood also
exists about the solution for any given - ? 0 under our nondegeneracy assumption. In this
section, we introduce the notion of the sphere of convergence for Newton's method. We analyze
the behavior of the radius of the sphere of convergence associated with systems (3) and (5)
by considering Newton's method applied to these equivalent systems as - ! 0. Under the
nondegeneracy assumption, our analysis shows that the radius of the sphere of convergence of
Newton's method on system (3) decreases to zero in the same order as - ! 0. However, we
show the radius of the sphere of convergence of Newton's method applied to system (5) has a
lower-bound estimate independent of -. These results provide a theoretical explanation on why
Newton's method is more efficient on system (5) than on system (3) at least for small values of
3.1. Preliminaries
We introduce the notion of the sphere of convergence for Newton's method. Then, we present
lemmas to be used in our analysis for the radius of the sphere of convergence of Newton's
method on system (3).
We remark that the notion of the sphere of convergence is not new. Several references can
be found in the literature where this notion or similar concept is used, see (Ref. 1, p. 91), for
example. To conduct a rigorous study on the radius of convergence for Newton's method, we
give a formal definition for the sphere of convergence below.
Definition 3.1. We define the closed ball with radius r centered at v as B(v
rg.
Definition 3.2. For a given nonlinear system, F (v) = 0, and a solution v , the sphere of
convergence of Newton's method at v is defined as the largest closed ball centered at v such
that starting from any interior point in the sphere, excluding v , Newton's method (with unit
steplength) is well-defined and generates a sequence that converges to v .
Lemma 3.1. Consider - ? 0 and (x
contained in CB . Then under the nondegeneracy
assumption, there exists -
so that for -
- there is a ball B(x
such that for any
for constants C
Proof. Since x
strictly positive, and x
exist -
constants such that for -
-, we have G 1 - (x
-.
- is an interior point of R n
Such a point must also satisfy
First we show that x i for are bounded away from zero. From (11) for
are bounded above and below and ffi - G 1 =2, from (12) we obtain
Thus
Now, we show the second part of the proof. By the nondegeneracy assumption, strict complementarity
(8) holds at the solution, which, together with the definition (4), implies
lim
Hence, for sufficiently small - and for some constants G 3
Consequently,
By (12) and (13) for i 2 N we obtain
Therefore,
where We note that fi can be chosen so that C 3 ?
Lemma 3.2. Define
where
under the nondegeneracy assumption, there exists ~
-, and for any
is such that Lemma 3.1 holds,
for constants C
Proof. Consider
- is such that Lemma 3.1 holds. Without loss of generality,
consists of the columns A i of A with i 2 B. Similarly, we can define
Substituting in the definition of P we obtain
Now, introduce the m \Theta (n \Gamma m) matrix R where
Then P can be partitioned as follows
Applying the bounds in (9) to (16), we obtain kRk - C 5 - for a constant C 5 ? 0. Since
- such that for
all - ~
-, we obtain kRR T k ! 1. Then using the Neumann series on (I m +RR
from (17) that
for - and constants C
3.2. Sphere of Convergence for System (3)
We provide a tight result showing that the radius of the sphere of convergence of Newton's
method on system (3) decreases to zero in the same order that - ! 0. Our result follows from
showing that a lower-bound and an upper-bound of order - exist for the radius of the sphere
of convergence.
Lemma 3.3. Under the nondegeneracy assumption, there exist ~
such that for any - ~
-, the radius of the sphere of convergence, r B (-), of Newton's method
satisfies
Proof. We will prove the above result by showing that the sequence of Newton iterates
converges to the solution (x
the initial point x 0 satisfies
Consider ~
- given in Lemma 3.2. Assume Newton's method is applied to system (3) for a
particular value of - ~
-. Denote (x; y) as the current Newton iterate where x 2 B(x
and x satisfies the conditions given in (9). Now, consider the next Newton iteration
x
y
Using the fact that FB (x
x
y
-(X
By Taylor's Theorem,
for some -
Substituting (20) into (19) we obtain
where
Making the above substitution for
and multiplying the right-hand-side of (21)
we obtain
Using the definition of P in (14), we rewrite
We now consider first the vector
- ) in (22). If we partition
its basic
and nonbasic components and use the notation for P in (14), then
which leads to
Applying the bounds given in (9) and (15) to the above, we obtain
for some constant C ? 0. Recall that
It follows from (23) that if the initial
iterate satisfies
ae
fi;C
oe
then the x-component of the Newton iteration sequence will converge to x
- .
Now, consider the remaining m components of (22). Taking the norm and partitioning
matrices, we obtain
Applying (9), we have
Then it follows that for some constant -
Thus, the y-component of the Newton iteration sequence converges to y
holds. In view
of (24) and (25), we conclude that the Newton iteration sequence converges to (x
the
initial iterate x 0 satisfies (18) for all - ~
- and for K 1 defined as the constant in the right-hand
side of (24). 2
The above lemma shows that the radius of the sphere of convergence of Newton's method
It establishes only a lower-bound result for the radius of the sphere of
convergence of system (3). To establish that the radius of the sphere of convergence decreases
to zero at exactly the same order as - ! 0, we need an upper-bound of the same order. The
following lemma establishes such an upper-bound.
Lemma 3.4. Consider Newton's method applied to system (3). There exist constants -
and such that for any given -
-, the radius of the sphere of convergence, r B (-),
corresponding to this - satisfies
Proof. It suffices to show the existence of a point x - 0 with
Newton's method does not converge or is not defined. From Lemma 3.1 there exist -
constant such that for -
- and for i 2 N , (x
Consider an i 2 N , and let
where e i is the ith canonical vector. Obviously,
Newton's method is not defined at x. Therefore, r B (- K 2 -. 2
Now we are ready to give the main result for system (3).
Theorem 3.1. There exist constants ~ - ? 0 and K 1 such that for - ~
-, the radius
for the sphere of convergence, r B (-), of Newton's method applied to system (3) satisfies
Proof. Application of Lemma 3.3 and Lemma 3.4 produces the result. 2
Since system (3) is not well-defined in a neighborhood of the solution for it is not
surprising that as - ! 0, the sphere of convergence would decrease to zero. However, it was
previously not known that the radius of the sphere of convergence would decrease to zero at
exactly the same rate as - goes to zero. For the log-barrier formulation of the nonlinear program
with inequality constraints, S. Wright (Ref. lower-bound result for the radius of
the sphere of convergence. In (Ref. 9), it is shown that there exists a -
such that for -
convergence to the solution x
- can be obtained from any point x 0 that satisfies
C- ff (26)
for constant -
In the case of linear programming, our result for system (3)
is tight and shows that the radius of the sphere of convergence decreases in the same order as
our results.
3.3. Sphere of Convergence for System (5)
We now give a lower-bound estimate for the radius of the sphere of convergence of Newton's
method on system (5), which is independent of the value of -. This result shows that the sphere
of convergence is bounded away from zero as - ! 0.
Proposition 3.1. Under the nondegeneracy assumption, there exist constants R ? 0 and
~
such that for any - ~
-, the radius of the sphere of convergence, r P (-), of Newton's
method satisfies
Proof. We will show that Newton's method applied to system (5) generates iterates that
converge to the solution (x
the initial point
which then implies that r P (- R ? 0.
At a given value of -, let (x;
respectively the current iterate and
the solution of Newton's method applied to system (5). Since F 0
by continuity there exist positive constants j and D such that
is independent of -.
choose ~
- such that for all - ~
-,
x
y
z
Now let (x;
-. Then
x
y
z
Hence, for such chosen - and (x; y; z), the Jacobian F 0
is nonsingular and satisfies
in view of (28).
The Newton iterates are of the formB B @
z +C C A =B B @
x
y
Hence,
It follows from (29) that if the initial iterate
for any value of - 2 (0; ~ -), Newton's method converges to the solution (x
a lower bound estimate for the radius of the sphere of convergence of Newton's method is
which is independent of -. 2
Our analysis shows that the radius of the sphere of convergence is independent of - and thus
stays bounded away from zero as - ! 0. This result indicates that the sphere of convergence
associated with system (5) would eventually be larger than the sphere of convergence associated
with system (3); that is, at least for small - values, r B (- r P (-). In the next section, we show
numerically that this is indeed the case.
4. Sphere of Convergence: Numerical Results
In Section 3, we provided bounds on the radii of the spheres of convergence of Newton's method
on systems (3) and (5) under the nondegeneracy assumption. Our analysis shows that at least
for small values of -, the sphere of convergence for system (5) is larger than that for system (3).
In this section, we try to compute numerical upper-bound estimates on the radii of the spheres
of convergence for Newton's method on systems (3) and (5). The purpose of these computations
is not only to confirm our theory for nondegenerate problems for small values of -, but also to
obtain empirical information on degenerate problems and for relatively large values of -.
4.1. Description of the Numerical Experiments
We note that (a) the variables are (x; y) for system (3), and (x;
(z
(c) the variable y appears linearly in both systems (3) and (5). For the sake
of comparison, we will only estimate the radii of the spheres of convergence for both systems
in the x-space, using a fixed initial point for y. More specifically, for any given - ? 0 and any
chosen initial point x 0 , we set y In the rest of the section, the term
"sphere of convergence" is always restricted to the x-space only.
Our upper-bound estimates are based on the following simple idea. Let x ff 2 R n be an
arbitrary unit vector and - ? 0 be a scalar. Consider applying Newton's method to systems (3)
and (5) starting from initial points of the form
and with z (5). If for - ff ? 0, Newton's method does not converge
to
(or to (x
-(X
system (5)), then obviously - ff is an
upper bound for the radius of the sphere of convergence of Newton's method at x
- . This upper
bound is the tightest possible in this particular direction if Newton's method converges to x
- for
any - 2 (0; - ff ). Numerically, this upper bound - ff can be approximated by gradually increasing
- from zero by a small increment until Newton's method fails to converge. We can generate a
tighter upper bound by calculating - ff for a set of random unit vectors fx ff g, and then taking
min ff f- ff g as an upper bound.
Under the nondegeneracy assumption, for system (5) Newton's method is well-defined in a
neighborhood of the solution to the linear program, which includes negative values for x and
z. Therefore, we can choose x ff to be any unit random vector. In our experiments on system
(5), ten unit random vectors x ff are selected using the Matlab function randn followed by a
normalization.
As we mentioned earlier, because of the presence of the term X \Gamma1 , system (3) is not well-defined
nor is Newton's method in any neighborhood of the solution to the linear
program. This fact implies that the sphere of convergence of Newton's method shrinks to zero
However, it is not clear at all that the largest half-sphere inside the positive orthant
where Newton's method is well-defined and convergent should also shrink to zero as - ! 0.
To be fair to system (3), we use only positive unit random vectors x ff . In this way, we
actually estimate an upper bound for the radius of the "half-sphere" of convergence instead of
the sphere of convergence. In our experiments on system (3), ten positive unit random vectors
x ff are selected using the Matlab function rand followed by a normalization.
To observe the behavior of the radii of the half-sphere of convergence for system (3) and of
the sphere of convergence for system (5) as - ! 0, the numerical procedure described above
was performed for a set of values of - ? 0:
We include large values of in order to see the behavior of the radius of the
sphere of convergence of Newton's method when far from the solution at The parameter
- in (30) was given an initial value of 10 \Gamma10 and was incremented when the convergence criteria
was satisfied at some iteration k, where v
Such residual definitions were designed to prevent the stopping
criterion from being in favor of one system or another. Nonconvergence was recorded for a
particular run with a given - value and initial point of the form given in (30) if the maximum
number of iterations, which we set to 50, was reached. The convergence tolerance was set to
. The numerical solution v
was obtained by solving system (5) with a given value of
- in (31) and with a stopping tolerance of 10 \Gamma8 . In particular, system (5) was solved using an
interior-point primal-dual method.
For the given set of - values, the estimates for the radii of the half-sphere or sphere of
convergence were recorded as min ff f- ff g, where by our construction -
- k, and
We emphasize that in these experiments, we always used the pure Newton's
method with the unit step-length.
In our implementation, we used a plain partial-pivoting Gaussian elimination (Matlab back-
slash) to solve all linear equations in computing Newton directions for both systems (3) and
(5). This should minimize the effect of ill-conditioning caused by different elimination schemes
that exploit sparsity.
4.2. Test Problems
Test problems consisted of six randomly generated problems r1-r6 which are all nondegenerate,
the Netlib nondegenerate problems: scagr7, sc50b, share1b and the Netlib degenerate prob-
lems: adlittle, afiro, blend, sc50a, and share2b. For the random problems, the data
were generated from a uniform distribution on the interval (0; 1) using the Matlab function
rand. For a given problem, the same ten unit random vectors x ff were used for all values of - in
(31). The problems were run on a Sun Ultra Sparc workstation using Matlab version 5.1. Test
problem dimensions can be found in Table I, where the first nine problems are nondegenerate
and the last five are degenerate. We mention that some Netlib problems are not in the standard
form and have inequality constraints, and the numbers of variables shown are before the addition
of slack variables.
4.3. Results for Nondegenerate Problems
We present numerical results for only four nondegenerate problems since we obtained similar
results for the remaining problems. Figures 1-2 show the radii of the half-sphere of convergence
associated with system (3) and the sphere of convergence associated with system (5) graphed
against the values of - given in (31). Figure 1 contains the graph for a random problem, and
the remaining graphs show results for the Netlib problems. The results show that the radius of
the sphere of convergence of Newton's method on system (5) is bounded away from zero even
for - sufficiently small, but the radius of the half-sphere of convergence of Newton's method
on system (3) appears to decrease to zero as - ! 0 in a linear fashion. Furthermore, our tests
show a larger radius of the sphere of convergence of Newton's method on system (5) than on
system (3) even before - becomes small. In the case of problem r2, the radius for system (3) is
noticeably larger than that for system (5) only when - is very large.
4.4. Results for Degenerate Problems
For degenerate problems, we do not have a theory for the radius of the sphere of convergence of
Newton's method on either of the two systems. We hope that numerical results would provide
some empirical information on the behavior of Newton's method applied to these problems.
We present numerical results on four of the five degenerate problems, as shown in Figures 3-4,
omitting results for the problem adlittle because they are similar to the presented results.
The results show that the radius of the half-sphere of convergence of Newton's method on
system (3) appears to decrease to zero as - approaches zero, as in the case with the nondegenerate
problems. We observe that unlike the case of nondegenerate problems, the radius of the
sphere of convergence on system (5) also appears to decrease to zero with -.
In these tests, we observe that the radius of the sphere of convergence of system (5) is always
larger than or equal to the radius of the sphere of convergence of system (3) for all the - values
given in (31). In particular, the radius associated with system (5) stays well above that for
system (3), by at least an order of magnitude, as - ! 0.
5. Conclusions
In this paper, we studied the local behavior of Newton's method on two equivalent systems
from linear programming: the optimality system (3) for the log-barrier formulation of the linear
program and the perturbed optimality system (5) for the linear program itself.
For nondegenerate problems, we have shown that the radius of the sphere of convergence of
Newton's method on system (3) decreases to zero at exactly the same order as - ! 0, while
the radius of the sphere of convergence associated with system (5) stays bounded away from
zero as - ! 0. These theoretical results are established for exact arithmetics and hence are
independent of the numerical conditioning of the Jacobian matrices for systems (3) and (5).
The numerical experiments have confirmed our theoretical results. Interestingly, on the majority
of our test problems the estimated radius of the sphere of convergence of Newton's method
was consistently larger on system (5) than on system (3); not only for small values of -, but also
for medium and large values of - for which numerical ill-conditioning does not play a critical
role.
There are multiple reasons why Newton's method performs more favorably on system (5)
than on system (3) (see (Ref. 10) for a recent work on this subject). Contrary to previous belief,
M. Wright (Ref. 11, 12) has shown that numerical ill-conditioning is not a determining factor.
We believe that the results in this paper provide another fundamental reason why system (5)
should be the system of choice to be used in an interior-point path-following framework. Similar
results have been extended to the nonlinear program and will be reported in a subsequent paper.
--R
Numerical Methods for Unconstrained Optimization and Nonlinear Equations
The Logarithmic Potential Method of Convex Programming
Sequential Unconstrained Minimization Techniques
Hessian Matrices of Penalty Functions for Solving Constrained Optimization Problems.
Analytic Expressions for the Eigenvalues and Eigenvectors of Hessian Matrices of Barrier and Penalty Functions.
A Primal-Dual Interior Point Algorithm for Linear Pro- gramming
An Analogue of Moreau's Proximation Theorem
On the Convergence of the Newton/Log-Barrier Method
Why a Pure Primal Newton Barrier Step May be Infeasible.
Some Properties of the Hessian in the Logarithmic Barrier Function
--TR
A primal-dual interior point algorithm for linear programming
Some properties of the Hessian of the logarithmic barrier function
On the formulation and theory of the Newton interior-point method for nonlinear programming
Ill-Conditioning and Computational Error in Interior Methods for Nonlinear Programming
--CTR
D. C. Jamrog , R. A. Tapia , Y. Zhang, Comparison of two sets of first-order conditions as bases of interior-point Newton methods for optimization with simple bounds, Journal of Optimization Theory and Applications, v.113 n.1, p.21-40, April 2002 | equivalent systems;linear programming;newton's method;sphere of convergence |
589926 | The dyadic stream merging algorithm. | We study the stream merging problem for media-on-demand servers. Clients requesting media from the server arrive by a Poisson process, and delivery to the clients starts immediately. Clients are prepared to receive up to two streams at any time, one or both being fed into a buffer cache. We present an on-line algorithm, the dyadic stream merging algorithm, whose recursive structure allows us to derive a tight asymptotic bound on stream merging performance. In particular, let be the Poisson request arrival rate, and let L be the fixed media length. Then the long-time ratio of the expected total stream length under the dyadic algorithm to that under an algorithm with no merging is asymptotically equal to 3/log(L)2L;. Furthermore, we establish the near-optimality of the dyadic algorithm by comparisons with experimental results obtained for an optimal algorithm constructed as a dynamic program. The dyadic algorithm and the best on-line algorithm of those recently proposed differ by less than a percent in their comparison with an off-line optimal algorithm. Finally, the worst-case performance of our algorithm is shown to be no worse than that of earlier algorithms. Thus, the dyadic algorithm appears to be the first near optimal algorithm that admits a rigorous average-case analysis. | INTRODUCTION
At a sequence of random times, clients request content streaming from a
given media server, e.g., videos from a video-on-demand server, with delivery
for each client to begin immediately. To reduce the potentially heavy
tra-c burden created by these media streams, it is clearly desirable to
combine streams of the same content; this can be implemented in practice
by using multicast protocols (e.g., see [28]). With a multicast protocol
This work is supported by the NSF Grant No. 0092113.
Journal of Algorithms, 2002, to appear.
CILOVI
in place, a stream sent to a client can be received by all other clients at
a minimal possible usage of network resources. To see how this can be
done and still preserve immediate-start delivery, we need the following as-
sumptions: clients can receive two streams in parallel and each has a cache
for buering stream content. Although multimedia streaming embraces
video, audio, and data streaming, we will stay with video terminology for
simplicity.
The basic idea of stream merging can be explained on the following
example. Consider a situation in which (i) client C 1 arrives at t 1 and
requests a video of duration L and (ii) client C 0 is currently playing the
same video from a stream S 0 that began at time t 0 < t 1 . Client C 1 missed
the rst := t 1 t 0 time units of the video from S 0 and that part of the
video needs to be sent to C 1 by the server in stream S 1 . However, C 1 can
make use of stream S 0 by buering its content for later playback. In that
way the stream S 1 can be terminated after time units. This process is
called stream merging; in the present case, S 1 was discontinued after being
"merged" at time with the earlier starting
Note that the total streaming time has been reduced from 2L; with no
merging, to a minimum achievable value of streaming
time is a simple and eective measure of bandwidth consumption that we
will retain throughout the paper.
merging becomes much more involved as we increase the number
of streams that are candidates for merging, because then the number of
ways in which merging can be done also increases. For example, consider
the case of three clients C 0 , C 1 , and C 2 arriving at times t 0 <
and initiating streams S 0 , S 1 , and S 2 for a video of duration L. Let
be the interarrival times. Figure 1 illustrates an example in which
the t i 's are given by 0, 3 and 4, and Consider the ways in which
we can merge the streams for all three clients. For the given setup, the two
possible merging patterns are shown in Fig. 1. In Fig. 1(a), S 1 and S 2 are
merged independently with S 0 as described earlier: C 1 caches S 0 during
during at the end of the
respective intervals S 1 and S 2 are merged with
The second possibility is rst to merge S 2 with S 1 and then S 1 with S 0 .
This scenario is illustrated in Fig. 1 (b). Figure 2 breaks down Fig. 1 (b)
into the individual schedules for C 1 and C 2 . Client C 1 plays S 1 and caches
during its buer which is only
fed by S 0 during the last L 2 1 time units of the video. Client C 2 caches
and plays from S 2 during at which point S 2 is discontinued,
and play proceeds from C 2 's buer. Client C 2 continues to cache S 1 ; but in
addition, it caches the remainder of S 0 (in a suitably chosen region of the
cache where the two buering operations can not overlap). This continues
2 at which point S 1 is shut down and S 0 becomes the only
(a) (b)
FIG. 1. Stream merging examples. The position of the video runs diagonally. The
x-axis represents time. By following the zig-zag lines one obtains which part of the video
is being played from which stream. The dashed lines show where the play of the video
changes from one stream to another.
schedule C schedule
FIG. 2. Individual schedules for the clients C 1
and C 2
in Fig. 1 (b).
active stream while it is supplying the last L units of
the video to the buer of C 2 . In this process, C 2 has played the rst 2
time units of the video directly from S 2 , the next time units from
a cached segment of S 1 and the last L time units from a cached
segment of S 0 .
At any given time, a vertical line in Fig. 2 crosses each of the streams
currently being received. Accordingly, in the schedules for C 1 and C 2 the
bold lines incident to the vertical lines at time t indicate that the buer
content at time t consists of the corresponding segments of S 0 and S 1 .
CILOVI
Note that, although the streaming at C 1 is the same as in the rst merging
example, does not terminate at time no longer needed
by the media server must still send S 1 to C 2 until C 2 can switch to
which occurs at in order to
facilitate the requirements of C 2 . Without such an extension C 2 would not
be in a position to receive all parts of the movie. Note also that the cost
(sum of stream durations) of the second merging pattern is 16 as compared
to the cost 17 of the rst pattern. In general, the best merge pattern for an
arrival at time t depends not only on arrival times before t; but also on the
arrival times after t. As it will become clear in the next section, for this
example the solution, that our dyadic tree algorithm yields, corresponds to
Figure
(b).
The technique of stream merging originated with Eager, Vernon, and Zahorjan
[11, 12] as a model of the pyramid broadcasting scheme introduced
by Viswanathan and Imielinski [36,37]. This paradigm provides the multi-cast
basis for sharing streams and is built upon the assumption that clients
can receive more bandwidth than they need for play-out. The skyscraper
broadcasting scheme [15,22,31] is another example of these new techniques.
A number of related techniques go under the names of batching [1, 9, 10],
patching [6, 16, 21], tapping [7, 8], and piggy-backing [2, 18, 19, 29] and the
general problem has several parameters and useful performance metrics.
Other parameters include delay guarantees, receiving bandwidth, server
bandwidth, and buer size [5, 13{15, 17, 20, 23{27, 30{35]. The maximum
number of streams is another metric that is of greater interest in certain
circumstances. In this setting, the algorithm of this paper has the properties
It is on-line, i.e., the media server does not know arrival times in advance
It gives a zero-delay guarantee, i.e., all video requests are satised
immediately.
It restricts the number of streams being received by a client at any one
time to at most two { the receive-two model.
The buer size can accommodate up to half of the video.
The last two assumptions are justied in the papers by Bar-Noy and Ladner
[3, 4], which supply the primary motivation for the work here. In
particular, most of the improvement of merging streams is already present
in the receive-two model. The L=2 buer size limit comes about because
our algorithm does not attempt merging with an existing stream that is
already at least half over. As Bar-Noy and Ladner argue, this convenience
does not lead to increased average cost even for only moderately large arrival
rates. For further discussion of the literature on stream merging, we
refer the reader to the mini-survey of [3].
Many excellent numerical/experimental studies have appeared in the
stream-merging literature, but the absence of mathematical foundations
has stood out, at least until the work in [3, 4], which focuses on compet-
itive, or worst-case, analysis. Here, we give what appears to be the rst
rigorous average-case analysis of a near-optimal algorithm.
The paper is organized as follows. In Section 2 we present the dyadic
tree algorithm and state our main results. Section 3 contains numerical
experiments that verify the algorithm's performance and conclusions. The
proofs of the main results can be found in Section 4.
2. ALGORITHM AND RESULTS
The problem of stream merging can be posed as a problem on trees
(see [3, 4]). A merge tree is a representation of a stream merging diagram,
such as those shown in Figure 1. Each stream of the merging diagram
corresponds to a node in the associated merge tree. Thus, the number of
nodes in the merge tree is equal to the number of requests placed with the
server, i.e., the number of clients. If stream S j is merged directly to an
earlier starting stream S i ; then the node associated with S j is a child of
the node associated with S i . It is convenient to label the nodes with the
arrival times of the corresponding streams.
A root stream is merged with no other stream, i.e., it is the root in a
merge tree. The length of the root stream is always the full length of the
video, L. The start rule below provides a simple way to determine which
streams are roots, i.e. it denes a sequence of merge trees. Let t
the stream starting times.
Start rule: Node t 0 is a root. If t i is a root, then t
L=2g is a root.
In other words, the start rule says that a node will be in a given tree
only if the root stream of that tree started less the L=2 time units ago. As
noted earlier, this constraint simplies the algorithmics; there is a sacrice
in e-ciency, but only when tra-c is low. For example, suppose we have a
root stream starting at time t 0 and an arrival at time t 1 with
made a descendant of t 0 ; then no other node arriving
after can be merged with t 1 without extending its length to L.
Hence, some gain is achieved only if there are no arrivals in the interval
However, this is an unlikely scenario under high tra-c
load.
When the arrivals are Poisson, the sequence of merge trees becomes a
renewal process. This fact allows us to focus our analysis on a single merge
tree rooted at t 0 . Let ft n g 1
n=0 be a sample path of a Poisson process with
rate on the non-negative reals, and assume for convenience that t
6 COFFMAN, JELENKOVI
CILOVI
I
I
I 14
I 2
I 5FIG. 3. Dyadic partition of the interval.
The total length of all streams in a merge tree is dened as
l n 1ft n < L=2g; (1)
where l n denotes the length of the stream initiated by the arrival at time
the indicator function 1fAg is equal to 1 if A is true and 0 otherwise.
By denition l L. The quantity T will measure the eectiveness of
stream merging algorithms.
Our new stream merging algorithm is implicit in the following algorithm
for constructing merge trees from a given root.
The Dyadic Tree Algorithm: The input is a sequence of n > 0 arrival
times and the output is a tree of n nodes. The
arrival at time 0 determines the root. To nd the children of the root, rst
divide the interval [0; L=2) into dyadic subintervals I
with lengths shown in Figure 3. If I i contains at
least one arrival time, then t (i) denotes the earliest such time; otherwise,
Each t (i) > 0 is made a child of the root. Then for each t (i) > 0;
the algorithm is applied recursively to the interval [t (i) to determine
the subtree rooted at t (i) .
It is not di-cult to verify that this can be formulated as an on-line
algorithm, as we show at the end of this section. In particular, the decision
as to where a node t i should be attached to an existing tree is unaected by
arrivals after time t i . The following theorem gives our rst result, a uniform
bound on total stream length. We postpone the proof until Section 4.
Throughout the paper we use log to denote log 2 .
Theorem 2.1. The total cost of the dyadic tree algorithm satises4 L log(L)4 L ET (L;
Furthermore, it can be shown that the upper bound of the preceding
theorem is asymptotically tight for large values of L. A detailed proof of
the next theorem is given in Section 4.
Theorem 2.2. The total cost of the dyadic tree algorithm satises
lim
Observe that, by Theorem 2.2, the long-time ratio of the expected total
stream length under the dyadic algorithm to that under an algorithm with
no merging is asymptotically equal to 3 log(L)=(2L).
Here we point out that, by Lemma 1 of [3], the length l of the non-root
stream initiated at time t > 0 is given by
where t p is its parent and t l is the last stream that merges with it. If t is
a leaf then t l = t, i.e.,
In order to consider the worst-case performance we examine a slightly
dierent model. This modication is necessary owing to the fact that in
the original model the number of requests in [0; L=2) is unbounded, so that
the worst case performance is meaningless. Let time be slotted and let the
video have a length of 2n time slots. We assume that in each of the slots at
most one stream can be initiated. According to the start rule a merge tree
is being built on n slots. The total stream length achieves its maximum
when a stream is initiated in every time slot. In [4] it is proved that the
worst-case performance of the optimal algorithm is (n log n).
Let T (2n) denote the total stream length for the worst-case merge tree
built on n slots, 1. It is easy to show by induction that T (n)
is monotonic in n; hence, one can assume that n is a power of 2. Next,
consider two merge trees built on n=2 slots each, i.e.,
1. The key fact is that in these two cases only the lengths of
streams initiated in the 0th and n=2-th slot dier. This follows from the fact
that the length of the stream initiated at t depends only on t and starting
times of the parent stream and the last stream that merges with it (see (2)).
In the rst case the lengths of streams initiated at are
2n and 3n=2, respectively. In the second case the lengths are equal to
n. Thus, the dierence is 3n=2 and one obtains T
The solution to this recurrence has the form T n). Thus, the
dyadic algorithm is within a constant factor of optimal in the worst-case. A
more detailed numeric comparison of the dyadic algorithm and the optimal
algorithm is made in the next section.
We conclude this section with a straightforward on-line implementation
of the algorithm.
8 COFFMAN, JELENKOVI
CILOVI
On-line Dyadic Stream Merging: Let S be a stack with push and pop
operations dened for triples of numbers (t a Each triple corresponds
to a stream: t a is the time at which the stream was initiated, t r is the time
after which newly arrived streams will not be allowed to merge with it and
t e is the time when the stream terminates.
push the root triple (0; L=2; L) onto S. At time t < L=2 of
a new request:
1. pop the triples (t a t; at this point let
be the top of the stack,
2. for all but the root triple in S increase the last component by 2t
is the arrival time of the parent of
3. add the new stream to the stack by performing push (t; t
a g; the stream
started at t is the child of the stream started at ^ t a .
This procedure uniquely and explicitly denes the merge tree as well as the
stream termination times.
3. NUMERICAL RESULTS AND CONCLUSIONS
This section provides a numerical validation of the asymptotic approxi-
mation
The rst example investigates the dependency of the total cost on the
length of the stream for xed values of the arrival rate . The parameter
values are set within the regions that are plausible for real-life systems. In
particular, we set plot the ratio ET=T 0 in
Figure
4, where ET is obtained by simulating 10,000 trees for each set of
values. Points marked with "o", "+" and "x" correspond to 1 equal to
5, 20, and 60 sec, respectively. Note that for the
merge tree consists of only 11 nodes on average.
In the second example we x L and look at ET (; L) as a function of the
rst argument. The simulation results of ET=T 0 are plotted in Figure 5. As
in the previous case we simulated 10,000 trees for each point. Values of L
are set to 120, 60, and 30 min and denoted respectively by the symbols "o",
"+", and "x". Using approximation T 0 with the appropriate multiplicative
factor yields excellent engineering estimates for all reasonable values of L
and .
Finally, we compare the performance of the dyadic tree algorithm to the
performance of the optimal o-line algorithm. The cost of the latter can be
2.5 30.10.30.50.70.9Length of the stream, hours
FIG. 4. ET=T 0 as a function of the stream length for three values of the arrival
rate. Expected interarrival times are 5 sec ("o"), 20 sec ("+") and 60 sec ("x").
FIG. 5. ET=T 0 as a function of the arrival rate for three values of the stream
length. The stream length is set to 120 min ("o"),
("x").
determined by a dynamic program (see [2]). Let T opt (i; j) be the optimal
cost of the merge tree for streams initiated at 0 t i < < t j < L=2. The
optimal merge tree satises the preorder traversal property [4] and, hence,
1kn
CILOVI
Increase
in
cost,
FIG. 6. Performance of the algorithm in comparison with the optimal o-line
algorithm. The length of the stream is equal to 2 hours.
with T opt (i; L. The last term represents the gain from a merge of
optimal trees rooted at t 0 and t k . We used the fact that the length of the
stream t is given by (2).
For numerical comparison, let the length of the video be 2 hours and let
the value of the expected interarrival time vary from 5 sec to 60 sec in steps
of 5 sec. For every pair (; L) we simulated 1,000 trees and based on that
computed the average cost for two algorithms. The increase in expected
cost when using the dyadic tree algorithm instead of the optimal o-line
algorithm is rather small as shown in Figure 6. For all parameter values
the increase did not exceed 8%.
In summary, we have been able to prove the tight asymptotic average-case
behavior 3
4 L log(L) for the dyadic stream merging algorithm, and
to show in addition that its average-case and worst-case performance are
comparable to those of the best on-line algorithms known to date.
4. PROOFS
We start by introducing a recursive procedure for labeling the arrival
times in [0; L=2). For the purposes of the proof these labels replace the t i
labels. The procedure can be thought of as a function EL : T 7! ! that
maps a set T of arrival times to the space of indices !. Each index !
consists of a number of digits equal to the depth of the node in the merge
tree that corresponds to the given arrival. In general,
2:::, and the parent of the node labeled ! is a
FIG. 7. An illustration of the labeling algorithm. In this example there are seven
points that need to be labeled. On the rst call of the procedure three points are assigned
labels (1,2 and 4). The recursive algorithm is applied until all points are labeled.
node labeled with the prex . The algorithm labels the
arrivals as follows. The interval [0; L=2) is divided into dyadic intervals in
increasing order from the root as shown in Figure 3. If a point t is the rst
point in the subinterval I i then its label is i. Label the rest of the points
in [t; 2 i L) recursively by using the parent's label as a prex for childrens'
labels. An example of how the points are labeled is shown in Figure 7.
4.1. Proof of Theorem 2.1
Lower bound: By applying the above labeling procedure, it is not hard to
verify that (1) becomes
l n 1ft n <
l
where l !1 :::! n is the length of the stream starting at the point labeled
. If for a particular realization of the Poisson process there is no
point with label
Next, we estimate the expected values of l !1 :::! n . Let ; fn g 1
n=1 be a set
of i.i.d. exponential random variables with mean 1 , and consider rst
the streams that are children of the root, i.e., the streams whose indices
consist of a single digit. Given that, for a particular realization of the
Poisson process, there exists a stream with label ! 1 , its length must be at
least 2 !1 L=2 according to (2). Therefore,
l !1 L=2
L=2
CILOVI
taking into account the memoryless property
of the Poisson process, we conclude that
L=2
A node with label of form a child of the node with label ! 1 . Considering
the preceding inequality, the recursive nature of the merging algorithm
and the size of the problem in which node ! 1 is the root one obtains
L=2
L=2
The recursive structure of the merging algorithm shows that for a stream
with an arbitrary index
with the understanding that the sum in the above expression is equal to
zero if
and, hence, the expected value of an individual stream length is further
lower bounded by
L=2
Now observe that the number of indices with a digit sum equal to k is
since the above sum is equal to the number of ways one can partition a set
of cardinality k. Rearrange the sum in (3), use the bound (4) and apply (5)
to nd
L +X
L=2
L +X
L=2
where the last step follows from Jensen's inequality. Finally, simple manipulations
of the preceding bound yield
blog L
Llog(L) L
from which we conclude that the lower bound holds.
Upper bound: Consider the streams that are children of the root. For such
streams we have by (2)
l !1 3 L=2
The inequality is tight when there is an arrival right after time 2 !1 L=2
and an arrival just before time 2 !1 L. Next we examine the streams that
can be reached from the root in exactly two steps. An upper bound on
their length is
l !1!2 3
whereupon the memoryless property of the Poisson process gives
14 COFFMAN, JELENKOVI
CILOVI
Note that (6) and (7) are of the same form. In the rst inequality the size
of the problem is L=2 while in the second the size is (2 !1 L=2 infft n
. Since the merging algorithm is recursive, for
streams that have depth n 2 in the merge tree one can conclude that
3E
L=2
Recall (5) in order to verify that the number of indices with the digit sum
k and last digit i is equal to 2 k
The length of the root stream is always L so (3), (6), (8) and (9) yield
L=2
A simple computation shows that E (1
therefore, by changing the order of summation and setting
one obtains
Finally, straightforward but tedious calculations show
which in conjunction with bound (10) and the monotonicity of the function
log(L)je
This concludes the proof.
4.2. Proof of Theorem 2.2
The upper bound is a direct consequence of Theorem 2.1. Below we
provide the proof of the lower bound. Let P P (;
the probability of having at least one Poisson arrival in an interval
of length . By conditioning on an arrival in both (2
and one obtains from (2)
3L=2
Extending the above reasoning to the streams with two-digit labels yields
a lower bound on their expected lengths
3L=2
In the above inequality we conditioned on the position of the stream
its parent and the last stream that will merge to it. Due to the recursive
structure of the algorithm, for a stream with an arbitrary label
the lower bound has the following form
3L=2
CILOVI
Next, the preceding inequality, (1) and (5) result in
3L=2
6
3L
Lblog(L)c 6L:
Finally, setting log log(L) and using log e > 1 yield
lim
log(L)
and, therefore,
blog(L)c
log(L)log log(L)
log(L)
as L !1:
This concludes our proof.
ACKNOWLEDGMENT
The authors would like to thank the anonymous reviewer for helpful comments.
--R
On optimal batching policies for video- on-demand storage servers
On optimal piggyback merging policies for video-on-demand systems
Competitive on-line stream merging algorithms for media-on-demand
Optimizing patching performance.
Improving video-on-demand server e-ciency through stream tapping
Improving bandwidth e-ciency of video-on-demand servers
A periodic broadcasting approach to video-on-demand service
Dynamic batching policies for an on-demand video server
Minimizing bandwidth requirements for on-demand data delivery
Optimal and e-cient mergind schedules for video-on-demand servers
Optimized regional caching for on-demand data delivery
Dynamic skyscraper broadcasts for video-on-demand
Supplying instantaneous video-on-demand services using controlled multicast
Reducing I/O demand in video-on- demand storage servers
Adaptive piggybacking: A novel technique for data sharing in video-on-demand storage servers
Exploiting client bandwidth for more e-cient video broadcast
Patching: a multicast technique for true video- on-demand services
Skyscraper broadcasting: A new broadcasting scheme for metropolitan video-on-demand systems
Fast broadcasting for hot video access.
Harmonic broadcasting for video-on-demand service
Staircase data broadcasting and receiving scheme for hot video service.
Enhancing harmonic data broadcasting and receiving scheme fo popular video service.
Fast data broadcasting and receiving scheme for popular video service.
Computer Networking: A Top-Down Approach Featuring the Internet
Merging video streams in a multimedia storage server: complexity and heuristics.
Data broadcasting and seamless channel transition for highly-demanded videos
Pyramid broadcasting for video-on-demand ser- vice
Metropolitan area for video-on-demand service using pyramid broadcasting
--TR
Reducing I/O demand in video-on-demand storage servers
Dynamic batching policies for an on-demand video server
Adaptive piggybacking
On optimal piggyback merging policies for video-on-demand systems
Metropolitan area video-on-demand service using pyramid broadcasting
Skyscraper broadcasting
Merging video streams in a multimedia storage server
<italic>Patching</italic>
Improving bandwidth efficiency of video-on-demand servers
Zero-delay broadcasting protocols for video-on-demand
Optimal and efficient merging schedules for video-on-demand servers
Catching and selective catching
An efficient bandwidth-sharing technique for true video on demand systems
Competitive on-line stream merging algorithms for media-on-demand
Computer Networking
Dynamic Skyscraper Broadcasts for Video-on-Demand
Fast broadcasting for hot video access
Supplying Instantaneous Video-on-Demand Services Using Controlled Multicast
A Low Bandwidth Broadcasting Protocol for Video on Demand
Exploiting Client Bandwidth for More Efficient Video Broadcast
On Optimal Batching Policies for Video-on-Demand Storage Servers
Video-on-Demand Server Efficiency through Stream Tapping
--CTR
Marcus Rocha , Marcelo Maia , talo Cunha , Jussara Almeida , Srgio Campos, Scalable media streaming to interactive users, Proceedings of the 13th annual ACM international conference on Multimedia, November 06-11, 2005, Hilton, Singapore
Marcelo Maia , Marcus Rocha , talo Cunha , Jussara Almeida , Srgio Campos, Network bandwidth requirements for optimized streaming media transmission to interactive users, Proceedings of the 12th Brazilian symposium on Multimedia and the web, November 19-22, 2006, Natal, Rio Grande do Norte, Brazil
Wun-Tat Chan , Tak-Wah Lam , Hing-Fung Ting , Prudence W. H. Wong, On-line stream merging in a general setting, Theoretical Computer Science, v.296 n.1, p.27-46, 4 March
Amotz Bar-Noy , Richard E. Ladner, Competitive on-line stream merging algorithms for media-on-demand, Journal of Algorithms, v.48 n.1, p.59-90, August
Raj Kumar Rajendran , Dan Rubenstein, Optimizing the quality of scalable video streams on P2P networks, Computer Networks: The International Journal of Computer and Telecommunications Networking, v.50 n.15, p.2641-2658, October 2006 | video-on-demand;average-case analysis;stream merging |
590354 | Symbolic representation of user-defined time granularities. | In the recent literature on time representation, an effort has been made to characterize the notion of time granularity and the relationships between granularities. The main goals are having a common framework for their specification, and allowing the interoperability of systems adopting different time granularities. This paper considers the mathematical characterization of finite and periodic time granularities, and investigates the requirements for a user-friendly symbolic formalism that could be used for their specification. Instead of proposing yet another formalism, the paper analyzes the expressiveness of known symbolic formalisms for the representation of granularities, using the mathematical characterization as a reference model. Based on this analysis, a significant extension to the collection formalism defined in [15] is proposed, in order to capture a practically interesting class of periodic granularities. | Introduction
There is a wide agreement in the AI and database
community on the requirement for a data/knowledge
representation system of supporting standard as well
as user-defined time granularities. Examples of standard
time granularities are days, weeks, months,
while user defined granularities may include business-
weeks, trading-days, working-shifts, school-terms,
with these granularities having different definitions in
different application contexts. The work in [3, 4] represents
an effort to formally characterize the notion of
time granularity and the relationships between granu-
larities, in order to have a common framework for their
specification and to allow the interoperability of systems
adopting different time granularities. The formal
definition, however, is essentially a mathematical characterization
of the granules, and it is not suitable for
presentation and manipulation by the common user.
The goal of this paper is identifying an intuitive formalism
which can capture a significant class of granularities
within the formal framework and which is
closed for this class with respect to the operations it
allows. This class can be intuitively described as containing
all finite granularities, as well as all periodical
ones. Instead of inventing yet another symbolic formalism
for this purpose, in this work we consider some
existing proposals, analyzing their expressiveness with
respect to our goal.
A symbolic formalism, based on collections of
temporal intervals, was proposed in [11] to represent
temporal expressions occurring in natural language
and used in several application domains like appointment
scheduling and time management. This formalism
has been adopted with some extensions by many
researchers in the AI [9, 15, 6] and Database area
[8, 5]. From the deductive database community, a
second influential proposal is the slice formalism introduced
in [14], and adopted, among others, in [2].
None of these formalisms and extensions seems to
have the expressive power we are seeking, despite
some of the proposals include features that go beyond
what is needed in our framework. For example, [6]
provides a powerful formalism to represent calendars
and time repetition, including existential and universal
quantification, which supports the representation
of uncertainty, a feature not considered in our frame-
work. Moreover, some calendar expressions in [6] go
beyond the specification of granularities, as defined in
[4, 3] and in this paper, allowing the representation of
overlapping granules of time. The formalism can represent
recurring events in the form of non-convex in-
tervals, but it does not seem to be able to represent
what in the following we call gap-granularities, where
gaps may not only occur between one granule and the
next, but also within granules. A business-month seen
as an indivisible time granule defined as the union of
all business-days within a month is an example.
Relevant work on non-convex intervals and repetition
includes [10, 13], but the emphasis in these
works is more on reasoning with qualitative relations
than on calendar expression representation. In addition
to the research cited above, significant work on
time granularity includes [16, 12, 7].
The contribution of this paper is twofold: on one
side we give results on the expressiveness of the formalisms
proposed in [11] and [14] which we identify
as the two basic approaches to symbolic representa-
tion, while, on the other side, we propose an extension
to one of these formalisms that allows to capture exactly
the class of finite and infinite periodical granularities
we defined in [3].
In the next section we introduce the formal notion
of time granularity. In Section 3 we briefly describe
the collection and slice symbolic representation
formalisms, and we evaluate their expressiveness
and formal properties. In Section 4, we propose an
extension to the collection formalism to capture gap-
granularities, and we conclude the paper in Section 5.
Appendix
A summarizes the syntax of the symbolic
formalism, and Appendix B contains the proofs of the
results in the paper.
2. Characterization of time granularities
In this section we introduce the mathematical
characterization of time granularities as proposed in
[4] and further refined and summarized in [3]. Granularities
are defined with respect to an underlying time
domain, which can be formally characterized simply
as a set whose elements are ordered by a relation-
ship. For example, integers (Z; ), natural numbers
rational (Q; ), and real numbers (R; ) are
all possible choices for the time domain.
granularity is a mapping G from the
integers (the index set) to subsets of the time domain
such and G(j) are non-
empty, then each element of G(i) is less than all elements
of G(j), and (2) if
are non-empty, then G(k) is non-empty.
The first condition states that granules in a granularity
do not overlap and that their index order is the
same as their time domain order. The second condition
states that the subset of the index set that maps to
non-empty subsets of the time domain is contiguous.
While the time domain can be discrete, dense, or con-
tinuous, a granularity defines a countable set of gran-
ules, each one identified by an integer. The index set
can thereby provide an "encoding" of the granularity
in a computer.
The definition covers standard granularities like
Days, Months, Weeks and Years, bounded granularities
like Years-since-2000, granularities with
non-contiguous granules like Business-Days, and
gap-granularities, i.e., granularities with non-convex
intervals as granules like Business-Months. As
an example of the encoding, Years-since-2000
can be defined as a mapping G, with G(1) mapped
to the subset of the time domain corresponding to the
year 2000, G(i + 1) to the one corresponding to the
year 2001, and so on, with
Independently from the integer encoding, there
may be a "textual representation" of each non-empty
granule, termed its label, that is used for input and
output. This representation is generally a string
that is more descriptive than the granule's index
(e.g.,"August 1997", "1/2/2000", etc.
Among the many relationships between time
granularities (see [4]), the following defines an essential
concept for this paper.
periodical with respect
to a granularity G if
1. For each i 2 Z there exists a (possibly infi-
nite) subset S of the integers such that
2. There exist R; P than the
number of non-empty granules of H , such that for
j2S G(j) and H(i+R) 6=
The first condition states that any non-empty
granule H(i) is the union of some granules of G;
for instance, assume H(i) is the union of the granules
The periodicity property
(condition 2 in the definition) ensures that the R th
granule after H(i), i.e., H(i non-empty, is
the union of G(a
This results in a periodic "pattern" of the composition
of R granules of H in terms of granules of G. The pattern
repeats along the time domain by "shifting" each
granule of H by P granules of G. P is also called the
"period" of H . The condition on R enforces that at
least one granule of H is a periodic repetition of another
granule.
A granularity H which is periodical with respect
to G is specified by: (i) the R sets of indexes of G
describing the non-empty granules of
H within one period; (ii) the value of P ; (iii) the indexes
of first and last non-empty granules in H , if their
value is not infinite. Then, if S are the sets
of indexes of G describing
spectively, then the description of an arbitrary granule
H(j) is given by 1 S
Many common granularities are in this kind
of relationship, for example, Years is periodical
with respect to both Days and Months.
Business-Months is periodical with respect to
Business-Days, which in turn is periodical with
respect to Days. Most practical problems seem to require
only a granularity system containing a set of time
granularities which are all periodical with respect to a
basic granularity. Usually Days, Hours, Seconds
or Microseconds take this role, depending on the
accuracy required in each application context. In this
paper, for simplicity, we assume there is a fixed basic
granularity covering the whole time domain.
Definition 3 We say that a granularity G is periodical
if it is periodical with respect to the basic granularity.
In
Figure
we represent the whole set of granu-
larities, according to Definition 1, partitioned in two
main subsets: those having all granules with contiguous
values (NO-GAP) and those admitting granules
with non-contiguous values (GAP). The inner
circle identifies finite and periodical granularities: finite
granularities are divided (dash line) into finite irregular
and finite periodical 2 while infinite periodical
granularities are divided into those having a first
non-empty granule and no last granule (INFINITE-R),
those having a last non-empty granule and no first
granule (INFINITE-L), and those infinite on both sides
(INFINITE). This classification will be useful when
considering the expressive power of symbolic formalisms
3. Two approaches to symbolic representa-
tion
In this section we first remind the syntax and semantics
of collection and slice formalisms, and then
analyze their expressiveness with respect to the class
of periodical granularities.
3.1. Collections and slices
The temporal intervals collection formalism was
proposed in [11]. A collection is a structured set of intervals
where the order of the collection gives a measure
of the structure depth: an order 1 collection is
This formula is correct provided that no granule of H is empty,
but it can be easily adapted to the case with finite index for first and
last non-empty granules.
2 Despite this formal distinction, finite granularities will be
treated uniformly in the results.
GAP NO-GAP
INFINITE-R
INFINITE-L
Figure
1. A classification of time granularities
an ordered list of intervals, and an order n (n ? 1)
collection is an ordered list of collections having order
Each interval denotes a set of contiguous
moments of time. For example, the collection
of Months, where each month is represented as the
collection of days in that month, is a collection of order
2. In order to provide a user-friendly representation
of collections, the authors introduce two classes
of operators on collections and the notion of calen-
dar, as a primitive collection. A calendar is defined as
an order 1 collection formed by an infinite number of
meeting 3 intervals which may start from a specific one.
The two classes of operators are called dicing and slic-
ing. A dicing operator allows to further divide each
interval within a collection into another collection.
For example, Weeks:during:January1998 divides
the interval corresponding to January1998
into the intervals corresponding to the weeks that are
fully contained in that month. Other dicing operators
are allowed, adopting a subset of Allen's interval
relations [1]. Slicing operators provide means
of selecting intervals from collections. For example,
selects
the first and last week from those identified by the dicing
operator above. In general, slicing can be done using
a list of integers, as well as with the keyword the,
which identifies the single interval of the collection (if
it is single), and the keyword any, which gives non-deterministically
one of the intervals. Collection expressions
can be arbitrarily composed using these two
classes of operators starting from calendars, which are
explicitly specified either by a periodic set of intervals,
or as a grouping of intervals from previously defined
meets interval I 2
if I 2
starts when I 1
finishes.
calendars.
The slice formalism was introduced in [14] as an
alternative to the collection formalism in order to have
an underlying evaluation procedure for the symbolic
expressions. It is based on the notions of calendar
and slice. Similarly to the collection formalism, calendars
are periodic infinite sets of consecutive inter-
vals, but there is no first nor last interval. Intervals in
a calendar are indexed by consecutive integers. Once
a basic calendar is given in terms of the time domain,
other calendars can be defined dynamically from existing
ones by the construct Generate(sp; C; l
which generates a new calendar with m intervals in
each period, the first one obtained grouping l 1 granules
of calendar C, starting from C(sp), the second
grouping the successive l 2 granules, and so on, with
treated as a circular list. A calendar C 1 is
a subcalendar of C each interval of
C 2 is exactly covered by finite number of intervals
of C 1 . Weeks, Days, Months are calendars with
DaysvMonths, DaysvWeeks, Weeks6vMonths.
A slice is a symbolic expression built from calendars
and denoting a (finite or infinite) set of not necessarily
consecutive intervals. It has the form
where the sum identifies the starting points of the intervals
and D their duration. Each C i is a symbol denoting
a calendar and O i is either a set of natural numbers
or the keyword all. If the sum is simply O 1 :C 1 , it denotes
the starting points of the intervals of C 1 whose
index belongs to O 1 , or the starting points of all intervals
all. If the sum is
On :Cn with On = fon g it denotes the starting points
of the on -th interval of Cn following each point in
. For example, the sum all.Years
denotes the set of
points corresponding to the beginning of the first day
of February and April of each year. The duration D
has the form h:C d where C d is a symbol denoting a
calendar such that C d v Cn , and h is the number
of successive intervals of C d specifying the duration.
Hence, the slice all.Years
f1g.Days . 2.Days denotes a set of intervals
corresponding to the first 2 days of February and April
of each year.
3.2. Expressiveness and relationships
Both collections and slices essentially characterize
periodic sets. Similarly to granularities, even in
these formalisms there is the notion of a basic cal-
endar, which defines the finest time units in the do-
main. Without loss of generality, in the following of
the paper we assume that this basic calendar (denoted
by C) is the basic granularity we mentioned in Section
2. A period, in terms of C can be associated
with each slice expression S as well as with any collection
expression E. Intuitively, the period indicates
the number of instants of C after which the same pattern
of intervals denoted by the expression is repeat-
ing; each interval in a period can be obtained by a constant
shift of the corresponding interval in another pe-
riod. If C are the calendars appearing in the
expression, then the period is the least common multiple
of P eriod(C i =C). Technically, P eriod(C i =C), is
defined as
is a list of integers, each one denoting
the duration of an interval of C i in terms of
returns
the j th element of the list, and length(list) returns
the number of elements in the list. For example,
and, hence, P
We now consider the expressiveness of slice expressions
with respect to the formal notion of granularity
introduced in Section 2. If all the intervals denoted
by a slice S are disjoint, we call S a disjoint slice. We
also say that a granularity G is equivalent to a slice S,
if each granule of G is formed by the union of a set of
granules of the basic granularity (C) and this set is represented
by one of the intervals denoted by the slice;
moreover, each of the intervals must describe one of
these sets.
Theorem 1 Given a disjoint slice S, there exists a no-
gap finite granularity, or a no-gap infinite periodical
granularity G equivalent to S.
Technically, if
is an infinite
slice we have an algorithm to derive
the intervals f[r
is the length in terms of the basic calendar
C corresponding to h granules of C d , starting at
r i . These intervals are the ones denoted by S within
a slice period. Then, a periodical granularity G can
be defined by taking
P eriod is the slice period in terms of C, and
C(x) for each It is shown that
G is equivalent to S. When S is finite, the same algorithm
can be easily adapted to derive all the intervals S
denotes. Then, the equivalent granularity is simply defined
explicitly mapping each granule to one of these
intervals. Disjointness ensures that the result of this
mapping is indeed a granularity.
Ignoring exceptions to leap years.
Example 1 Let S=all.Weeks
. 12.Hours be an infinite slice and Hours be
the basic calendar. The slice P eriod is 168 hours
(the number of hours in a week) and in the period
containing Hours(1) the slice denotes the
set of intervals f[25; 36]; [49; 60]g. The periodical
G, equivalent to S, is defined by taking
(the number of intervals in a period),
x=25 Hours(x) and
The following example shows that if a slice is
non-disjoint, then there is no equivalent granularity.
Example 2 Let S=all.Weeks
. 3.Days. According to the slice semantics, this expression
denotes all intervals spanning from Tuesday
to Thursday and all intervals from Wednesday through
Friday. By Definition 1, no pair of granules of the
same granularity can overlap. Hence, no granularity
can be found which is equivalent to S. 2
To understand the expressiveness of the slice formalism
with respect to granularities, we still need to
check if any granularity in the identified classes is representable
by a disjoint slice.
Theorem 2 Given a no-gap finite granularity or a
no-gap infinite periodical granularity, there exists an
equivalent slice.
The theorem states that any finite (periodical or
not) granularity can be represented by a slice, and that
the same holds for periodical granularities which are
unbounded on both sides. INFINITE-R and INFINITE-
granularities cannot be represented by a slice, since
the only way to denote an infinite set of intervals with
a slice is to have O all, and there is no way within
the slice formalism to impose a minimum or a maximum
on that set. 5
From the above results we can conclude that disjoint
slices can represent exactly the set of granularities
identified in Figure 2, while non-disjoint ones
do not represent granularities at all. Unfortunately, it
seems that there is no way to enforce disjointness by
simple syntax restrictions.
We now consider the collection formalism.
Proposition 1 Any collection E resulting from the application
of a dicing or slicing operator is such that
5 Note however, that the addition of a reference interval (bound)
to each slice, as used in [2], provides an easy extension to capture
all no-gap periodical granularities.0000000000000000001111111111111111111111110000000000001111111111111111110000000000000000001111111111111111111111110000000000111111111111111000000111111
GAP NO-GAP
INFINITE-R
INFINITE-L
Disjoint slice expressions
Figure
2. The subset of the granularities
captured by the slice formalism
any two intervals t and u contained in E are either
equal or disjoint.
Proposition 1 follows from the semantics of the
operators, and from the fact that each calendar contains
only disjoint intervals. Similarly to slices, we
say that a granularity G is equivalent to a collection
E, if each granule of G is formed by the union of the
granules of C represented by one of the intervals in
the collection; moreover, each interval in the collection
describes the composition of one of the granules
of G.
Theorem 3 Given a collection expression, there exists
an equivalent no-gap periodical or finite non periodical
granularity.
Similarly to Theorem 1, we developed an algorithm
to parse the expression, to derive its period, the
intervals it denotes within the period 6 , and lower/upper
bounds if present. Once the intervals are derived, we
have all the data that is needed to define the granularity
G, since it will have the same period, the intervals
within the period define the corresponding granules,
and the lower/upper bounds are used to impose a start-
ing/ending non-empty granule.
Example 3 Consider
E=f1/Mondays:during:Years.2000g.
This collection expression identifies an order 1
6 The intervals may be structured in a collection of order higher
than 1, but this is irrelevant with respect to the time granules that the
expression denotes.
collection that contains all first Mondays of each
year starting since Monday, January 1st 2001. We
assume Days is the basic calendar with
Saturday, Jan 1st 2000. We first have to compute
the expression period. Since Mondays is defined
as 1/Days:during:Weeks with the periods
of Days and Weeks equal to 1 and 7 respec-
tively, is the period computed for
Mondays. Similarly, since the period for Years
with respect to the basic calendar is 1461 (4 years
in Days), the whole expression period is computed
as lcm(7; years in Days).
Then, G is defined as having period
28 (the number of granules in each period),
(7/1/2002), . ,
and To obtain these intervals the
algorithm first restricts years to those after 2000, then
it represents all Mondays within those years, and in
the end it extracts the intervals corresponding to the
first Monday. 2
We also have the counterpart of Theorem 3.
Theorem 4 Given a no-gap periodical or finite non-
periodical granularity, there exists an equivalent collection
GAP NO-GAP
INFINITE-R
INFINITE-L
collection expressions
Figure
3. The subset of the granularities
captured by the collection formalism
Note that in this case, all granularities in the right
side of the inner circle of Figure 3 are captured. We
can conclude that slices and collections have incomparable
expressiveness, since slices can represent sets
of overlapping intervals, and collections can represent
INFINITE-R and INFINITE-L periodical granularities.
From the above results, it is clearly possible to translate
from one formalism to the other, when considering
expressions denoting FINITE or INFINITE granular-
ities, but it seems to be difficult to devise general rules
to translate at the symbolic level, preserving the intuitiveness
of the expression. Indeed, despite the
in slices may be intuitively interpreted as equivalent
to :during: in collections, they actually have a
different semantics.
The collection formalism has been extended with
some additional operators in [8]. In particular, control
statements if-then-else and while are introduced
to facilitate the representation of certain sets
of intervals, as for example, the fourth Saturday of
April if not an holiday, and the previous business-day
otherwise. Unfortunately, the syntax allows the user
to define collections which contain overlapping intervals
7 . This implies that there are collection expressions
in the extended formalism for which there does
not exist an equivalent granularity.
4. An extension proposal
Both the collection and slice formalisms as well
as their known extensions cannot represent gap gran-
ularities. Indeed, this requires a non-convex interval
representation for each granule which is formed by
non-contiguous instants. For example, they cannot
represent Business-Months, where each granule
is defined as the set of Business-Days within a
month, and it is perceived as an indivisible unit. We
propose an extension to the collection formalism in order
to capture the whole set of periodical granularities.
We introduce the notion of primitive collection,
which includes calendars as defined in the collection
formalism as well as order 1 collections of non-convex
intervals, where each of the intervals represents a gran-
ule. A primitive collection PC can be specified by
sp is a synchronization
point with respect to an existing calendar
is the period expressed in terms of C 0 , and X is
the set of non-convex intervals 8 identifying the position
of granules of PC within a period. The synchronization
point sp says that PC(1) will start at the same
instant as C 0 (sp).
7 For example, consider an expression representing a semester
following the last day of the month, if it is a Sunday, otherwise
the week following that day. Considering 31/5/1998 and 30/6/1998,
both the semester starting 1/6/1998 and the week starting
will be denoted, with the first properly containing the second.
8 Each x is the non-convex interval representing the i-th
granule.
Example 4 Suppose a company has 2 weekly working
shifts for its employees:
shift1=fMonday, Wednesday, Saturdayg and
shift2=fTuesday, Thursday, Fridayg. It may
be useful to consider these as two periodic
granularities, where each shift is treated as a
single time granule within a week. If Thursday
1/1/1998 is taken as Days(1), shift1 =
Generate(5; Days; 7; fh[1; 1]; [3; 3]; [6; 6]ig). In-
deed, the synchronization point is 5, since the first
granule of shift1 following Days(1) starts on
Monday January 5th 1998 which is 5 days later. C 0
is Days, the period P is 7 days and X is composed
by x which identifies the
single granule within the period, formed by the
first, third, and sixth day, starting from 5/1/1998,
and repeating every 7 days. Similarly,
Generate(6; Days; 7; fh[1; 1]; [3; 3]; [4; 4]ig) denotes
the first, third and fourth day, starting from 6/1/1998,
and repeating every 7 days. 2
The user can specify collection expressions by
arbitrarily applying dicing and slicing operators starting
from primitive collections. Since operators now
apply to non-convex intervals, we need to revise
their definition. Let t and u be non convex in-
tervals, with
an x b n g and S
be the sets of values represented
by t and u respectively. Dicing operators are
based on the following binary relations on non-convex
intervals: 9
during u iff S ' S 0
intersects u
starts u iff (a
A dicing operator :rel: takes an order 1 collection
as its left operand and an interval u
as its right operand, and it returns an order 1 collection
rel ug. If
the strict form : rel : is used, then
rel ug, i.e., only the portion
of t which is contained in u is part of the resulting
9 This set of relations is similar to the one chosen in [11] for convex
intervals. We consider it only as a good basic set which allows
the representation of most common granularities while having a simple
implementation. It can be extended to a richer set considering,
for example, the taxonomy of relations given in [10].
collection. When the right operand is a collection, instead
of a single interval, the same procedure is applied
for each of its intervals, resulting in a collection of one
order higher. A slicing operator k=E replaces each
collection contained in E with the k-th non-convex
interval in that collection, while
replaces it with the collection made of the subset of intervals
whose position in the collection is specified by
g.
Example 5 Consider the collection expression
Weeks:?:2/shift1:during:1998/Years
where shift1 was defined in Example 4. This
expression denotes all weeks following the end of
the second work-shift of 1998. Years is the order
ity, we assume the interval [1::365] corresponds
to year 1998. Then, the slicing 1998/Years
returns the interval h[1::365]i, and the dicing
shift1:during:1998/Years returns the finite
collection of order 1 composed by all the work-shifts
during 1998: fh[5; 5], [7; 7], [10; 10]i, . , h[355; 355],
360]ig. The selection of the second
of those work-shifts returns the non-convex interval
Finally, the dicing
Weeks:>:h[12; 12]; [14; 14]; [17; 17]i generates the
collection of all the weeks that start after
January 17-th, i.e., fh[19; 23]i; h[26;
We state a formal property of the proposed extension
Theorem 5 The extended collection formalism can
represent all and only the granularities which are either
periodical or finite non-periodical.
To support this result, the algorithm used in the proof
of Theorem 3 has been extended to consider non-convex
intervals. The granularities captured by the
proposed extension are shown in Figure 4.
5. Conclusions
In this paper we have considered a recently proposed
theoretical framework for time granularities and
we have analyzed two of the most influential proposals
for calendar symbolic representation. On one side,
we have shown that the theoretical framework is general
enough to capture all the sets of disjoint intervals
representable by those formalisms. On the other side
we have shown exactly which subclass of granularities
can be represented by each formalism. From this
GAP NO-GAP
INFINITE-L
INFINITE-R
extended collection expressions
Figure
4. The subset of granularities captured
by the proposed extension
analysis, we have proposed an extension of the collection
formalism which captures a well-defined and
large class of granularities, providing a good coverage
of granularities that may be found in database and temporal
reasoning applications.
We are currently working at the definition and implementation
of set operations, performed at the symbolic
level, among extended collection expressions.
This problem has interesting applications (see e.g.,
[2]) but it is not addressed in [11] and derivative work
for collections, and only briefly investigated in [14] for
slices.
--R
Maintaining Knowledge about Temporal Intervals
An Access Control Model Supporting Periodicity Constraints and Temporal Reasoning
A General Framework for Time Granularity and its Application to Temporal Reasoning
in book Database support for workflow management: the WIDE project
Expressing Time Intervals and Repetition within a Formalization of Calendars
Temporal Granularity for Unanchored Temporal Data
Implementing calendars and temporal rules in next generation databases
a taxonomy of interval relations
A representation for collections of temporal intervals
Metric and Layered Temporal Logic for Time Granularity
An efficient symbolic representation of periodic time
Reasoning About Periodic Events
The TSQL2 Temporal Query Language
for any integer s and 1
is one of those denoted by S.
is among the intervals denoted by the slice.
"!"
--TR
--CTR
Claudio Bettini , X. Sean Wang , Sushil Jajodia, Temporal Reasoning in Workflow Systems, Distributed and Parallel Databases, v.11 n.3, p.269-306, May 2002
flexible approach to user-defined symbolic granularities in temporal databases, Proceedings of the 2005 ACM symposium on Applied computing, March 13-17, 2005, Santa Fe, New Mexico
Lavinia Egidi , Paolo Terenziani, A mathematical framework for the semantics of symbolic languages representing periodic time, Annals of Mathematics and Artificial Intelligence, v.46 n.3, p.317-347, March 2006 | time granularity;knowledge representation;temporal reasoning;time representation |
590507 | Probabilistic Default Reasoning with Conditional Constraints. | We present an approach to reasoning from statistical and subjective knowledge, which is based on a combination of probabilistic reasoning from conditional constraints with approaches to default reasoning from conditional knowledge bases. More precisely, we introduce the notions of i>z-, lexicographic, and conditional entailment for conditional constraints, which are probabilistic generalizations of Pearl's entailment in system i>Z, Lehmann's lexicographic entailment, and Geffner's conditional entailment, respectively. We show that the new formalisms have nice properties. In particular, they show a similar behavior as reference-class reasoning in a number of uncontroversial examples. The new formalisms, however, also avoid many drawbacks of reference-class reasoning. More precisely, they can handle complex scenarios and even purely probabilistic subjective knowledge as input. Moreover, conclusions are drawn in a global way from all the available knowledge as a whole. We then show that the new formalisms also have nice general nonmonotonic properties. In detail, the new notions of i>z-, lexicographic, and conditional entailment have similar properties as their classical counterparts. In particular, they all satisfy the rationality postulates proposed by Kraus, Lehmann, and Magidor, and they have some general irrelevance and direct inference properties. Moreover, the new notions of i>z- and lexicographic entailment satisfy the property of rational monotonicity. Furthermore, the new notions of i>z-, lexicographic, and conditional entailment are proper generalizations of both their classical counterparts and the classical notion of logical entailment for conditional constraints. Finally, we provide algorithms for reasoning under the new formalisms, and we analyze its computational complexity. | Introduction
In this paper, we elaborate a combination of probabilistic
reasoning from conditional constraints with approaches to
default reasoning from conditional knowledge bases. As a
main result, this combination provides new notions of entailment
for conditional constraints, which respect the ideas
of classical default reasoning from conditional knowledge
bases, and which are generally much stronger than the classical
notion of logical entailment based on conditioning.
Moreover, the results of this paper can also be applied for
handling inconsistencies in probabilistic knowledge bases.
Informally, the ideas behind this paper can be described as
follows. Assume that we have the following knowledge at
hand: "all penguins are birds" (G1), "between 90 and 95%
of all birds fly" (G2), and "at most 5% of all penguins fly"
(G3). Moreover, assume a first scenario in which "Tweety is
a bird" (E1) and second one in which "Tweety is a penguin"
(E2). What do we conclude about Tweety's ability to fly?
A closer look at this example shows that the statements
G1-G3 describe statistical knowledge (or objective knowl-
while E1 and E2 express degrees of belief (or subjective
knowledge). One way of handling such combinations of
statistical knowledge and degrees of belief is reference class
reasoning, which goes back to Reichenbach (1949) and was
further refined by Kyburg (1974; 1983) and Pollock (1990).
Another related field is default reasoning from conditional
knowledge bases, where we have generic statements of the
form "all penguins are birds", "generally, all birds fly", and
"generally, no penguin flies" in addition to some concrete
evidence as E1 and E2. The literature contains several different
approaches to default reasoning and extensive work on
the desired properties. The core of these properties are the
rationality postulates proposed by Kraus et al. (1990). These
rationality postulates constitute a sound and complete axiom
system for several classical model-theoretic entailment
relations under uncertainty measures on worlds. In detail,
they characterize classical model-theoretic entailment under
preferential structures (Shoham 1987; Kraus et al. 1990),
infinitesimal probabilities (Adams 1975; Pearl 1989), possibility
measures (Dubois & Prade 1991), and world rankings
(Spohn 1988; Goldszmidt & Pearl 1992). They also characterize
an entailment relation based on conditional objects
(Dubois & Prade 1994). A survey of all these relationships
is given in (Benferhat et al. 1997). Recently, Friedman and
Halpern (2000) showed that many approaches describe to
the same notion of inference, since they are all expressible
as plausibility measures.
Mainly to solve problems with irrelevant information, the
notion of rational closure as a more adventurous notion
of entailment has been introduced by Lehmann (Lehmann
1989; Lehmann & Magidor 1992). This notion of entailment
is equivalent to entailment in system Z by Pearl (1990), to
the least specific possibility entailment by Benferhat et al.
(1992), and to a conditional (modal) logic-based entailment
by Lamarre (1992). Finally, mainly in order to solve problems
with property inheritance from classes to exceptional
subclasses, the maximum entropy approach to default entailment
was proposed by Goldszmidt et al. (1993); the notion
of lexicographic entailment was introduced by Lehmann
(1995) and Benferhat et al. (1993); the notion of conditional
entailment was proposed by Geffner (Geffner 1992; Geffner
Pearl 1992); and an infinitesimal belief function approach
was suggested by Benferhat et al. (1995).
Coming back to our introductory example, we realize
that G1-G3 and E1-E2 represent interval restrictions for
conditional probabilities, also called conditional constraints
(Lukasiewicz 1999b). The literature contains extensive
work on reasoning about conditional constraints (Dubois &
Prade 1988; Dubois et al. 1990; 1993; Amarger et al. 1991;
Jaumard et al. 1991; Th-one et al. 1992; Frisch & Haddawy
1994; Heinsohn 1994; Luo et al. 1996; Lukasiewicz 1999a;
1999b) and their generalizations, for example, to probabilistic
logic programs (Lukasiewicz 1998).
Now, the main idea of this paper is to use techniques for
default reasoning from conditional knowledge bases in order
to perform probabilistic reasoning from statistical knowledge
and degrees of beliefs. More precisely, we extend
the notions of entailment in system Z, Lehmann's lexicographic
entailment, and Geffner's conditional entailment to
the framework of conditional constraints.
Informally, in our introductory example, the statements
G2 and G3 are interpreted as "generally, a bird flies with
a probability between 0.9 and 0.95" (G2 ? ) and "generally,
a penguin flies with a probability of at most 0.05" (G3 ? ),
respectively. In the first scenario, we then simply use the
whole probabilistic knowledge to conclude
under classical logical entailment that "Tweety flies
with a probability between 0.9 and 0.95". In the second
scenario, it turns out that the whole probabilistic knowledge
precisely,
is inconsistent in the context of a pen-
guin. In fact, the main problem is that G2 ? should not be
applied anymore to penguins. That is, we can easily re-solve
the inconsistency by removing G2 ? , and then conclude
from classical logical entailment that
"Tweety flies with a probability of at most 0.05".
Hence, the results of this paper can also be used for
handling inconsistencies in probabilistic knowledge bases.
More precisely, the new notions of nonmonotonic entailment
coincide with the classical notion of logical entailment
as far as satisfiable sets of conditional constraints are con-
cerned. Furthermore, they allow desirable conclusions from
certain kinds of unsatisfiable sets of conditional constraints.
We remark that this inconsistency handling is guided by
the principles of default reasoning from conditional knowledge
bases. It is thus based on a natural preference relation
on conditional constraints, and not on the assumption that
all conditional constraints are equally weighted (as, for ex-
ample, in the work by Jaumard et al. (1991)).
The work closest in spirit to this paper is perhaps the one
by Bacchus et al. (1996), which suggests to use the random
worlds method (Grove et al. 1994) to induce degrees
of beliefs from quite rich statistical knowledge bases. How-
ever, differently from (Bacchus et al. 1996), we do not make
use of a strong principle such as the random worlds method
(which is closely related to probabilistic reasoning under
maximum entropy). Moreover, we restrict our considerations
to the propositional setting.
The main contributions of this paper are as follows:
We illustrate that the classical notion of logical entailment
for conditional constraints is not very well-suited for default
reasoning with conditional constraints.
We introduce the notions of z-entailment, lexicographic
entailment, and conditional entailment for conditional
constraints, which are a combination of the classical notions
of entailment in system Z (Pearl 1990), Lehmann's
lexicographic entailment (Lehmann 1995), and Geffner's
conditional entailment (Geffner 1992; Geffner & Pearl
1992), respectively, with the classical notion of logical
entailment for conditional constraints.
We give some examples that analyze the nonmonotonic
properties of the new notions of entailment for default reasoning
with conditional constraints. It turns out that the
new notions of z-entailment, lexicographic entailment,
and conditional entailment have similar properties like
their classical counterparts.
We show that the new notions of z-entailment, lexicographic
entailment, and conditional entailment for conditional
constraints properly extend the classical notions
of entailment in system Z, lexicographic entailment, and
conditional entailment, respectively.
We show that the new notions of z-entailment, lexicographic
entailment, and conditional entailment for conditional
constraints properly extend the classical notion of
logical entailment for conditional constraints.
Note that all proofs are given in (Lukasiewicz 2000).
Preliminaries
We now introduce some necessary technical background.
We assume a finite nonempty set of basic propositions
(or atoms) . We use ? and > to denote the propositional
constants false and true, respectively. The set of classical
formulas is the closure of [f?;>g under the Boolean
operations : and ^. A strict conditional constraint is an expression
real numbers l; u2 [0; 1] and classical
formulas and . A defeasible conditional constraint
(or default) is an expression ( k)[l; u] with real numbers
classical formulas and . A conditional
constraint is a strict or defeasible conditional constraint.
The set of strict probabilistic formulas (resp., probabilistic
formulas) is the closure of the set of all strict conditional
constraints (resp., conditional constraints) under the
Boolean operations : and ^. We use
and to abbreviate :(:F ^:G), :(F ^:G), and
(:(:F ^G))^ (:(F ^:G)), respectively, and adopt the
usual conventions to eliminate parentheses.
A probabilistic default theory is a pair
P is a finite set of strict conditional constraints and D is
a finite set of defeasible conditional constraints. A probabilistic
knowledge base KB is a strict probabilistic formula.
Informally, default theories represent strict and defeasible
generic knowledge, while probabilistic knowledge bases express
some concrete evidence.
A possible world is a truth assignment I : ! ftrue,
falseg, which is extended to classical formulas as usual. We
use I to denote the set of all possible worlds for . A possible
world I satisfies a classical formula , or I is a model
of , denoted I
A probabilistic interpretation Pr is a probability function
on I (that is, a mapping Pr : I ! [0; 1] such that all
Pr(I) with I 2 I sum up to 1). The probability of a classical
formula in the probabilistic interpretation Pr , denoted
Pr(), is defined as follows:
For classical formulas and with Pr () > 0, we use
Pr( to abbreviate Pr(). The truth of
probabilistic formulas F in a probabilistic interpretation Pr ,
denoted Pr inductively defined as follows:
Pr
Pr
Pr
Pr G.
We remark that there is no difference between strict and
defeasible conditional constraints as far as the notion of truth
in probabilistic interpretations is concerned.
A probabilistic interpretation Pr satisfies a probabilistic
formula F , or Pr is a model of F , iff Pr
a set of probabilistic formulas F , or Pr is a model of F ,
denoted Pr is a model of all F 2F . We say F
is satisfiable iff a model of F exists.
We next define the notion of logical entailment as fol-
lows. A strict probabilistic formula F is a logical consequence
of a set of probabilistic formulas F , denoted F
iff each model of F is also a model of F . A strict conditional
constraint ( j)[l; u] is a tight logical consequence
of F , denoted F is the infimum
(resp., supremum) of Pr( j) subject to all models
Pr of F with Pr() > 0 (note that we canonically define
We remark that every notion of entailment for conditional
constraints is associated with a notion of consequence and a
notion of tight consequence. Informally, the notion of consequence
describes entailed intervals, while the notion of
tight consequence characterizes the tightest entailed inter-
val. That is, if ( j)[l; u] is a tight consequence of F , then
Motivating Examples
What should a probabilistic knowledge base entail under a
probabilistic default theory? To get a rough idea on the reply
to this question, we now introduce two natural notions of
entailment and analyze their properties. It will turn out that
neither of these two notions is fully adequate for probabilistic
default reasoning with conditional constraints.
In the sequel, let D) be a probabilistic default
theory. We first define the notion of 0-entailment,
which applies to probabilistic knowledge bases of the
In detail, a strict conditional constraint
( j)[l; u] is a 0-consequence of KB , denoted
It is a tight
0-consequence of KB , denoted KB k 0
tight ( j)[l; u], iff
Informally, we use the concrete
evidence in KB to fix our "point of interest" and the
generic knowledge in T to draw the requested conclusion.
That is, we perform classical conditioning.
We next define the notion of 1-entailment, which applies
to all probabilistic knowledge bases KB . A strict probabilistic
formula F is a 1-consequence of KB , denoted
strict conditional constraint
( j)[l; u] is a tight 1-consequence of KB , denoted
tight ( j)[l; u], iff P[D[fKBg
Informally, we draw our conclusion from the union of the
concrete evidence in KB and the generic knowledge in T .
We now analyze the properties of these two notions of
entailment. Our first example concentrates on the aspects of
ignoring irrelevant information and property inheritance.
Example 1 The knowledge "all penguins are birds" and
"at least 95% of all birds have legs" can be expressed by
the following probabilistic default theory
should entail that "generally, birds have legs
with a probability of at least 0.95" (that is, e.g., if we know
that Tweety is a bird, and we do not have any other knowl-
edge, then we should conclude that the probability of Tweety
having legs is at least 0.95). Indeed, this conclusion is drawn
under both 0- and 1-entailment (see item (1) in Table 1).
should entail that "generally, yellow birds
have legs with a probability of at least 0.95" (as the property
"yellow" is not mentioned at all in T 1 and thus irrelevant),
and that "generally, penguins have legs with a probability of
at least 0.95" (as the set of all penguins is a nonexceptional
subclass of the set of all birds, and thus penguins should
inherit all properties of birds). However, while 1-entailment
still allows the desired conclusions, 0-entailment just yields
the interval [0; 1] (see items (2)-(3) in Table 1). 2
We next concentrate on the principle of specificity and the
problem of inheritance blocking.
Example 2 Let us consider the following probabilistic default
theory
(fly k bird)[:9; :95];
(fly k penguin)[0;
This default theory should entail that "generally, penguins
fly with a probability of at most 0.05" (as properties of
more specific classes should override inherited properties of
less specific classes). Indeed, 0-entailment yields the desired
conclusion, while 1-entailment reports an unsatisfiability
(see item (7) in Table 1).
Moreover, should entail that "generally, penguins have
legs with a probability of at least 0.95", since penguins are
exceptional birds w.r.t. to the ability of being able to fly, but
not w.r.t. the property of having legs. However, 0-entailment
provides only the interval [0; 1], and 1-entailment reports
even an unsatisfiability (see item (5) in Table 1). 2
The following example deals with the drowning problem
(Benferhat et al. 1993).
Example 3 Let us consider the following probabilistic default
theory
f(fly k bird)[:9; :95]; (fly k penguin)[0; :05];
(easy to see k
This default theory should entail that "generally, yellow
penguins are easy to see", as the set of all yellow penguins
Table
1: Examples of 0- and 1-entailed tight intervals.
tight
(easy to seej>) [0; 1] [1; 0]
undefined [:86; :91]
undefined [1; 0]
is a nonexceptional subclass of the set of all yellow objects.
But, 0-entailment gives only the interval [0; 1], and 1-entail-
ment reports an unsatisfiability (see item (8) in Table 1). 2
The next example is taken from (Bacchus et al. 1996).
Example 4 Let us consider the following probabilistic default
theory
This default theory should entail "generally, the probability
that magpies chirp is between 0.7 and 0.8", since we
know more about birds w.r.t. the property of being able to
chirp than about magpies. Indeed, both 0- and 1-entailment
yield the desired conclusion (see item (9) in Table 1). 2
The following example concerns ambiguity preservation
(Benferhat et al. 1995).
Example 5 Let us consider the following probabilistic default
theory
f(fly k metal wings)[:95; 1]; (fly k bird)[:95; 1];
(fly k penguin)[0;
Assume now that Oscar is a penguin with metal wings. As
Oscar is a penguin, we should conclude that the probability
that Oscar flies is at most 0.05. However, as Oscar has also
metal wings, we should conclude that the probability that
Oscar flies is at least 0.95. As argued in the literature on
default reasoning (Benferhat et al. 1995), such ambiguities
should be preserved. Indeed, 0-entailment yields the desired
interval [0; 1], while 1-entailment reports an unsatisfiability
(see item (10) in Table 1). 2
What about handling purely probabilistic evidence?
Example 6 Let us consider again the probabilistic default
theory T 2 of Example 2. Assume a first scenario in which
our belief is "the probability that Tweety is a bird is at least
0.9" and "the probability that Tweety is a penguin is at least
0.1" and a second scenario in which our belief is "the probability
that Tweety is a bird is at least 0.9" and "the probability
that Tweety is a penguin is at least 0.9". What do we
conclude about Tweety's ability to fly in these scenarios?
The notion of 0-entailment is undefined for such purely
probabilistic evidence, whereas the notion of 1-entailment
yields the probability interval [:86; :91] in the first scenario,
and reports an unsatisfiability in the second scenario (see
items (11)-(12) in Table 1). 2
Summarizing the results, 0-entailment is too weak, while
1-entailment is too strong. In detail, 0-entailment often
yields the trivial interval [0; 1] and is even undefined for
purely probabilistic evidence, while 1-entailment often reports
unsatisfiabilities (in fact, in the most interesting sce-
narios, as 1-entailment is actually monotonic).
Roughly speaking, our ideal notion of entailment for
probabilistic knowledge bases under probabilistic default
theories should lie somewhere between 0- and 1-entailment.
One idea to obtain such a notion could be to strengthen 0-
entailment by adding some inheritance mechanism. Another
idea is to weaken 1-entailment by handling unsatisfiabilities.
In the rest of this paper, we will focus on the second idea.
Probabilistic Reasoning
In this section, we extend the classical notions of entailment
in system Z (Pearl 1990), Lehmann's lexicographic entailment
(1995), and Geffner's conditional entailment (Geffner
1992; Geffner & Pearl 1992) to conditional constraints.
The main idea behind these extensions is to use the following
two interpretations of defaults. As far as default
rankings and priority orderings are concerned, we interpret a
"generally, if is true, then the probability
of is between l and u". Whereas, as far as notions
of entailment are concerned, we interpret ( k)[l; u] as "the
conditional probability of given is between l and u".
Preliminaries
A probabilistic interpretation Pr verifies a default ( k)[l;u]
It falsifies a default
set of defaults
D tolerates a default d under a set of strict conditional
constraints has a model that verifies d. A set
of defaults D is under P in conflict with d iff no model of
verifies d.
A default ranking on D maps each d 2D to a nonnegative
integer. It is admissible with iff each set of
defaults D 0 D that is under P in conflict with some default
d 2D contains a default d 0 such that (d 0 ) <(d). A probabilistic
is -consistent iff there
exists a default ranking on D that is admissible with T . It is
-inconsistent iff no such default ranking exists.
A probability ranking maps each probabilistic interpretation
on I to a member of f0;
for at least one interpretation Pr . It is extended
to all strict probabilistic formulas F as follows. If F is sat-
isfiable, then
We say is admissible with F iff (:F
It is admissible with a default ( k)[l; u] iff
Roughly speaking, the intuition behind this definition is to
"generally, if is true, then the probability
of is between l and u". A probability ranking is
admissible with a probabilistic default theory
is admissible with all F 2P and all d 2D.
System Z
We now extend the notion of entailment in system Z (Pearl
1990; Goldszmidt & Pearl 1996) to conditional constraints.
In the sequel, let D) be a -consistent probabilistic
default theory. The notion of z-entailment is linked to an
ordered partition of D, a default ranking z, and a probability
ranking z .
We first define the z-partition of D. Let (D
the unique ordered partition of D such that, for
each D i is the set of all defaults in D
that are tolerated under P by D
that we define D
call this (D the z-partition of D.
Example 7 The z-partition for the probabilistic default theory
is given as follows:
(f(legs k bird)[:95; 1]; (fly k bird)[:9; :95]g;
f(fly k penguin)[0;
We now define the default ranking z. For
each d 2D j is assigned the value j under z. The probability
ranking z on all probabilistic interpretations Pr is then
defined as follows:
z(d) otherwise.
The following result shows that, in fact, z is a default
ranking that is admissible with T , and z is a probability
ranking that is admissible with T .
Lemma 8 a) z is a default ranking admissible with T .
b) z is a probability ranking admissible with T .
We next define a preference relation on probabilistic in-
terpretations. For probabilistic interpretations Pr and Pr 0 ,
we say Pr is z-preferable to Pr 0 iff z (Pr) < z (Pr 0
A model Pr of a set of probabilistic formulas F is a z-
minimal model of F iff no model of F is z-preferable to Pr .
We are now ready to define the notion of z-entailment
as follows. A strict probabilistic formula F is a z-con-
sequence of KB , denoted KB k z F , iff each z-minimal
model of P [ fKBg satisfies F . A strict conditional constraint
( j)[l; u] is a tight z-consequence of KB , denoted
tight ( j)[l; u], iff l (resp., u) is the infimum (resp.,
supremum) of Pr( j) subject to all z-minimal models Pr
of P [ fKBg with Pr() > 0.
Coming back to Examples 1-6, it turns out that the non-monotonic
properties of z-entailment differ from the ones of
0- and 1-entailment (see Table 2).
In detail, in the given examples, z-entailment ignores irrelevant
information, shows property inheritance to globally
nonexceptional subclasses, and respects the principle
of specificity. Moreover, it may also handle purely probabilistic
evidence. However, properties are still not inherited
to more specific classes that are exceptional with respect to
some other properties. Moreover, z-entailment still has the
drowning problem and does not preserve ambiguities.
The following examples illustrate how z-entailed tight intervals
are determined.
Example 9 Given T 2 of Example 2, we get:
tight (legs j >)[0; 1]
Here, the interval "[0; 1]" comes from the tight logical consequence
Given T 5 of Example 5, we get:
tight (fly j
Here, the interval "[0; :05]" comes from the tight logical
consequence
metal wingsj>)[1; 1]g
Lexicographic Entailment
We now extend Lehmann's lexicographic entailment
(Lehmann 1995) to conditional constraints.
In the sequel, let D) be a -consistent probabilistic
default theory. We now use the z-partition (D
of D to define a lexicographic preference relation on probabilistic
interpretations.
Table
2: Examples of z-, lexicographically, and conditionally entailed tight intervals.
tight k lex
tight k ce
tight
(fly j>) [:9; :95] [:9; :95] [:9; :95]
(fly j>) [0; :05] [0; :05] [0; :05]
(easy to see j>) [0; 1] [:95; 1] [:95; 1]
(fly j>) [0; :05] [0; :05] [0; 1]
(fly j>) [:86; :91] [:86; :91] [:86; :91]
(fly j>) [0; :15] [0; :15] [0; :15]
For probabilistic interpretations Pr and Pr 0 , we say Pr
is lexicographically preferable to Pr 0 iff there exists some
for all i < j k. A model Pr of a set of probabilistic formulas
F is a lexicographically minimal model of F iff no
model of F is lexicographically preferable to Pr .
We now define the notion of lexicographic entailment as
follows. A strict probabilistic formula F is a lexicographic
consequence of KB , denoted KB k lex F , iff each lexicographically
minimal model of P [fKBg satisfies F . A strict
conditional constraint ( j)[l; u] is a tight lexicographic
consequence of KB , denoted KB k lex
tight ( j)[l; u], iff l
(resp., u) is the infimum (resp., supremum) of Pr( subject
to all lexicographically minimal models Pr of P[fKBg
with Pr () > 0.
Coming back to Examples 1-6, it turns out that lexicographic
entailment has nicer nonmonotonic features than
z-entailment (see Table 2).
In detail, in the given examples, lexicographic entailment
ignores irrelevant information, shows property inheritance
to nonexceptional subclasses, and respects the principle of
specificity. Moreover, it does not block property inheritance,
it does not have the drowning problem, and it may also handle
purely probabilistic evidence. However, lexicographic
entailment still does not preserve ambiguities.
The following examples illustrate how lexicographically
entailed tight intervals are determined.
Example 11 Given T 2 of Example 2, we get:
tight (legs j >)[:95;
Here, the interval "[:95; 1]" comes from the tight logical consequence
(fly k penguin)[0; :05],
Example 12 Given T 5 of Example 5, we get:
tight (fly j
Here, the interval "[0; :05]" comes from the tight logical
consequence
metal wingsj>)[1; 1]g
Conditional Entailment
We next extend Geffner's conditional entailment (Geffner
1992; Geffner & Pearl 1992) to conditional constraints.
In the sequel, let D) be a probabilistic default
theory.
We first define priority orderings on D as follows. A priority
ordering on D is an irreflexive and transitive binary
relation on D. We say is admissible with T iff each set
of defaults D 0 D that is under P in conflict with some default
d 2D contains a default d 0 such that d 0 d. We say T
is -consistent iff there exists a priority ordering on D that
is admissible with T .
Example 13 Consider the probabilistic default theory
2. A priority ordering on D 2
that is admissible with T 2 is given by (fly k bird)[:9; :95]
(fly k penguin)[0; :05]. 2
The existence of an admissible default ranking implies the
existence of an admissible priority ordering.
Lemma 14 If T is -consistent, then T is -consistent.
We next define a preference ordering on probabilistic interpretations
as follows. Let Pr and Pr 0 be two probabilistic
interpretations and let be a priority ordering on D. We
say that Pr is -preferable to Pr 0 iff fd 2D j Pr 6j= dg 6=
fd 2D j Pr 0 6j= dg and for each d 2D such that Pr 6j= d and
there exists some default d 0 2D such that d d 0 ,
Pr model Pr of a set of probabilistic
formulas F is a -minimal model of F iff no model of F
is -preferable to Pr . A model Pr of a set of probabilistic
formulas F is a conditionally minimal model of F iff Pr is
a -minimal model of F for some priority ordering admissible
with T .
We finally define the notion of conditional entailment. A
strict probabilistic formula F is a conditional consequence
of KB , denoted KB k ce
F , iff each conditionally minimal
model of P [ fKBg satisfies F . A strict conditional constraint
( j)[l; u] is a tight conditional consequence of KB ,
denoted KB k ce
tight ( j)[l; u], iff l (resp., u) is the infimum
(resp., supremum) of Pr( j) subject to all conditionally
minimal models Pr of P [ fKBg with Pr() > 0.
Coming back to Examples 1-6, we see that among all introduced
notions of entailment, conditional entailment is the
one with the nicest nonmonotonic properties (see Table 2).
In detail, in the given examples, conditional entailment
ignores irrelevant information, shows property inheritance
to nonexceptional subclasses, and respects the principle of
specificity. Moreover, it does not block property inheritance,
and it does not have the drowning problem. Finally, conditional
entailment preserves ambiguities and may also handle
purely probabilistic evidence.
The following examples illustrate how conditionally entailed
tight intervals are determined.
Example 15 Given T 2 of Example 2, we get:
ce
tight (legs j >)[:95;
Here, the interval "[:95; 1]" comes from the tight logical consequence
(fly k penguin)[0; :05],
Example Given T 5 of Example 5, we get:
ce
tight (fly j >)[0;
Here, the interval "[0; 1]" is the convex hull of the intervals
"[0; :05]" and "[:95; 1]", which come from the tight logical
consequences
metal wings j>)[1; 1]gj= tight (fly j >)[0; :05] and P 5 [f(fly k
bird)[:95; 1], (fly k metal wings)[:95; 1], (penguin ^ metal
wings
Relationship to Classical Formalisms
We now analyze the relationship to classical default reasoning
from conditional knowledge bases and to classical probabilistic
reasoning with conditional constraints.
logical formula is a probabilistic formula that contains
only conditional constraints of the kind ( j)[1;1] or
strict logical formula is a strict probabilistic
formula that contains only strict conditional constraints of
the form ( j)[1; 1]. A logical default theory T is a probabilistic
default theory that contains only logical formulas. A
logical knowledge base KB is a strict logical formula.
We use the operator
on logical formulas, sets of logical
formulas, and logical default theories, which replaces
each strict conditional constraint ( j)[1; 1] (resp., defeasible
conditional constraint ( k)[1; 1]) by the classical implication
Given a
logical ce ) to
denote the classical notion of z-, (resp., lexicographic, con-
ditional) entailment with respect to
The following result shows that the introduced notions of
z-, lexicographic, and conditional entailment are generalizations
of their classical counterparts.
Theorem 17 Let D) be a logical default theory
and let KB be a logical knowledge base. Then, for every
semantics s 2 fz; lex ; ceg:
The next result shows that, when the union of generic
and concrete probabilistic knowledge is satisfiable, the notions
of z-, lexicographic, and conditional entailment coincide
with the notion of 1-entailment.
Theorem D) be a probabilistic default theory
and let KB be a probabilistic knowledge base such
that P [D[ fKBg is satisfiable. Then, for every semantics
1. KB k s F iff P [D[ fKBg
2. KB k s
tight ( j)[l; u] iff P[D[fKBgj= tight ( j)[l; u].
Summary and Outlook
We presented the notions of z-entailment, lexicographic en-
tailment, and conditional entailment for conditional con-
straints, which combine the classical notions of entailment
in system Z, Lehmann's lexicographic entailment, and
Geffner's conditional entailment with the classical notion of
logical entailment for conditional constraints. We showed
that the introduced notions for probabilistic default reasoning
with conditional constraints have similar properties like
their classical counterparts. Moreover, they properly extend
both their classical counterparts and the classical notion of
logical entailment for conditional constraints.
An interesting topic of future research is to extend other
formalisms for classical default reasoning to the probabilistic
framework of conditional constraints.
Acknowledgments
I am very grateful to the referees for their useful comments.
This work was supported by a DFG grant and the Austrian
Science Fund Project N Z29-INF.
--R
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590522 | Coloration Neighbourhood Search With Forward Checking. | Two contrasting search paradigms for solving combinatorial problems are i>systematic backtracking and i>local search. The former is often effective on highly structured problems because of its ability to exploit consistency techniques, while the latter tends to scale better on very large problems. Neither approach is ideal for all problems, and a current trend in artificial intelligence is the hybridisation of search techniques. This paper describes a use of forward checking in local search: pruning coloration neighbourhoods for graph colouring. The approach is evaluated on standard benchmarks and compared with several other algorithms. Good results are obtained; in particular, one variant finds improved colourings on geometric graphs, while another is very effective on equipartite graphs. Its application to other combinatorial problems is discussed. | Introduction
Graph colouring is an NP-hard combinatorial optimisation problem with
real-world applications such as timetabling, scheduling, frequency assign-
ment, computer register allocation, printed circuit board testing and pattern
matching. A graph E) consists of a set V of vertices and a set E
of edges between vertices. Two vertices connected by an edge are said to be
adjacent . The aim is to assign a colour to each vertex in such a way that no
two adjacent vertices have the same colour. A graph colouring problem for a
graph G is the problem of nding a k-colouring with k as small as possible.
The chromatic number (G) of a graph is the minimum number of colours
required to colour it.
Many algorithms have been proposed for the graph colouring problem.
Systematic backtracking gives good results on small graphs but scales poorly
to large problems. Most colouring algorithms are stochastic in nature, searching
in a non-systematic way with a variety of heuristics. The simplest type of
stochastic search is local search: hill climbing, often augmented with heuristics
for escaping local minima. Local search explores the neighbourhood of
a point in a space by making local moves. The neighbourhood consists of
the set of points 0 that can be reached by a single local move. The aim is
to minimise (or equivalently to maximise) some objective function f() on
the space. A local move ! 0 can be classied as backward , forward or
sideways, depending on whether f( 0 ) f() is positive, negative or zero.
Some algorithms choose a forward move that yields the greatest reduction
in value, a strategy sometimes called greedy or steepest descent . A draw-back
of local (and other stochastic) search is that it may converge on a local
minimum: a point that has lower value than all its neighbours but is not a
global minimum. The aim of backward moves is to escape from local minima
by providing noise, while sideways moves are often used to traverse function
plateaus.
We might classify colouring algorithms by the search spaces they explore.
The space of total colorations consists of the possible colour assignments to all
the vertices of a graph, using a xed number of colours. This approach is used
by the TABU algorithm, which tries to minimise the number of con
icts. A
con
ict is an adjacent pair of vertices with the same colour. TABU generates
a selection of possible single-vertex re-assignments and selects the best, even
if this leads to more con
icts. It also maintains a list of recent moves and
avoids reversing them, which helps it to escape local minima.
The consistent total colorations are the total colorations that contain no
con
icts. This space is explored by the Greedy (or sequential) algorithm,
which tries to colour each vertex with a colour already used for a previous
vertex; if this is not possible then a new colour is used. Heuristics control the
vertex order and colour selection. The Iterative Greedy algorithm iteratively
applies the Greedy algorithm, using vertex orderings that are guaranteed to
generate a sequence of colorations using a non-increasing number of colours.
Brelaz's DSATUR algorithm [2] explores a similar space. It orders vertices
dynamically by maximum saturation (number of distinct colours assigned
to adjacent vertices), breaking ties by choosing a vertex of greatest forward
degree. The degree of a vertex is the number of its adjacent vertices, and its
forward degree is the number of its uncoloured adjacent vertices. DSATUR
has also been extended by backtracking. Mehrotra & Trick's version [15] uses
full (exhaustive) search and begins by computing a clique which is never re-
coloured. Another version described by Culberson, Beacham and Papp [3]
uses limited backtracking.
The space of consistent partial colorations consists of consistent colorations
of subsets of the vertices that do not use more than a specied number of
colours. This space is explored by the IMPASSE colouring algorithms. The
objective function to be minimised is the sum of the forward degrees of
the uncoloured vertices. Two such algorithms are Morgenstern's distributed
IMPASSE [17] and Lewandowski & Condon's parallel IMPASSE [14]. Distributed
IMPASSE performs limited searches on a distributed architecture,
each search starting from previous good coloration, which are maintained in
a pool. Parallel IMPASSE is a hybrid of IMPASSE and systematic search;
the two execute in parallel and communicate colouring improvements to each
other.
Finally, the independent sets of a graph are the sets of pairwise non-adjacent
vertices. They exploit the fact that all the vertices of an independent
set can be assigned the same colour. The space of independent sets
is explored by algorithms such as Johnson et al.'s XRLF [10] and Culberson
using (exhaustive or restricted) backtracking and
iteration. Mehrotra & Trick's LPCOLOR [15] uses column generation and
branch-and-bound to explore this space.
The motivation for this classication is that we shall describe a local
search algorithm that explores a new space. This is a subspace of the consistent
partial colorations, reduced by applying the Constraint Programming
technique of forward checking . We describe the new algorithm in Section 2
and evaluate its performance in Section 3. The algorithm, results and related
work are discussed in Section 4. This paper is an extension of earlier
work [19], but the algorithm is described in greater detail and evaluated more
systematically.
FC-consistent partial coloration neighbourhood
search
We rst discuss a simple consistency technique from constraint programming:
forward checking (FC). FC is commonly used with systematic backtracking
[9], and this combination can be applied to graph colouring as follows. Each
vertex has an associated domain of possible colours, initialised to the full
set of available colours. On colouring a vertex with one of the colours in its
domain, that colour is deleted from the domains of the adjacent vertices. No
colour assignment is permitted if it causes the domain of some uncoloured
vertex to become empty. This domain wipe-out often occurs long before the
vertex in question is due to be coloured, greatly reducing the search space.
On backtracking, a vertex is uncoloured and its assigned colour restored to
the domains of adjacent vertices. An analogue to the Brelaz heuristic may
be used to select vertices for colouring: select a vertex with smallest domain,
breaking ties by selecting a vertex with greatest forward degree.
FC is a simple and inexpensive algorithm, and sometimes out-performs
theoretically more powerful techniques. FC with backtracking also has the
advantage of completeness. That is, all k-colourings will eventually be found,
and if there is no k-colouring then the algorithm will eventually prove this
by terminating without nding a solution. However, systematic backtracking
algorithms often suer from poor scalability. For example the FC algorithm
is eective on small N-queens problems, but cannot solve problems with
more than approximately 100 queens; in contrast, a local search algorithm
solves up to 10 6 queens in linear time [16]. Backtracking and local search
are complementary search techniques for solving colouring and other combinatorial
problems, and considerable research has been devoted to combining
advantages of the two.
We describe how to exploit FC within a local search algorithm for graph
colouring. The idea is to explore the subspace of the consistent partial col-
orations that are also consistent under forward checking. That is, for each
of the currently uncoloured vertices, there is at least one available colour
that can be consistently assigned to it; colorations causing vertex domain
wipe-out are avoided. We shall call this subspace the FC-consistent partial
colorations. Our reasoning is that by reducing the search space we may avoid
some local minima.
Before describing the particular algorithm used to explore this space, we
discuss a complication that arises when applying local search to it. In systematic
backtracking it is simple to maintain vertex domains: the order in which
colours are restored from domains on backtracking is the reverse of the order
in which they were deleted during assignment. It is su-cient to maintain a
boolean variable for each colour in each domain, denoting whether or not the
colour is currently in the domain. However, local search is non-systematic,
and from a given coloration we may wish to uncolour any vertex, not just
the most recently coloured one. To do this we need a new way of maintaining
domains. A number we shall call a con
ict count is maintained for each
vertex-colour pair (v,c) recording how many currently coloured vertices the
assignment would con
ict with; initially all con
ict counts are zero. A
colour is classed as deleted from a vertex domain if and only if its con
ict
count is greater than zero. A domain's size is the number of its non-deleted
colours. The memory requirement for con
ict counts is not excessive: for n
vertices and k colours k n con
ict counts are needed, which is roughly the
amount of memory required to represent the problem. They may be updated
incrementally: on colouring/uncolouring a vertex, the con
ict count for that
colour in each adjacent vertex is incremented/decremented. However, they
cause a signicant runtime overhead compared to standard forward check-
ing, because they are updated for the domains of uncoloured and coloured
vertices.
We can now design a local search algorithm on FC-consistent partial col-
orations. The algorithm chosen is rather simple, starting exactly as standard
FC: it selects a vertex for colouring, nds a colour that can be used without
causing domain wipe-out, colours the vertex, updates the domains of adjacent
vertices, and repeats by selecting another vertex. The only dierence so
far is that domains are maintained by con
ict counts. However, on reaching
a dead-end , where the selected vertex cannot be coloured, the new algorithm
behaves dierently to standard FC. It heuristically selects a vertex to be un-
coloured, instead of selecting the most recently coloured vertex. No attempt
is made to backtrack systematically, so completeness is lost. Because there
is now no obvious criterion for deciding when to stop backtracking and start
colouring vertices again, we introduce a parameter B 1. On reaching a
dead-end B vertices are uncoloured, and colouring resumes. Note that the
vertices selected for colouring and uncolouring may follow dierent heuris-
tics, so that the set of coloured vertices may change rapidly during search.
B plays the part of a noise parameter (or the temperature in simulated an-
nealing), controlling the permitted disruption to the state on reaching a local
minimum.
It remains to ll in details by describing three heuristics: selecting B
coloured vertices for uncolouring (CVERTEX), selecting an uncoloured vertex
for colouring (UVERTEX), and selecting colours to try when colouring
a vertex (COLOUR). We consider two alternative UVERTEX rules:
select the vertex with smallest current domain; break ties by
selecting the vertex adjacent to the greatest number of uncoloured ver-
tices; break further ties randomly.
Nonsingleton: randomly select a vertex with more than one colour in
its current domain; if none exists then select a vertex randomly.
The Brelaz heuristic is an obvious choice. The idea behind the Nonsingleton
heuristic is to emulate the MAXIS algorithm, which constructs independent
sets of vertices, whereas DSATUR constructs cliques. By selecting vertices
using an inverse of the Brelaz heuristic, and thus focusing on vertices that
are as independent as possible from those currently coloured, we might expect
to obtain a forward-checking analogue to MAXIS. This was tested and,
perhaps surprisingly, performed rather well on random graphs, whereas the
Brelaz heuristic performed poorly. However, the weaker Nonsingleton heuristic
performs better, possibly because of its greater
exibility in selecting a
vertex. It is discussed further in Section 4.
Given a free choice of vertices for uncolouring, which should be selected?
An obvious idea is to use an inverse of Brelaz: uncolour a vertex with large
domain and small degree (note that because con
ict counts are updated irrespective
of whether a vertex is coloured, coloured vertices also have domain
sizes). In tests this often caused stagnation, but the weaker Nonsingleton
heuristic (applied to coloured vertices) works well. To further reduce stagna-
tion, with probability 1=n (where n is the number of vertices in the graph)
the CVERTEX rule selects a vertex randomly instead of by domain size.
A random ordering on domain values works well, but performance can
be improved by remembering the previous colour of each vertex (if it was
coloured earlier). The COLOUR rule
ips between two modes: initially it
prefers dierent colours to those remembered for each vertex; if a dierent
colour is successfully used, the rule
ips to preferring the remembered colour;
when CVERTEX is next invoked it
ips back to preferring a dierent colour.
The aim of this rule is to minimise disruption to colorations as the set of
coloured vertices changes, while avoiding null local moves.
function
for
while U 6= fg
let
let colouring u to d
does not cause domain wipe-outg
for i=1 to min(B; jCj)
uncolour c and update domains
else
colour u to COLOUR(D) and update domains
return coloration
Figure
1: FC partial coloration neighbourhood search for xed k
The new algorithm FCNS (FC-consistent partial coloration Neighbourhood
is shown in Figure 1. k 1 is the permitted number of colours
and B 1 is the noise parameter. C is the current set of coloured vertices,
initialised to fg. U is the current set of uncoloured vertices, initialised to the
full set of n vertices g. Each vertex has a domain of colours that
are FC-consistent with the current partial coloration, initialised to the full
set of colours kg. The algorithm proceeds by selecting uncoloured vertices
using the UVERTEX rule, and colours them using the COLOUR rule.
On reaching a dead-end (D = fg) it uncolours B vertices, each selected by
the CVERTEX rule. Termination is not guaranteed but occurs if all vertices
are coloured
The algorithm can be used to nd a near-optimal colouring by applying
it iteratively in an obvious way: start with large k (for example
and apply the algorithm; on nding a total coloration using k 0 k colours,
restart the algorithm with k 0 1 colours; repeat until reaching a target number
of colours or a specied time. Performance is improved by starting each
iteration with a coloration similar to the previous one: colour assignments
are stored between iterations, and until the rst dead-end occurs each vertex
is assigned its previous colour where possible. It is also possible to speculatively
reduce k further in the hope of nding better colourings more quickly.
However, this aspiration approach does not always speed up search, because
inadvertently choosing k less than the chromatic number of the graph runs
the risk of spending a long time in fruitless search. Aspiration is not used in
current FCNS implementations.
3 Experimental results
FCNS is now evaluated using published results for several other algorithms
on the well-known DIMACS [11] 1 benchmarks. They are Culberson &
Luo's Iterated Greedy (IG) [4], Morgenstern's distributed IMPASSE [17],
Wheel Optimization (SWO) [12] and Glover, Parker & Ryan's TABU [8].
The TABU algorithm combines the TABU meta-heuristic with branch-and-
bound. SWO operates in two search spaces: a solution space and a prioritisation
space. Both searches in
uence each other: each solution is analysed
and used to change the prioritisation, which guides the search strategy used
to nd the next solution, found by restarting the search.
We use a standard set of benchmarks taken from the DIMACS web site.
Geometric graphs Rx.y and DSJRx.y are generated by randomly placing
x vertices in a unit square, then assigning edges between any two vertices
with Euclidean distance less than y=10 between them; a graph denoted by
Gc is the complement of the graph G. The names R and DSJR re
ect
dierent sources, but are (we believe) the same type of graph. Random
graphs Cn.p and DSJCn.p have n vertices, an edge being assigned between
any two vertices with a xed probability p=10. The names C and DSJC again
re
ect dierent sources. Flat graphs contain colorations that are hidden in
such a way as to mislead Brelaz-style heuristics; a graph
atn c x contains
vertices and a known hidden (though not necessarily optimal) c-colouring.
Leighton graphs le450 15x are derived from scheduling, and
have 450 edges and known chromatic number 15. Graph colouring is closely
related to the timetabling problem and there are two timetabling graphs; the
school1 problem is derived from timetabling data from a real high school with
around 500 students; the school1 nsh problem is derived from the same data
but ignores study halls. Register allocation graphs are used in compilers
to assign variables to registers, with the aim of avoiding the placement of
two variables in the same register when both may be active; there is one
such graph, mulsol.i.1. The latin square graph latin sqr 10 is derived from a
standard problem in design theory.
Figure
reproduces published results for SWO, IG, d-IMP (distributed
IMPASSE), p-IMP (parallel IMPASSE) and TABU, and Figure 3 shows results
for FCNS with the Brelaz (FCNS-b) and Nonsingleton (fCNS-n) heuris-
tics. All times are normalised to our machine (a 300 MHz DEC Alphaserver
1000A 5/300 under Unix) using benchmark timings from [11]; the DIMACS
benchmark program dfmax r500.5 takes 46.2 seconds on our machine. The
times for parallel IMPASSE were not normalised because of its parallel platform
(a 32-processor CM-5). In both tables k is the number of colours used
and t is the time taken in seconds. In Figure 3 B is the value used for the
parameter. Its value was chosen after a few runs to nd an appropriate
setting. This ad hoc approach is unfortunately necessary with many local
search algorithms; TABU has a list length parameter, and some algorithms
have more than one parameter (for example several local search algorithms
for the satisability problem). The initial number of colours k 0
for FCNS was
set to the worst k found by the other algorithms in each case (except where
our algorithms were even worse, when higher values were used). FCNS was
halted on reaching the target k, which was selected after a few experimental
runs. Times shown for FCNS are mean times taken to reach k from k 0
averaged
runs (more for short times). Experimental details for the other
algorithms vary (for details see the cited papers). Brie
y, SWO was terminated
after 1000 iterations, IG after 1000 iterations without improvement,
TABU after an hour or sooner if a lack-of-progress condition was satised,
distributed IMPASSE used conditions depending on the problem but always
halted on reaching a specied target k, and parallel IMPASSE ran for 3 hours
then reported the time taken to nd the best solution. The use of a time
limit instead of a target number of colours explains the occasional fractional
values of k.
First we discuss FCNS-b, which is clearly the best algorithm on the geometric
graphs. On R1000.5 and DSJR500.5 it nds the best reported colour-
ings, and on most of the others it nds equally good colourings in shorter
times. The geometric graphs are randomly generated but closely related to
a real problem: frequency allocation [10]. FCNS-b is therefore a promising
algorithm for solving such problems, and this is an area for future research.
It also performs very well on the school and mulsol graphs, roughly matching
SWO IG d-IMP p-IMP TABU
problem
school1 nsh 14 3.9 14.1 4.8 14 <0.24 14 66.4 26 16.8
mulsol.i.1
at300 26 0 35.8 6.4 37.1 4.1 26 5.4 32.4 6637 41 1849
at300 28 0 35.7 6.4 37 5.2 31 1028 33 1914 41 1849
Figure
2: Previous results for DIMACS benchmarks
FCNS-b FCNS-n
problem
Figure
3: FNCS results for DIMACS benchmarks
the performance of distributed IMPASSE. We also tested Mehrotra & Trick's
implementation 2 on the geometric graphs because it is known to
perform well on such graphs. On those with edge probability 0.1 it found the
same colourings in a slightly shorted time than FCNS-b. On those with edge
probability 0.5 it quickly found good colourings but then made no further
progress for a long time. For R125.5 it found a 36-colouring in 63.4 seconds,
for R250.5 a 66-colouring in 2.9 seconds, for DSJR500.5 a 130-colouring in
17.6 seconds, and for R1000.5 a 246-colouring in 75.8 seconds; no further
progress was made after several minutes. FCNS-b clearly scales better than
DSATUR, nding better colourings on the larger graphs. However, it is
very poor on the random,
at and latin square graphs, and mediocre on the
Leighton graphs.
Next we discuss FCNS-n. On the geometric and school graphs it is poor,
sometimes the worst algorithm, and (like FCNS-b) mediocre on the Leighton
graphs, but on the random,
at and latin square graphs it is beaten only
by distributed IMPASSE. This is presumably due to the use by distributed
IMPASSE of the XRLF algorithm [10] to generate initial colorations: parallel
IMPASSE does not use XRLF and is beaten by FCNS-n on random and
at
graphs. However, other algorithms are also better than FCNS-n on random
graphs. For example on G 1000;0:5
graphs the best algorithms nd colourings
in the low- or mid-80s.
To further investigate FCNS-n we applied it to equipartite graphs, which
have been studied by several researchers. A k-colourable equipartite graph
is generated by partitioning its vertices into k subsets, which are as equally-sized
as possible, the smallest being no more than 1 vertex smaller than the
largest. Edges are assigned with probability p, disallowing edges between vertices
in the same subset. This guarantees a k-colouring but does not preclude
better colourings. Eiben, van der Hauw & van Hemert [5] apply evolutionary
algorithms to 3-colourable equipartite graphs with 200 vertices. They report
low success rates on graphs with low density, especially around
where a phase transition occurs. Minton et al. [16] also report that the Min-
Con
icts local search algorithm has di-culties with similar problems, but
that a backtracking version of DSATUR solves them easily for 3 n 180.
Yugami et al. [27] apply local search with constraint propagation to the same
problems and obtain improved results. FCNS-n solves these problems easily:
nds 3-colourings in approximately 3 seconds. Moving to larger
problems, Culberson et al. [4] show that IG can nd hidden k-colourings for
G 1000;0:5
equipartite graphs with k 60. FCNS-n is also able to do this
and can go a little further. The algorithm was quite insensitive to B until
k 55, after which it became more sensitive. The optimal value for
was approximately the problems rapidly became harder
and the optimal value of B fell. It found a hidden 67-colouring after several
hours computation with but failed to nd a hidden 68-colouring. So
far as we known 67 is the highest value of k solved for this class of graph. In
further experiments FCNS-n also managed to nd the hidden colourings in
at1000 f50,60g 0, by setting starting with the target colouring
(50 or 60) specied as the initial colouring. However, these results took much
trial and error to achieve, so they were not included in our table.
It is perhaps surprising that the Nonsingleton heuristic should be successful
at all, let alone competitive. In particular, if a vertex has domain size 1
(hence only one possible colour) then Brelaz will select it before a vertex with
larger domain (hence more than one possibility), but Nonsingleton will delay
colouring such vertices as long as possible. To investigate the eect of adding
\forced moves" another variant was tried: select a vertex with domain size
1 if one exists, otherwise select one with maximum domain size. However,
this variant was inferior to both Brelaz and Nonsingleton. We speculate that
Nonsingleton causes FCNS to behave in a similar way to independent set-based
algorithms such as MAXIS, by focusing search on low-degree vertices.
A better algorithm might be obtained by explicitly designing it to nd independent
sets, and applying forward checking. Another research direction
is the design of new vertex orderings, with the aim of improving FCNS's
performance on random, Leighton and latin square graphs.
The noise parameter B unfortunately requires tuning to each graph. As
with any noise parameter, the eect of setting B too low is stagnation: FCNS
will never nd a colouring because it becomes trapped in a local minimum.
The eect of setting B too high is less serious, simply increasing the time
taken to nd a solution, but the increase depends on the problem. The
performance of FCNS-n seems to be fairly insensitive to the value of B when
nding an 18-colouring for DSJC125.5, while on DSJC1000.5 increasing B
from 1 to 2 slows it down greatly | or equivalently, prevents it from nding
good colourings in the same time. FCNS-b seems less sensitive, but can
still be slowed down signicantly by too much noise. We experimented with
variable noise levels to try to reduce sensitivity to noise, but with inconclusive
results. A slightly surprising feature of B is that, on several graphs (for
example R250.5), best results were obtained for FCNS-b and FCNS-n using
dierent values of B. However, perhaps this should not be surprising, because
the two algorithms focus on dierent regions of the graphs and therefore
might be expected to encounter local minima of dierent depths.
The main dierence between FCNS and other stochastic colouring algorithms
is that it performs forward checking. It also has an additional advantage over
IG and SWO: incrementality . IG and SWO are not incremental because
restarting is an expensive move, whereas IMPASSE, TABU and FCNS make
small, cheap moves in the search space. This is pointed out as a source of
ine-ciency by Joslin & Clements [12], and they propose hybrids of SWO
with local search for future work.
Graph colouring is a binary constraint satisfaction problem (given xed
k), and the use of con
ict counts to perform forward checking in local search
is easily generalised to other such problems. It can be further generalised
to non-binary constraint problems, and this has been done for propositional
satisability (SAT). Experimental results are very promising: on some large,
structured SAT problems it out-performs current systematic and local search
algorithms [20]. In fact our colouring and SAT algorithms are instances
of a general-purpose approach to combinatorial optimisation and constraint
satisfaction, which we call Constrained Local Search (CLS). The aim of CLS
is to enhance local search with constraint programming techniques used in
systematic search. It has also given good results on other SAT problems [20],
maximum clique problems [22], Golomb rulers [22] and a hard optimisation
problem (the generation of low-autocorrelation binary sequences) [21]. The
general approach is to take an eective backtracking algorithm and replace
systematic by randomised backtracking, usually improving its scalability at
the expense of completeness.
It might be argued that FCNS (or more generally CLS) is not a local
search algorithm, but simply a randomised backtracker. It certainly is a randomised
backtracker and has much in common with Dynamic Backtracking
(DB), which also allows the removal of early assignments without aecting
the assignments made since. This increased
exibility of backtracking was a
stated aim in the design of DB, and a later hybrid algorithm called Partial
Order Dynamic Backtracking [7] achieved even greater
exibility. Is FCNS
simply an inferior version of DB, sacricing completeness to no good pur-
pose? A counter-example to this view is the random 3-SAT problem, on
which DB is slower than depth-rst search [1] while CLS scales precisely as
local search [21]. Our view is that FCNS stochastically explores a space of
FC-consistent partial colorations by local search; the objective function it
minimises is the number of uncoloured vertices. However, to some extent
the question is academic: even if FCNS is not local search, experimental
results show that it captures its essence, successfully solving problems that
are beyond the range of systematic backtracking.
There are several other hybrids of local search and constraint techniques.
The simplest hybrid is a parallel or distributed implementation of more than
one algorithm, as in the IMPASSE algorithms used for colouring. Schaerf's
timetabling algorithm [24], extended to constraint satisfaction problems,
searches the space of all partial assignments (not only the consistent ones)
using an objective function that includes a measure of constraint violation.
This is a dierent space again than those searched by current colouring algorithms
and FCNS. In graph colouring terms this space may be called the
partial colorations as opposed to the consistent partial colorations explored
by IMPASSE. Jussien & Lhomme's Path-Repair Algorithm [13] is described
as a generalisation of Schaerf's approach that includes learning (allowing
complete versions to be devised) and a TABU list. Yugami, Ohta & Hara's
EFLOP algorithm [27] uses constraint propagation to escape local minima,
while allowing some constraint violation. However, forward checking is not
maintained throughout the search, as it is in FCNS. Ginsberg & McAllester's
Partial-order Dynamic Backtracking [7] is a hybrid of the Dynamic Back-tracking
algorithm with a local search algorithm [6], enabling it to follow
local gradients in the search space. Pesant & Gendreau [18] apply systematic
branch-and-bound search to e-ciently explore local search neighbourhoods.
The two-phase algorithm of Zhang & Zhang [28] searches a space of partial
variable assignments, alternating backtracking search with stochastic local
search on the same data structure. It can be tuned to dierent problems
by spending more time in either phase. Yokoo's Weak Commitment Search
[26] (WCS) greedily constructs consistent partial assignments. On reaching
a dead-end it randomly restarts, and uses learning to maintain complete-
ness. Richards & Richards [23] describe a SAT algorithm called learn-SAT
based on WCS. Shaw [25] describes a vehicle routing algorithm called Large
Neighbourhood Search. It performs local search and uses backtracking with
constraint propagation to test the legality of moves.
Each of these algorithms either permits constraint violation or uses learn-
ing, or both. Constraint violation implies, in the view of this author, that
constraints are being under-used. This may be a drawback when solving
highly structured problems: the best graph colouring results for structured
problems are obtained by algorithms such as IMPASSE and FCNS, which
do not violate constraints. The use of learning is a drawback when solving
large problems. It can be restricted to use only polynomial memory, but
combinatorial problems may be very large. The FCNS approach combines
constraint handling and local search, making cheap local moves and avoiding
memory-intensive learning techniques. We believe that this combination of
features makes it ideal for large, highly constrained problems.
--R
The hazards of fancy backtracking
Exploring the k-colorable landscape with iterated greedy
Graph coloring with adaptive evolutionary algorithms
Journal of Arti
GSAT and dynamic backtracking
Coloring by tabu branch and bound
Increasing tree search e-ciency for constraint satisfaction problems
Optimization by simulated annealing: an experimental evaluation
Journal of Arti
The path-repair algorithm
Experiments with parallel graph coloring heuristics and applications of graph coloring
A column generation approach to graph colouring
Minimizing con- icts: a heuristic repair method for constraint satisfaction and scheduling problems
Distributed coloration neighborhood search
A view of local search in constraint pro- gramming
Using an incomplete version of dynamic backtracking for graph colouring
Stochastic local search in constrained spaces
A hybrid search architecture applied to hard random 3-SAT and low-autocorrelation binary sequences
Trading completeness for scalability: hybrid search for cliques and rulers
Combining local search and look-ahead for scheduling and constraint satisfaction problems
Using constraint programming and local search methods to solve vehicle routing problems
Improving repair-based constraint satisfaction methods by value propagation
Combining local search and backtracking techniques for constraint satisfaction
--TR
--CTR
Steven Prestwich, SAT problems with chains of dependent variables, Discrete Applied Mathematics, v.130 n.2, p.329-350, 15 August
Marco Chiarandini , Thomas Sttzle, Stochastic Local Search Algorithms for Graph Set T-colouring and Frequency Assignment, Constraints, v.12 n.3, p.371-403, September 2007 | forward checking;graph colouring;coloration neighbourhood |
590526 | Approximate Qualitative Temporal Reasoning. | We partition the time-line in different ways, for example, into minutes, hours, days, etc. When reasoning about relations between events and processes we often reason about their location within such partitions. For example, i>x happened yesterday and i>y happened today, consequently i>x and i>y are disjoint. Reasoning about these temporal granularities so far has focussed on temporal units (relations between minute, hour slots). I shall argue in this paper that in our representations and reasoning procedures we need into account that events and processes often lie skew to the cells of our partitions. For example, happened yesterday does not mean that i>x started at 12 a.m. and ended 0 p.m. This has the consequence that our descriptions of temporal location of events and processes are often approximate and rough in nature rather than exact and crisp. In this paper I describe representation and reasoning methods that take the approximate character of our descriptions and the resulting limits (granularity) of our knowledge explicitly into account. | Introduction
Every temporal object and every spatio-temporal object is
located at a unique region of time bounded by the begin
and the end of its existence. In every moment of time
a spatio-temporal object, o, is exactly located at a single
region, x, of space (Casati & Varzi 1995). This region
is the exact or precise location of o at the time point t,
at t. Spatio-temporal wholes have temporal
parts, which are located at parts of the temporal regions
occupied by their wholes. Consider, for example, the
region of time, y, where the object, o, is located tempo-
rally, while being spatially located at the spatial region x.
If y is a maximal connected temporal region, i.e.,
once spatially located at x for a while, left and never came
back, then y is bounded by the time instances (points) t 1
and t 2 . Since time is a totally ordered set of time points
forming a directed one-dimensional space (McTaggart 1927;
Geach 1966), we have t 1 < t 2 .
In knowledge representation we are interested in representing
spatio-temporal reality as experienced by human
beings. In this context it is essential to represent spatio-temporal
location (Galton 1997). In this paper I concentrate
on representing temporal location. One way of representing
temporal location is to represent qualitative relationships
between temporal regions occupied by temporal and spatio-temporal
objects and their parts (Allen 1983).
Copyright c
2000, American Association for Artificial Intelligence
(www.aaai.org). All rights reserved.
Human knowledge is gained by observations and reasoning
about observations. (Bittner 1999) argued 1 that by
means of observation and measurement (a precise form of
humans cannot know the exact spatial and,
hence, exact temporal location. Observations and measurement
yield knowledge about approximate spatio-temporal
location, i.e., knowledge about relations between spatio-temporal
objects and cells of regional partitions of space and
time. Regional partitions are sets of regions (cells) that do
not overlap but sum up the whole space. Regional partitions
are created by measurement and observation processes. Approximate
location can be known by observing (qualitative)
relations between objects and the cells of the underlying regional
partitions.
Consider, for example, the measurement of temporal loca-
tion. Measurement of temporal location is based on clocks.
A clock creates a regional partition of the time-line. The
cells forming this partition are time intervals separated by
'clock ticks'. Measurement of temporal location involves
counting time intervals and observing relationships between
time intervals and (temporal parts of) temporal or spatio-temporal
objects. No matter how fine the resolution of the
partition there are always partition cells that are disjoint to
(the exact region of) the observed object, there may be partition
cells that are completely covered by (the exact region
of) the observed object, and there always are partition cells
that are partly covered by (the exact region of) the observed
object. Consequently, observing spatio-temporal location
means observing relations between partition cells and regions
occupied by spatio-temporal objects, i.e., observing
approximate location rather than exact location. Other examples
of regional partition in which approximate temporal
location is observed is the partition of the time line into
past and future separated by the present moment, the hours
of the day, forenoon and afternoon, the four seasons. Con-
sequently, in the context of knowledge representation reasoning
about approximate spatio-temporal location, i.e., approximations
of spatial and temporal regions, is more important
than reasoning about exact location, i.e., spatial and
temporal regions themselves.
In the remainder of this paper I omit the distinction between
objects and spatial and temporal regions and the
1 Based on (Carnap 1966).
(functional) relation of (exact) location between them and
concentrate on the approximation of the exact regions (of
objects) with respect to an underlying regional partition.
Moreover, I concentrate on temporal regions and approximations
of temporal regions. This paper builds on (Bittner
& Stell 1998) and (Bittner & Stell 2000), in which various
ways of providing qualitative approximations of regions
with respect to a partition of the plane as well as reasoning
about those approximations were described.
(Bittner & Stell 2000) showed that approximate qualitative
reasoning is based on: (1) Jointly exhaustive and pair-wise
disjoint sets of qualitative relations between exact re-
gions. These relations need to be defined in terms of the
meet operation of the underlying Boolean algebra structure
of the domain of regions. As a set these relations must form
a lattice with bottom and top element. (2) Approximations
of regions with respect to a regional partition of the underlying
space. (3) Pairs of join and meet operations on those ap-
proximations, which approximate join and meet operations
on exact regions. This this is reflected by the structure of
this paper:
I start with the definition of qualitative relations between
temporal regions. I distinguish boundary sensitive
and boundary insensitive sets of relations and relations between
regions in a directed and non-directed underlying one-dimensional
space. Based on the definition of approximations
of temporal regions with respect to an underlying regional
partition I then generalize the definitions of relations
between temporal regions to definitions of relations between
approximations of regions. This provides the formal basis
for qualitative temporal reasoning about approximate location
in time. The conclusions are given in the end.
Relations between one dimensional regions
Boundary insensitive relations
RCC5 relations Given two regions x and y boundary
insensitive topological relation (RCC5 relations 2
them can be determined by considering the triple of boolean
values (Bittner & Stell 2000):
The correspondence between such triples and boundary insensitive
relations between regions on an undirected line is
given in the following table (Bittner & Stell 2000).
I use the notion RCC in order to stress the correspondence between
the relations defined in this paper and relations defined by
Cohn and his co-workers in terms of the region connection calculus
(RCC) (Cohn et al. 1997). Correspondence in this context
means that I am talking about regular regions that satisfy the
RCC-axioms (Randell, Cui, & Cohn 1992) and that similar relations
could be defined or have been defined in terms of RCC,
e.g., (Randell, Cui, & Cohn 1992; Cohn, Gooday, & Bennett 1994;
Cohn et al. 1997). I am going to use sub- and superscripts (e.g.,
RCC where the superscript refers to the number of relations
in the denoted set and the subscript refers to the dimension of the
regions and the embedding space.
The set of triples is partially ordered by setting
where the Boolean values are ordered by F < T. The
Hasse diagram is given in the diagram below. (Bittner &
Stell 2000) call this graph the RCC5 lattice to distinguish it
from the conceptual neighborhood graph (Goodday & Cohn
1994).
I@ @ @
I@ @ @
relations Given two one dimensional regions x
and y. I assume that x and y are maximal connected one
dimensional regions, i.e., intervals. Boundary insensitive
topological relation between intervals x and y on a directed
line (RCC 9
1 relations) can be determined by considering the
triple of truth values:
where
where
x y
:(y x)
x y
:(y x)
and where
x y
x y
with
L(x) (R(y)) is the one dimensional region occupying the
whole line left (right) 3 of x. The intuition behind FLO and
FLI (FRO and FRI) is "false because of parts 'sticking
out' to the left (right)'' 4 .
The triples formally describe jointly exhaustive and pair-wise
disjoint relations under the assumption that x and y are
intervals in a one dimensional directed space. The correspondence
between the triples and the boundary insensitive
relations between intervals is given in the following table.
FRO FRO FRO DRR
For example. The intuition behind DRL(x; y) is that x and
y do not overlap and x is left of y. The intuition behind
POL(x; y) is that x and y do overlap without containing
each other and the non overlapping parts of x are left of y.
The intuition behind PPL(x; y) is that x is contained in y but
x does not cover the very right parts of y. Possible geometric
interpretations of the relations defined above are given in
Figure
1.
Assuming the ordering FLO < FLI < T < FRI <
FRO a lattice is formed, which has (FLO;FLO;FLO)
as minimal element and (FRO;FRO;FRO) as maximal
element.
x y
POR(x,y)
Figure
1: Possible geometric interpretations of the
RCC 9
relations.
Boundary sensitive relations
RCC8 relations In order to describe boundary sensitive
relations between regions x and y (Bittner & Stell 2000) use
3 I use the spatial metaphor of a line extending from the left to
the right rather than the time-line extending from the past to the
future in order to focus on the aspects of the time-line as a one-dimensional
directed space. Time itself is much more difficult. For
example, it is not clear if the future already exists yet (Broad 1923).
4 Notice that in the case of FLI and FRI this dose not exclude
that there are also 'parts sticking out' to the opposite side.
a triple, where the three entries may take one of three truth
values rather than the two Boolean ones. The scheme has
the form:
where
T if the interiors of x and y overlap;
if only the boundaries x and y overlap;
F if there is no overlap between x and
and where
x^y x => > > > > > > > > > > > <
T if x is contained in the interior of y and
the boundaries are either disjoint
or identical, i.e.,x
if x is contained in y and the boundaries
are not disjoint and not identical,
F if x is not contained within
and similarly for x y. The correspondence between
such triples and boundary sensitive topological relations is
given in the following table (Bittner & Stell 2000).
(Bittner & Stell 2000) define F < M < T and call the
corresponding Hasse diagram (diagram below) RCC8 lattice
to distinguish it from the conceptual neighborhood graph
(Goodday & Cohn 1994).
I@ @ @
In this paper I concentrate on regions of one-dimensional
space and relations between them. In order to distinguish
sets of relations between one dimensional regions from relations
between regions of higher dimension I use the notion
rather than RCC8. Possible geometric interpretations
of their RCC 8
1 relations are given in Figure 2.
y
x
Figure
2: Geometric interpretations of RCC 8
relations between
one-dimensional regions of a non-directed line.
1 relations In order to describe boundary sensitive
relations between intervals on a directed line (RCC 15
define the relationship between x and y by using a triple,
where the three entries may take one of four truth values.
The scheme has the form
where
and where
and similarly for x y.
The correspondence between such triples, boundary sensitive
topological relations between intervals on a directed
line, and the 13 relations defined by (Allen 1983) is given in
the table below.
5 To be distinguished from RCC15 relations (Cohn et al. 1997)
between concave regions of higher dimension.
FLO FLO FLO DCL before
FRO FRO FRO DCR after
MLO FLO FLO ECL meets
MRO FRO FRO ECR meets i
starts
during
during
during i
during i
We define FLO < MLO < FLI < MLI < T <
MRI < FRI < MRO < FRO and call the corresponding
Hasse diagram RCC 15
1 lattice to distinguish it from the
conceptual neighborhood graph (Freksa 1992). Possible geometric
interpretations of the lower RCC 15
1 relations are
given in Figure 3.
x y
Figure
3: Geometric interpretations of the lower
relations between connected intervals.
Approximations
Approximating regions
Boundary insensitive approximation Consider the set of
regions, R, of a one-dimensional space. By imposing a
partition, G, on R we can approximate elements of R by
elements
of
G
3 (Bittner & Stell 1998). That is, we approximate
regions in R by functions from G to the set
nog. The function which assigns to each region
r 2 R its approximation will be denoted 3
G
3 .
The value of ( 3 r)g is fo if r covers all the of the cell g, it
is po if r covers some but not all of the interior of g, and it
is no if there is no overlap between r and g. (Bittner & Stell
1998) call the elements
of
G
3 the overlap & containment
sensitive approximations of regions r 2 R with respect to
the underlying regional partition G.
Boundary sensitive approximation Consider one dimensional
non-directed space. We can further refine the approximation
of regions R with respect to the partition G by taking
boundary points shared by neighboring partition regions into
account. That is, we approximate regions in R by functions
from GG to the
set
nog. The function
which assigns to each region r 2 R its boundary sensitive
approximation will be denoted 4
GG
4 . The value
of ( 4 r)(g covers all of the cell g i , it is bo if
r covers the boundary point, (g shared by the cell g i
and g j and some but not all of the interior of g i , it is nbo
if r does not cover the boundary point (g
some but not all of the interior of g i , and it is no if there is
no overlap between r and g i .
The Semantic of approximate regions Each approximate
region XG
stands for a set of precise
regions, i.e., all those precise regions having the approximation
X . This set which will be denoted [[X
a semantic for approximate regions.
Where ever the context is clear the superscript is omitted.
Approximate operations
The domain of regions is equipped with join and meet op-
erations, _ and ^. (Bittner & Stell 1998) showed that join
meet operations on regions can be approximated by pairs of
greatest minimal and least maximal operations on approx-
imations. In this paper I discuss the operations
on boundary insensitive approximations and boundary sensitive
approximations. A detailed discussion can be found in
(Bittner & Stell 1998).
Boundary insensitive operations Firstly we define operations
on the
set
nog.
no no no no
po no no po
no po fo
no no no no
po no po po
no po fo
These operations extend to elements
of
G
3 (i.e. the set of
functions from G
to
and similarly for ^ .
Boundary sensitive operations We define the operations
on the
set
no no no no no
nbo no nbo nbo nbo
bo no nbo bo bo
no nbo bo fo
These operations extend to elements
of
GG
4 (i.e. the set
of functions from G G
to
The definition of the operations ^ is slightly more com-
plicated. In this case we need to take the approximation
values referring to both boundary points (g
account. Let
be the set of pairs of approximation values of X and Y with
respect to g i . We define the operation X ^Y as follows:
defined as
no no no no no
nbo no
bo no
no nbo bo fo
and
(N) is defined as
no if (bo; bo) 62 N
nbo if (bo; bo)
This definition corresponds to the definitions of operations
on boundary sensitive approximations of two-dimensional
regions in the plane discussed in (Bittner & Stell 1998).
Semantic and Syntactic Generalizations of
RCC*
(Bittner & Stell 2000) showed that there are two approaches
to generalizing RCC relations between precise regions to
approximate ones: the semantic and the syntactic.
Semantic We can define the RCC relationship between
approximate regions X and Y to be the set of relationships
which occur between any pair of precise regions approximated
by X and Y . That is, we can define
Syntactic We can take a formal definition of RCC in the
precise case, which uses operations on R, and generalize
this to work with approximate regions by replacing the
operations on R by analogous ones
for
G
or
GG .
In the remainder of this section I discuss syntactic and
semantic generalizations for RCC5 , RCC 8
1 , and
1 .
Generalization of RCC5 relations
Syntactic generalization If X and Y are approximate regions
(i.e. functions from G
to
3 ) we can consider the two
triples of Boolean values (Bittner & Stell 2000):
In the context of approximate regions, the bottom element,
?, is the function from G
to
3 which takes the value no
for every element of G. Each of the above triples defines
an RCC5 relation, so the relation between X and Y can be
measured by a pair of RCC5 relations. These relations will
be denoted by R(X;Y ) and R(X;Y ).
Theorem 1 ((Bittner & Stell 2000)) The pairs
(R(X; Y ); R(X;Y )) which can occur are all pairs
(a; b) where a b with the exception of (PP; EQ) and
Correspondence of semantic and syntactic generalization
Let the syntactic generalization of RCC5 be defined
by
where R and R are as defined above.
Theorem 2 ((Bittner & Stell 2000)) For any approximate
regions X and Y syntactic and semantic generalization of
RCC5 are equivalent in the sense that
where RCC5 is the set fEQ; PP; PPi; PO;DRg, and is
the ordering in the RCC5 lattice.
Generalization of RCC 8relations
Syntactic generalization Let X and Y be boundary sensitive
approximations of regions x and y. The generalized
scheme has the form
where
and where
and similarly for X ^Y Y ,
X,and X ^Y Y . In this context the bottom element,
?, is either the value no or the function from G G
towhich takes the value no for every element of GG.
Assume the partial order of the RCC 8
only if the least relation
1 -relation that can hold between x 2 [[X
boundary intersection of -(x) and -(y)
at a boundary point, (g of the underlying partition G.
only if the greatest RCC 8
relation that can hold between x 2 [[X
boundary intersection at a boundary point in G. For a
detailed discussion of the 2D case see (Bittner & Stell 2000).
Each of the above triples defines a RCC 8
the relation between X and Y can be measured by a pair
of RCC 8
1 relations. These relations will be denoted by
R 8 (X; Y ) and R 8 (X; Y ). Let X and Y be approximations
of one dimensional regions in one dimensional space. Then
the following holds:
Theorem 3 The pairs (R 8 (X; Y
can occur are all pairs (a; b) where a b with the
exception of (TPP; EQ), (TPPi; EQ),(NTPP; EQ),
(DC; EC), (DC; TPP), (DC; TPPi), EC; NTPP),
6 This is an application of theorem 5 in (Bittner & Stell 2000) to
the one-dimensional case.
Correspondence of syntactic and semantic generalization
Let SEM(X;Y ) be a set of RCC 8
relations defined
as
]]g.
Theorem 4 If there are G such that (X(g
Assume (X(g
possibly
This conflicts with
We define the semantically corrected
syntactic generalization of RCC 8
as:
(R 8
there are
that (X(g
otherwise. The semantic generalization of
1 relations is defined as SEM(X;Y
R 8
)g.
Theorem 5 For any boundary sensitive approximations X
and Y of regular one dimensional regions, the syntactic and
semantic generalization of RCC 8
are equivalent in the sense
that SYN(X;Y
Generalization of RCC 9relations
Syntactic generalization Let X and Y be boundary sensitive
9 approximatons of regions x and y. Then we can consider
the two triples of Boolean values:
where
and where
7 This is an application of theorem 6 in (Bittner & Stell 2000) to
the one-dimensional case.
8 This is an application of theorem 7 in (Bittner & Stell 2000) to
the one-dimensional case.
9 We need boundary sensitive approximations since we need to
approximate intervals, i.e., maximally connected temporal regions.
and similarly for
and X ^Y Y . We define X Y as
and similarly X Y using R(X) and R(Y ), where L and
R are defined as follows. Firstly, we define the complement
operation
no nbo bo fo
Secondly, assuming that partition cells g i are numbered in
increasing order in direction of the underlying space, we define
L(Y ) as
no otherwise
and R(Y ) is defined as
(R(Y
no otherwise
Each of the above triples defines an RCC 9
the relation between X and Y can be measured by a pair
of RCC 9
1 relations. These relations will be denoted by
R 9 (X; Y ) and R 9 (X; Y ).
Theorem 6 The pairs
that can occur are all pairs (a; b) where a b EQ and
EQ a b with the exception of (PPL; EQ), (PPR; EQ),
The pairs (PPL; EQ), (PPR; EQ), (PPiL; EQ),
(PPiR; EQ) cannot occur, since they are refinements of the
relations (PP; EQ), (PPi; EQ), which cannot occur in the
undirected case. The pair (EQ; DRR) cannot occur due to
the non-symmetric definition of FL and FR.
The pair (DRL; EQ) represents the most indeterminate
case. Since (DRL; EQ) is consistent with (EQ; DRR) and
(DRL; EQ) was chosen arbitrarily, (DRL; EQ) is corrected
syntactically to (DRL; DRR). The corrected relation will be
denoted by R 9
Correspondence of semantic and syntactic generalization
Let the syntactic generalization of RCC 9
1 be defined
by
R 9 and R 9
c are defined as discussed above.
Proposition 1 For approximations X and Y syntactic and
semantic generalization of RCC 9
1 relations are equivalent
in the sense that
1 is the set RCC 9
1 relations and is the ordering
in the RCC 9
lattice.
Generalization of RCC 15relations
Syntactic generalization If X and Y are boundary sensitive
approximations of intervals x and y in a directed one-dimensional
space then we can consider the two triples of
Boolean values:
where
and similarly for
Each of the above triples provides a RCC 15
the relation between X and Y can be measured by a pair of
1 relations. These relations will be denoted by R 15
and R 15 (X; Y ). The pairs of relations
that can occur are all pairs (a; b) where a b EQ and
EQ a b with the exception of pairs of relations that
are refinements of pairs of relations that cannot occur in the
undirected case (RCC 9
theorem or that cannot occur in
the boundary insensitive case (RCC 8
theorem 3).
Correspondence of semantic and syntactic generalization
Corresponding to the generalization of the RCC 8
1 and
the RCC 9
relations syntactic corrections are needed in order
to generalize RCC 15
relations between intervals, x and
y, to pairs of RCC 15
relations between approximations X
Firstly. Corresponding to the RCC 8
1 case we define
R 15
are
G such that (X(g
R 15
to the RCC 9
1 case the pair (DCL; EQ) represents
the most indeterminate case. Since (DCL; EQ) is consistent
with (EQ; DCR) and (DCL; EQ) was chosen arbitrar-
ily, (DCL; EQ) is corrected syntactically to (DCL; DCR).
The corrected relation will be denoted by R 15
Let the syntactic generalization of RCC 15
1 be defined by
c and R 15
c are defined as discussed above.
Proposition 2 For approximations X and Y syntactic and
semantic generalization of RCC 15
1 relations are equivalent
in the sense that
maxfR 15
where RCC 15
1 is the set RCC 15
1 relations and is the ordering
in the RCC 15
lattice.
Conclusions
In this paper I defined methods of approximate qualitative
temporal reasoning. Approximate qualitative temporal reasoning
is based on:
1. Jointly exhaustive and pair-wise disjoint sets of qualitative
relations between exact regions, which are defined in
terms of the meet operation of the underlying Boolean algebra
structure of the domain of regions. As a set these
relations must form a lattice with bottom and top element.
2. Approximations of regions with respect to a regional partition
of the underlying space. Semantically, an approximation
corresponds to the set of regions it approximates.
3. Pairs of meet operations on those approximations, which
approximate the meet operation on exact regions.
Based on those 'ingredients' syntactic and semantic generalizations
of jointly exhaustive and pair-wise disjoint relations
between exact one-dimensional regions were defined.
Generalized relations hold between approximations of regions
rather than between (exact) regions themselves. Syntactic
generalization is based on replacing the meet operation
defining relations between exact regions by its minimal
and maximal counterparts on approximations. Se-
mantically, syntactic generalizations yield upper and lower
bounds (within the underlying lattice structure) on relations
that can hold between the corresponding approximated exact
regions.
In the temporal domain I defined four sets of topological
relations between one dimensional regions:
RCC5 Boundary insensitive binary topological relations
between regions in a non-directed one-dimensional space.
RCC 9
Boundary insensitive binary topological relations
between maximally connected regions (intervals) in a directed
one-dimensional space.
Boundary sensitive binary topological relations between
regions in a non-directed one-dimensional space.
Boundary sensitive binary topological relations between
maximally connected regions (intervals) in a directed
one-dimensional space.
For each of these sets of relations between exact regions I
discussed the syntactic and semantic generalization for the
corresponding approximations and showed the equivalence
of syntactic and semantic generalization. This provides the
formal basis for qualitative temporal reasoning about approximate
location in time.
Acknowledgements
This research was financed by the Canadian GEOID net-
work. This support is gratefully acknowledged.
--R
Maintaining knowledge about temporal intervals.
A boundary-sensitive approach to qualitative location
Approximate qualitative spatial reasoning.
On ontology and epistemology of rough location.
Scientific Thought.
An Introduction to the Philosophy of Science.
The structure of spatial localization.
Qualitative spatial representation and reasoning with the region connection calculus.
A comparison of structures in spatial and temporal logics.
Temporal reasoning based on semi- intervals
Some problems about time.
Conceptual neighborhoods in temporal and spatial reasoning.
A spatial logic based on regions and connection.
--TR
--CTR
Thomas Bittner , John G. Stell, Approximate qualitative spatial reasoning, Spatial Cognition and Computation, v.2 n.4, p.435-466, 2001 | approximate reasoning;ontology;temporal relations;qualitative reasoning;granularity |
590552 | Refreshment policies for web content caches. | Web content caches are often placed between end users and origin servers as a mean to reduce server load, network usage, and ultimately, user-perceived latency. Cached objects typically have associated expiration times, after which they are considered stale and must be validated with a remote server (origin or another cache) before they can be sent to a client. A considerable fraction of cache "hits" involve stale copies that turned out to be current. These validations of current objects have small message size, but nonetheless, often induce latency comparable to full-fledged cache misses. Thus, the functionality of caches as a latency-reducing mechanism highly depends not only on content availability but also on its freshness. We propose policies for caches to proactively validate selected objects as they become stale, and thus allow for more client requests to be processed locally. Our policies operate within the existing protocols and exploit natural properties of request patterns such as frequency and recency. We evaluated and compared different policies using trace-based simulations. | INTRODUCTION
Caches are often placed between end-users and origin servers
as a mean to reduce user-perceived latency, server load, and
network usage (see Figure 1). Among these dierent performance
objectives of caches, improving end-user Web experience
is gradually becoming the most pronounced. Many
organizations are deploying caching servers in front of their
LANs, mainly as a way to speed up users Web access. Gen-
erally, available bandwidth between end-users and their Internet
Service Providers (ISPs) is increasing and is complemented
by short round trip times. Thus, the latency bottleneck
is shifting from being between end-users and cache
to being between cache and origin servers. From the view-point
of Web sites and Content Distribution Networks (like
decreasing costs of server-machines and back-
An earlier version of the paper appeared in the Proceedings
of Infocom '01 [1].
bone connectivity bandwidth along with increasing use of
the Web for commercial purposes imply that server and net-work
load are gradually becoming a lesser issue relative to
end-user quality of service. At the limit, these trends indicate
that communication time between local caches and
remote servers increasingly dominates cache service-times
and user-perceived latency, and that technologies which provide
tradeos between tra-c-increase and latency-decrease
would become increasingly worthwhile for both Web sites
and ISPs.
Servicing of a request by a cache involves remote communication
if the requested object is not cached (in which case
the request constitutes a content miss). Remote communication
is also required if the cache contains a copy of the
object, but the copy is stale, that is, its freshness lifetime
had expired and it must be validated (by the origin server
or a cache with a fresh copy) prior to being served. If the
cached copy turns out to be modied, the request constitutes
a content miss. Otherwise, the cached copy is valid and we
refer to the request as a freshness miss. Validation requests
that turn out as freshness misses typically have small-size
responses but due to communication overhead with remote
servers, often their contribution to user-perceived latency is
comparable to that of full-edged
content misses.
Thus, cache service times can be improved by reducing both
content and freshness misses. The content hit rate is measured
per object or per byte and sometimes weighted by
estimated object fetching cost. It is dictated by the available
cache storage and the replacement policy used. Replacement
policies for Web caches were extensively studied
(e.g. [3, 4, 5, 6, 7, 8, 9, 10]). Policies that seem to perform
well are Least Recently Used (lru, which evicts the
least recently requested object when the cache is full), Least
Frequently Used (lfu, which evicts the least-frequently requested
object), and Greedy-Dual-Size (which accounts for
varying object sizes and fetching costs). Squid [11], a popular
caching server software, implements the lru policy. Under
these replacement policies, however, and due to decreasing
storage cost, cache hit rate is already at a level where
it would not signicantly improve even if unbounded storage
is made available. Content availability can be improved
by prefetching [12, 13], but prefetching of content involves
more involved predictions and induces signicant bandwidth
overhead. The freshness hit rate of a cache is not directly
addressed by replacement policies or captured by the content
hit rate metric.
clients origin servers
cache
Figure
1: Schematic conguration of Cache, clients,
and origin servers
The expiration time of each object is determined when it
is brought into the cache, according to attached HTTP response
headers provided by the origin server. Expired content
must be validated before being served. Most current
caching platforms validate their content passively i.e. only
when a client request arrives and the cached copy of the object
is stale. They perform validation via a conditional GET
request (typically this is an If-Modified-Since (IMS) Get
request). This means that validation requests are always
performed \online," while the end-user is waiting. Here we
promote proactive refreshment where the cache initiates unsolicited
validation requests for selected content. Such \of-
ine" validations extend freshness time of cached objects
and more client requests can be served directly from the
cache. Our motivation is that the most signicant cost issue
associated with freshness misses is their direct eect on
user-perceived latency rather than their eect on server and
network load, and thus it is worth performing more than
one \oine" validation in order to avoid one performed \on-
line." We formalize a cost model for proactive refreshment,
where overhead cost of additional validation requests to origin
servers is balanced against the increase in freshness hits.
We propose and evaluate refreshment policies, which extend
freshness periods of selected cached objects. The
decision of which objects to renew upon expiration varies
between policies and is guided by natural properties of the
request history of each object such as time-to-live (TTL)
values, popularity, and recency of previous requests. Refreshment
policies can also be viewed as prefetching fresh-
ness. Their methodology and implementation, however, is
closer to replacement policies than object prefetching al-
gorithms. Our refreshment policies resemble common replacement
policies (such as lru and lfu) in the way objects
are prioritized. First, policies prefer renewing recently-
or frequently-requested objects. Second, implementation is
similar since object value is determined only by the request-
history of the object rather than by considering request
history of related objects. Another dierence of refreshment
and document prefetching is that validations typically
have considerably smaller response-sizes than complete doc-
uments, but due to communication overhead, the latency
gap is not nearly as pronounced. Hence, refreshment potentially
provides considerably better tradeos of bandwidth
vs. reduced latency compared to object prefetching.
Our experimental study indicates that the best among the
refreshment policies we have studied can eliminate about
half of the freshness misses at a cost of 2 additional validation
requests per eliminated freshness miss. Freshness
themselves constitute a large fraction (30%-50%) of
cache hits in a typical cache today. Therefore we conclude
that one can considerably improve cache performance by
incorporating a refreshment policy in it.
Overview
The paper proceeds as follows. We discuss related work
in Section 2. We then provide a brief overview of HTTP
freshness control in Section 3. In Section 4 we discuss and
analyze our log data. In Section 5 we present the dierent
refreshment policies. In Section 6 we describe our methodology
for trace-based simulations and the simulation results
for the performance of the dierent policies. We conclude in
Section 7 with a summary and future research directions.
2. RELATED WORK
Recent work addressed validation latency incurred on freshness
misses, including transferring stale cached data from
the cache to the client's browser while the data validity is being
veried [14] or while the modied portion (the \delta")
is being computed [15]. These schemes, however, may require
browser support and are eective only when there is
limited bandwidth between end-user and cache (such as with
modem users).
Some or all freshness misses can be eliminated via stronger
cache consistency. Cache consistency architectures include:
server-driven mechanisms where clients are notied by the
server when objects are modied (e.g. [16, 17]); client-driven
mechanisms, where the cache validates with the server objects
with a stale cached copy; and hybrid approaches where
validations are initiated either at the server or the client.
Hybrid approaches include the server piggybacking validations
on responses to requests for related objects [18, 19, 20],
which are used by the cache to update the freshness status
of its content. Another hybrid approach is leases where the
server commits to notify a cache of modication, but only for
a limited pre-agreed period [21, 22, 23, 24]. Server-driven
mechanisms provide strong consistency and can eliminate
all freshness misses. Hybrid approaches can provide a good
balance of validation overhead and reduced validation traf-
c. None of these mechanisms, however, is deployed or even
standardized. Implementation requires protocol enhancements
and software changes not only at the cache, but at
all participating Web servers, and may also require Web
servers to maintain per-client state. Hence, it is unlikely
they become widely deployed in the near future. Mean-
while, except for some proprietary coherence mechanisms
deployed for hosted or mirrored content [2, 25] (which require
control of both endpoints), the only coherence mechanism
currently widely deployed and supported by HTTP/1.1
is client-driven based on TTL (time-to-live). Our refreshment
approach utilizes this mechanism.
We considered refreshment policies similar to the ones proposed
here in [26] for caches of Domain Name System (DNS)
servers. Each resource record (RR) in the domain name system
has a TTL value initially set by its authoritative server.
A cached RR becomes stale when its TTL expires. Client
queries can be answered much faster when the information
is available in the cache, and hence, refreshment policies
which renew cached RRs oine increase the cache hit rate
and decrease user-perceived latency.
Although not yet supported by some widely deployed Web
caching platforms (e.g., Squid [11]), proactive refreshment is
oered by some cache vendors [27, 28]. Such products allow
refreshment selections to be congured manually by the administrator
or integrate policies based on object popularity.
Our work formalizes the issues and policies and systematically
evaluates them.
3. FRESHNESS CONTROL
We provide a simplied overview of the freshness control
mechanism specied by HTTP and supported by compliant
caches. For further details see [29, 30, 11, 31, 32]. Caches
compute for each object a time-to-live (TTL) value during
which it is considered fresh and beyond which it becomes
stale. When a request arrives for a stale object, the cache
must validate it before serving it, by communication either
with an entity with a fresh copy (such as another cache) or
with the origin server. The cachability and TTL computation
is performed using directives and values found in the
object's HTTP response headers.
When the cache receives a client request for an object then
it acts as follows:
If the object is cached and fresh, the request constitutes
content and freshness hit (chit and fhit respec-
tively) and the cached copy is immediately returned
to the client.
If the object is cached, but stale, the cache issues a
conditional HTTP GET request to the origin server
(or another appropriately-selected cache). The conditional
GET uses the entity tag of the cached copy
(with HTTP/1.1 and if an E-tag value was provided
with the response) or it issues an If-Modified-Since
request with the Last-Modified response header
value (indicating last modication time of the object).
If the source response is Not-Modified then the request
constitutes a content hit (chit) but a freshness
miss (fmiss). If the object was modied, the request
is a content miss (cmiss-r).
If the item is not found in the cache, then it is fetched
from the origin (or another cache) and the request constitutes
a content miss (cmiss-d).
If the request arrives from the client with a no-cache
header then the cache forwards it to the origin server.
The cache must forward the request even if it has a
fresh copy. The cache uses the response to replace or
refresh its older copy of the object. We refer to such
requests as no-cache requests.
The TTL calculation for a cachable object as specied by
HTTP/1.1 compares the age of the object with its freshness
lifetime. If the age is smaller than the freshness lifetime
the object is considered fresh and otherwise it is considered
stale. The TTL is the freshness lifetime minus the age (or
zero if negative).
The age of an object is the dierence between the current
time (according to the cache's own clock) and the timestamp
specied by the object's Date response header (which is
supposed to indicate when the response was generated at
the origin). If an age header is present, the age is taken to
be the maximum of the above and what is implied by the
age header.
Freshness lifetime calculation then proceeds as follows. First,
if a max-age directive is present, the value is taken to be the
freshness lifetime. Otherwise, if Expires header (indicating
absolute expiration time) is present, the freshness lifetime
is the dierence between the time specied by the Expires
header and the time specied by the Date header (zero if
this dierence is negative). Thus, the TTL is the dierence
between the value of the Expires header and the current
time (as specied by the cache's clock). Otherwise, no explicit
freshness lifetime is provided by the origin server and
a heuristic is used: The freshness lifetime is assigned to be
a fraction (HTTP/1.1 mentions 10% as an example) of the
time dierence between the timestamp at the Date header
and the time specied by the Last-Modified header, subject
to a maximum allowed value (usually 24 hours, since
HTTP/1.1 requires that the cache must attach a warning if
heuristic expiration is used and the object's age exceeds 24
hours).
Before concluding this overview, we point on two qualitative
issues with the actual use of freshness control and their
relation to our refreshment approach. In [33] we analyze
the distribution of dierent freshness control mechanisms
for objects in the traces we used and it shows that the large
majority of cachable objects do not have explicit directives.
For these objects, the heuristic calculation is used to determine
the freshness lifetime, and thus, tradeos between
freshness and coherence can be controlled by tuning parameter
values and URL lters [31]. Our refreshment policies
are applied on top of this heuristic, take it as a given, and
attempt to reduce freshness misses without further compromising
coherence. Another important issue suggested by
recent studies is that cache control directives and response
header timestamp values are often not set carefully or ac-
curately. These practices may skew freshness calculations
away from the original intent [32, 34, 33]. This issue is also
orthogonal to our approach since our policies, like caches,
take these settings at face value.
4.
We used two 6 days NLANR cache traces [35] collected from
the UC and SD caches from January 20th till January 25th,
2000. The NLANR caches run Squid [11] which logs and
labels each request with several attributes such as the request
time, service time, cache action taken, the response
code returned to the client, and the response size.
The data analysis below considered all HTTP GET requests
such that a 200 or 304 response codes (OK or Not-Modified)
were returned to the client. We classied each of these requests
as fhit, fmiss, cmiss-r, cmiss-d, or no-cache using the
Squid logging labels as follows.
Content hits:
{ fhit: (freshness hit) the cache had a fresh cached
copy. Squid labels TCP HIT, TCP MEM HIT,
{ fmiss: (freshness miss) the cache had a stale cached
copy and validated it. Squid label TCP REFRESH HIT.
Content misses:
{ cmiss-r: the cache had a stale cached copy, issued
an IMS request, and got a new copy with a Modied
response. Squid label TCP REFRESH MISS.
{ cmiss-d: there was no cached copy of the object.
Squid label TCP MISS.
no-cache: the request arrived with a no-cache request
header. Squid label TCP CLIENT REFRESH MISS.
The table of Figure 2 shows the fraction of requests of each
type.
Requests classied as fmisses, cmisses, or no-cache involve
communication with the origin server or another cache. Freshness
misses, targeted by refreshment policies, constitute 13%
(UC) and 19% (SD) of all such requests. Moreover, it is evident
that freshness misses constitute 31% (UC) and 43%
(SD) of all content hits (requests classied as fmisses or
fhits). These NLANR caches directed most validation requests
(fmisses and cmisses-r) to the origin server (100% in
the UC cache and 99.3% in the SD cache). It is also apparent
that the vast majority (90% for UC and 95% for SD)
of validation requests return Not-Modified (are classied as
fmisses).
The NLANR traces also recorded the service time of each
request. That is, the time from when the HTTP request
is received to when the last byte of the response is written
to the client's socket. Note that this is usually one Round
Trip Time (RTT) less than from the client's viewpoint. Figure
3 plots the Cumulative Distribution Function (CDF)
of service time of requests, broken down by the cache ac-
tion. The gap between freshness misses to freshness hits
indicates the potential benet, in terms of latency, of eliminating
fmisses. The gap between freshness misses and content
misses is in part due to the additional time required to
transfer a larger-size response. Another explaining factor is
that content misses exhibit less locality of reference in the
sense that the elapsed time since the preceding request to
the server is more likely to be longer. The decreased locality
implies longer DNS resolutions of the server's hostname,
since the required DNS records are less likely to be cached,
and longer response time for the HTTP request, since the
origin server is more likely to be \cold" with respect to the
cache 1 . The similar service time distribution for no-cache
and freshness misses suggests that most no-cache requests
are made to popular cached content.
Figure
4 plots the CDF of service times on freshness misses
and freshness hits, further broken down by the response code
that the cache returned to the client. HTTP response code
200 indicates that content was returned whereas response
code 304 (Not-Modified) indicates that the client issued
an IMS GET request and that its copy was validated by the
cache. Responses with code 304 are typically smaller-size
than responses with code 200. We can see that freshness
hits with a 304 response to the client had minimal service
time whereas freshness hits with 200 responses re
ect RTTs
Our study in [36] indicates that a rst HTTP request to
a server in a time period is more likely to take longer than
subsequent ones.
between the cache and its clients and additional processing.
The gap for freshness misses between 200 and 304 responses
is also similar and re
ects the additional communication between
cache and clients due to the larger transmitted response
size.
5. REFRESHMENT POLICIES
Refreshment policies associate with every cached object a
renewal credit (a nonnegative integer). When a cached copy
is about to expire (according to its respective TTL interval),
and it has nonzero renewal credit, a renewal request is sent
to the respective authoritative server, and the renewal credit
is decremented.
The association of renewal credits to objects is governed by
the particular policy. The policies we consider may increment
the renewal credit only upon a client request. Renewal
scheduling can be implemented via a priority queue,
grouping together objects with close-together expiration, or
possibly by an event-triggered process.
We discuss two types of renewal requests:
1. Conditional fetch: The cache noties the server of
last modication times or entity tag(s) of cached ver-
sion(s) of the object, and requests either a validation
of its current version or a new valid version. (This
is supported by HTTP/1.0 by an If-Modified-Since
GET request)
2. Pure validation request: Test whether the cached
version of the object is valid, without requesting a valid
copy if it is no longer valid. (Under HTTP/1.0 this
is performed by issuing an If-Modified-Since HEAD
request. HTTP/1.1 provides additional mechanisms,
e.g., range requests.) Policies that use pure validation
requests stop renewing a copy once it is invalidated,
even if its renewal credit is positive.
Pure validation renewals generally use less bandwidth than
conditional fetches, since if the object was modied, only the
object header is transmitted. On the other hand, conditional
fetches result in a fresh cached copy even when the object
was modied. Thus, policies that only perform pure validation
renewals target only freshness misses whereas policies
that allow conditional fetches also address some content
misses (cmisses-r). For small-size objects that can t on a
single packet, however, the overhead of pure validations is
similar to that of conditional fetches so conditional fetches
would be more eective. Ultimately, the type of renewal
request can be determined by the policy on a per-request
or per-object basis, according to (previous) content length
and modication patterns. The data analysis in Section 4
shows that only a small fraction (5%-10%) of IMS requests
result in invalidations and content transmission. This suggests
that the additional overhead of performing conditional
fetches over pure validations is typically amortized over 10-
renewals. Therefore, for moderate size objects, the total
bandwidth usage is similar under both choices of renewal
actions. For the sake of simplicity, we evaluated only policies
with pure validation renewals. Since the likelihood of
modication is low, pure validation renewals capture most
trace total req. 200+304 req. fhits fmisses cmisses-d cmisses-r no-cache
UC 7.5M 6.3M 23% 10% 56% 1% 10%
SD 5.6M 4.4M 19% 15% 56% 3% 7%
Figure
2: Classication of the requests in the UC and SD traces.0.10.30.50.70.90 500 1000 1500 2000 2500 3000 3500 4000
fraction
below
x
milliseconds
UC: CDF of request service times
fhits
no-cache
cmisses-r
cmisses-d0.10.30.50.70.90 500 1000 1500 2000 2500 3000 3500 4000
fraction
below
x
milliseconds
SD: CDF of request service times
fhits
no-cache
cmisses-r
cmisses-d
Figure
3: CDF of the service times, broken down by cache action
of the potential of refreshments and provide a good indication
for the full potential. We believe, however, that the
incorporation of conditional fetches in the policies deserves
further study.
Another design issue that may require explaining is that
we chose not to consider predictive policies which renew
or prefetch long-expired objects. Predictive renewals and
document prefetching are typically eective if activity is
traced at a per-user basis, where future requests are predicted
according to current requests made by the same user
to related objects. Our refreshment approach diers from
predictive-renewal in that we consider the aggregate behavior
across users on each object. Our policies use minimal
book-keeping, simple implementation, and do not require
Web server support. Ultimately, it may be eective for refreshment
policies to co-exist with predictive renewals and
content prefetching, but we believe that the basic dierences
between these techniques call for separate initial evaluations.
One last important point that we would like to make explicit
is that our policies were designed (and evaluated) for
caches that forward requests to origin servers. For exam-
ple, top-level caches in a hierarchy and caches that are not
attached to a hierarchy. Directing renewal requests to an
authoritative source assures a maximum freshness time for
the response. Caches that are congured to forward requests
to a parent cache may also benet from deploying a refreshment
policy. The potential gain, however, is limited since
renewals would often obtain aged responses (i.e. objects that
have already been cached for a time period at the higher level
cache). We studied age eects on performance of cascaded
caches in [37, 38].
We proceed by describing the dierent policies. An illustrated
example is provided in Figure 5. In the next section
we evaluate and compare their performance using trace-based
simulations.
Policies
passive: passive validation, objects are validated only
as a result of a freshness miss i.e. when a request arrives
and there is a stale cached copy. This is the way most
caches work today.
opt(i): An approximation to the optimal omniscient
oine policy. 2 This policy assumes knowledge of the
time of the subsequent request for the object and whether
the object copy would still be valid then. If the subsequent
request is such that the current copy remains
valid and it is issued within c i freshness-lifetime-
durations after the expiration of the current copy, then
the renewal credit is set to c. Otherwise, no renewals
are performed.
recency(k): The renewal credit is reset to k following
any request for the object, including no-cache requests.
recency(k), similarly to the cache replacement policy
lru, exploits the recency property of request se-
quences, which states that future requests are more
likely to be issued to recently-requested objects.
recency (k): A variant of recency(k) that resets
the renewal credit to be k following any request for
the object, except for no-cache requests.
increment the renewal credit by j for any
request that would have been a freshness miss under
passive. In other words, we add j to the renewal credit
We specify optimal oine algorithm in [26].
fraction
below
x
milliseconds
UC: CDF of request service times
304 fhits
200 fhits
200 fmisses0.10.30.50.70.90 500 1000 1500 2000 2500 3000 3500 4000
fraction
below
x
milliseconds
SD: CDF of request service times
304 fhits
200 fhits
200 fmisses
Figure
4: CDF of the service times for content-hits, broken down by response code to the client
recency(2)
time
request no-cache request refresh miss hit
m2.501.671.80overhead
passive
Figure
5: Behavior of dierent refreshment policies on an example sequence of 9 requests. The time line is
in units of freshness-lifetime durations. All policies incur at least two misses: The rst request is a cold-start
miss and the 8th request, which has a no-cache request header, is always a miss. The gure also summarizes
the number of misses and renewals performed by each policy. passive, for example, incurs 7 misses and
performs no renewals, while th-freq(0:5; incurs only 2 misses and performs 13 renewals. The policy opt(1)
is the most e-cient in the following sense: it performs the least renewals among all policies that incur
at most 4 misses (e.g., recency(1) and freq(1; 0)). This example illustrates how the coverage (fraction of
misses eliminated) of the policies recency(i) and freq(i) increases with i. The overhead (number of \extra"
requests per eliminated miss), however, typically (but not always) increases as well. For example, recency(1)
eliminates 3 misses with respect to passive but performs 8 renewals. Thus, it issues 8+4
(renewals and misses) than passive and has overhead of 5=3 1:67. Similarly, recency(2) eliminates 4 misses
while issuing 3 than passive, and thus, has overhead of
upon any request which is issued more than freshness-
lifetime-duration units of time after a previous request
that caused the passive policy to contact the origin
server. In addition, upon any request (except for no-cache
requests) the renewal credit is set to m if it is less
than m. With policy freq(j; purely exploits
the frequency property that states that objects
that were frequently requested in the past are more
likely to be requested soon. replacement policy
that exploits the frequency property is lfu.) For
the policy freq(j; m) is a hybrid of freq(j;
and recency(m).
th-freq(th; m) keep renewing objects until the ratio
of would-have been passive freshness misses to number
of freshness-lifetime-durations since beginning of
log drops below a threshold. In other words, upon
each request which would have been a freshness miss
for passive we increment the renewal credit such that
we would keep renewing until the ratio drops below
a threshold. In addition, upon any request (except
for no-cache requests) the renewal credit is increased
to m if it was less than m. This policy exploits the
frequency property, and normalizes it by freshness-
lifetime-duration. It also naturally provides a more
continuous range of tradeos, since th is not necessarily
an integer. With the policy is purely
frequency-based whereas higher values of m correspond
to hybrids with recency(m) policy.
6. EXPERIMENTAL EVALUATION
We conducted trace-based simulations in order to evaluate
cache performance under the dierent refreshment policies.
We outline our methodology and then proceed to present
and discuss the simulation results.
6.1 Methodology
The traces included the cache action taken on each request
for the currently-implemented passive refreshment. In order
to simulate other policies, however, we had to obtain
response header values or TTL values for requested objects.
Unfortunately, this data is not available in the recorded
trace. We therefore separately issued GET requests for the
URLs in the trace shortly after downloading it. We processed
the response headers and extracted cache directives
and values of relevant header elds. For cachable objects
(objects without a no-cache directive in the response header),
we applied the Squid object freshness model [11] (HTTP/1.1
compliant) described in Section 3 to calculate TTLs using
the values extracted above. When an explicit freshness-
lifetime duration was not provided by an Expires or max-age
header, we applied Squid's heuristic. We used 10% of
the time dierence between the time specied by the Date
header and the time specied by the Last-Modified header
subject to a maximum value of one day.
We issued more than a single GET for a sample of the objects
and repeated the TTL calculation. We found that cache control
directives and freshness-lifetime values do not change
frequently. This indicated that our estimates re
ected reasonably
well values that would have been obtained from the
origin server at the time requests were logged.
We then ran simulations using the original sequence of requests
and the extracted TTL values. To put all policies on
equal ground and eliminate the eect of boundary conditions
we also simulated passive, so the performance gures provided
later correspond to the simulated passive rather than
the one re
ected by the original labels (given in Section 4).
We rst discarded all requests that were not labeled as
HTTP GET. The simulation was only applied to requests
for URLs on which we obtained a 200 (OK) response on our
separately-issued requests. Note that 302 responses (HTTP
redirect) are not cachable and hence requests of URLs for
which we obtained such a response were discarded. The
simulation then utilized the logged information in the following
way: (i) All logged requests for each considered URL
were used along with the request time to determine the status
of the object in the simulated cache and the resulting
cache actions, (ii) the original cache-action label was used
to identify requests which arrived with a no-cache header
(our simulation accounted for such requests by resetting the
TTL to the freshness-lifetime duration even if a fresh copy
was present in the simulated cache), and (iii) the original
labels were also used in a heuristic that estimated at which
points objects were modied.
The modication heuristic considered various recorded labels
of the requests. Clearly, when a successful request was
classied by the labels in the trace as a cmiss-r (i.e. labeled
TCP REFRESH MISS in the trace, see Section contents
had changed. For requests labeled cmisses-d we did
not have an explicit indication whether content had changed
so we used a heuristic based on the logged size of the response
to the client. In order to simulate performance of
refreshment policies, we also had to estimate at which point
in the interval between requests the modication had actually
occurred (since refreshment policies stop refreshing once
the server invalidates the object). The simulation assumed
that when a modication had occurred between two consecutive
requests (and therefore incur a content miss on the
later it happened at the midpoint of the time interval
between the two requests. If more modications had
happened, it is likely that the rst one occurred earlier in
the interval, and hence our assumption means that more unproductive
renewals occur in the simulation than in reality,
and thus, the simulated policies would exhibit somewhat
worse tradeos. Since the majority of validations return
not-modied, however, this assumption could not have had
a signicant eect on our results.
Since we could not issue GET requests for all the URLs present
in the trace in a reasonable time without adversely aect-
ing our environment, we selected a subset and then scaled
the results up to factor out the sampling. We applied non-uniform
sampling with denser samples from more frequently-
requested URLs. In particular, we included all URLs that
were requested more than 12 times. In total we fetched
about 224K distinct URLs. The reason for non-uniform
sampling is the Zipf-low relation between requests and URLs
where many requests are issued to a small set of popular
URLs. The original logs had about 5 million dierent
URLs, most of them requested only once. Thus, a same-size
sample obtained through uniform sampling over URLs
would have yielded lower-condence estimates than the non-uniform
sample we used. The sampling bias was factored out
by scaling each frequency group individually.
The simulations assumed innite cache storage capacity. This
is consistent both with current industry trends and with the
actual traces we used, since objects requested twice or more
in the 6 day period were not likely to be discarded by the
replacement policy used in Squid.
In the performance metric used in the simulations we counted
all requests that constituted content hits. Content hits that
occurred more than a freshness-lifetime duration past the
previous (simulated) server contact were counted as freshness
misses and requests occurring within the duration were
counted as freshness hits. Content hits exclude requests to
explicit noncachable objects, requests with no-cache request
headers, and requests when the content had changed. Since
we had no information on such requests, we did not classify
the rst request of each object. Appropriate requests for objects
with freshness-lifetime-duration of 0 were counted as
content hits, but were all considered freshness misses. Note
that renewals are not performed on such objects and hence,
the number of freshness misses incurred on these objects is
not reduced by refreshment policies.
6.2 Simulation results
Under the simulated baseline policy passive (where objects
are refreshed only as a result of client requests), 48% of
content hits constituted freshness misses on the UC trace,
and 53% were freshness misses on the SD trace. We recall
that the respective numbers according to labels on the
recorded trace provided in Section 4 are 31% and 43%. The
gap is mainly due to simulating a shorter trace, dierent
boundary conditions, and the conservative heuristic used to
identify content hits. These factors should aect all policies
in a similar manner. So in order to put passive on equal
grounds with all other policies we chose to simulate it rather
than using the labels of the trace.
We evaluated the performance of the dierent refreshment
policies by the tradeo of overhead vs. coverage. The coverage
(reduction in freshness-misses) is calculated as the fraction
of the number of freshness misses according to passive
that are eliminated by the respective policy. More precisely,
if x denotes the number of freshness misses of passive, and
y the number of freshness misses of a policy P then the
coverage of P is the fraction (x y)=x.
We calculated the overhead of policy P as the dierence
between the number of validation requests issued by P and
the number of validation requests issued by passive. Recall
that the cache issues a validation request for each freshness
miss. Hence, the request overhead is the total number of
renewals performed minus the number of freshness misses
eliminated (converted to freshness hits). We normalize the
overhead by dividing it with the total number of freshness
misses that were converted to freshness hits.
To obtain the coverage/overhead tradeo for each type of
policy, we swept the value of its respective parameter. For
example, the points on the curve of recency correspond
to runs with recency(1); recency(2);
the points for freq were obtained on runs with freq(1;
. Note that opt(0), recency(0), recency (0),
are in fact passive. 3 These
tradeos are shown in Figure 6.
The performance of recency and recency policies was almost
identical, hence we omitted the recency curve. This
similarity shows that requests without a no-cache header are
about as likely to follow a request with a no-cache header
as one without a no-cache header.
Under all types of policies, the coverage peaks at about 63%-
67%. The remaining 33%-37% of freshness misses mostly
occur on objects with freshness-lifetime-duration of 0, on
which refreshment is not eective. The opt policies eliminates
all \addressable" freshness misses with overhead of
about 1.3 requests per miss, and eliminates the bulk of these
misses with an overhead of 0.5. These numbers bound the
potential of refreshment policies. They also indicate on the
performance loss by the restriction to the refreshment frame-
work. 4
The simulation results for opt and recency show the locality
eect, where most freshness misses that can be eliminated
occur within a small number of freshness-lifetime-
durations after a previous request. The left-most point on
the curves of recency and opt, which correspond to recency(1)
and opt(1), show that about 30% of freshness misses occur
within one freshness-lifetime-duration after the expiration of
the cached copy. The second points on the left correspond
to recency(2) and opt(2) and indicate that about additional
15% of freshness misses occur between one and two
freshness-lifetime durations passed the expiration. We note
that the observed fact that a very small number of freshness
occur more than 10 freshness-lifetime-durations
passed the expiration is not only due to locality but also
re
ects the interaction of the log duration of 6 days and the
most common freshness-lifetime-duration of 24 hours [33].
The fact that coverage of recency and the frequency-based
policies peaks at about the same place indicates that a very
small fraction of freshness misses are incurred on very infrequently
requested objects (since the frequency-based policies
do not perform any renewals on the rst request and
thus can not eliminate misses incurred on the second re-
quest. The correspondence in peak coverage of opt and
other policy-types is due to a \threshold phenomenon" where
most freshness misses occur on objects with a freshness-
lifetime-duration of 0 or occur within a small number of
freshness-lifetime-durations following a previous request.
The frequency-based policies freq(j; 0) and th-freq(th;
signicantly outperformed recency(k). This re
ects the
fact that the vast majority of freshness misses which can
be eliminated occur on the more popular URLs. The gap is
caused by the very large number of cachable URLs that were
3 We remark that these policies are able to achieve \contin-
uous" tradeos by mixing two consecutive integral values of
the respective parameter, for example, applying recency(1)
on some URLs and recency(2) on others.
4 Recall that under the refreshment framework, objects must
be kept fresh continuously till the following \hit." The optimal
unrestricted policy, which validates objects just before
they are requested, incurs no overhead.
request-overhead
per
eliminated
fmiss
fraction of fmisses eliminated
UC trace: performance of refreshment policies
OPT
RECENCY
request-overhead
per
eliminated
fmiss
fraction of fmisses eliminated
SD trace: performance of refreshment policies
OPT
RECENCY
Figure
Performance of the dierent refreshment policies when simulated on the UC and SD traces
requested only once. The recency(k) policy performed up
to k unproductive renewals 5 on each such request. The hybrid
policies freq and th-freq with m > 0 performed considerably
worse than the pure frequency-based policies (that
correspond to hence only results for are
shown. This behavior is not surprising given that recency
yielded much worse tradeos. Overall, the results indicate
that frequency-based object prioritization is more eective
than recency-based prioritization.
The domination of frequency-based policies is also consistent
with studies of cache replacement policies for Web contents
[8, 7], since dierent URLs tend to have a wide range of
characteristic popularities, a property that is captured better
by frequency-based policies. It is interesting to contrast
these results with a related study of refreshment policies that
we performed for DNS records. In contrast with our nd-
ings here, for DNS caches the recency and frequency-based
policies exhibited similar performance [26]. Our explanation
is that at the hostname level, there is signicantly smaller
fraction of \objects" that are resolved only once.
The performance of freq and th-freq is similar, although
th-freq, which normalizes the frequency by freshness-lifetime-
duration, performed somewhat better. The similarity is
mostly explained by the fact that the large majority of freshness-
lifetime-durations are of the same length (one day) and also
because for shorter durations, frequency of requests is correlated
with the duration. The policy th-freq provides a
spectrum of tradeos and better performance, particularly
in the low-overhead range. th-freq, however, may require
more book-keeping than freq. The particular tradeos obtained
by the frequency-based policies shows that signi-
cant improvements can be obtained with fairly low overhead.
About 10% of freshness misses can be eliminated with the
overhead of half of a validation request per eliminated freshness
miss; 25% of freshness misses can be eliminated with
overhead of a single request per eliminated miss; 50% can
be eliminated with overhead of two; and 65% of freshness
misses can be eliminated with overhead of three.
5 It could be less than k since we did not perform renewals
passed the termination time of the log.
7. CONCLUSION
A large fraction (30%-50%) of cache hits constitute freshness
misses, that is, the cached copy was not fresh, but
turned out to be valid after communication with the origin
server. Validations are performed prior to sending responses
to users, and signicantly extend cache service time. There-
fore, freshness misses impede the cache ability to speed-up
Web access.
An emerging challenge for Web content caches is to reduce
the number of freshness misses by proactively maintaining
fresher content. It seems that some cache vendors had already
implemented ad-hoc refreshment mechanisms. Our
proposed refreshment policies are a relatively low-overhead
systematic solution. Refreshment policies extend freshness
lifetime by selectively validating cached objects upon their
expiration. Since cache freshness is increased, requested objects
are more likely to be fresh and thereby are serviced
faster. We demonstrated that a good refreshment policy can
eliminate about 25% of freshness misses with an overhead of
a single validation requests per eliminated miss, that is, two
\oine" validation requests replace one \online" request.
For future work, we propose ways to further reduce renewal
overhead. We rst propose that renewals of objects located
at the same host are batched together, and thus, decrease
overhead by sharing the same persistent connection. Batching
can be natural as co-located objects often share the same
cache-control mechanism and subsets of such objects (that
are embedded on the same page or related pages) are often
requested together. A second proposal is to perform
renewals at o-peak hours. The most common freshness-
lifetime duration of 24 hours provides su-cient scheduling
exibility to do so (e.g., performing renewals due at 10am
EST at the signicantly less-busy time of 7am EST).
As a next step in the evaluation of the eectiveness of a
refreshment policy we hope to incorporate one such policy
in a popular caching server software such as Squid. Our
results indicate that the integration of refreshment would
not impose a signicant computational overhead, and would
boost performance in terms of user-perceived latency.
Acknowledgment
Our experiments would not have been possible without the
collection and timely availability of the NLANR cache traces.
We thank Duane Wessels for answering questions with regard
to a Squid logging bug.
8.
--R
--TR
Scale and performance in a distributed file system
Leases: an efficient fault-tolerant mechanism for distributed file cache consistency
Web cache coherence
Removal policies in network caches for World-Wide Web documents
Improving end-to-end performance of the Web using server volumes and proxy filters
Exploiting regularities in Web traffic patterns for cache replacement
On-line file caching
Aging through cascaded caches
Volume Leases for Consistency in Large-Scale Systems
Image-based Rendering with Controllable Illumination
Evaluating Server-Assisted Cache Replacement in the Web
Proactive Caching of DNS Records
Application-level document caching in the Internet
Maintaining Strong Cache Consistency in the World-Wide Web
--CTR
Edith Cohen , Haim Kaplan, Proactive caching of DNS records: addressing a performance bottleneck, Computer Networks: The International Journal of Computer and Telecommunications Networking, v.41 n.6, p.707-726, 22 April
Timo Koskela , Jukka Heikkonen , Kimmo Kaski, Web cache optimization with nonlinear model using object features, Computer Networks: The International Journal of Computer and Telecommunications Networking, v.43 n.6, p.805-817, 20 December | validation requests;HTTP;web caching;user-perceived latency;refreshment policies |
590776 | Overcoming the Myopia of Inductive Learning Algorithms with RELIEFF. | Current inductive machine learning algorithms typically use greedy search with limited lookahead. This prevents them to detect significant conditional dependencies between the attributes that describe training objects. Instead of myopic impurity functions and lookahead, we propose to use RELIEFF, an extension of RELIEF developed by Kira and Rendell [10, 11], for heuristic guidance of inductive learning algorithms. We have reimplemented Assistant, a system for top down induction of decision trees, using RELIEFF as an estimator of attributes at each selection step. The algorithm is tested on several artificial and several real world problems and the results are compared with some other well known machine learning algorithms. Excellent results on artificial data sets and two real world problems show the advantage of the presented approach to inductive learning. | Introduction
Inductive learning algorithms typically use a
greedy search strategy to overcome the combinatorial
explosion during the search for good hy-
potheses. The heuristic function that estimates
the potential successors of the current state in
the search space has a major role in the greedy
search. Current inductive learning algorithms
use variants of impurity functions like information
gain, gain ratio[25], gini-index[1], distance
ever, all these measures assume that attributes
are conditionally independent given the class and
therefore in domains with strong conditional dependencies
between attributes the greedy search
has poor chances of revealing a good hypothesis.
Kira and Rendell [10], [11] developed an algorithm
called which seems to be very
powerful in estimating the quality of attributes.
For example, in the parity problems of various degrees
with a significant number of irrelevant (ran-
dom) additional attributes RELIEF is able to correctly
estimate the relevance of all attributes in
a time proportional to the number of attributes
and the square of the number of training instances
(this can be further reduced by limiting the number
of iterations in RELIEF). While the original
RELIEF can deal with discrete and continuous at-
tributes, it can not deal with incomplete data and
is limited to two-class problems only. We developed
an extension of RELIEF called RELIEFF
that improves the original algorithm by estimating
probabilities more reliably and extends it to
handle incomplete and multi-class data sets while
the complexity remains the same.
RELIEFF seems to be a promising heuristic
function that may overcome the myopia of current
inductive learning algorithms. Kira and Rendell
used RELIEF as a preprocessor to eliminate irrelevant
attributes from data description before learn-
ing. RELIEFF is general, relatively efficient, and
reliable enough to guide the search in the learning
process. In this paper a reimplementation of Assistant
learning algorithm for top down induction
of decision trees [4] is described, named Assistant-
R. Instead of information gain, Assistant-R uses
RELIEFF as a heuristic function for estimating
the attributes' quality at each step during the tree
generation. Experiments on a series of artificial
and real-world data sets are described and the results
obtained using RELIEFF as a selection criterion
are compared to results of some other ap-
proaches. The following approaches are compared:
ffl the use of information gain as a selection criterion
ffl LFC [27], [28] that tries to overcome the myopia
of information gain with a limited lookahead;
ffl the naive Bayesian classifier, that assumes conditional
independence of attributes;
ffl the k-nearest neighbors algorithm.
The paper is organized as follows. In the next
section, the original RELIEF is briefly described
along with its interpretation and its extended version
RELIEFF. In Section 3, we present the reimplementation
of Assistant called Assistant-R. In
Section 4.1 we briefly describe the other algorithms
used in our experiments. In Section 4.2
we describe the experimental methodology. Section
5 describes experiments, and compare the results
of the different algorithms. We show that
Assistant-R performs at least as well as Assistant-
I and sometimes much better. In conclusion, the
potential breakthroughs are discussed on the basis
of the excellent results on artificial data sets.
Finally, integration of the compared algorithms is
proposed.
2. RELIEFF
2.1. RELIEF
The key idea of RELIEF is to estimate attributes
according to how well their values distinguish
among the instances that are near to each other.
For that purpose, given an instance, RELIEF
1. set all weights W[A] := 0.0;
2. for i := 1 to n do
3. begin
4. randomly select an instance R;
5. find nearest hit H and nearest miss M;
6. for A := 1 to #all attributes do
7. W[A] := W[A] - diff(A,R,H)/n
8.
9. end;
Figure
1 The basic algorithm of RELIEF
searches for its two nearest neighbors: one from
the same class (called nearest hit) and the other
from a different class (called nearest miss). The
original algorithm of RELIEF [10], [11] randomly
selects n training instances, where n is the user-defined
parameter. The algorithm is given in Figure
1.
Function diff(Attribute,Instance1,Instance2)
calculates the difference between the values of Attribute
for two instances. For discrete attributes
the difference is either 1 (the values are different)
or 0 (the values are equal), while for continuous
attributes the difference is the actual difference
normalized to the interval [0; 1]. Normalization
with n guarantees all weights W [A] to be in the
interval [\Gamma1; 1], however, normalization with n
is an unnecessary step if W [A] is to be used for
relative comparison among attributes.
The weights are estimates of the quality
of attributes. The rationale of the formula
for updating the weights is that a good attribute
should have the same value for instances
from the same class (subtracting the difference
should differentiate between
instances from different classes (adding the difference
The function diff is used also for calculating
the distance between instances to find the nearest
neighbors. The total distance is simply the
sum of differences over all attributes. In fact original
RELIEF uses the squared difference, which
for discrete attributes is equivalent to diff. In all
our experiments, there was no significant difference
between results using diff or squared differ-
ence. If N is the number of all training instances
then the complexity of the above algorithm is
O(n \Theta N \Theta #all attributes).
2.2. Interpretation of RELIEF's estimates
The following derivation shows that RELIEF's estimates
are strongly related to impurity functions.
It is obvious that RELIEF's estimate W [A] of attribute
A is an approximation of the following difference
of probabilities:
(different value of Aj
nearest instance from different class)
\GammaP (different value of Aj
nearest instance from same class) (1)
If we eliminate from (1) the requirement that
the selected instance is the nearest, the formula
becomes:
(different value of Ajdifferent class)
\GammaP (different value of Ajsame class)
(2)
If we rewrite
(equal value of
classjequal value of
we obtain using Bayes rule:
For sampling with replacement in strict sense
the following equalities hold:
Using the above equalities we obtain:
const \Theta
where
is highly correlated with the gini-index gain [1]
for classes C and values V of attribute A. The
difference is that instead of factor
the gini-index gain uses
Equation (3) shows strong relation of RE-
LIEF's weights with the gini-index gain. The
probability
that two instances have
the same value of attribute A in eq. (3) is a
kind of normalization factor for multi-valued at-
tributes. Impurity functions tend to overestimate
multi-valued attributes and various normalization
heuristics are needed to avoid this tendency (e.g.
gain ratio [25], distance measure [16], and binarization
of attributes [4]). Equation (3) shows that
RELIEF exhibits an implicit normalization effect.
Another deficiency of gini-index gain is that its
values tend to decrease with the increasing number
of classes [14]. Denominator which is constant
factor in equation (3) for a given attribute again
serves as a kind of normalization and therefore
RELIEF's estimates do not exhibit such strange
behavior as gini-index gain does.
The above derivation eliminated the "nearest
instance" condition from the probabilities. If we
put it back we can interpret RELIEF's estimates
as the average over local estimates in smaller
parts of the instance space. This enables RELIEF
to take into account the context of other
attributes, i.e. the conditional dependencies between
attributes given the class value which can
be detected in the context of locality. From the
global point of view, these dependencies are hidden
due to the effect of averaging over all training
instances, and exactly this makes impurity functions
myopic. Impurity functions use correlation
between the attribute and the class disregarding
the context of other attributes. This is the same
as using the global point of view and disregarding
the local peculiarities.
The example data set given in Table 1 illustrates
the difference between myopic estimation
functions and RELIEF. We have three attributes
and eight training instances. The class value is
determined with XOR function on attributes A1
and A2, while the third attribute A3 is randomly
generated. RELIEF (equation (1)) correctly estimates
that attributes A1 and A2 are the most
important while the contribution of attribute A3
is poor. On the other hand, W'[A] (equation (3)),
Table
Example data set and the estimated quality of
attributes
function A1 A2 A3 Class
information gain [9] 0.000 0.000 0.049
gain-ratio [25] 0.000 0.000 0.051
distance [16] 0.000 0.000 0.026
Ginigain' (equation (4)), original gini-index gain
[1], information gain [9], gain ratio [25], and distance
measure [16] estimate that the contribution
of A3 is the highest while attributes A1 and A2
are estimated as completely irrelevant.
Hong [8] developed a procedure similar to RELIEF
for estimating the quality of attributes,
where he directly emphasizes the use of contextual
information. The difference to RELIEF is that
his approach uses only information from nearest
misses and ignores nearest hits. Besides, Hong
uses the normalization to penalize the contribution
of nearest misses that are far away from a
given instance.
2.3. Extensions of RELIEF
The original RELIEF can deal with discrete and
continuous attributes. However, it can not deal
with incomplete data and is limited to two-class
problems only. Equation (1) is of crucial importance
for any extensions of RELIEF. It turned out
that the extensions of RELIEF are not straightforward
unless we realized that RELIEF in fact approximates
probabilities. The extensions should
be designed in such a way that those probabilities
are reliably approximated. We developed
an extension of RELIEF, called RELIEFF, that
improves the original algorithm by estimating
probabilities more reliably and extends it to deal
with incomplete and multi-class data sets. A brief
description of the extensions follows.
Reliable probability approxima-
tion: The parameter n in the algorithm RE-
LIEF, described in Section 2.1, represents the
number of instances for approximating probabilities
in eq. (1). The larger n implies more reliable
approximation. The obvious choice, adopted
in RELIEFF for relatively small number of training
instances (up to one thousand), is to run the
outer loop of RELIEF over all available training
instances.
The selection of the nearest neighbors is of crucial
importance in RELIEF. The purpose is to
find the nearest neighbors with respect to important
attributes. Redundant and noisy attributes
may strongly affect the selection of the nearest
neighbors and therefore the estimation of probabilities
with noisy data becomes unreliable. To increase
the reliability of the probability approximation
RELIEFF searches for k nearest hits/misses
instead of only one near hit/miss and averages
the contribution of all k nearest hits/misses. It
was shown that this extension significantly improves
the reliability of estimates of attributes'
qualities[13]. To overcome the problem of parameter
tuning, in all our experiments k was set to 10
which, empirically, gives satisfactory results. In
some problems significantly better results can be
obtained with tuning (as is typical for the majority
of machine learning algorithms).
Incomplete data: To enable RELIEF
to deal with incomplete data sets, the function
diff(Attribute,Instance1, Instance2) in RELIEFF
is extended to missing values of attributes by calculating
the probability that two given instances
have different values for the given attribute:
ffl if one instance (e.g. I1) has unknown value:
ffl if both instances have unknown value:
The conditional probabilities are approximated
with relative frequencies from the training set.
nearest neighbors
correlation
coefficient
independent atts
parity problems
Figure
2 The correlation of the RELIEFF's
estimates with the intended quality of attributes
on data sets with conditionally independent and
strongly dependent attributes.
This approach assumes that conditional probabilities
of attribute-values given the class are
applicable without the context of any other at-
tribute. This may in some cases be too naive,
however including the context of other atributes
is far too inefficient.
Multi-class problems: Kira and Rendell
that RELIEF can be used to estimate
the attributes' qualities in data sets with
more than two classes by splitting the problem
into a series of 2-class problems. This solution
seems unsatisfactory (in Section 4.1 we discuss the
performance of this approach and compare it with
the extension described below). To use it in prac-
tice, RELIEF should be able to deal with multi-class
problems without any prior changes in the
knowledge representation that could affect the final
outcomes.
Instead of finding one near miss M from a different
class, RELIEFF searches for k near misses
for each different class C and averages
their contribution for updating the estimate
W [A]. The average is weighted with the prior
probability of each class:
The idea is that the algorithm should estimate the
ability of attributes to separate each pair of classes
regardless of which two classes are closest to each
other. The normalization if prior probabilities of
classes is necessary as k near misses from each different
class would tend to exaggerate the influence
of classes with small number of cases.
Note that the time complexity of RELIEFF is
O(N 2 \Theta #attributes), where N is the number of
training instances.
2.4. RELIEFF's estimates and attribute's qualit
To estimate the contribution of parameter k (#
nearest hits/misses) on RELIEFF's estimates of
attribute's quality Kononenko [13] compared the
intended information gain of attributes with the
estimates, generated by RELIEFF, by calculating
the standard linear correlation coefficient. The
correlation coefficient can show how is the intended
quality and the estimated quality of attributes
related.
A typical graph for data sets with conditionally
independent attributes and with strongly dependent
attributes (parity problems of various de-
grees) is shown in Figure 2. For conditionally
independent attributes, the quality of the estimate
monotonically increases with the number of
nearest neighbors. For conditionaly dependent at-
tributes, the quality increases up to a maximum
but later decreases as the number of nearest neighbors
exceeds the number of instances that belong
to the same peak in the distribution space for a
given class.
Note that, if attributes were evaluated with
the myopic impurity functions, like the gini-index
and the information gain, the quality of the estimates
would be high for conditionally independent
attributes and poor for strongly dependent
attributes. This corresponds to the estimates
by RELIEFF with very large number of nearest
hits/misses.
To test the effect of the normalization factor in
eq. (3) we run RELIEFF also on one well known
medical data set, "primary tumor", described in
6 THE AUTHORS???
Section 5.3. The major difference between the estimates
by impurity functions and the estimates
by RELIEFF in the "primary tumor" problem is
in the estimates of two most significant attributes.
Information gain and gini-index overestimate one
attribute with 3 values (by the opinion of physicians
specialists). RELIEFF and normalized versions
of impurity functions correctly estimate this
attribute as less important.
3. Assistant-R
Assistant-R is a reimplementation of the Assistant
learning system for top down induction of decision
trees[4]. The basic algorithm goes back to CLS
(Concept Learning System) developed by Hunt et
al. [9] and reimplemented by several authors (see
[25] for an overview). In the following we describe
the main features of Assistant.
Binarization of attributes: The algorithm
generates binary decision trees. At each
decision step the binarized version of each attribute
is selected that maximizes the information
gain of the attribute. For continuous attributes a
decision point is selected that maximizes the at-
tribute's information gain. For discrete attributes
a heuristic greedy algorithm is used to find the
locally best split of attribute's values into two
subsets. The purpose of the binarization is to
reduce the replication problem and to strengthen
the statistical support for generated rules.
Decision tree pruning: Prepruning and
postpruning techniques are used for pruning off
unreliable parts of decision trees. For preprun-
ing, three user-defined thresholds are provided:
minimal number of training instances, minimal
attributes information gain and maximal probability
of majority class in the current node. For
postpruning, the method developed by Niblett
and Bratko [22] is used that uses Laplace's law
of succession for estimating the expected classification
error of the current node commited by
pruning/not pruning its subtree.
Incomplete data handling: During
learning, training instances with a missing value
of the selected attribute are weighted with probabilities
of each attribute's value conditioned with
a class label. During classification, instances with
missing values are weighted with unconditional
probabilities of attribute's values.
Naive Bayesian classifier: For each internal
node in a decision tree eventually a third
successor appears labeled with attribute's values
for which no training instances are available. For
such "null leaves", the naive Bayesian formula is
used to calculate the probability distribution in
the leaf by using only attributes that appear in
the path from the root to the leaf:
Y
A
Note that this calculation is done off-line, i.e.
during the learning phase. For classification, the
"null" leaves are already labeled with the calculated
class probability distribution and are used
for classification in the same manner as ordinary
leaves.
The main difference between Assistant and
its reimplementation Assistant-R is that RELI-
EFF is used for attribute selection. In addi-
tion, wherever appropriate, instead of the relative
frequency, Assistant-R uses the m-estimate
of probabilities, which was shown to often significantly
increase the performance of machine
learning algorithms[2], [3]. For prior probabilities
Laplace's law of succession is used:
of possible outcomes
where N is the number of all trials and N (X) the
number of trials with the outcome X. These prior
probabilities are then used in the m-estimate of
conditional probabilities:
The parameter m trades off between the contributions
of the relative frequency and the prior probability
In our experiments, the parameter m was set to
(this setting is usually used as default and, em-
pirically, gives satisfactory results [2], [3] although
with tuning in some problem domains better results
may be expected). The m-estimate is used
in the naive Bayesian formula (5), for postpruning
instead of Laplace's law of succession as proposed
by Cestnik and Bratko[3], and for RELIEFF's estimates
of probabilities. In eq. (1) we can use
probabilities from the root of the tree as an estimate
of prior probabilities for a lower internal
node t with n(t) corresponding training instances:
of Ajnearest miss;
Ajnearest miss; root)
of Ajnearest hit;
Ajnearest hit; root)
4. Experimental environment
4.1. Algorithms for comparison
We performed a series of experiments with
Assistant-R and compared its performance to the
following algorithms:
Assistant-I: A variant of Assistant-R that instead
of RELIEFF uses information gain for
the selection criterion, as does Assistant. How-
ever, the other differences to Assistant remain
(m-estimate of probabilities). This algorithm
enables us to evaluate the contribution of RE-
LIEFF. The parameters for Assistant-I and
Assistant-R were fixed throughout the experiments
(no prepruning, postpruning with
2).
LFC: Ragavan et al. [27], [28] use limited lookahead
in their LFC (Lookahead Feature Con-
struction) algorithm for top down induction of
decision trees to detect significant conditional
dependencies between attributes for constructive
induction. They show interesting results
on some data sets. We reimplemented their algorithm
[29] and tested its performance. Our
results, presented in this paper, show some
drawbacks of the experimental comparison described
by Ragavan and Rendell and confirm
the advantage of the limited lookahead for constructive
induction.
LFC generates binary decision trees. At each
node, the algorithm constructs new binary attributes
from the original attributes, using logical
operators (conjunction, disjunction, and
negation). From the constructed binary at-
tributes, the best attribute is selected and the
process is recursively repeated on two sub-sets
of training instances, corresponding to the
two values of the selected attribute. For constructive
induction a limited lookahead is used.
The space of possible useful constructs is re-
stricted, due to the geometrical representation
of the conditional entropy which is the estimator
of the attributes' quality. To further reduce
the search space, the algorithm also limits the
breadth and the depth of search.
AS LFC uses lookahead it is less myopic than
the greedy algorithm of Assistant. The comparison
of results may show the performance of
the greedy search in combination with RELI-
EFF versus the lookahead strategy. To make
results comparable to Assistant-R we equipped
LFC with pruning and probability estimation
facilities as described in Section 3. All tests
were performed with a default set of parameters
(depth of the lookahead 3, beam size 20),
although in some domains better results may
be obtained by parameter tuning. However,
higher values of the parameters may combinatorially
increase the search space of LFC, which
makes the algorithm impractical.
Naive Bayesian Classifier: A classifier that
uses the naive Bayesian formula (5) to calculate
the probability of each class given the values of
all attributes and assuming the conditional independence
of the attributes. A new instance
is classified into the class with maximal calculated
probability. The m-estimate of probabilities
was used and the parameter m was set
to 2 in all experiments. The performance of
the naive Bayesian classifier can serve as an estimate
of the conditional independence of attributes
k-NN: The k-nearest neighbor algorithm. For a
given new instance the algorithm searches for
nearest training instances and classifies the
instance into the most frequent class of these
k instances. For the k-NN algorithm the same
distance measure was used as for RELIEFF (see
Section 2.1).
The presented results were obtained with
Manhattan-distance. The results using Euclidian
distance are practically the same. The best
results with respect to parameter k are pre-
sented, although for fair comparison such parameter
tuning should be allowed only on the
training and not the testing sets.
We selected the naive Bayesian classifier and
the k-NN algorithm for comparison because they
are both well known, simple, and they both perform
well in many real-world problems. The performance
of these two algorithms may show the
nature of the classification problems.
4.2. Experimental methodology
Each experiment on each data set was performed
times by randomly selecting 70% of instances
for learning and 30% for testing and the results
were averaged. Each system used the same subsets
of instances for learning and for testing in order to
provide the same experimental conditions. To verify
the significance of differences we used the one-tailed
t-test with confidence
level) and the null hypothesis stating that the difference
is zero[5]. All the differences in results
having the value of statistic t above the threshold
are considered significant.
The exception from the above methodology
were the experiments in the finite element mesh
design problem, where the experimental methodology
was dictated by previous published results,
as described in Section 5.4.
Besides the classification accuracy, we measured
also the average information score[15]. This
measure eliminates the influence of prior probabilities
and appropriately treats probabilistic answers
of the classifier. The average information
score is defined as:
#testing instances
#testing instances
where the information score of the classification of
i-th testing instance is defined by:
is the class of the i-th testing instance,
P (Cl) is the prior probability of class Cl and
the probability returned by a classifier.
If the returned probability of the correct class is
greater than the prior probability the information
score is positive, as the obtained information is
correct. It can be interpreted as the prior information
necessary for correct classification minus the
posterior information necessary for correct classi-
fication. If the returned probability of the correct
class is lower than the prior probability the
information score is negative, as the obtained information
is wrong. It can be interpreted as the
prior information necessary for incorrect classification
minus the posterior information necessary
for incorrect classification.
The main difference between the classification
accuracy and the information score can be illustrated
with the following example. Let the
prior distribution of classes be P
let the posterior distribution
returned by the classifier be P
If the correct class is C 1 then the
information score is positive while the classification
accuracy treats the given posterior distribution
as wrong answer. If the correct class is C 2
then the information score is negative while the
classification accuracy treats the given posterior
distribution as correct answer.
Classification accuracy may in some special
cases exhibit high variance while information score
is much more stable. In a very special case
where we have a data set with irrelevant attributes
and exactly 50% of instances from one class and
50% of instances from the other class, the leave-
one-out testing for a probabilistic classifier would
give the approximate accuracy of 50%, while for
the"default" classifier, that classifies every instance
into the majority class, the accuracy would
be 0%. A slight modification of the distribution
of training instances would drastically change the
latter accuracy to approximately 50%. A more
drastic modification of the distribution, say 80%
of cases for one class and 20% for the other, would
increase the accuracy of the "default" classifier to
80%, while the accuracy of the probabilistic classifier
would be approximately 0:8 \Theta 0:8+0:2 \Theta 0:2 =
68%. However, for both classifiers the information
score would in all scenarios remain approximately
0 bits which would indicate, that both classifiers
are unable to extract any useful information from
attributes.
5. Experimental results
In this section we give results on several artificial
and real-world data sets. The presentation
of the experiments is divided into four parts according
to four groups of data sets: artificial data
sets with the controlled conditional dependency
between attributes, some other benchmark artificial
data sets, medical data sets, and other real-world
data sets. For each group we give a brief
description of data sets followed by the results.
The results in tables include averages over several
runs and standard errors.
5.1. Artificial data sets
We generated several data sets in order to compare
the performance of various algorithms:
INF1: Domain with three conditionally independent
informative binary attributes for each of
the three classes and with three random binary
attributes. The learner should detect which
three attributes are informative which is a relatively
easy task. All five algorithms should be
able to solve this problem.
INF2: Domain obtained from INF1 by replacing
each informative attribute with two attributes
whose values define the value of the original
attribute with XOR relation. For this prob-
lem, the learner should detect six important
attributes and the fact that attributes are pair-wise
strongly conditionally dependent. This
is a fairly complex problem and cannot be
solved with the myopic heuristics. This data
set should show the advantage of LFC and
Assistant-R.
TREE: Domain whose instances were generated
from a decision tree with 6 internal nodes, each
containing a different binary attribute. 5 random
binary attributes were added to the description
of instances. This problem should
be easy for greedy decision tree learning algorithms
while other approaches may have difficulties
due to an inappropriate knowledge representation
of the target concept.
PAR2: Parity problem with two significant binary
attributes and 10 random binary at-
tributes. 5% of randomly selected instances
were labeled with wrong class. This problem
is hard as there is a lot of attributes with equal
score when evaluated with a myopic evaluation
function, such as information gain.
PAR3: Same as PAR2 except that there were
three significant attributes for the parity relation
which makes the problem harder.
PAR4: Same as PAR2 except that there were
four significant attributes for the parity relation
which makes the problem the hardest among
the parity problems used in our experiments.
The basic characteristics of the artificial data
sets are listed in Table 2. Characteristics include
the percentage of the majority class (which can
be interpreted as "default accuracy") and the class
entropy which gives an impression of the complexity
of the classification problem.
The results of the learning algorithms LFC,
Assistant-I and Assistant-R, as well as the naive
Bayesian classifier and the k-NN algorithm, are
given in Table 3 (classification accuracy) and Table
(information score). The results are as expected
and show that:
ffl All classifiers perform well in a (relatively sim-
ple) domain with conditionally independent attributes
ffl Both versions of Assistant perform well in the
problem of the reconstruction of a decision tree
(TREE), while the other classifiers are significantly
worse.
Only Assistant-R and LFC are able to successfully
solve the problems with strong conditional
dependencies between attributes (INF2, PAR2-
4). However, of these two, Assistant-R performs
better, especially in the case of the hardest
problem (PAR4). Note that LFC can solve
PAR4 if the depth of the lookahead is increased,
Table
Basic description of artificial data sets
domain #class #atts. #val/att. # instances maj.class (%) entropy(bit)
INF2 3 21 2.0 200 36 1.58
Table
3 Classification accuracy of the learning systems on artificial data sets
domain LFC Assistant-I Assistant-R naive Bayes k-NN
INF1 86.0\Sigma5.1 90.1\Sigma3.5 88.8\Sigma3.8 91.6\Sigma3.1 89.0\Sigma3.6
INF2 67.1\Sigma6.3 55.4\Sigma9.8 68.7\Sigma7.8 32.1\Sigma4.5 56.8\Sigma6.3
TREE 75.8\Sigma5.4 79.2\Sigma5.7 78.8\Sigma6.2 69.0\Sigma5.9 68.2\Sigma5.3
93.6\Sigma3.3 74.9\Sigma7.9 95.7\Sigma2.8 56.7\Sigma5.7 79.4\Sigma4.3
however, the time complexity of the lookahead
increases exponentially with its depth. On the
other hand, Assistant-R solves all parity problems
equally quickly.
ffl The information score of the naive Bayesian
classifier in the problems with strong conditional
dependencies between attributes is poor
which indicates that this classifier failed to find
any regularity in these data sets.
5.2. Benchmark artificial data sets
Besides the artificial data sets from the previous
subsection, we used also the following benchmark
artificial data sets used by other authors (note
that results of other authors can not be directly
compared to our results as experimental conditions
(training/testing splits) were not the same):
BOOL: Boolean function defined on 6 attributes
with 10% of class noise (optimal recognition
rate is 90%). The target function is:
This data set was used by Smyth et al. [31]and
they report 67.2\Sigma1.7% of the classification accuracy
for naive Bayes, 82.5\Sigma1.1% for back-
propagation, and 85.9\Sigma0.9% for their rule-based
classifier.
LED: LED-digits problem with 10% of noise in
attribute values. The optimal recognition rate
is estimated to be 74%. Smyth et al. [31]
report 68.1\Sigma1.7% of the classification accuracy
for naive Bayes, 64.6\Sigma3.5 for backpropa-
gation, and 72.7\Sigma1.3 for their rule-based classi-
fier. This data set can be obtained from Irvine
database[21].
KRK1: The problem of legality of King-Rook-
King chess endgame positions. The attributes
describe the relevant relations between pieces,
such as "same rank" and "adjacent file". Originally
the data included five sets of 1000 examples
(1000 for learning and 4000 for testing) and
was used to test Inductive Logic Programming
algorithms[7]. The reported classification accuracy
is 99.7\Sigma0.1 %. We used only one set of
1000 examples (i.e. 700 instances for training).
KRK2: Same as KRK1 except that the only
available attributes are the coordinates of
pieces. The same data set was used by
Mladeni-c[19]. The reported results are about
69% accuracy for her ATRIS system and 64%
for Assistant.
The basic description of data sets is provided
in
Table
5 and results are given in Tables 6 and 7.
It is interesting that in the LED domain, the
naive Bayesian classifier and the k-NN algorithm
reach the estimated upper bound of the classification
accuracy. This suggests that all attributes
should be considered for optimal classification
in this domain. In this problem the attributes
are conditionally independent given the
class, therefore the good performance of the naive
Bayesian classifier is not surprising. However, in
the other three domains the performance of the
naive Bayesian classifier is poor, due to the strong
Table
4 Average information score of the learning systems on artificial data sets
domain LFC Assistant-I Assistant-R naive Bayes k-NN
Table
5 Basic description of some benchmark artificial data sets
domain #class #atts. #val/att. # instances maj.class (%) entropy(bit)
Table
6 Classification accuracy of the learning systems on artificial data sets
domain LFC Assistant-I Assistant-R naive Bayes k-NN
LED 70.8\Sigma2.3 71.1\Sigma2.4 71.7\Sigma2.2 73.9\Sigma2.1 73.9\Sigma2.1
KRK1 98.7\Sigma1.2 98.6\Sigma1.2 98.6\Sigma1.2 91.6\Sigma1.4 92.2\Sigma1.9
KRK2 86.0\Sigma2.1 66.6\Sigma3.1 70.1\Sigma3.3 64.8\Sigma2.1 70.7\Sigma1.7
Table
7 Average information score of the learning systems on artificial data sets
domain LFC Assistant-I Assistant-R naive Bayes k-NN
LED 2.13\Sigma0.07 2.11\Sigma0.06 2.12\Sigma0.07 2.33\Sigma0.05 2.22\Sigma0.05
conditional dependencies between attributes. The
information score (see Table 7) shows that the
naive Bayesian classifier provides (on the average)
no information in the BOOL and KRK2 domains.
The performance of the different variants of Assistant
is almost the same, except for the KRK2
domain, where the performance of Assistant-I is
poor (note that the default accuracy in KRK2
is 67%). The performance of Assistant-R and
the k-NN algorithm is significantly better (99.95%
confidence level). However, the information score
shows that both, Assistant-R and k-NN, are not
very successful in this problem. As expected,
without constructive induction it is not possible to
reveal regularities in the chess positions described
only with the coordinates of pieces. LFC is able
to construct important attributes in this domain,
which enables it to achieve significantly better results
than the other algorithms.
5.3. Medical data sets
We compared the performance of the algorithms
on several medical data sets:
ffl Data sets obtained from University Medical
Center in Ljubljana, Slovenia: the problem of
locating of primary tumour in patients with
metastases (PRIM), the problem of predicting
the recurrence of breast cancer five years after
the removal of the tumour (BREA), the
problem of determining the type of the cancer
in lymphography (LYMP), and diagnosis in
rheumatology (RHEU).
ffl HEPA: prognostics of survival for patients suffering
from hepatitis. The data was provided
by Gail Gong from Carnegie-Mellon University.
ffl Data sets obtained from the StatLog
database[18]: diagnosis of diabetes (DIAB) and
diagnosis of heart diseases (HEART). For the
DIAB data set, Ragavan & Rendell [27]report
78.8% classification accuracy with their LFC al-
gorithm. They also report poor performance of
Table
8 Basic description of the medical data sets
domain #class #atts. #val/att. # instances maj.class (%) entropy(bit)
PRIM 22 17 2.2 339 25 3.89
Table
9 Classification accuracy of the learning systems on medical data sets
domain LFC Assistant-I Assistant-R naive Bayes k-NN
BREA 76.1\Sigma4.3 76.8\Sigma4.6 78.5\Sigma3.9 78.7\Sigma4.5 79.5\Sigma2.7
LYMP 82.4\Sigma5.2 77.0\Sigma5.5 77.0\Sigma5.9 84.7\Sigma4.2 82.6\Sigma5.7
HEPA 79.0\Sigma5.3 77.2\Sigma5.3 82.3\Sigma5.4 86.1\Sigma3.9 82.6\Sigma4.9
HEART 77.3\Sigma5.2 75.4\Sigma4.0 77.6\Sigma4.5 84.5\Sigma3.0 82.9\Sigma3.7
Table
Average information score of the learning systems on medical data sets
domain LFC Assistant-I Assistant-R naive Bayes k-NN
several other algorithms without constructive
induction (up to 58%). However, our results
(see below) and results of the StatLog project
[18] show that the poor results of the other algorithms
in this domain are not due to the lack
of constructive induction. In our experiments,
on DIAB dataset, all classifiers perform equally
well, with the exception of the naive Bayesian
classifier which is significantly better.
The basic characteristics of the above medical
data sets are given in Table 8. The results of experiments
on these data sets are provided in Tables
9 and 10.
In medical data sets, attributes are typically
conditionally independent given the class . There-
fore, it is not surprising that the naive Bayesian
classifier shows clear advantage on these data
sets[12]. It is interesting that the performance of
the k-NN algorithm is good in these domains, although
worse than the performance of the naive
Bayesian classifier.
The information score (Table 10) for BREA
data set indicates that no learning algorithm was
able to solve this problem. This suggests that the
attributes are not relevant.
Both versions of Assistant have similar per-
formance, except in the HEPA domain where,
Assistant-R has significantly better performance
confidence level). A detailed analysis
showed that in this problem RELIEFF discovered
a significant conditional interdependency between
two attributes given the class. These two
attributes score poorly when considered indepen-
dently. That is why Assistant-I was not able to
discover this regularity in data.
On the other hand, other attributes
are available that contain similar information
as these two attributes together. This is the
reason why the naive Bayesian classifier performs
better. We tried to provide the naive Bayesian
classifier with an additional attribute by joining
the two conditionally dependent attributes. How-
ever, the performance remained the same.
achieved significantly better results than
the other two inductive algorithms in the LYMP
domain, where constructive induction seems to
be useful. However, LFC performed significantly
worse in the RHEU domain while in the other
domains the three inductive algorithms perform
equally well.
5.4. Non-medical real-world data sets
We compared the performance of the algorithms
also on the following non-medical real world data
sets (SOYB, IRIS, and VOTE are obtained from
the Irvine database[21], SAT is obtained from the
StatLog database [18]):
SOYB: The famous soybean data set used by
IRIS: The well known Fisher's problem of determining
the type of iris flower.
MESH3,MESH15: The problem of determining
the number of elements for each of the edges
of an object in the finite element mesh design
problem[6]. There are five objects for which
experts have constructed appropriate meshes.
In each of five experiments one object is used
for testing and the other four for learning and
the results are averaged. The results reported
by D-zeroski [7] for various ILP systems are
12% classification accuracy for FOIL, 22% for
mFOIL and 29% for GOLEM and the result reported
by Pompe et al. [23] is 28% for SFOIL.
The description of the MESH problem is appropriate
for ILP systems. For attribute learners
only relations with arity 1 (i.e. attributes) can
be used to describe the problem. Note that in
this domain the training/testing splits are the
same for all algorithms. The testing methodology
is a special case of leave-one-out, therefore,
the results in the tables for this problem have
no standard deviations.
Quinlan [26] reports results of some ILP systems
that achieved over 90% in that domain
testing on positive and negative instances.
However, those results are misleading. Each
positive instance has ten negative instances in
average. Therefore we have 11 copies of the
same instance and any classification of this instance
is correct at least for 9 out of 11 copies
which gives 82% classification accuracy for a
classifier that always classifies into wrong class.
MESH3 contains the three basic attributes
from the original database and ignores the relational
description of objects. Therefore, in the
domain attribute learners are given less
information than ILP learners.
contains, besides the 3 original at-
tributes, 12 attributes derived from the relational
background knowledge. In this prob-
lem, attribute learners have advantage as they
are already provided with additional attributes.
The provided description of objects for ILP
learners is actually more informative. In princi-
ple, the same attributes and a number of additional
attributes could be derived by (extremely
cleaver) ILP learners from the relational description
of the background knowledge. How-
ever, this is a fairly complex task. Therefore
attribute learners with MESH15 data set have
better chances than ILP learners to reveal a
good hypothesis.
SAT: The database consists of multi-class spectral
values of pixels in 3 \Theta 3 neighborhoods in
a satellite image, and the classification of the
central pixel in each neighborhood. The results
of the StatLog project[18] are 90.6% classification
accuracy for the k-NN algorithm, 86.1%
for backpropagation, 85.0% for C4.5, 84.8% for
CN2 and 69.3% for the naive Bayesian classifier
(using relative frequencies and not the m-estimate
of probabilities).
VOTE: The voting records are from a session
of the 1984 United States Congress. Smyth
et al. [31] report 88.9% of classification accuracy
for the naive Bayesian classifier, 93.0% for
backpropagation and 94.9% for their rule-based
classifier.
The basic characteristics of non-medical real
world data sets are presented in Table 11. Tables
12 and 13 give the results. On SOYB and
IRIS data sets, all classifiers perform equally well.
The results of the naive Bayesian classifier indicate
that the attributes are conditionally relatively independent
in these data sets, which is in agreement
with previously published results.
On the SAT data set, k-NN significantly out-performs
other algorithms which is in agreement
with the results of the StatLog project [18]. How-
ever, the naive Bayesian classifier with the m-estimate
of probabilities reaches the classification
accuracy of inductive learning algorithms. The
results of the naive Bayesian classifier used in the
14 THE AUTHORS???
Table
Basic description of the non-medical real-world data sets
domain #class #atts. #val/att. # instances maj.class (%) entropy(bit)
Table
Classification accuracy of the learning systems on non-medical real-world data sets
domain LFC Assistant-I Assistant-R naive Bayes k-NN
IRIS 95.0\Sigma3.8 95.7\Sigma3.7 95.2\Sigma2.6 96.6\Sigma2.6 97.0\Sigma2.1
Table
Average information score of the learning systems on non-medical real-world data sets
domain LFC Assistant-I Assistant-R naive Bayes k-NN
StatLog project are much worse. Cestnik [2] has
shown that the m-estimate significantly increases
the performance of the naive Bayesian classifier
which is also confirmed with our experiments.
Both versions of Assistant perform the same
on all data sets except on the SAT data set where
Assistant-R and LFC achieve significantly better
result (99.95% confidence level). This result confirms
that RELIEFF estimates the quality of attributes
better than the information gain.
On the VOTE data set the naive Bayesian classifier
is the worst, while both versions of Assistant
are comparable to the rule based classifier by
Smyth et al. [31].
The most interesting results appear in the
domains. Although attribute learners in
MESH3 have less information than ILP systems,
they all outperform the results by ILP systems
as reported by D-zeroski [7] and Pompe et al.
[23]. With 12 additional attributes in MESH15,
the results of inductive learners are significantly
improved. All inductive learning systems significantly
outperform the naive Bayesian classifier
and the k-NN algorithm.
A detailed analysis showed that this excellent
result by both versions of Assistant is due to the
use of the naive Bayesian formula to calculate the
class probability distribution in "null" leaves (see
Section 3). Namely, for this problem it often happens
that testing instances fall into a "null" leaf
because there are no training instances that have
the same values of significant attributes as the
testing instances. The naive Bayesian classifier
efficiently solves this problem.
LFC generates no "null" leaves as all constructed
attributes are strictly binary with values
true and false. Therefore, the classification of objects
with a different value of the original attribute
than all training instances always proceeds to the
branch labeled false. The effect of this strategy is
that, for a given testing instance, the corresponding
leaf contains training instances with same or
similar values for most of the attributes that appear
on the path from the root to the leaf. This
strategy also works well in MESH problems.
6. Discussion
Note that the null leaves of both versions of Assistant
had no influence on the performance on arti-
ficial data sets as there is no missing values in the
data. Also, in MESH15 problem the performance
of LFC is good although it does not generate null
leaves. Therefore, the use of null leaves is not the
crucial difference between Assistant and LFC.
Equation (3) shows an interesting relation between
the RELIEF's estimates and impurity func-
tion. RELIEF can efficiently estimate continuous
and discrete attributes. The implicit normalization
in eq. (3) enables RELIEF to appropriately
deal with multivalued attributes. However,
if Assistant-I would use eq. (3) instead of the information
gain, it would still be myopic. For ex-
ample, in PAR2-4 problems, eq. (3) would estimate
all attributes as equally non-important.
Therefore, the reason of the success of
Assistant-R is in the "nearest instances" heuristic
which influences the estimation of probabil-
ities. This heuristic enables RELIEF to detect
strong conditional dependencies between the attributes
which would be overlooked if the estimates
of probabilities would be done on randomly
selected instances instead of the nearest instances.
RELIEFF is an efficient heuristic estimator of
attribute quality that is able to deal with data sets
with conditionally dependent and independent at-
tributes. The extensions in RELIEFF enable it to
deal with noisy, incomplete, and multi-class data
sets. With increasing the number (k) of nearest
hits/misses the correlation of RELIEFF's estimates
with other impurity functions also increases
unless k is greater than the number of instances
in the same peak of the instance space. The study
reported in [14] showed that RELIEFF has an acceptable
bias with respect to other measures when
estimating attributes with different number of values
The myopia of current inductive learning systems
can be partially overcome by replacing
the existing heuristic functions with RELIEFF.
Assistant-R, a variant of top down induction of
decision trees algorithms that uses RELIEFF for
estimating the quality of the attributes, significantly
outperforms other classifiers in domains
with strong conditional dependencies between at-
tributes. The myopia of other inductive learners
may cause them to overlook significant relations.
While this can be easily demonstrated with artificial
data sets, it was also shown in two real
world problems: HEPA and SAT. In these data
sets RELIEFF detected significant conditional interdependencies
between attributes, that resulted
in a significantly better result by Assistant-R than
the result by Assistant-I.
One feature of RELIEF not addressed in this
paper is that if the same attribute is replicated
in a data set, all replications will get the same
estimate. With the increasing number of replications
the quality of estimates will descrease as the
replicated attribute affects the distances between
instances.
For constructive induction LFC uses a limited
lookahead to detect significant conditional dependencies
between the attributes. LFC shows similar
advantage over other algorithms as Assistant-
R does. In one artificial problem (KRK2) and
one real world problem (LYMP) LFC performs
significantly better due to constructive induction.
However, in some cases the constructive induction
may spoil the results as is the case with RHEU
data set. LFC performs well in most of the prob-
lems, which suggests that the limited lookahead
is a good search strategy in most real-world prob-
lems. The lookahead, however, should have a reasonable
limit as the time complexity exponentialy
increases with the lookahead depth.
Although RELIEFF may overcome the myopia,
it is useless in Assistant-R when the change of representation
is required. In such cases the constructive
induction should be applied. For example, in
the KRK2 problem, Assistant-R achieves good result
which can not be further improved without
constructive induction. A good idea for constructive
induction may be to use RELIEFF instead of
or in the combination with the lookahead.
The naive Bayesian classifier has obvious advantage
in domains with conditionally relatively
independent attributes, such as medical diagnostic
problems. In such domains, the naive Bayesian
classifier is able to reliably estimate the conditional
probabilities and is also able to use all at-
tributes, i.e all available information. It would be
interesting to appropriately combine the power of
RELIEFF and the naive Bayesian classifier.
Current ILP systems [20] are not able to use the
attributes appropriately. This was demonstrated
in the MESH3 domain where all attribute learn-
ers outperformed existing ILP systems. To enable
ILP systems to deal with the attribute-value rep-
resentation, a combination with the (semi) naive
Bayesian classifier could be useful. On the other
hand, current ILP systems use greedy search techniques
and the heuristics that guide the search
are myopic. Pompe and Kononenko [24] implemented
an adapted version of RELIEFF in the
FOIL like ILP system called ILP-R and prelemi-
nary experiments show similar advantages of this
system over other ILP systems as Assistant-R has
over Assistant-I.
7. Conclusion
RELIEFF is an efficient heuristic estimator of attribute
quality that is able to deal with data sets
with conditionally dependent and independent at-
tributes, with noisy, incomplete, and multi-class
data sets. The myopia of current inductive learning
systems can be partially overcome by replacing
the existing heuristic functions with RELI-
EFF. The acceptable increase in computational
complexity may in certain domains payoff with
eventual discovery of strong conditional dependencies
between attributes, which cannot be detected
using the myopic impurity measure to guide
the greedy search.
The experimental results indicate that in the
majority of real world problems the myopia has
no or only marginal effect. One may wonder
whether myopia is really worth much attention
at all. However, when faced with a new data
set it is unreasonable to try only myopic algorithm
unless it is know in advance that in the
data set there are no strong conditional dependencies
between attributes. Any serious application
of machine learning on new data should try
to discover as much regularities in the data as pos-
sible. Therefore, non-myopic approaches, such as
one described in this paper, should be used as indispensable
tools for analysing the data.
Acknowledgements
The use of m-estimate in equation (1) was proposed
by Bojan Cestnik. We thank Matja-z Zwitter for the
PRIM and BREA data sets, Milan Sokli-c for LYMP,
Gail Gong for HEPA, Padhraic Smyth for BOOL and
LED, Sa-so D-zeroski for KRK1 and MESH, Bob Hen-
ery for the DIAB, HEART, and SAT data sets from
the StatLog database at Strathclyde University, and
Patrick Murphy and David Aha for the data sets from
the Irvine database. We are grateful to our colleagues
Sa-so D-zeroski, Matev-z Kova-ci-c, Matja-z Kukar, Uro-s
Pompe, and Tanja Urban-ci-c and to anonymous reviewers
for their comments on earlier drafts that significantly
improved the paper. This work was supported
by the Slovenian Ministry of Science and Technology
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590793 | Learning Structure from Data and Its Application to Ozone Prediction. | In this paper we propose an algorithm for structure learning in predictive expert systems based on a probabilistic network representation. The idea is to have the simplest structure (minimum number of links) with acceptable predictive capability. The algorithm starts by building a tree structure based on measuring mutual information between pairs of variables, and then it adds links as necessary to obtain certain predictive performance. We have applied this method for ozone prediction in Mxico City, where the ozone level is used as a global indicator for the air quality in different parts of the city. It is important to predict the ozone level a day, or at least several hours in advance, to reduce the health hazards and industrial losses that occur when the ozone reaches emergency levels. We obtained as a first approximation a tree-structured dependency model for predicting ozone in one part of the city. We observe that even with only three parameters, its estimations are acceptable.A causal network representation and the structure learning techniques produced some very interesting results for the ozone prediction problem. Firstly, we got some insight into the dependence structure of the phenomena. Secondly, we got an indication of which are the important and not so important variables for ozone forecasting. Taking this into account, the measurement and computational costs for ozone prediction could be reduced. And thirdly, we have obtained satisfactory short term ozone predictions based on a small set of the most important parameters. | Introduction
Learning is defined as "any process by which a
system improves its performance" [1]. Since the
first days of research in artificial intelligence, the
ability to learn has been considered as one of the
essential attributes of an "intelligent system", and
a considerable amount of research has been done
in this area. Learning has focused in acquiring
concepts from examples in what is called inductive
learning. The development of expert systems
has motivated further research in learning to automate
the process of knowledge acquisition. This is
considered one of the main problems for the construction
of knowledge-based systems.
An important aspect in inductive learning is
to obtain a model which represents the domain
knowledge, and is accessible to the user. In par-
ticular, it is useful to obtain the dependency information
between the variables involved in the
phenomena. That is, those factors that are important
for certain variable, and those that are not.
This is of particular interest in predictive expert
systems, when we want to forecast some variables
based on known parameters. It is useful to know
the parameters which have more incidence in the
unknowns, and the ones that have not much influ-
ence. A knowledge representation paradigma that
captures this dependency information is a probabilistic
network.
Probabilistic networks (PN) [3], also known as
Bayesian networks, causal networks or probabilistic
influence diagrams, are graphical structures
used for representing expert knowledge, drawing
conclusions from input data and explaining the
reasoning process to the user. A PN is a directed
acyclic graph (DAG) whose structure corresponds
to the dependency relations of the set of variables
represented in the network (nodes), and which
is parameterized by the conditional probabilities
(links) required to specify the underlying distri-
bution. The structure of the network makes explicit
the dependence and independence relations
between the variables, which are important: (i)
in representing the knowledge of the domain, and
(ii) for efficient probability propagation.
Fig. 1. A probabilistic network.
If we use a PN representation, learning is divided
naturally into two aspects: parameter learning
and structure learning [2]. Parameter learning
has to do with obtaining the required probability
distributions for a certain structure. Structure
learning has to do with obtaining the topology of
the network, including which variables are relevant
for a particular problem, and their depen-
dencies. We are interested in this second aspect,
that is in obtaining the dependency structure of
certain phenomena, to get a better understanding
of it and to use it as a predictive tool.
In section 2 we give a brief introduction to
probabilistic networks. Section 3 reviews previous
work on structure learning, and section 4 introduces
our methodology for obtaining a dependency
structure for predictive systems. In section
5 we describe the problem of Ozone prediction in
Mexico City, and we present some experimental
results in section 6. Finally, we give some conclusions
and possible directions for future work.
2. Probabilistic Networks
A probabilistic network is a graphical representation
of dependencies and independencies for probabilistic
reasoning in expert systems. Each node
represents a discrete random variable and each arc
a probabilistic dependency. The variable at the
end of a link is dependent on the variable(s) at
its origin, e.g. C is dependent on A in the PN
in figure 1, as indicated by link 1. We can think
of the graph in figure 1 as representing the joint
probability distribution of the variables A; B; :::; G
as:
Equation (1) is obtained by applying the chain
rule and using the dependency information represented
in the network.
The topology of a PN gives direct information
about the dependency relationships between
the variables involved. In particular, it represents
which variables are conditionally independent
given another variable. By definition, A is
conditionally independent of B, given C, if:
This is represented graphically by node C "sep-
arating" A from B in the network. In general, C
will be a subset of nodes from the network that if
removed will make the subsets of nodes A and B
"disconnected". Independence in a PN network is
tested with a criteria called D-separation [2].
A DAG representation G of a probability distribution
P is an I-map [2] if all the independencies
represented in G are present in P . It is a minimal
I-map if it is an I-map with the minimum number
of links, that is, if any link is removed, there
will be an independency relation in G that is not
present in P . Formally, a probabilistic network is
minimal I-map for a joint probability distribution
P [2]. In other words, it is a graph with the minimum
number of links that faithfully represents all
the probabilistic independencies for a set of random
variables.
Given a knowledge base represented as a probabilistic
network, it can be used to reason about
the consequences of specific input data, by what
is called probabilistic reasoning. This consists in
instantiating the input variables, and propagating
their effect through the network to update the
probability of the hypothesis variables. In contrast
with previous approaches (e.g., MYCIN and
(c)
(a)
(b)
Fig. 2. Network structures: (a) tree, (b) polytree, and (c)
multiply-connected.
Prospector [4]), the updating of the certainty measures
is consistent with probability theory, based
on the application of Bayesian calculus and the
independencies represented in the network.
Probability propagation in a general network
is a complex problem, but there are efficient algorithms
for certain restricted structures, and
alternative approaches for more complex net-
works. Pearl [3] developed a method for propagating
probabilities in networks which are tree-
structured, i.e. each node has only one incoming
link or one parent. An example of a probabilistic
tree is shown in figure 2 (a).
In a probabilistic tree every node has only one
parent except one node, denoted root, which has
no incoming links. Given certain evidence V , represented
by the instantiation of the corresponding
variables, the posterior probability of any variable
taking its i value (B i ), by Bayes' theorem will
be:
Given the dependencies represented in the tree,
separates it into two independent subtrees, one
formed by all the descendants of B and the other
by every other node. Thus, we can decompose the
evidence variables into two sets, V \GammaB which represents
all the data rooted at B, and V +B for the
data contained in the rest of the network. So (3)
can be written as:
Given that the two subtrees are conditionally independent
given B:
Substituting (5) in (4), and applying some alge-
bra, we obtain:
Where ff is a normalizing constant. Equation (6)
provides a product rule for updating the probability
on every node in the network by combining the
evidence coming from its descendants with the one
coming from its parent. It shows that a prior probability
is not required, except for the root node
(A) for which P
the following terms [2]:
Then we can write (6) as:
Equation constitutes the basis for the propagation
mechanism in a probabilistic tree. For
this we only need to store the vectors - and - in
each node, and update them with the corresponding
parameters from its neighbors; and the fixed
conditional probability matrix P for that node.
This can be implemented by a message passing
scheme in which each node acts as a simple process
which communicates with its neighbours (fa-
ther and sons). Initially the network is in equi-
librium. When information arrives, some nodes
called data nodes, are instantiated and the information
is propagated through the network by
each node sending messages to its parent and sons.
Each node uses this information to update its local
parameters, and update its posterior probability if
required. After the messages reach the root node,
they will propagate top-down until they reach the
leaf nodes, where the propagation terminates and
the network comes to a new equilibrium. So the
information propagates through the tree in a single
pass, in a time (in parallel) proportional to the
diameter of the network.
An extension for polytrees, was proposed by
Kim and Pearl [5]. In a polytree each node can
have multiple parents, but it is still a singly connected
graph. A polytree is depicted in figure 2
(b). The main difference with the algorithm for
trees, is that for multi-parent nodes, the conditional
distribution given all their parents is re-
quired. The time for propagation is still linearly
proportional to the diameter of the network.
For more complex, multiply connected net-
works, see fig. 2 (c), there are alternative techniques
for probability propagation, such as clustering
[6], conditioning [2], and stochastic simulation
[2]. These methods are efficient for certain
types of structures, mainly sparse networks. But
in general, probability propagation in a complex
network is an NP-hard problem [7].
Thus, for efficiency reasons, and also for clarity
and expressiveness, it is important to obtain the
simplest structure, with the minimum number of
links, which models appropriately the phenomena
of interest. A complete graph will be a trivial Imap
of any probability distribution, but it would
not be useful in terms of knowledge representation
or computational efficiency.
3. Structure Learning Approaches
Structure learning consists in finding the topology
of the network, that is the dependency relationships
between the variables involved. Most expert
systems obtain this structure from the expert, representing
in the network the expert's knowledge
about the causal relations in the domain. But for
complex problems there might be no expert that
has a complete understanding of the domain to
obtain all these dependency (and independency)
relations, and if so, her/his knowledge could be de-
ceiving. Also, knowledge acquisition could be an
expensive and time consuming process. So we are
interested in using empirical observations to obtain
and improve the structure of a probabilistic
network. Some previous research has been done
on inducing the structure of a PN from statistical
data. Chow and Liu [8] presented an algorithm for
representing a probability distribution as dependency
tree, and this was later extended by Rebane
and Pearl [9] for recovering causal polytrees.
Chow and Liu's [8] motivation was reducing the
memory requirements for storing a n-dimensional
discrete probability distribution. For this they developed
a method for approximating a probability
distribution by a product of second-order dis-
tributions, which is equivalent to a probabilistic
tree. Thus, the joint probability distribution will
be represented as:
Y
Where X j(i) is the cause or parent of X i . Each
variable has one parent, except one (the root)
which has no parent, so the method is restricted
to a tree structure. They considered the approximation
of the original distribution by a dependency
tree as an optimization problem, and used
a quantity that measures the difference in information
contained in the two distributions. That
is:
x
Where the problem is reduced
to finding the tree dependency distribution
that approximates the original distribution P
such that I(P; P ) is minimal. To find the optimum
tree, they use the entropy measure for the
mutual information between two variables, defined
as:
x
log(P
If we assign to every branch in the tree the
weight that corresponds to the mutual information
between the variables connected by that link,
then the weight of the tree is the sum of the
weights for all the branches. It can be shown
[8] that maximizing the total branch weight is
equivalent to minimizing the closeness measure
so the tree with the maximum weight
will be the optimum tree dependency approximation
of P . This result makes it possible to find the
optimum tree structure by a simple algorithm that
uses only the n(n \Gamma 1)=2 second-order distributions
that correspond to all the possible branches for n
variables. These are ordered according to their
weight, and the two with maximum weight are selected
as the first two branches in the tree. Then
the other branches are selected in decreasing order
whenever they do not form a cycle with the
previously selected ones, until all the variables are
covered branches). Thus, to obtain a tree-structured
PN from sample data, we just need to
estimate the joint frequencies and mutual information
for all pairs of variables, and then construct
the optimum tree by the previous algorithm.
Rebane and Pearl [9] extended Chow's method,
developing a similar algorithm for the construction
of a polytree from statistical data. A polytree
is a singly connected network in which each
each node can have multiple parents. So the joint
probability distribution can be expressed as:
Y
Where fX is the set of parents
of the variable X i . The algorithm for constructing
a polytree starts by using the tree recovering
algorithm for constructing the skeleton,
that is the network without the directionality in
the links. Then it checks for the local dependencies
between variables and uses this information to
determine the directionality of the branches. The
local dependency tests is applied to all connected
variable triples and, by checking if
all variable pairs are dependent or independent,
it can partially determine the directionality of the
corresponding links. This test is applied to all
nodes starting from the outermost ones (leafs) in-
wards, until all possible directionalities are found.
In general, it is not possible to find the direction
of all the branches, and external semantics are
needed for completion [2].
Recent work has focused in two aspects: to
combine statistical data with expert knowledge;
and to induce multiply-connected networks from
data. The first approach is based on combing expert
knowledge and data to overcome the limitations
of the previous techniques, and obtain a
more general and complete dependence structure.
Sucar et al. [10] start from a structure derived
from subjective rules as an initial topology. Then
they develop a methodology based on statistical
techniques to improve the structure by testing the
independence assumptions, and altering the structure
if any of them is not satisfied. Kwoh and
Gillies [11] extended this work, by creating hidden
nodes to improve the structure of a Bayesian tree
when the independence assumptions do not hold.
Srinivas et al. [12] combine expert knowledge and
dependence information (which can be obtained
from statistical tests) in an iterative algorithm for
approximating the structure of a Bayesian net-
work. The expert knowledge they use includes
which variables are hypothesis (root nodes), which
are evidence (leaf nodes), and partial knowledge
6 THE AUTHORS???
about causality and independence between some
of the variables.
In the second approach, Cooper and Herskovits
[13] developed a Bayesian Method for the induction
of probabilistic networks from data. Given
certain assumptions about the probability distri-
butions, they developed an algorithm for obtaining
the most probable Bayesian network given a
database of cases. With this algorithm the probability
of certain structure given the data can
also be obtained, and it can handle missing data
and hidden variables. Recently, Lam and Bacchus
[14] developed and alternative technique for inducing
multiply-connected networks based on Rissa-
nen's minimal description length (MDL) principle.
Their algorithm tries to make a trade off between
the accuracy and complexity of the structure ob-
tained. That is, favoring simpler structures even
if they are not as accurate as a more complex one.
Chow and Liu's algorithm has two important
limitations: (i) it is restricted to tree structures,
and (ii) it does not obtain the directions of the
links (causality). Rebane and Pearl's extension
is still restricted in both aspects, generality and
directionality. As Lam and Bacchus [14] point
out, the approach in [13] assumes a uniform distribution
over all possible network structures, so it
could favor a much more complex structure even
if it is only slightly more accurate. The approach
based on the MDL principle [14] overcomes this
difficulty by considering both, accuracy and com-
plexity. Still, it is based on certain heuristics so
that it can not always obtain the optimum solu-
tion. The other approaches assume the existence
of expert knowledge which is not always available.
The special case of predictive systems have certain
characteristics, as we explain in the following
section, that make the previous algorithms inap-
propriate. In particular, most previous techniques
consider all the variables at the same level, while
in predictive systems accuracy in terms of the un-
is the most important factor.
4. Structure Learning for Predictive System
In a predictive system there is one (or a few) variable
whose value is unknown, and which is estimated
based on other known variables. It is
possible to have spatial or temporal predictions.
In the first case, the unknown is not observable
and is estimated from other measured parameters.
In the second case, the unknown is in the future
and is predicted form present, and past, measure-
ments. For instance, in pollution prediction we
might want to estimate the pollution level in some
part where there are no measurements, or one day
in advance.
We are interested in obtaining dependence
structures for predictive systems, which have some
special characterists:
ffl There is usually only one variable which we
want to predict, so it can be considered the
hypothesis or root node.
ffl The other variables are evidence nodes which
can have different levels of influence in the hypothesis
ffl Not all the evidence nodes have direct influence
in the hypothesis, but there influence could be
through other evidence nodes.
Thus, we propose an algorithm for structure
learning in predictive expert systems based on the
previous observations. The idea is to have the
"simplest" structure (minimum number of links)
with acceptable predictive capability. Our approach
is to start with a PN with the minimum
possible number of links that connects all the variables
involved. For N variables, the smallest connect
graph is a tree, with arcs. This will
constitute the "skeleton" of the network. If the
predictive accuracy of a tree is good enough then
we consider this structure. Otherwise, we start to
add links, according to certain criteria, until we
obtain the desired performance.
The algorithm is the following:
1. Obtain an initial tree structure by Chow and
Liu's algorithm.
2. Make the hypothesis variable the root node.
This fixes the directions of the links.
3. Produce an ordering of the variables
Xng starting from the root, and
following the tree according to the order of
mutual information between variables.
4. Test the predictive capability of the network:
4.1 If it is satisfactory, stop(1).
play
outlook
temperature humidity
windy
Tree links
Other links
Fig. 4. Initial probabilistic tree for the golf example.
4.2 If not, and the number of links is less than
a maximum, add a link to the network and
to 4. A link is added with the highest
mutual information such that: (i) it
does not produce a cycle, (ii) the node at
the start of the link is a predecessor of the
node at the end, according to the previous
ordering.
4.3 If not, and the number of links is equal to
the maximum, stop(0).
Stop (1) indicates successful termination, and
stop (0) that it could not achieve the desired predictive
performance. The predictive capability is
tested statistically, by performing predictions on
different data than the training set, and comparing
the predictions with the actual values for the
unkown. Maximum is the number of links for a
completely connected graph, N (N \Gamma 1)=2.
The theoretical justification for step (4.2) in the
algorithm is based on a general procedure for obtaining
a minimal I-Map (a PN in which every
independence relation represented in the network
is valid) [15]. It consists on defining an ordering of
the variables, and constructing a graph such that
the "parents" of each variable are a subset of its
predecessors that makes it independent from the
rest of its predecessors.
If the number of arcs reaches the maximum, we
obtain a totally connected graph, which represents
the joint probability distribution of the N variables
without any independence assumptions. As
we mention before, this will be a trivial I-map for
any distribution. If the complete graph still does
not have the desired predictive accuracy, it means
that the training data is inadequate for generating
an appropriate structure. Either more cases
are required (larger sample), or other parameters
not included in the set of variables need to be considered
To illustrate the procedure, we use a small, hypothetical
example for "predicting when to play
Table
1 shows the variables and their
values for this examples, table 2 shows the set of
examples used for training, and table 3 the dependency
links (variable pairs) ordered by mutual in-
Table
1. Variables for the golf prediction example.
Variable Values
play Play, Don't Play
outlook sunny, overcast, rain
temperature continuous
humidity continuous
windy true, false
Table
2. Set of training data for the golf prediction example
outlook temp. hum. windy play
sunny 85 85 false false
sunny
overcast
rain 70 96 false true
rain 68 80 false true
rain
overcast
sunny 72 95 false false
sunny 69 70 false true
rain
sunny
overcast 72 90 true true
overcast 81 75 false true
rain 71 96 true false
Table
3. Set of links (variable pairs) ordered by mutual
information for the data in table ??.
Link Variables
9 windy - play
formation. This is a very small data set just used
to illustrate the ideas. The initial tree structure,
overimposed in the complete graph is depicted in
figure 4. A possible ordering for the variables in
these case will be: fplay, outlook, temperature,
humidity, windyg.
In figure 3, the next 3 steps in the algorithm
are shown, assuming that the tree structure is not
good enough (for this small example we did not
test the predictive accuracy). In each one a new
link is added according to the mutual informa-
tion, and its direction is determined by the variable
ordering. The algorithm will terminate when
we obtain the desired accuracy, or generate the
complete graph (10 links in this example).
An interesting aspect to notice is that, unless
we need the complete graph, we can usually eliminate
some variables for predicting the unknown.
For a tree structure, we can eliminate all the variables
but the ones directly connected to the hypothesis
(root) node. This is because of the independence
relations that are represented in the
PN. In a tree, a node is independent of all the
other variables given its direct parent and sons.
For the root node, these are only its direct sons.
In general, if the network is not complete (all links
present), a subset of nodes will make a node independent
of the remaining nodes. Thus, if all the
variables but one are known, we can use these independence
information to eliminate parameters
and simplify the estimation problem.
In the following section we introduce the problem
of Ozone prediction in Mexico City, and apply
the previous algorithm to obtain the dependence
structure of this phenomena.
play
outlook
temperature
humidity
windy
play
outlook
temperature
humidity
windy
play
outlook
temperature
humidity
windy
(a) (b) (c)
Fig. 3. Structures produced by the second stage in the structure learning algorithm for the golf example: (a) first additional
link, (b) second, (c) third.
5. Ozone Prediction in Mexico City
Air quality in M'exico City is a major problem.
Air pollution is one of the highest in the world,
with high average daily emissions of several primary
pollutants, such as hydrocarbons, nitrogen
oxides, carbon monoxide and others. The pollution
is due primarily to transportation and industrial
emissions. When the primary pollutants
are exposed to sunshine, they undergo chemical
reactions and yield a variety of secondary pollu-
tants, ozone being the most important. Besides
the health problems it may cause, ozone is considered
as an indicator of the air quality in urban
areas.
The air quality is monitored in M'exico City in
stations, with five of these being the most com-
plete. Nine variables are measured in each of the
5 main stations, including: wind direction and
velocity, temperature, relative humidity, sulphur
dioxide, carbon monoxide, nitrogen dioxide and
ozone. These are measured every minute 24 hours
a day, and are averaged every hour.
It is important to be able to forecast the pollution
level several hours, or even a day in advance
for several reasons, including:
1. To be able to take emergency measures if the
pollution level is going to be above certain
threshold.
2. To help industry to make contingency plans in
advance to minimize the cost of the emergency
measures.
3. To estimate the pollution in an area where
there are no measurements.
4. To take preventive actions in some places, as in
schools, to reduce the health hazards produced
by high pollution levels.
In M'exico City, the ozone level is used as a
global indicator for the air quality in the different
parts of the city. The concentrations of ozone are
given in IMECA (Mexican air quality index). So
it is important to predict the ozone level a day, or
at least several hours in advance using the other
variables measured in different stations.
Previous work [16] has been done in using neural
network techniques to forecast ozone in M'exico
City. The results are encouraging for estimating
the ozone level up to 4 hours in advance. The
problem with these techniques is that we do not
get any insight into the structure of the phenom-
ena. It will be useful to know the dependencies
between the different variables that are measured,
and specially their influence in the ozone concen-
tration. This will provide a better understanding
of the problem with several potential benefits:
ffl Determine which factors are more important
for the ozone concentration in M'exico City.
ffl Simplify the estimation problem, by taking into
account only the relevant information.
ffl Find out which are the most critical primary
causes of pollution in M'exico City which could
help for future plans to reduce it.
6. Experimental Results
We started by applying the learning algorithm to
obtain an initial structure of the phenomena. For
this we considered 47 variables [17]: 9 measurements
in 5 stations, plus the hour and month in
which they were taken. We used nearly 400 random
samples, and applied the first step in our
algorithm to obtain the tree structure that best
approximates the data distribution. This tree-structured
Bayesian network is shown in figure 5.
We then considered the ozone in one station
(Pedregal) as unknown, to estimate it one hour
in advance using the other measurements. So we
make ozone-Pedregal the hypothesis variable and
consider it as the root in the probabilistic tree,
as shown in figure 5. From this initial structure
we can get an idea of the relevance or influence of
the other variables for estimating ozone-Pedregal.
The nodes "closest" to the root are the most important
ones, and the "far-away" nodes are not so
important.
In this case we observe that there are 3 variables
(ozone-Merced, ozone-Xalostoc, and wind
velocity in Pedregal) that have the greatest influence
in ozone-Pedregal. What is more, if the tree
structure is a good approximation to the "real"
structure, these 3 nodes make ozone-Pedregal independent
from the rest of the variables (see figure
7). Thus, as a first test of this structure, we estimated
ozone-Pedregal using only these 3 variables.
The estimation is done with the probability propagation
algorithm for trees presented in section 2.
This algorithm works with discrete variables only,
so continuos variables are discretized in fixed size
intervals. We made two experiments: (1) estimate
ozone-Pedregal using 100 random samples taken
from the training data, and (2) estimate ozone-
Pedregal with other 100 samples taken from other
data, not used for training. The results for a sub-set
of 20 representative samples in each case are
shown in figures 6 and 8.
We observe that even with only three param-
eters, the estimations are quite good. For training
data the average error (absolute difference between
real and estimated ozone concentration) is
11.2 IMECA or 12.1%, and for not-training data
it is 26.8 IMECA or 22.1%. This results should
be judged taking into account that this is the first
approximation to a dependency model, and that
we are only considering 3 variables for estimating
the ozone at Pedregal. The neural network model
[16] with 46 inputs, has an average error of
with a similar set of test (not-training) data.
O3_T
O3_L O3_Q VV_T
CO_T
HORA
RH_F
TMP-T
O3_F
RH_L
RH_T
CO_Q
CO_L
CO_F
Fig. 5. A Bayesian tree that represents the ozone phenomena in 5 stations in M'exico City. The nodes represent the
variables according to the following nomenclature. For the measured variables, each name is formed by two parts,
"measurement-station", using the following abbreviations: the measurements, O3-ozone, SO2-sulphur dioxide, CO-carbon
monoxide, NO2-nitrogen dioxide, NOX-nitrogen oxides, VV-wind velocity, DV-wind direction, TMP-temperature, RH-
relative humidity; the monitoring stations, T-Pedregal, F-Tlanepantla, Q-Merced, L-Xalostoc, X-Cerro de la Estrella. The
other two variables correspond to the time when the measurements were taken, and they are: HORA-hour, MES-month.
The same data was used to train and test C4.5
[18]. The error on the test set was of 17.64%. The
tree produced by C4.5 is given in figure 9. It is
interesting to note that C4.5 considered the wind
direction in Pedregal as its principal attribute. A
north-south wind direction (? 120 increases the
levels of ozone, whether in a south-north direction
the ozone levels are mainly located within the first
intervals (below 70 IMECAS). We then tested the
accuracy of C4.5 by pruning its tree at different
depths, that is, considering only the most relevant
attributes. The leaves were labeled with the may-
ority class of the training set at that level. At
depth 4, the accuracy of C4.5 is 21.86%. Considering
only the three most important attributes
(i.e., at depth 2), and the mayority class for that
branch of the tree, C4.5 has an error of 24.80%.
Ozone
Pedregal
Ozone
Merced
Ozone
Xalostoc
Pedregal
Fig. 7. Reduced tree for predicting Ozone-Pedregal.
The accuracy of C4.5 with the complete tree is
higher than with our reduced dependency model,
with the tree pruned at depth 4 it is about the
same, and it is lower with 3 attributes. It is difficult
to compare both algorithms because they use
a different representation, so a decision node in
a decision tree and a variable node in a Bayesian
network are not the same. Still, we can consider
that the accuracy is similar with these two different
representations and learning algorithms, and
will expect a higher accuracy if the dependency
model is extended with more variables and relations
An advantage of the dependency model is that
it is generally easier to understand. The relevance
of each attribute for predicting certain variable is
explicitly represented. This is more difficult to obtain
from a decision tree, where an attribute can
be repeated as different nodes at different depths.
A second advantage is that it gives a probability
measure for each value (range) of the hypothesis,
which is, in general, not available with a decision
tree. Finally, a Bayesian network can be used to
predict any variable with any subset of attributes
known, while a decision tree is for one variable and
with all (or most) of the attributes known.
For practical purposes, the ozone measured
in IMECAS is divided in several intervals, each
of size 50. The air quality and corresponding
emergency measures are based on these intervals.
In our experiments with the probabilistic model,
aprox. 90% of the predictions fall in the same 50
IMECAS interval as the measured ozone level.
Fig. 6. Real vs. estimated levels for ozone-Pedregal using 3 variables and training data.
7. Conclusions and Future Work
A causal network representation and the structure
learning techniques produced some very interesting
results for the ozone prediction problem.
Firstly, we got some insight into the dependence
structure of the phenomena. For example, the
ozone in Pedregal is influenced by the ozone in
other stations and the wind velocity. This is due
to the fact that the pollution in the south (Pedre-
gal) of M'exico City, is, in large part, produced by
the industrial plants in the north and a dominant
north-south wind direction. Secondly, we got an
indication of which are the important and not so
important variables for ozone forecasting. Taking
this into account could reduce the measurement
and computational costs for ozone predic-
tion. Thirdly, this dependency information could
be used for improving other alternative prediction
techniques, such as neural networks.
With respect to ozone prediction in M'exico
City, we plan to continue this work in several aspects
ffl Improve the structure of the Bayesian network
by using the second part of our algorithm.
ffl Obtain the dependence structure for other variables
of interest, in particular the ozone in other
stations.
ffl Test its predictive capability using other vari-
ables, assuming that the most influential ones
are unknown or not reliable.
ffl Improve the longer term predictions by using
additional information, such as weather forecasting
variables.
In structure learning in Bayesian networks in
general, there are several research issues which remain
to be addressed. Firstly, there is the problem
of obtaining the optimal structure in the general
case, considering the model's accuracy and
computational complexity. Secondly, there is no
general algorithm for obtaining the directions of
all the links in the network. And thirdly, most
structure learning algorithms only consider the observable
variables. But, in many cases, the introduction
of other variables (called hidden or virtual
nodes) can produce simpler structures with an improved
predictive capability. We will be addressing
these issues in our future research work.
Fig. 8. Real vs. estimated levels for ozone-Pedregal using 3 variables and not-training data.
RH_F
RH_L
HORA
484 28687219227104323177Fig. 9. Decision tree for ozone-Pedregal produced with C4.5. Each node represents a decision over the value of a variable: if
the value is less than the value shown under the node, then the left branch is followed, otherwise the right one. The leaves
represent different classes for ozone-Pedregal (from C0 to C19), with the same discretization used in the Bayesian network.
Notes
1. Data taken from public domain file
--R
"Why Machines should Learn?"
Probabilistic Reasoning in Intelligent Sys- tems
"On Evidential Reasoning on a Hierarchy of Hypothesis"
"Uncertainty Management in Expert Systems"
"A Computational Model for Combined Causal and Diagnostic Reasoning in Inference Systems"
"Local Computations with Probabilities on Graphical Structures and their Application to Expert Systems"
"The Computational Complexity of Probabilistic Inference Using Bayesian Networks"
"Approximating Probability Distributions with Dependence Trees"
"Objective Probabilities in Expert Systems"
"Using Hidden Nodes in Bayesian Networks"
"Automated Construction of Sparse Bayesian Networks from Unstructured Probabilistic Models and Domain Information"
"A Bayesian Method for the Induction of Probabilistic Networks from Data"
"Learning Bayesian Net- works: An Approach Based on the MDL Principle"
"Causal Networks: Semantics and Expressiveness"
"In- duction of Dependence Structures from Data and its Application to Ozone Prediction"
--TR
--CTR
Elias Kalapanidas , Nikolaos Avouris, Feature selection for air quality forecasting: a genetic algorithm approach, AI Communications, v.16 n.4, p.235-251, December
Ciprian-Daniel Neagu , Nikolaos Avouris , Elias Kalapanidas , Vasile Palade, Neural and Neuro-Fuzzy Integration in a Knowledge-Based System for Air Quality Prediction, Applied Intelligence, v.17 n.2, p.141-169, September-October 2002 | structure learning;bayesian networks;predictive systems;decision trees;atmospheric pollution |
590817 | Evolving Neural Networks to Play Go. | Go is a difficult game for computers to master, and the best go programs are still weaker than the average human player. Since the traditional game playing techniques have proven inadequate, new approaches to computer go need to be studied. This paper presents a new approach to learning to play go. The SANE (Symbiotic, Adaptive Neuro-Evolution) method was used to evolve networks capable of playing go on small boards with no pre-programmed go knowledge. On a 9 9 go board, networks that were able to defeat a simple computer opponent were evolved within a few hundred generations. Most significantly, the networks exhibited several aspects of general go playing, which suggests the approach could scale up well. | Introduction
Go is hard. For computers at least, this is true. Though the game has not received the level
of attention that computer chess, for example, has received, considerable effort has gone
into trying to create strong go playing programs. Yet, despite this effort, the best computer
programs are still relatively weak.
There are a number of reasons why go is hard for traditional computer game playing
techniques: the branching factor is prohibitively large, the game is pattern oriented, and
there are multiple interacting goals. In fact, the game is so difficult that new techniques are
probably going to be needed before go programs are as strong as those that play checkers,
chess, or Othello.
This paper explores the usefulness of neuro-evolution as a mechanism for learning to play
go. The SANE (Symbiotic, Adaptive Neuro-Evolution [7, 8, 9]) algorithm demonstrates that
networks that display a general ability in playing go on small boards can be evolved without
To appear in Applied Intelligence.
y This research was supported in part by NSF under grant #IRI-9504317.
\Omega \Gamma\Omega\Gamma(a) Four liberties.
\Theta\Gamma\Omega \Gamma\Omega\Gamma\Gamma\Omega \Gamma\Omega \Gamma\Omega \Theta\Gamma\Omega \Gamma\Omega\Gamma(b) One liberty.
\Gammaff\Delta\Delta\Gamma\Omega \Theta\Gamma\Omega \Gamma\Omega\Gamma(c) No liberties.
Figure
1: The group in (a) has four liberties, or adjacent free positions, while the group in (b) has
one. After that last liberty is lost (c), the group is said to be captured and is removed from the
board.
any prior knowledge about the game. This result forms a promising foundation for scaling
up to full-scale go.
The paper begins with an introduction to the rules of go followed by a brief word on
computer go and why neural network techniques might be useful for go programs. Next
the SANE neuro-evolution algorithm is is reviewed, and details of the architecture and the
experiments given. The paper concludes with an analysis of the strategies evolved and
suggestions for future research.
2 The Game of Go
Although the term go is taken from the Japanese word for the game, go is believed to have
originated in China more than 3,000 years ago, making it one of the oldest board games still
actively played in modern times. Go is an appealing game because it appears simple yet
features strategy and tactics that rival games such as chess.
Go is played on a square grid 19 intersections across. Smaller boards are often used
for teaching purposes. The two players, black and white, alternate placing stones of their
respective colors on the intersections of the grid. Game play continues until both players
pass, at which time the score is calculated and a winner is determined.
Game play is deceivingly simple. Stones can be placed on empty intersections. Once
played, a stone cannot be moved to another location. However, a stone or a group of stones
can be captured and removed from the board.
A liberty is an empty point adjacent to a group of stones. Any group that has no
liberties is said to be dead and the stones in that group will be removed from the board. For
example, the black group in figure 1a has 4 liberties. Figure 1b shows the same black group
with further white stones placed such that the black group now has only 1 liberty. If white
were to play an additional move at 1, the black stones would be reduced to 0 liberties. They
would be considered dead and removed from the board, as shown in figure 1c.
The liberty rule gives rise to the simple concept of an eye. Any group of stones that
completely surrounds some interior space is said to have an eye. In figure 2a, the black
group has one eye, at point "a". The black group in figure 2b has 2 eyes, at points "b" and
"c". The first group can easily be captured if white plays at point "a". However, the second
black group cannot be captured by white as white would need to simultaneously occupy
\Omega \Gamma\Omega\Gamma
\Omega \Gamma\Omega\Gamma
\Omega \Gamma\Omega\Gamma
\Omega \Gamma\Omega\Gamma
\Omega \Gamma\Omega\Gamma
\Delta\Delta\Delta\Delta\Delta\Delta\Delta(a) A group with one eye.
\Delta\Gamma\Omega \Gamma\Omega \Gamma\Omega \Gamma\Omega \Gamma\Omega \Gamma\Omega \Gamma\Omega\Gamma
\Delta\Gamma\Omega \Gamma\Omega \Gamma\Omega \Gamma\Omega \Gamma\Omega \Gamma\Omega \Gamma\Omega\Gamma
\Delta\Gamma\Omega \Gamma\Omega b
\Gamma\Omega c
\Gamma\Omega \Gamma\Omega\Gamma
\Delta\Gamma\Omega \Gamma\Omega \Gamma\Omega \Gamma\Omega \Gamma\Omega \Gamma\Omega \Gamma\Omega\Gamma
\Delta\Gamma\Omega \Gamma\Omega \Gamma\Omega \Gamma\Omega \Gamma\Omega \Gamma\Omega \Gamma\Omega\Gamma
\Delta\Delta\Delta\Delta\Delta\Delta\Delta\Delta\Delta(b) A group with two eyes.
Figure
2: Eye space determines life and death of a group of stones. The group in (a) has one eye
and can be killed by placing a stone in the eye (position "a"), while the group in (b) has two eyes,
"b" and "c", and cannot be killed.
\Omega \Gamma\Omega\Gamma
\Omega a
-\Omega \Gamma\Omega \Phi\Pi\Gammaffi\Gammaffi\Pi
Figure
3: Repetition of full board positions is not allowed. If black has just played the marked
stone to capture a white stone at "a", white would not be allowed to play at "a" immediately
to capture the black stone, because that would recreate the board full board position before the
black's move.
both points "b" and "c" to reduce the black group to 0 liberties. This is not possible, and
therefore the group cannot be killed. This example demonstrates the most common and
simplest form of a living group. Forming living groups is one of the primary goals of the
game.
Previous full-board positions cannot be repeated. This rule is known as the ko rule.
Figure
3 shows an example of how the rule applies. Suppose black has just played the marked
stone to capture a white stone at point "a". White would not be allowed to immediately
play at point "a" to capture the marked stone because it would recreate the board position
before black's move. White must instead play elsewhere, and thereby create a new full-board
position. If black does not move to point "a", white would then be allowed to play at "a"
on a subsequent move because the full-board position would no longer be repeated.
Play continues until there are no more moves of value to be made and both players pass.
Each side receives a score where all stones and all locations completely surrounded by groups
of the same color are counted as points. A komi, 5.5 points in a typical even game, is added
to white's score to offset black's first move advantage. The player with the highest score
wins. Because komi is not an integer, a tie cannot occur.
Unlike many other board games, go provides an easy way to handicap games so that
players of different ranks can play an even game. White will give black a certain number of
free moves at the beginning of the game. This advantage is generally well defined, and go
ranks are based on it. A player who is ranked 2 stones above another player should be able
to give the other player 2 free moves in order to play an even game.
3 Computers and Go
Despite the relatively simple game play, computers have had little success mastering the
game. Whereas in games like chess, Othello and checkers the traditional game playing
techniques such as minimax search and its variations are competitive even at the master level
[4, 5, 13], those techniques, when applied to go, have not been able to produce programs
that can challenge even weak amateur players. The best computer programs in the world
are ranked 6-8 stones below what would be considered master level. Progress continues to
be made; however, the gap is so large that traditional techniques are unlikely to reach even
the weakest master levels for some time to come.
A game of go can be divided into three general phases: the opening, the midgame and
the endgame. Computers have had varying success in each of the these stages, revealing
insight into what can and cannot be achieved with computational methods.
3.1 The Opening
The opening stage of the game is referred to as the fuseki. Typically play starts in the
corners. Specific sequences in the corners are referred to as joseki, and they are similar to
book openings in chess. However, the fuseki typically refers to the direction of play as it
relates to all 4 corners. Good go programs tend to have joseki move databases that range
from 5,000 to 50,000 moves, and current programs do not have any difficulty in playing
through a database of joseki sequences. Choice of joseki and choosing between variations is
more troublesome; however, play by the computers is not advanced enough to consider such
issues. Even with such limited techniques, some programs are capable of playing very good
openings.
3.2 The Midgame
As play moves into the midgame, search-based programs begin to have difficulty. One reason
is the sheer size of the game. On a 19 \Theta 19 board, there are typically 200-300 potential moves
available from a midgame position, so brute force searching of the game tree is not a viable
option.
Current go programs apply a wide variety of techniques to control the complexity of the
midgame. Typically, a move generation facility is used to generate a number of candidate
moves from a position using techniques such as pattern libraries, tactical analysis, and rule-based
expert systems. Then, the candidate moves are evaluated, usually through static
board evaluation functions. Some programs rely solely on the evaluation function for choice
of moves while others attempt limited global search using traditional search techniques.
Search plays a more prominent role in local tactical analysis where the number of moves and
the size of the search tree are significantly smaller.
There are several problems with the current midgame techniques. They tend to be
difficult to apply and error-prone. Because most evaluation techniques are static, it is difficult
to achieve general play, and advance can only be made by laborous human tweaking of rules,
patterns and databases.
3.3 The Endgame
The endgame is typically significantly easier for computers. During the endgame, programs
are capable of playing very well because the branching factor is significantly reduced and
patterns are local in nature. However, in the endgame the score is often already mostly
decided and fewer points are usually at stake. Even perfect endgame play might only change
the score by a small number of points. So, the most important and most difficult part of
playing go is in the midgame where the traditional techniques are the weakest.
4 Neural Networks and Go
Go is largely pattern based; as a matter of fact, go players often refer to board positions
as shape. Groups of stones can be said to have good shape or bad shape depending on the
shape's potential of creating a living group and of efficiently capturing territory. Human
players instinctively know where to search for moves based largely on learned knowledge of
shape. Although there are many techniques that highly skilled players use and computer
programs do not, viewing the board as a search node instead of a collection of shapes and
patterns is probably the most significant factor holding computer go programs back.
Neural networks are very good at pattern transformation tasks, and thus could well be
applicable to go. A network could be trained to compute a mapping between the input
space, that is, the current board position, and the output indicating the next move. The
main problem with this approach is the credit assignment problem. Suppose a standard
backpropagation neural network [12] were being trained to play go. For backpropagation
to work, advance knowledge about the best move at any given position would be required.
Such knowledge is difficult to come by. In reality, only the final game result is available. The
credit assignment problem is the problem of determining which of the many moves played
were good and deserve credit for a win, and which were bad and deserve to be blamed
for a loss. In go, this problem is severe enough that standard learning techniques such as
backpropagation cannot be effectively applied.
Adaptive Neuro-Evolution [7, 8, 9]) solves the credit assignment problem
by using evolutionary algorithms to search for effective neural networks. Instead of punishing
or rewarding individual moves, networks are evaluated, selected, and recombined based on
their overall performance in the game. Evolutionary algorithms perform a global, parallel
1 SANE is described in more detail in [8], and the source code can be obtained from
http://www.cs.utexas.edu/users/nn/.
Label Weight-0.7-0.6
-1.2
Input Layer
Output Layer
Figure
4: A three-layer feedforward network is created from 3 neurons. The neurons are
shown on the left, and the corresponding network is shown on the right. Labels indicate
which input or output unit a connection corresponds to, while the weight indicates the
strength of the connection.
search and are guided by a fitness function that measures the goodness of a particular
solution. The search tries to maximize the goodness level throughout the search space to
find the best solution.
SANE differs from other approaches to neuro-evolution systems where each individual in
the population represents a complete neural network. In SANE, two separate populations
are maintained and evolved: a population of neurons and a population of neural network
blueprints. The neuron level evolution explicitly decomposes the search space and maintains
a high level of diversity throughout evolution. The blueprint population maintains and
exploits effective combinations of individuals in the neuron population. Conjunctively, the
two levels of evolution provide an efficient genetic search that is capable of solving difficult
real-world decision problems with minimal domain information [8].
In the neuron population, SANE evolves a large population of hidden neuron definitions
for a three-layer feedforward network (figure 4). A neuron is represented by a series of
connection definitions that describe the weighted connections of the neuron from the input
layer and to the output layer. Each neuron has a fixed number of connections, but may
allocate them arbitrarily among the units in the input and output layers. A connection
definition consists of a label and weight pair. The label is an integer value that specifies a
specific input or output unit, and the weight is a floating point number that specifies the
strength of the connection. Figure 4 gives three example hidden neuron definitions and the
resulting neural network.
The activation of a neuron is computed as the sum of all the connected input units
multiplied by their weights and passed through the sigmoid activation function
application, the output units are linear so that both positive and
negative values and be generated.
Neural networks could be formed by randomly choosing neurons from the neuron popu-
lation. In fact, this approach performs well in simpler problems [7, 10]. However, random
participation does not retain knowledge of the best combinations of neurons and can often
l
l l
l
l l
l
l l
l
l l
l
l l
Network Blueprint Population
l
l l
l
l l
l
l l
Neuron Population
Figure
5: The network blueprints consist of a set of neurons in the neuron population. A
neural network is formed from a blueprint by following its neuron pointers and decoding the
respective neurons.
stall the search in more difficult problems [8]. To focus the search on the best neuron com-
binations, SANE maintains and evolves a separate population of good neuron combinations
called neural network blueprints. The blueprints are made up of a series of pointers to
members of the neuron population and define an effective neural network from a previous
generation. Figure 5 shows how the network blueprint population and the neuron population
are related.
SANE integrates the neuron and blueprint populations in a generational evolutionary
algorithm that iterates over two phases: an evaluation phase and a reproduction phase.
In the evaluation phase, SANE simultaneously evaluates the blueprints and the neurons.
A blueprint is evaluated by the performance of the network that it specifies. A neuron is
evaluated based on the performance of the networks in which it participates. The basic steps
in the SANE evaluation phase are shown in the following pseudo code:
for each neuron n in population P n
n:fitness
n:participation
for each blueprint b in population P b
neuralnet / decode(b)
b:fitness / task(neuralnet)
for each neuron n in b
n:fitness b:fitness
n:participation
for each neuron n in population P n
n:fitness / n:fitness / n:participation
Neural networks are formed from each blueprint and evaluated in the task environment. The
evaluation score is given to each blueprint and is added to each neuron's fitness variable.
After all blueprints have been evaluated, each neuron's fitness is normalized by dividing the
sum of its scores by the total number of networks in which it was a participant.
In the reproduction phase, SANE uses common genetic operators such as selection,
crossover, and mutation to obtain new blueprints and neurons. Each population is ranked
based on fitness and a mate is selected for each of the elite individuals. In this application,
the elite parameter is defined as the top 15% in the blueprint population and the top 25%
in the neuron population. The mate for each elite individual is selected from the other elite
individuals. SANE uses a one-point crossover to mate two individuals, which creates two
offspring. The offspring from each of the crossover operations replace the worst performing
individuals (according to fitness) in the population. All individuals that are not explicitly
replaced by offspring remain in the population, although they may be mutated.
A conservative mutation rate of 1% per chromosome position is used on the neuron pop-
ulation, because neuron evolution automatically maintains high diversity (good networks
require serveral different types of neurons). A more aggressive, two-tiered strategy is used
on the blueprint level. First, a small number (approximately 1%) of neuron pointers in each
blueprint are swapped with randomly selected neurons in the neuron population. Second,
pointers to breeding neurons are replaced by pointers to their offspring with a 50% prob-
ability. The second mutation component promotes utilization of offspring neurons, which
has two advantages. First, it creates diversity in the blueprint population, and second, it
explores new structures created by the neuron population.
6 Applying SANE to Go
SANE has previously been shown effective in several sequential decision tasks including robot
control [7, 8, 9], constraint satisfaction [10], pursuit and evasion [3], and the game of Othello
[6, 8, 10]. This paper will evaluate the usefulness of SANE in learning to play go. SANE is
used to evolve networks to play on small boards against a simple computer opponent, and
the scale-up properties are evaluated.
In order to apply SANE, three aspects of the architecture must be specified: the network
parameters, evolution parameters and the evaluation function. Let us look at each one of
these in turn in the go task.
6.1 Network Parameters
SANE evolves standard three-layer feed-forward networks. The network architecture is fixed;
only the associated weights and connections are evolved. The number of units depends on
the board size. There are 2 input units and one output unit for each board position. The
Board size Neuron Population Blueprint Population Number of neurons per network
5 \Theta 5 2000 200 100
7 \Theta 7 3000 200 300
9 \Theta 9 4000 200 500
Table
1: Network definitions used for evolving networks for various small board sizes.
first input unit indicates whether a black stone is present at that location, and the second
unit whether a white stone is present. Since only one stone can occupy any given board
position at a time, both input units cannot be active simultaneously for any position.
The output units are signed floating point values. A positive value indicates a good move.
The larger the value, the better the move. Negative (or output indicates that the move
is not suggested.
6.2 Evolution Parameters
Most aspects of SANE are easily tunable. Some experimentation was done to find good
values, however, it was not necessary to find optimum values as SANE operates well as long
as the values are withing reasonable ranges.
The neurons evolved are 312 bits long and represent a set of 12 weights connecting either
from input layer to hidden layer or from the hidden layer to the output layer. Table 1 shows
the population and network sizes used in conjunction with the various board sizes. Each
generation, 200 networks were formed. This allowed each neuron on average to participate in
10-25 networks per generation. Mutation occurred at a rate of 0.1% The crossover operation
was a one-point crossover between neurons or networks in the breeding population. The top
25-30% of the population were allowed to breed.
6.3 Evaluation Function
The most difficult aspect of the evaluation function was deciding on a set of evaluation
criteria that could be computed completely without human intervention. The first difficulty
is in determining the end of the game. When humans play, the end of the game is decided
by agreement. When the players feel the game is over, they pass their turn. Stones that
are mutually agreed to be dead are removed from the board. If there is a dispute, play can
be resumed to settle the issue. After the status of each group is determined, a final score is
calculated.
Since there is no separate output unit for pass, the network can pass only when none of
its positive output units (if any) correspond to a legal move. Because there is no arbitration
phase for disputed groups, a series of 3 passes is required to end the game. This simplifies
certain endgame situations where ko (i.e. repetition) might occur.
Removing dead stones is more difficult. Rather than defining a separate protocol for this
task, the evaluation function requires a player to explicitly kill any stones it thinks are dead.
Human players find this process tedious, so those groups that are obviously dead are simply
removed from the board at the end of the game. For computer opponents, killing groups is
not so tedious. If all stones on the board are considered alive, the need for settling disputes
after the game is over is eliminated, and the task of scoring is greatly simplified.
An upper bound is placed on the number of moves, so that it is not actually necessary
to check for repetition of entire board positions. It is enough that only the simple ko
(demonstrated in figure 3) is checked. An upper bound also ensures that non-repeating but
prohibitively long sequences are not followed. Games between human players do not involve
such sequences. However, they may occur in a game by an unskilled program. An example of
such a sequence would be the filling in of a player's own eye space, which allows a previously
alive group to be killed. If two unskilled opponents play in this manner, excessively long
sequences of play might result. Such play is punished by counting excessively long games as
a loss for the network. Because this behavior is selected against, the networks become less
likely to develop it.
Since all stones still on the board are presumed to be alive, determining the score becomes
a straightforward task. Simple Chinese scoring, where all stones of each color and all locations
completely surrounded by stones of that color are counted as points, is used.
The evaluation function must produce a fitness level for the network, and it should be a
fine-grained value so that slight improvements in the network's play can be rewarded. In our
experiments, the difference in score between SANE and its opponent (for example +10.5 or
-7.5) is summed over N games, which allows for good resolution in determining improvement.
7 Results
SANE was tested with various board sizes. The opponent used was wally (written by Bill
Newman), a simple publicly available go program. Wally is a good choice for an opponent for
several reasons. First, wally is one of the few go playing programs available in source code.
This turns out to be particularly helpful when trying to adjust parameters, like the degree of
randomness, to make the opponent more useful as a training partner. Second, wally's skill
level is appropriate for a first training partner. It is strong enough to be a challenge to an
unskilled network without being so strong that progress cannot be made.
7.1 Evolution Efficiency
was able to evolve a network that could defeat wally on small boards. On a 5 \Theta 5
board, SANE needed only 20 generations, on a 7 \Theta 7 board, 50 generations, and for a 9 \Theta 9
game, 260 generations. These numbers are averages over 100 - 1,000 simulations, requiring
the ability to beat the opponent 75% of the time. The network was playing black without a
komi, which is an equivalent to a 1-stone handicap for the network.
Although these results were relatively easy to get, they take a lot of CPU time (up to
5 days for the 9 \Theta 9 board). Moreover, the training times increase with board size quite
rapidly. It can be estimated that for a 13 \Theta 13 board, several thousand generations would
be required, and for a full-size 19 \Theta 19 board, perhaps tens of thousands. The CPU time
for such simulations could be more than a year with the current CPU speeds, and was not
available. However, it is still possible to get an idea of how well such a network plays go by
studying the effect of nondeterminism and handicap in the opponent.
7.2 Effect of Nondeterminism
An important issue with developing general go playing is the degree of determinism of the
opponent. SANE actually manages to learn to defeat more deterministic opponents very
rapidly. However, in those cases the network learned little about playing go and only learned
what was necessary to win against that particular opponent.
To force the network to learn more diverse solutions, 10% non-determinism were applied
to wally. This means that 10% of the time, instead of making the normal move, a random
legal move would be played instead. The 10% value was chosen experimentally to be a
reasonable value. Smaller values did not significantly increase the diversity of the games
played nor the solutions learned, and larger values made the opponent behave too randomly
and easy to beat.
As a test of generality, one network was evolved against the original wally on a 7 \Theta 7
board, while another was evolved against wally playing with 10% randomness. An otherwise
deterministic player playing occasional random moves should be weaker in absolute terms.
However, when playing a series of games against a learning opponent, the deterministic
player turned out to be easier to beat. The first network learned to defeat wally very
rapidly. However, it would be defeated easily by the weaker but less-deterministic wally. In
fact, it would even lose some games against the randomly moving opponent. The network's
behavior in this case was not diverse enough to be useful against other opponents. Instead
of learning moves that represent general go-playing ability, the network simply learned tricks
and simple sequences that utilize flaws in the static opponent.
The network playing against the less deterministic opponent required more generations
to train. However, the solution evolved was capable of defeating wally at various levels of
determinism, including its normal mode of play, and did not lose to the random opponent.
7.3 Effect of Handicaps
Since few stronger go playing programs are freely available, there was no good opportunity
to evolve networks against other opponents. However, the go handicapping mechanism does
provide a way to modify the difficulty of the game against a given opponent.
Networks were evolved on the 7 \Theta 7 board while giving wally differing handicaps. Initially,
the networks were evolved to play black and make the first move. After about 50 generations,
a network evolves to defeat wally. The networks were then evolved with wally playing the
first move. After 130 more generations, a network was able to beat wally 75% of the time.
\Omega
\Omega
\Omega \Gammaj\Pi\Pi-ffi\Gammaffi(a)
\Gammaff\Gamma\Omega \Gamma\Omega\Gamma4
a
\Gamma\Omega \Gamma\Omega\Gamma4
\Gammaff\Gamma\Omega \Gamma\Omega \Gamma\Omega \Gamma\Omega -ff\Gamma\Omega \Gamma\Omega \Gamma\Omega \Gamma\Omega \Theta\Gamma\Omega \Gamma\Omega \Gamma\Omega \Gamma\Omega \Gammaj\Pi b \Gammaffi\Gammaffi(b)
Figure
This position is from a game played between wally (white) and the network (black).
After white plays the marked stone in (a), black should be dead in the left corner as it should not
be able to make the two eyes necessary to live. But the network has learned a trick. It plays the
marked stone in (b), threatening to capture the white stone below "a". Instead of playing at "b",
which would ensure the death of the black group, it plays the move at "a" to defend the single
stone below it. As a result, the network can move to "b" and live.
At this level, it could give wally a 2-stone handicap and still win 45% of the games.
With these results, we can get a rough idea of the level of play evolved. Handicap stones
on a small board represent a larger difference in skill than on a larger board. For example, a
single handicap stone on a 13\Theta13 board represents approximately 3 stones on a 19\Theta19 board.
On a 9 \Theta 9 board, the difference is about 4-5 stones. Thus, the 2 stone handicap on a 7 \Theta 7
board may represent a difference in skill of about 10 stones, which is quite significant and
would allow a good amateur compete with master-level player on the full board. Although
this is just a rough estimate, it shows that quite powerful go play can be achieved through
neuro-evolution methods.
8 Strategies Evolved
Given that the networks started with no prior knowledge on how to play go, an important
question is: what kind of strategies did they evolve?
One peculiar problem with the evolutionary approach is that the strategy evolved often
exploits weaknesses found in the particular opponent rather than representing good general
go playing abilities. Figure 6 is an example of such a situation taken from an actual game
played by a network against wally on a 7 \Theta 7 board. The network is playing black and wally
is white.
In this situation, white plays the marked stone in 6a. This is a move that should effectively
kill the black group in the lower left corner because the black group would no longer be able
to make two eyes. However, black has learned that it can actually win in this situation
against wally. It plays the marked move in 6b, which makes a single eye and threatens
to capture the single white stone above it at "a". The correct move for white is to play
"b" next. Allowing black to play at "b" would give black life. However, this particular
computer opponent does not realize this and picks the tiny defensive move at "a" over the
large offensive move at "b". The network has learned to take advantage of this weakness
and moves to "b" as its next move.
That such strategies would evolve is understandable considering that the opponent is the
only source of information the network has about the game. The network is never explicitly
taught about living or dead groups. It's concept of a living group is any group that the
opponent cannot kill. In this case, since the opponent cannot kill this group, the network
learns it as a favorable position. This example emphasizes the importance the training
partner has on the strategies learned.
One possible method for forcing the network to rely less on exploiting this type of weakness
in specific opponents is to evolve against a variety of opponents. The network would
be less likely to learn these kinds of techniques because it is less likely that the same flaw
will be present in multiple opponents. This type of evolution would have a better chance of
producing a well-rounded go playing program. However, the lack of multiple freely available
go playing programs makes this approach impractical at present time.
The neuro-evolution system is clearly learning enough to defeat a simple opponent, but
are the networks evolving to play go in any general sense? A closer inspection of game transcripts
shows that especially when evolved against a nondeterministic opponent, the networks
demonstrated a reasonable amount of diversity and were able to cope with variations in play
from the opponent.
At the beginning of evolution, the networks' outputs are essentially random. After a few
generations, they start to make simple living groups. Typically, they evolve the capability
to make one or two such groups along the edges or in the corners, and to extend them from
there. As evolution continues, the networks become more flexible and capable of developing
a greater variety of living positions. Such a strategy is valid, although not particularly
strong. Since this type of strategy is all that is required to defeat the computer opponent,
the network really does not need to develop more advanced strategies. Against more powerful
opponents, the situation would be different. The experiments with handicaps show that in
such cases, more powerful strategies are likely to develop.
Some well-known general go strategies were also evolved. For example, consider choosing
the first move. In the first few generations, the network plays quite randomly, and therefore
its first move tends to be on the edge or the outer lines of the board since they comprise
56 of the 81 positions on a 9 \Theta 9 board. Such a move is not a good idea, however, because
it is likely to lead to a losing position. Indeed, in a few generations the networks start to
make more opening moves near the center of the board. Since the earlier strategy led to
losses, the networks that did not use that strategy are now more prevalent in the population.
Later on in evolution all the best networks open at or near the center of the board, which
is exactly the strategy good go players use. Remarkably, the evolution system discovered
it entirely on its own, based on what moves led to wins and losses. This result suggests
that the neuro-evolution method is capable of developing good go playing strategies without
preprogrammed knowledge, directed by the sparse reinforcement of the game outcomes only.
\Omega \Gamma\Omega\Gamma
\Phi\Pi\Pi\Pi\Pi\Pi\Pi(a)
\Omega \Upsilon\Omega \Xi\Omega\Gamma3
\Omega \Pi\Omega \Omega \Gamma\Omega\Gamma
\Theta\Delta\Delta\Theta\Omega \Delta\Omega \Gamma\Omega \Gamma\Omega \Theta\Delta\Delta\Delta\Gamma\Omega \Gamma\Omega\Gamma
jffffl\Omega fi\Omega\Gamma443
ffifffl\Omega ff\Omega \Phi\Omega\Gamma45
\Theta\Omega \Omega \Psi\Omega \Upsilon\Omega \Xi\Omega\Gamma4
\Theta\Delta\Sigma\Omega \Pi\Omega \Omega \Gamma\Omega\Gamma
\Theta\Delta\Delta\Theta\Omega \Delta\Omega \Gamma\Omega \Gamma\Omega \Theta\Delta\Delta\Delta\Gamma\Omega \Gamma\Omega\Gamma
\Phi\Pi\Pi\Pi\Pi\Pi\Pi(c)
Figure
7: Figure (a) shows a position known as a ladder, which retains its shape as it is grown in
successive moves. The life or death of the white stone depends on groups far away on the board.
Figures (b) and (c) show two such situations: in (b) the white stone in location "b" allows white to
escape. In (c) the ladder reaches the edge of the board and the white group is killed. In real games,
such variations can span the whole board and are difficult to evaluate with only local methods.
9 Future Work
There are three main issues for future work: scaling up to larger boards, enhancing the
network architecture, and evolving against stronger opponents.
9.1 Larger Boards
Ideally, a network should be able to play on any board size. Currently the networks can only
play on the board on which they were evolved. For example, a network that was evolved on
a 9 \Theta 9 board is not able to play on a 7 \Theta 7 board. One possibility would be to design a
representation that is independent of board size. Another would be to evolve solutions that
only consider a portion of the board at a time. This type of evaluation function could then
be extended to cover boards of varying sizes.
However, considering only local board positions instead of the whole board tends to result
in weaker play. To see why, consider the position shown in figure 7, known as a ladder. Based
on only the local position, it is impossible to tell whether or not the white group can escape.
White can play at point "a", and may live or die depending on stones that are on the other
side of the board. If there is a white stone at point "b", for example, white can easily live.
The extra stone allows white to break out of the ladder, as can be seen in figure 7b. However,
if there are no stones on the area, white cannot live. Eventually the ladder position faces
the edge of the board, where it is a losing position for white as can be seen in figure 7c. This
way stones that are far away from the current play can transform the position drastically.
These types of positions can span the entire board, not merely one corner. Recognizing
such distance relationships is essential for playing go on larger boards, yet they cannot be
captured with methods that consider only part of the board at a time.
It is likely that other types of network architectures need to be employed before play on
full boards becomes practical. Possibilities include architectures that use preprogrammed
features or are hierarchically organized,
9.2 Advanced Architectures
Evolving simple unstructured neural network architectures without any prior knowledge
demonstrates the feasibility of the neuro-evolution approach. There are several ways the
architecture could be enhanced to make it more effective, including preprogrammed feature
detectors and hierarchies of networks.
Since the networks are not given any prior knowledge about what features are relevant
to playing go, they are forced to discover useful features themselves. Allowing the network
to access a pre-defined feature space instead of looking at the raw board might make the
task easier [2]. Such features could include common patterns and positions such as an eye
or a group or even complicated constructs such as the ladder. These features would then
be used as inputs to the neural network [1, 11]. It would still probably be useful to let the
network develop its own features as well, but the pre-programmed features might allow it to
learn faster and deal with more complex patterns.
SANE demonstrates the feasibility of evolving structures on more than one level at the
same time. It should be possible to extend this idea and evolve a hierarchy of networks,
where the lower levels would provide the inputs for networks at higher levels. In effect,
evolution would be searching for an effective combination of networks, much the same way
it is searching for an effective combination of weights and neurons now. When the task of
playing go is broken into such subtasks, it may be the case that the number of generations
required will increase with the number and size of the networks evolved and not with the
size of the board. If this is the case, then evolving networks that play on full-size boards
would no longer be computationally prohibitive.
9.3 Stronger Opponents
Even with more sophisticated architectures, stronger opponents are necessary in order to
achieve truly high levels of play. The ability to use handicaps to simulate stronger opponents
is a useful technique but not enough alone. The techniques used in handicap games are
different than those that would be used against a stronger player in an even game. Handicap
stones allow the weaker player to build stronger positions, but it still continues weak play
from these positions. On the other hand, in an even game the opponent may play brilliant
moves that would never be seen in a handicap game. If evolution is never exposed to such
moves, it cannot develop comprehensive go skills.
A variety of stronger opponents would allow for a greater generality of play to evolve.
However, it is not known how great the difference in play would be nor what the effect on the
time required to evolve the networks would be. It is also not yet clear how much diversity is
necessary to achieve general play.
One problem with using stronger opponents is that they tend to take considerably longer
to generate moves than weaker programs. Given the large number of trial games generated
every generation, it may not be possible to evolve against a slow opponent in a reasonable
amount of time. However, the evaluation function might be modified to compensate for the
lack of time. Each generation, a significant number of the networks evolved are significantly
weaker than the networks of the previous generation. It should be possible to distinguish the
weaker networks from the stronger networks by the use of a faster but weaker opponent. Only
those networks that appear promising need be evaluated fully against the slower opponent.
Even if stronger computer opponents are used, eventually they would be exhausted. It
would be necessary to find a way to evolve networks against actual human players. Given
the popularity of internet-based go servers, there is no shortage of human players. However,
there would be difficulties, particularly in the generation of fitness values. Fitness is used
to distinguish the better networks from the worse networks in any given generation. It
requires that the evaluation function be consistent for all networks evaluated. Since it would
be unlikely for many different networks from the same generation to play the same human
opponent, it would be difficult to assign a fair fitness value. The problem is compounded in
that the strength of the human opponent is not always known and cannot be reliably used
to weight game results against the strength of the human opponent. Nevertheless, good
results have been reported in neuro-evolution with noisy evaluation functions[8], suggesting
that the problems could be overcome. This way perhaps go-playing programs could finally
be evolved that were able to compete with the best humans.
Conclusions
Traditional artificial intelligence techniques have been insufficient for building go programs
that would be competitive at high levels of play. It appears new techniques based on pattern
recognition and learning will be required to reach these levels. The SANE neuro-evolution
approach is one such promising direction. Networks were evolved to defeat a publicly available
go program on small boards with no pre-programmed knowledge of the game, and they
exhibited several aspects of general go strategies.
--R
The Golem go program.
The integration of a priori knowledge into a go playing neural network.
Incremental evolution of complex general behavior.
A grandmaster chess machine.
The development of a world-class othello program
Discovering complex Othello strategies through evolutionary neural networks.
Efficient reinforcement learning through symbiotic evolution.
Symbiotic Evolution of Neural Networks in Sequential Decision Tasks.
Evolving obstacle avoidance behavior in a robot arm.
Learning sequential decision tasks.
Exploratory learning in the game of go.
Learning internal representations by error propagation.
CHINOOK: The world man-machine checkers champion
--TR
--CTR
Karen T. Sutherland, Book reviews, intelligence, v.11 n.3, p.47-54, Sept. 2000
Khosrow Kaikhah , Ryan Garlick, Variable Hidden Layer Sizing in Elman Recurrent Neuro-Evolution, Applied Intelligence, v.12 n.3, p.193-205, May-June 2000
A. Agogino , K. Stanley , R. Miikkulainen, Online Interactive Neuro-evolution, Neural Processing Letters, v.11 n.1, p.29-38, Feb. 2000 | neuro-evolution;game playing;sequential decision making;symbiotic evolution |
590834 | Multiple Adaptive Agents for Tactical Driving. | Recent research in automated highway systems has ranged from low-level vision-based controllers to high-level route-guidance software. However, there is currently no system for tactical-level reasoning. Such a system should address tasks such as passing cars, making exits on time, and merging into a traffic stream. Many previous approaches have attempted to hand construct large rule-based systems which capture the interactions between multiple input sensors, dynamic and potentially conflicting subgoals, and changing roadway conditions. However, these systems are extremely difficult to design due to the large number of rules, the manual tuning of parameters within the rules, and the complex interactions between the rules. Our approach to this intermediate-level planning is a system which consists of a collection of autonomous agents, each of which specializes in a particular aspect of tactical driving. Each agent examines a subset of the intelligent vehicles sensors and independently recommends driving decisions based on their local assessment of the tactical situation. This distributed framework allows different reasoning agents to be implemented using different algorithms.When using a collection of agents to solve a single task, it is vital to carefully consider the interactions between the agents. Since each reasoning object contains several internal parameters, manually finding values for these parameters while accounting for the agents possible interactions is a tedious and error-prone task. In our system, these parameters, and the systems overall dependence on each agent, is automatically tuned using a novel evolutionary optimization strategy, termed Population-Based Incremental Learning (PBIL).Our system, which employs multiple automatically trained agents, can competently drive a vehicle, both in terms of the user-defined evaluation metric, and as measured by their behavior on several driving situations culled from real-life experience. In this article, we describe a method for multiple agent integration which is applied to the automated highway system domain. However, it also generalizes to many complex robotics tasks where multiple interacting modules must simultaneously be configured without individual module feedback. | Introduction
The task of driving can be characterized as consisting
of three levels: strategic, tactical and operational
[13]. At the highest (strategic) level,
a route is planned and goals are determined; at
the intermediate (tactical) level, maneuvers are selected
to achieve short-term objectives - such as
deciding whether to pass a blocking vehicle; and
at the lowest (operational) level, these maneuvers
are translated into control operations.
Mobile robot research has successfully addressed
the three levels to different degrees.
Strategic-level planners [18, 24] have advanced
from research projects to commercial products.
The operational level has been investigated for
many decades, resulting in systems that range
from semi-autonomous vehicle control [7, 11] to
autonomous driving in a variety of situations [4,
15]. Substantial progress in autonomous navigation
in simulated domains has also been reported
in recent years [17, 3, 16]. However, the decisions
required at the tactical level are difficult and a
general solution remains elusive.
Consider the situation depicted in Figure 1.
Car A, under computer control, is approaching
its desired exit when it comes upon a slow moving
blocker (Car B), in its lane. Car A's tactical
reasoning system must determine whether to pass
Car B and risk missing the exit. Obviously, the
correct decision depends on a number of factors
such as the distance to the exit, Car A's desired
velocity, and the density and speed of surrounding
traffic.
Such scenarios are of particular relevance to
intelligent vehicles operating in mixed-traffic en-
vironments. In these environments, computer-
and human-controlled cars share the roadway, and
tactical decisions must be made without relying
on communication-based protocols. This short-term
planning problem is challenging because real-time
decisions must be made based on incomplete,
noisy information about the state of the world.
Furthermore, the penalty for bad decisions is severe
since errors in judgment may result in high-speed
collisions.
SAPIENT, described in Section 3, is a tactical
reasoning system designed to drive intelligent ve-
hicles, such as the Carnegie Mellon Navlab [23],
in mixed-traffic environments. In SAPIENT, decisions
are made by a collection of independent
agents, termed reasoning agents, each of which
specializes in a particular aspect of the tactical
driving task. This article focuses on how these
agents automatically configure themselves to optimize
a user- specified evaluation function using
a novel evolutionary algorithm termed Population
Based Incremental Learning (PBIL).
This article is organized as follows. Section 2
presents the simulated highway environment used
to train the SAPIENT agents. Section 3 details
the SAPIENT architecture, describing the reasoning
agents and their voting language. Section 4 introduces
PBIL, and explains the encoding scheme
used to represent agent parameters. Subsequent
sections present our results, both on small-scale
tactical scenarios (such as the one shown in Figure
1), and on larger highway configurations. Finally,
Section 8 summarizes the research and outlines
areas for further research.
A
GOAL
Fig. 1. An example of tactical-level reasoning. Car A is approaching its desired exit behind a slow vehicle B. Should Car
A attempt to pass?
Agents for Tactical Driving 3
Fig. 2. SHIVA: A design and simulation tool for developing intelligent vehicle algorithms.
2. The SHIVA Simulator
Simulation is essential in developing intelligent
vehicle systems because testing new algorithms
in real traffic is expensive, risky and potentially
disastrous. SHIVA 1 (Simulated Highways for
Intelligent Vehicle Algorithms) [22, 21] is a kinematic
micro-simulation of vehicles moving and interacting
on a user-defined stretch of roadway that
models the elements of the tactical driving domain
most useful to intelligent vehicle designers. The
vehicles can be equipped with simulated human
drivers as well as sensors and algorithms for automated
control. These algorithms direct the vehi-
cles' motion through simulated commands to the
accelerator, brake, and steering wheel. SHIVA's
user interface provides facilities for visualizing and
specifying the interactions between vehicles (see
Figure
2). The internal structure of the simulator
is comprehensively covered in [22], and details of
the design tools may be found in [21].
All simulated vehicles are composed of three
subsystems: perception, cognition, and control.
The perception subsystem consists of a suite of
simulated functional sensors (e.g., global positioning
systems, range-sensors, lane-trackers), whose
outputs are similar to real perception modules
implemented on the Navlab vehicles. Simulated
vehicles use these sensors to obtain information
about the road geometry and surrounding traffic.
Vehicles may control the sensors directly, scanning
and panning the sensors as needed, encouraging
active perception. Some sensors also model occlusion
and noise, forcing cognition routines to be
realistic in their input assumptions.
While a variety of cognition modules have been
developed in SHIVA, this article only discusses
two types: rule-based reasoning and SAPIENT.
The rule-based reasoning system, which was manually
designed, is implemented as a monolithic decision
tree. It consists of a collection of tactical
driving rules such as:
"Initiate a left lane change if the vehicle
ahead is moving slower than f(v) m/s, and
is closer than h(v), and if the lane to your
left is marked for legal travel, and if there are
4 Sukthankar, Baluja, Hancock
no vehicles in that lane within g(v) meters,
and if the desired right-exit is further than
where: f(v) is the desired car following velocity,
h(v) is the desired car following distance (head-
way), g(v) is the required gap size for entering an
adjacent lane, and e(x; y; v) is a distance threshold
to the exit based on current lane, distance to
exit and velocity. While this system performs well
on many scenarios, it suffers from four disadvan-
tages: 1) as the example above illustrates, realistic
rules require the designer to account for many fac-
tors; 2) modification of the rules is difficult since a
small change in desired behavior can require many
non-local modifications; hand-coded rules perform
poorly in unanticipated situations; implementing
new features requires one to consider an
exponential number of interactions with existing
rules. Similar problems were reported by Cremer
et al. [3] in their monolithic state-machine implementation
for scenario control. The SAPIENT
distributed architecture, discussed in the next sec-
tion, was developed to address some of these problems
The control subsystem is compatible with the
controller available on the Carnegie Mellon Navlab
II robot testbed vehicle. Commands to the controller
are issued by the cognition modules at a
rate of 10 Hz.
3. SAPIENT
SAPIENT (Situation Awareness Planner Implementing
Effective Navigation in Traffic) [20]
consists of a collection of independent modules
(termed reasoning agents), each of which is an expert
on a specific aspect of the tactical driving
task. Each agent is assigned to monitor a relevant
physical entity in the environment and is responsible
for assessing the repercussions of that entity
on the intelligent vehicle's upcoming actions (see
Figure
3). For example, the reasoning agent associated
with a vehicle ahead monitors the motion of
that vehicle and determines whether to continue
car following, initiate a lane change, or begin brak-
ing. Similarly, a reasoning agent associated with
an upcoming exit is concerned with recommending
the lane changes and speed changes needed
to successfully maneuver the intelligent vehicle to
the off-ramp.
3.1. System Overview
The SAPIENT architecture is shown in Figure 4.
The perception modules (depicted as ellipses) are
connected to the intelligent vehicle's sensors, and
perform functions such as lane tracking or vehicle
detection. Wherever possible, they correspond to
existing systems available on the Carnegie Mellon
Navlab (e.g., the lane tracker is based on
ALVINN [15]). Each reasoning agent (shown as
a dark rectangle) obtains information about the
situation from one or two perception modules,
and independently calculates the utility of various
courses of action. This information is then sent to
the voting arbiter, which integrates these recommendations
and selects the appropriate response.
Finally, the tactical action is translated into steering
and velocity commands and executed by the
operational-level controller.
As seen in Figure 4, reasoning agents can be
classified into classes based on their area of spe-
cialization. SAPIENT's loosely-coupled architecture
allows new classes to be developed without
modifying the existing reasoning agents. Our current
implementation spans the following tactical-
level aspects:
ffl Road properties: local geometry, legal lanes,
speed limits, etc.
ffl Nearby vehicles: sizes, positions, and veloci-
ties
ffl Exits: distance, exit lane, speed constraints
ffl Self-state: current velocity, lateral position,
explicit goals
Each reasoning agent tracks the associated
physical entity's attributes by monitoring the appropriate
sensors. For example, a reasoning agent
associated with a nearby vehicle normally tracks
its longitudinal position and velocity, and its lateral
position (mapped into road coordinates). The
tracking has two important implications. First, it
allows the reasoning agent to obtain a better estimate
of the relevant attribute. Second, the reasoning
agent can accumulate statistics that can help
influence decisions. For instance, based on the ir-
Agents for Tactical Driving 5
A
Front vehicle tracker
region of interest.
Exit Finder
Velocity Preference
Reasoning Agent
Exit Reasoning Agent
Lane Reasoning
Agent
Vehicle Reasoning
Agent
Fig. 3. SAPIENT reasoning agents are associated with relevant physical entities in the environment. In this situation, the
intelligent vehicle (A) is following a car and approaching its desired exit.
regular lane-keeping performance of a nearby vehicle
(an indication of an inexperienced or intoxicated
driver), the reasoning agent associated with
that vehicle could favor actions that maintain a
greater distance from that vehicle. Thus, SAPIENT
is not a purely reactive system; the local
state associated with each reasoning agent allows
SAPIENT to make decisions based on past his-
tory. The relevant history is maintained by each
agent.
Externally, all reasoning agents share a similar
structure - each agent accepts inputs from a
subset of the intelligent vehicle's perception modules
and sends outputs to the voting arbiter as a
set of votes over the entire action space (See Section
3.2). Internally, however, SAPIENT's reasoning
agents are heterogenous, maintaining local
state and using those representations that are
most applicable to the assigned subtask. For ex-
ample, the reasoning agents responsible for exit
management are rule-based while the reasoning
agent monitoring other vehicles use generalized
potential fields [9, 10]. The different reasoning
agent types and their associated algorithms are
detailed in [20].
reasoning agents are myopic in their
outlook. For example, the Exit Reasoning Ob-
ject's votes are not influenced by the presence
of the blocking vehicle; conversely, the reasoning
agent associated with the blocking vehicle is oblivious
to the exit. Finally, the arbiter is completely
ignorant of the driving task. Yet the combination
of these local reasoning schemes leads to a distributed
awareness of the tactical-level situation.
Before discussing how a knowledge-free arbiter can
combine these local views of the tactical driving
task, a closer look at the action space is warranted.
3.2. Actions
Tactical maneuvers (such as lane changing) are
composed by concatenating several basic actions.
Reasoning agents indicate their preference for a
basic action by assigning a vote to that action.
The magnitude of the vote corresponds to the
intensity of the preference and its sign indicates
approval or disapproval. Each reasoning agent
must assign some vote to every action in the action
space. All actions have velocity (longitudinal)
and lane-offset (lateral) components; for exam-
ple, "brake hard while changing left" or "increase
speed and maintain your current lane position".
Since different reasoning agents can return different
recommendations for the next action, conflicts
must be resolved. SAPIENT uses a voting
arbiter to perform this integration. At each time-
step, the reasoning agents synchronously submit
votes or vetoes for each action in the action space
(see
Table
1). During arbitration, all of the votes
for a given action are summed together (after
6 Sukthankar, Baluja, Hancock
Operational
Controller
Lane
Tracker
Exit
Finder
Car Detection
Modules
Perception Cognition Control
Voting
Arbiter
Velocity
Agent
Lane
Agent
Exit
Agent
Front Left
Car Agent
Front Right
Car Agent
Back Right
Car Agent
Back Left
Car Agent
Hysteresis
Agent
Front
Car Agent
Fig. 4. SAPIENT consists of a collection of reasoning agents which recommend actions based upon local considerations.
Each reasoning agent monitors a subset of the vehicle's sensors and votes upon a set of possible actions. The hysteresis
reasoning agent is responsible for maintaining consistency over time (especially in cases where multiple actions are equally
advantageous); this is done by voting in favor of the action selected in the previous time step. Action fusion is performed
by a domain-independent, voting arbiter.
Table
1. The action space is a 3 \Theta 3 discretization of the lateral/longitudinal space. The labels are translated at the
operational level into specific numbers. Thus, "left" and "right" map to lateral positions (e.g., move left/right by 0.1 lane
while "accelerate" and "decelerate" map to changes in velocity (e.g., speed up/slow down by 0.1 m/s).
accelerate/shift-left accelerate/straight accelerate/shift-right
coast/shift-left coast/straight coast/shift-right
decelerate/shift-left decelerate/straight decelerate/shift-right
being scaled by the reasoning agent's influence
weight), and the action with the most accumulated
votes (which has not been vetoed by any
agent) is executed. The actions used in the implementation
described in this article are summarized
in
Table
1. Finer discretizations and alternate action
spaces are discussed in [20]. Although the
action space restricts reasoning objects to voting
on their adjacent lanes, the reasoning agents can
internally plan longer-range courses of action. For
example, the exit agent can vote for lane changes
towards the exit, even when the exit is several
lanes away.
Agents for Tactical Driving 7
car following
(external)
exit weight
(external)
bits
parameters * 3
{
{
desire to exit
(internal)
={
car following
(internal)
011 . 010100101110 . 101
Fig. 5. The three-bit encoding scheme used to represent parameters in the search space: internal parameters are linearly
scaled while external ones are exponentially scaled.
3.3. Parameters
Different reasoning agents use different internal al-
gorithms. Each reasoning agent's output depends
on a variety of internal parameters (e.g., thresh-
olds, gains, etc. Before going to the arbiter, each
agent's outputs are scaled by its influence weight
(external parameters).
When a new reasoning agent is being imple-
mented, it is difficult to determine whether a ve-
hicle's poor performance should be attributed to
a bad choice of parameters in the new agent, a
bug in the logic of the new reasoning agent or,
more seriously, to a poor representation scheme
(inadequate configuration of reasoning agents).
To overcome this difficulty, we have implemented
a method for automatically configuring the parameter
space. A total of twenty parameters,
both internal and external, were selected for the
tests described here, and each parameter was discretized
into eight values (represented as a three-bit
string). For internal parameters, whose values
are expected to remain within a certain small
range, we selected a linear mapping (where the
three bit string represented integers from 0 to 7);
for the external parameters, we used an exponential
representation (with the three-bit string mapping
to eight values in the range 0 to 128). The
latter representation increases the range of possible
weights at the cost of sacrificing resolution
at the higher magnitudes. A representation with
more bits per parameter would allow finer tuning
but increase the training time. The encoding is
illustrated in Figure 5. In the next section, we
describe the evolutionary algorithm used for the
learning task.
4. Population-Based Incremental
Learning
Population-Based Incremental Learning (PBIL) is
a combination of genetic algorithms (GAs) [8] and
competitive learning [1, 2]. The PBIL algorithm
attempts to explicitly maintain statistics about
the search space and uses them to direct its ex-
ploration. The object of the algorithm is to create
a real valued probability vector which, when
sampled, reveals high quality solution vectors with
high probability. The full algorithm is presented
in
Figure
6.
Initially, each element of the PBIL probability
vector is initialized to 0.5. Sampling from this
vector yields random solution vectors because zeros
and ones are generated with equal probability
in each bit position. As training progresses, the
values in the probability vector gradually shift to
represent high evaluation solution vectors through
the following process. A number of solution vectors
are generated based upon the probabilities
specified in the probability vector. The probability
vector is pushed towards the generated solution
vector with the highest evaluation. After the
probability vector is updated, a new set of solution
vectors is produced by sampling from the updated
probability vector, and the cycle is continued. As
the search progresses, the entries in the probability
vector move away from their initial settings
of 0.5 towards either 0.0 or 1.0. The best solution
ever generated in the run is returned as the
final solution. Note that because the algorithm
only returns the best solution generated during
the run, convergence of the probability vector is
not a prerequisite for the success of the algorithm.
8 Sukthankar, Baluja, Hancock
for to LENGTH do
while (NOT termination condition)
* Generate Samples *
for to SAMPLES do
best_vector := find_vector_with_best_evaluation( sample_vectors, evaluations );
* Update Probability Vector towards best solution *
for to LENGTH do
* Mutate Probability Vector *
for to LENGTH do
if (random (0,1) < MUT_PROBABILITY) then
if (random (0,1) > 0.5) then mutate_direction :=
else mutate_direction := 0;
mutate_direction * (MUT_SHIFT);
return the best solution found in run;
USER DEFINED CONSTANTS (Values Used in this Study):
SAMPLES: the number of vectors generated before update of the probability vector (100)
LR: the learning rate, how fast to exploit the search performed (0.1).
LENGTH: the number of bits in a generated vector (3 * 20)
MUT_PROBABILITY: the probability of a mutation occuring in each position (0.02).
MUT_SHIFT: the amount a mutation alters the value in the bit position (0.05).
Fig. 6. The PBIL algorithm used to train SAPIENT reasoning agent parameters. Here, the explicit preservation of the
best solution from the previous generation (elitist selection) is not shown.
However, empirically, the probability vector has
converged in all of the runs conducted.
The probabilistic generation of solution vectors
does not guarantee the creation of a good solution
vector in every iteration. This problem is exacerbated
by the small population sizes used in our
experiments. Therefore, in order to avoid moving
towards unproductive areas of the search space,
the best vector from the previous population is
included in the current population (by replacing
the worst member of the current population) - in
GA literature, this is termed elitist selection [8].
Since space limitations preclude a complete discussion
about the relationship between GAs and
PBIL, we can only provide a brief intuition. Diversity
in the population is crucial for GAs. By
maintaining a population of solutions, the GA is
able - in theory at least - to maintain samples
in many different regions. In genetic algorithms,
crossover is used to merge these different solu-
tions. However, when the population converges,
crossover is deprived of the diversity that it needs
to be an effective search operator. When this hap-
pens, crossover begins to behave like a mutation
operator that is sensitive to the convergence of the
value of each bit. If all individuals in the population
converge at some bit position, crossover leaves
those bits unaltered. At bit positions where individuals
have not converged, crossover will effectively
mutate values in those positions. Therefore,
crossover creates new individuals that differ from
the individuals it combines only at the bit positions
where the mated individuals disagree. This
is analogous to PBIL which creates new trial solutions
that differ mainly in bit positions where
prior good performers have disagreed. More details
can be found in [1].
Our application challenges PBIL in a number of
ways. First, since a vehicle's decisions depend on
the behavior of other vehicles which are not under
Agents for Tactical Driving 9
its control, each simulation can produce a different
evaluation for the same bit string. We evaluate
each set of vehicle parameters multiple times to
compensate for the stochastic nature of the envi-
ronment. Second, the PBIL algorithm is never exposed
to all possible traffic situations (thus making
it impossible to estimate the "true" performance
of a PBIL string). Third, since each evaluation
takes considerable time to simulate, minimizing
the total number of training evaluations is
important.
5. Training Specifics
All of the tests described below were performed on
the track shown in Figure 8, known as the SHIVA
cyclotron. While this configuration does not resemble
a real highway, it has several benefits as a
testbed: 1) It is topologically identical to a highway
with equally spaced exits; 2) Taking the nth
exit is equivalent to traveling n laps of the course;
One can create challenging traffic interactions
at the entry and exit merges with only a small
number of vehicles.
During training, each scenario was initialized
with one SAPIENT/PBIL vehicle, and eight rule-based
cars (with hand-crafted decision trees). The
SAPIENT car was directed to take the second exit
while the other cars had goals of
zero to five laps. Whenever the total number of
vehicles on the track dropped below nine, a new
vehicle was injected at the entry ramp to maintain
the desired traffic density. Only one SAPIENT
vehicle was permitted on the course at a time.
At the start of the run, the PBIL algorithm suggested
a candidate bit-string which was converted
into SAPIENT parameters, and instantiated as a
simulated vehicle. Each evaluation of a PBIL parameter
string required one run of a simulated ve-
hicle. At the end of the vehicle's run, the score
that it received was sent to PBIL as the evaluation
of that candidate bit-string. It should be noted
that the population size in PBIL affected the number
of evaluations required in each generation of
the PBIL algorithm. The population size does not
correspond to the number of SAPIENT vehicles
present on the track since each candidate vehicle
was independently evaluated (as stated earlier,
only one SAPIENT vehicle was permitted on the
track at a time).
Whenever a SAPIENT vehicle left the scenario
(upon taking an exit, or crashing 10 times), its
evaluation was computed based on statistics collected
during its run. This score was used by the
PBIL algorithm to update the probability vector
thus creating better SAPIENT agents in the
next generation.
While the definition of good driving is largely
subjective, the following characteristics are
strongly correlated with bad driving: 1) collisions;
taking the wrong exit; 3) deviating from the
desired speed; weaving (poor lane tracking).
Many possible evaluation functions could be constructed
from these characteristics. For our evaluation
function, we combined them in a simple
weighted sum, to be maximized:
\Gamma(1000 \Theta num-crashes)
\Gamma(500 \Theta if-wrong-exit)
\Gamma(0:02 \Theta speed-deviation)
\Gamma(0:02 \Theta lane-deviation)
+(dist-traveled)
where: all-veto indicates that the SAPIENT
vehicle objects to all actions (with good param-
eters, this should never happen); num-crashes
is the number of collisions involving the SAPIENT
vehicle; if-wrong-exit is a flag - true if
and only if the SAPIENT vehicle exited prema-
turely, or otherwise missed its designated exit;
speed-deviation is the difference between desired
and actual velocities, integrated over the entire
run; lane-deviation is the deviation from
the center of a lane, integrated over the entire
run; dist-traveled is the longitudinal distance
covered by the vehicle, in meters (an incremental
reward for partial completion)
While the evaluation function is a reasonable
measure of performance, it is important to note
that there can be cases when a "good" driver
becomes involved in unavoidable accidents; con-
versely, favorable circumstances may enable "bad"
vehicles to score well on an easy scenario. To minimize
the effects of such cases, we tested each candidate
string in the population on a set of four sce-
narios. In addition to traffic, these test cases in-
cluded some pathological situations with broken-
vehicles obstructing one or more lanes.
6. Training
We performed a series of experiments using a variety
of PBIL population sizes, evaluation functions
and initial conditions. More details about individual
experiments are presented in the next section.
This section focuses on evaluation metrics for the
training algorithm.
Figure
7 shows the results of a training run
with the evaluation function described earlier, and
a PBIL population size of 100. These 3-D histograms
display the distribution of scores for each
generation. It is clear that as the parameters
evolve in successive generations, the average performance
of SAPIENT vehicles increases, and the
variance of evaluations within a generation de-
creases. In the experiments with population size
100, good performance of some vehicles in the population
is achieved early (by the fifth generation)
although consistently good evaluations are not observed
until generation 15. The number of vehicles
scoring poor evaluations drops rapidly until generation
10, after which there are only occasional
low scores. The PBIL strings converge to a stable
set of SAPIENT parameters, and by the last gen-
eration, the majority of the vehicles are able to
take the proper exit, and avoid crashes in all sce-
narios. The results of experiments with different
population sizes were similar.
Figure
8 shows a scenario on the cyclotron
track. This scenario is pathological, in that it contains
many broken-down vehicles, scattered over
the roadway. The trace shows a trained SAPIENT
vehicle successfully navigating the course by
avoiding the obstacles.
Above, we described the overall performance of
the SAPIENT vehicles in terms of a global evaluation
function. Here, we examine how the individual
components of the scoring metric improve
over time. Three observable quantities play a significant
role in the SAPIENT training evaluation
function: , the total number of near-collisions; fi,
whether the vehicle made its exit; and, i , the distance
traveled by the intelligent vehicle in the sce-
nario. Thus, for a given population of SAPIENT
vehicles, the quantities:
vehicles, v, in the popula-
tion), reflect the "goodness" of the population.
The three graphs in Figure 9 show how K, B, and
Z change over successive generations. Each PBIL
population contains 40 vehicles, and each vehicle
is evaluated on four different scenarios. The
graphs show that:
ffl The number of near-collisions, K, drops
steadily as PBIL tunes the SAPIENT reasoning
agent parameters. In the final generation,
none of the vehicles in the population are involved
in any near-collisions over the entire
set of four scenarios.
ffl The fraction of vehicles in the population
which missed their exit also decreases steadily
over time as the SAPIENT vehicles learn.
This too is zero in the final generation.
ffl The third quantity, Z, reflects the incremental
improvement in performance of the vehicles
during training. It can be seen that the early
vehicles are eliminated from the scenario (ei-
ther through timeout, taking the wrong exit,
or crashing) before they travel very far. Over
time, the vehicles are able to travel greater
distances. Note that Z has an upper bound
cannot be greater than the distance
to the desired exit).
To investigate the robustness of this training
method, two additional sets of experiments were
performed, where the coefficients in the evaluation
function were perturbed. In the first set, six
experiments were conducted, and in each experi-
ment, one coefficient was multiplied by 10. Some
of these results are shown in Figures 10 and 11.
Somewhat surprisingly, the SAPIENT reasoning
agents generated from these perturbed evaluation
functions are still successful. We hypothesize two
reasons for this: 1) the tactical-level sub-tasks
are closely linked: it is quite likely that a vehicle
which makes the correct exit has also learned
to avoid collisions - otherwise it would have been
eliminated in a collision earlier on the track; 2) although
PBIL is responsible for setting the internal
and external parameters for each reasoning agent,
the underlying algorithms are predefined; thus, a
small perturbation in reasoning agent parameters
does not cause catastrophic failures in the system.
Agents for Tactical Driving 11
-5000
Evaluation1030Generation2060Number of Cars
-5000
Evaluation
Fig. 7. 3-D Histogram showing increase of high-scoring PBIL strings over successive generations. Population size is 100
cars in each generation.
Fig. 8. A pathological scenario on the cyclotron track with 15 obstacles. The trace shows a SAPIENT vehicle successfully
navigating the course by avoiding the obstacles.
The second set of experiments explored the limits
of this robustness. This time, the coefficients
were perturbed by a factor of 1000. While this
tended to create bad cars in general, some of the
coefficients, even when multiplied by 1000, still
of
near-crashes
Generations
of
exits
missed
Generations
B60000100000140000180000
Distance
covered
Generations
Z
Fig. 9. This graph shows how the number of near-collisions (K), number of missed-exits (B), and distance traveled (Z)
in a population of learning SAPIENT vehicles varies with successive generations. In these tests, the population size was set
to 40, and each vehicle was evaluated on four scenarios. The graphs show the accumulated statistics for all of the vehicles
in the given generation, over all four scenarios. Note that both K and Z decrease to zero, while Z, the incremental reward,
rises.
Agents for Tactical Driving 13501502500
of
near-crashes
Generations
of
near-crashes
Generations
of
near-crashes
Generations
of
near-crashes
Generations
Fig. 10. This graph shows that the SAPIENT parameters learned by PBIL converge to competent vehicles despite variations
in the evaluation function used. Each of these graphs shows the total number of near-collisions (K), in a population
of
exits
missed
Generations
of
exits
missed
Generations
of
exits
missed
Generations
of
exits
missed
Generations
Fig. 11. This graph shows that the SAPIENT parameters learned by PBIL converge to competent vehicles despite variations
in the evaluation function used. Each of these graphs shows the total number of missed exits (B), in a population of
Agents for Tactical Driving 15
A
Fig. 12. This scenario tests if the tactical reasoning system can overtake a slower-moving vehicle.
Lateral
position
(lane
units)
Time (0.1 sec intervals)
Mono
Velocity
(m/sec)
Time (0.1 sec intervals)
Mono
Poly
Fig. 13. Lateral displacement (left) and velocity (right) as a function of time, for rule-based (denoted as Mono) and
SAPIENT (denoted as Poly) vehicles on the overtaking scenario (See Figure 12). See the text for a discussion of these
graphs.
generated competent vehicles. For example, increasing
the penalty of a collision from -1000 to
-1000000 does not affect vehicles since they learn
how to avoid all collisions. By contrast, radically
increasing the penalty for speed deviations in a
similar manner leads to vehicles that are willing
to collide with others in a desperate effort to avoid
the penalties incurred in dropping below the target
velocity.
7. Scenario-Based Evaluation of Tactical
Driving
Scenarios are widely used in driving research to
evaluate the performance of human subjects [12,
14]. Similar techniques are also used to measure
situation awareness in other domains [6, 5, 19].
Here, we use micro-scenarios to examine the performance
of SAPIENT's reasoning agents in situations
where tactical-level decisions are required. A
more comprehensive discussion of these scenarios
is available in [20].
In each of the following scenarios, we focus
on the vehicle marked A in the respective dia-
grams. SAPIENT's performance is compared to
the behavior of the default rule-based vehicle. In
the accompanying graphs, the monolithic, hand-
coded, rule-based vehicle is denoted as Mono,
while the multi-agent, adaptive, SAPIENT system
is marked Poly. It should be emphasized that
the SAPIENT vehicles have not been exposed to
any tactical scenarios - they were trained (us-
ing PBIL) exclusively on obstacle courses in the
cyclotron environment.
The first scenario (See Figure 12) involves a simple
overtaking maneuver, and is a common occurrence
on the highway. Initially, both vehicles are
moving at normal highway speeds, but the lead
vehicle begins braking (as it approaches its exit,
for example). There is no other traffic, so Car A
should safely overtake. As seen in the lateral displacement
and velocity profiles (See Figure 13),
both types of cognition module are able to solve
A
L32.03.04.0Fig. 14. Exit scenarios add complexity to the tactical driving domain because they introduce additional strategic-level
goals. The conflicts between two strategic-level goals leads to interesting tactical
Lateral
position
(lane
units)
Time (0.1 sec intervals)
Mono
Velocity
(m/sec)
Time (0.1 sec intervals)
Mono
Poly
Fig. 15. Lateral displacement (left) and velocity (right) as a function of time, for rule-based and SAPIENT vehicles on
the exit scenario (See Figure 14). See the text for a discussion of these graphs.
this scenario successfully. However, note that the
SAPIENT vehicle is more aggressive, maintaining
a smaller headway during the maneuver than
the hand-tuned, rule-based vehicle. This is because
the SAPIENT reasoning agent responsible
for car following has tuned its generalized potential
fields relative to the other vehicle based on
a time-to-impact metric, as opposed to using a
constant headway. The other notable feature is
the oscillation in the rule-based vehicle's velocity
profile. This is caused by a combination of two
factors: a discrete velocity controller and brittle
car-following rules. Note that the SAPIENT vehicle
is not perfectly centered in the passing lane
during the overtaking maneuver. This is because
the potential field surrounding the obstacle votes
for additional space, and since there is sufficient
space in the target lane, the SAPIENT vehicle is
able to drive off-center. This behavior can also be
observed in the other scenarios.
The second scenario (See Figure 14) introduces
a second (possibly conflicting) strategic goal: taking
an exit; also, ambient traffic is introduced. Vehicle
A must now change lanes to make its desired
exit without colliding with other cars. Figure 15
shows an interesting difference in driving behav-
ior. The rule-based car slows down until it can find
a gap in the exit lane, and then changes lanes. In
contrast, SAPIENT speeds up to overtake the vehicle
in the exit lane. This maneuver allows it to
maintain its desired speed while making the exit.
The final scenario, shown in Figure 16, is identical
to the one discussed in the Introduction. Recall
that Car A may take its desired exit by either
staying behind the slow blocker, or by passing.
Unlike the situation shown in Figure 14, chang-
Agents for Tactical Driving 17
A
GOAL
L22.03.04.0Fig. 16. This exit scenario is difficult because the lane changes are optional. To address the strategic-level goal of maintaining
speed, the intelligent vehicle must decide whether or not to attempt the overtaking maneuver at the risk of missing
the desired exit.1.622.42.83.2
Lateral
position
(lane
units)
Time (0.1 sec intervals)
Mono
Velocity
(m/sec)
Time (0.1 sec intervals)
Mono
Poly
Fig. 17. Lateral displacement (left) and velocity (right) as a function of time, for rule-based and SAPIENT vehicles on
the more difficult exit scenario (See Figure 16).
ing lanes is not mandatory; in fact, should Car A
decide to pass, it will have to complete two lane
changes before exiting. Once again, the two different
vehicle types choose differently. The rule-based
vehicle opts to stay in its lane, based solely
on a rule which depends on the distance to the
exit. On the other hand, the SAPIENT vehicle
chooses to overtake the blocker.
In the final set of experiments, vehicles were injected
into an initially empty cyclotron track from
the on-ramp at regular intervals, . Each vehicle
was given two strategic-level goals: 1) make exactly
one circuit of the track before exiting; 2)
maintain the speed at which it was injected whenever
possible. The aim of the experiment was to
see how the two tactical driving systems, rule-
based, and SAPIENT, would behave as the roadway
became more congested. Three sets of experiments
with different traffic configurations were
performed: all-rule-based cars, all-SAPIENT cars,
and a uniform mix of rule-based and SAPIENT
cars.
As expected, the number of vehicles on the
roadway increased until the rate of vehicles entering
the track was equal to the rate of vehicles
leaving (either because the vehicles had successfully
completed their circuit, or because the vehicles
were unable to merge into the traffic stream).
At low rates of traffic flow (e.g., ? 6 seconds),
all of the three traffic configurations safely negotiated
the scenario (with no missed exits). However,
once the traffic flow was increased, the behavior of
the three traffic configurations diverged.
of
cars
on
track
Time (secs)
Mono
Poly
Fig. 18. This graph shows how the number of vehicles on the cyclotron varies as a function of time in heavy traffic (traffic
injection rate cars). Note that only six rule-based vehicles were able to merge onto the cyclotron loop;
by contrast, all of the SAPIENT vehicles were able to merge and complete the scenario.
The graph in Figure shows how the number
of vehicles varies as a function of time when
seconds for the all-rule-based and all-SAPIENT
cases. Even in heavy traffic, neither of the pure
traffic types have any collisions. Although both
types of vehicles perform well initially (when the
roadway is clear), once the number of vehicles
on the track increases to about 6, the conservative
rule-based drivers are unable to merge into
the traffic stream (since they require a guaranteed
headway of two seconds on both sides of the
gap). Thus they are unable to change lanes, and
exit the scenario prematurely. To make matters
worse, the rule-based vehicles that were already
on the roadway become trapped in the inner loops
of the cyclotron (due to the high rate of traffic in
the entry/exit lane).
The all-SAPIENT traffic, on the other hand,
is able to drive successfully. This can probably
be attributed to two factors: 1) the aggressive
driving style, relying on time-to-impact reasoning
agents, is willing to merge into smaller gaps; 2) the
distributed reasoning system is better at making
tradeoffs - the negative votes for merging into a
potentially unsafe gap are tolerated since the alternative
(missing the exit) is seen to be worse.
The brittle decision tree used in the rule-based
cars, on the other hand, rejects these gaps outright
Interleaving rule-based and SAPIENT cars in
the heavy traffic scenario leads to a stable heterogenous
behavior with no collisions. While the
more aggressive SAPIENT vehicles still miss fewer
exits, even the rule-based vehicles perform better
than they did in the pure-rule-based case because
of the reduced congestion. This may have positive
implications for the deployment of automated vehicles
in mixed traffic conditions.
8. Conclusion and Future Directions
Our experiments have demonstrated: 1) The potential
for intelligent behavior in the tactical driving
domain using a set of distributed reasoning
agents; 2) The ability of evolutionary algorithms
to automatically configure a collection of
these modules for addressing their combined task.
While the evaluation sections compared SAPI-
ENT's performance with a rule-based vehicle, the
results should not be taken out of context: clearly
it is possible to encode SAPIENT's current knowledge
in the form of rules to create a more competent
rule-based vehicle. The difference is that
Agents for Tactical Driving 19
creating a monolithic rule-based vehicle is a much
more difficult task due to the interactions between
large number of rules, the manual tuning of parameters
within the rules, and the complex interactions
between the rules.
In this study, we used a simple evaluation func-
tion. By introducing alternative objective func-
tions, we plan to extend this study in at least
two directions. First, for automated highways,
we would like the cars to exhibit altruistic be-
havior. In a collection of PBIL vehicles, optimizing
a shared evaluation function (such as highway
may encourage cooperation. Second,
we are developing reasoning agents to address additional
complications which will arise when these
vehicles are deployed in the real world, such as
complex vehicle dynamics and noisy sensors.
Our system, which employs multiple automatically
trained agents, can competently drive a ve-
hicle, both in terms of the user-defined evaluation
metric, and as measured by their behavior on several
driving situations culled from real-life expe-
rience. In this article, we described a method for
multiple agent integration which is applied to the
automated highway system domain. However, it
also generalizes to many complex robotics tasks
where multiple interacting modules must simultaneously
be configured without individual module
feedback.
9.
Acknowledgments
The authors would like to acknowledge the valuable
discussions with Dean Pomerleau and Chuck
Thorpe which helped to shape this work. Thanks
also to Gita Sukthankar for the data processing
scripts and graphs. This research was partially
supported by the Automated Highway System
project, under agreement DTFH61-94-X-00001,
and was started while Shumeet Baluja was supported
by a graduate student fellowship from
NASA, administered by the Lyndon B. Johnson
Space Center. The views and conclusions contained
in this document are those of the authors
and should not be interpreted as representing the
official policies, either expressed or implied, of the
AHS Consortium or NASA.
Notes
1. More information and an interactive demo are available
at: !http://www.cs.cmu.edu/rahuls/shiva.html?
--R
Removing the genetics from the standard genetic algorithm.
The software architecture for scenario control in the Iowa driving simulator.
A curvature-based scheme for improving road vehicle guidance by computer vision
Towards a theory of situation awareness.
Automatic car controls for electronic highways.
Genetic Algorithms in Search
A generalized potential field approach to obstacle avoidance control.
Integrated path planning and dynamic steering control for autonomous vehi- cles
Driver education and task analysis Volume
A critical view of driver behavior models: What do we know
Coaching the experienced driver II
Neural Network Perception for Mobile Robot Guidance.
Using genetic algorithms to learn reactive control parameters for autonomous robotic navigation.
Selective Perception for Robot Driving.
Advanced driver information systems.
How in the world did we ever get into that mode?
Situation Awareness for Tactical Driving.
Also available as CMU Tech Report CMU- RI-TR-97-08
A simulation and design system for tactical driving algorithms.
SHIVA: Simulated highways for intelligent vehicle al- gorithms
Vision and navigation for the Carnegie Mellon Navlab.
Dynamic route guidance and interactive transport management with ALI-Scout
--TR
--CTR
Antonio Pellecchia , Christian Igel , Johann Edelbrunner , Gregor Schoner, Making Driver Modeling Attractive, IEEE Intelligent Systems, v.20 n.2, p.8-12, March 2005 | distributed AI;evolutionary algorithms;intelligent vehicles;simulation |
590846 | Incremental Feature Selection. | Feature selection is a problem of finding relevant features. When the number of features of a dataset is large and its number of patterns is huge, an effective method of feature selection can help in dimensionality reduction. An incremental probabilistic algorithm is designed and implemented as an alternative to the exhaustive and heuristic approaches. Theoretical analysis is given to support the idea of the probabilistic algorithm in finding an optimal or near-optimal subset of features. Experimental results suggest that (1) the probabilistic algorithm is effective in obtaining optimal/suboptimal feature subsets; (2) its incremental version expedites feature selection further when the number of patterns is large and can scale up without sacrificing the quality of selected features. | Introduction
Feature selection is about finding useful (relevant) features that describe an application
domain. The problem of feature selection can formally be defined as
selecting a minimum set of M relevant features from N original features where
M N such that the probability distribution of different classes given the values
for these M features is as close as possible to the original distribution given the
values for N features. Mathematically, if FN is the original feature set and FM
is the chosen feature subset, then the conditional probability, P(C j
should be as close as possible to P(C j possible classes, C. Here
f M and f N are value vectors of respective feature vectors FM and FN [11]. As
the dimensionality of a domain expands, the number of features increases. In
general, the role of feature selection is three-fold: 1. simplifying data descrip-
tion; 2. reducing the task of data collection; and 3. improving the quality of
problem solving. For the same problem, a representation by three features is
generally simpler than one by six features. The benefits of having a simple
representation are abundant such as easier understanding of problems, and better
and faster problem solving. In the context of data collection, having more
features means that more data should be collected. In many applications, this
could be time consuming and costly. The quality improvement of problem solving
resulting from feature selection can be illustrated via a classical supervised
learning task - pattern classification problem: given a training set of labeled
patterns, induce a classification model that can predict the class label for a
set of previously unseen patterns (the so-called testing set). Although having
more features enhances discriminating power in representation, having excessive
features would introduce many difficulties for induction algorithms [11]. First,
the time required by an induction algorithm often grows dramatically with the
number of features, rendering the algorithm impractical for problems with a
large number of features. Second, many learning algorithms can be viewed as
performing estimation of the probability of the class label given a set of fea-
tures. With many features, this distribution is of high dimension and becomes
very complex. Unless exponential amounts of data are available, it is difficult to
obtain a good estimation from a training dataset. Third, irrelevant and redundant
features may confuse a learning algorithm by obscuring the distribution of
the small set of truly relevant features. In addition, irrelevant and redundant
features require an exponential increase in data storage requirements [1]. This is
because with more features, much more data is required for effective induction.
For instance, in a binary domain, the extra m irrelevant/relevant features would
times more patterns to describe the whole data. For an induction
algorithm, the reduced features can also result in a simpler induction model
such as shorter and fewer classification rules.
For N features, if d of them are relevant, an exhaustive approach to finding
the optimal d features would require examining
subsets. The number
of possible subsets grows exponentially. Researchers have designed different
strategies in search of optimal subsets of d features (Branch and Bound [20] and
its variations [26], many heuristic and stochastic methods [5, 7]). If we view
these feature selection algorithms from the perspective of using an induction
algorithm, as pointed out in [8], the work on feature selection can be divided into
filter and wrapper models. In a filter model, a feature selector is independent
of an induction algorithm and serves as a filter to sieve the irrelevant and/or
redundant features; in a wrapper model, a feature selector wraps around an
inductive algorithm relying on which relevant features are determined. One
problem with the wrapper model is that it is restricted by the time complexity
of a learning algorithm [12]. This time complexity is dependent on the number
of features. Often the wrapper methods are prohibitively expensive to run and
can be intractable for a very large number of features. Recall that, in many
cases, feature selection is performed because of the excessive number of features
and because a favorite induction algorithm has difficulties in handling so many
features. Different models, however, suit for different applications. If a classifier
is chosen and it can run for an application at hand, then it may be wise to
choose a wrapper model since both feature selection and classifier induction
use the same bias. This work considers the cases in which learning a classifier
becomes cumbersome or ineffective due to the large size of a dataset. The
largeness can be defined by both the number of features (N ) and the number
of patterns (P ). It is the latter that makes some induction algorithms falter.
Hence, large datasets in terms of P are the main concern. Naturally, the filter
model is adopted. Our aim is to provide a simple and practical method that
can select features for large datasets. In the following, we first review related
work on feature selection.
Related Work
The problem of feature selection has long been an active research topic within
statistics and pattern recognition [30, 6, 7], but most work in this area has dealt
with linear regression [12] and is under assumptions that do not apply to most
machine learning algorithms [8]. Researchers [12, 8] pointed out that the most
common assumption is monotonicity that increasing the number of features can
only improve the performance of a learning algorithm 1 . Recently feature selection
has received considerable attention from researchers in machine learning
and knowledge discovery who are interested in improving the performance of
algorithms and in cleaning data. In handling large databases, feature selection
is even more important since many learning algorithms may falter or take too
long time to run before data is reduced.
Most feature selection methods [9, 12, 8] can be grouped into two categories:
exhaustive or heuristic search for an optimal set of M features. For example,
Almuallim and Dietterich's FOCUS algorithm [2] starts with an empty feature
set and carries out exhaustive search until it finds a minimal set of features that
is sufficient to construct a hypothesis consistent with a given set of examples.
It works on binary, noise-free data. Its time complexity is O(min(N
1 The monotonicity assumption is not valid for many induction algorithms used in machine
learning. An example is dataset 1 (CorrAL) in Section 5 which is reproduced from [8].
They proposed three heuristic algorithms to speed up the searching [2]. This
is because selecting a minimal subset is a known intractable problem, and in
practice, we often have to trade off the optimality of a solution for less time
spent on searching.
There are many heuristic feature selection algorithms. The Relief algorithm
[9] assigns a "relevance"weight to each feature, which is meant to denote
the relevance of a feature to the target concept. Relief samples patterns
randomly from the training set and updates the relevance values based on the
difference between the selected pattern and the two nearest patterns of the
same and opposite classes. According to [9], Relief assumes two-class classification
problems and does not help with redundant features. If most of the given
features are relevant to the concept (including redundant features), it would
select most of them even though only a fraction of them is necessary for concept
description. The PRESET algorithm [19] is another heuristic feature selector
that uses the theory of Rough Sets to rank the features heuristically, assuming
a noise-free binary domain. In order to consider higher order relations among
the features, Liu and Wen [16] suggest the use of high order information gains
to select features. Since the last two algorithms do not try to explore all the
combinations of features, it is certain that they fail on problems whose features
are highly interdependent such as the parity problem where combining a small
number of features does not help in finding the relevant ones. Another common
understanding is that some learning algorithms have built-in feature selection,
for example, ID3 [23], FRINGE [21] and C4.5 [24]. The results in [2] suggest
that one should not rely on ID3 or FRINGE to filter out irrelevant features.
A more detailed survey can be found in [5]. The latest development of feature
selection in pattern recognition can be found in [7].
To sum up, the exhaustive search approach is infeasible in practice; the
heuristic search approach can reduce the search time significantly, but will fail on
hard problems (e.g., the parity problem) or cannot remove redundant features.
A probabilistic approach is proposed as an alternative [15] in selecting the op-
timal/suboptimal subset(s) of features. In the context of large sized databases,
however, it would still take considerably long time to check if a subset is valid
or not 2 . We had first-hand experience of this problem when our probabilistic
system was dispatched to a local institute for on-site usage. All the evidence
showed that reducing data size can significantly speed up the selection of features
(see a case study in Section 3.3). Hence, the incremental probabilistic
method is designed and implemented. In the following, we describe the probabilistic
method first, then the incremental one, followed by an empirical study
in which the effectiveness of the algorithms is verified. At the end of the paper,
we provide relevant discussion.
2 The checking can be done in O(P ), where P is the number of patterns, by using a hashing
method.
3 Probabilistic Feature Selection
The proposed probabilistic approach is a Las Vegas Algorithm [4]. Las Vegas
algorithms make probabilistic choices to help guide them more quickly to a correct
solution. One kind of Las Vegas algorithms uses randomness to guide their
search in such a way that a correct solution is guaranteed even if unfortunate
choices are made. As we mentioned earlier, heuristic search methods are vulnerable
to datasets with high interdependency among their features. Las Vegas
algorithms free us from worrying about such situations by evening out the time
required on different situations. Another similar type of algorithms is Monte
Carlo algorithms in which it is often possible to reduce the error probability
arbitrarily at the cost of a slight increase in computing time (refer to page 341
in [4]). In this work, LVF (Las Vegas Filter) 3 is more suitable since probabilities
of generating distinct subsets are the same. The time performance of a Las
Vegas algorithm may not be better than that of some heuristic algorithms.
LVF algorithm
Input: MAX-TRIES, fl - allowed inconsistency rate,
dataset of N features;
Output: sets of M features satisfying the inconsistency criterion
best
for i=1 to MAX-TRIES
best and InconCheck(S; D) ! fl)
else if best ) and
printCurrentBest(S)
end for
The LVF algorithm generates a random subset, S, from N features in every
round. If the number of features (C) of S is less than the current best, i.e.,
best , the data D with the features prescribed in S is checked against
the inconsistency criterion. If its inconsistency rate (defined later) is below a
pre-specified one (fl), C best and S best are replaced by C and S respectively; the
new current best (S) is printed. If best and the inconsistency criterion is
satisfied, then an equally good current best is found and printed. MAX TRIES
in the algorithm is used to control the number of loops. A value of MAX TRIES
can be defined according to applications or based on the experience from exper-
imentation. Too small or too big a MAX TRIES will affect the performance of
LVF. The compromise is made for good and fast solutions. The longer LVF runs,
the better its results are. Refer to the analysis in Section 3.2. MAX TRIES is
set to 77\ThetaN in our experimental study 4 following the rule-of-thumb that the
Its counterpart is LVW - a wrapper feature selector applying a Las Vegas algorithm.
4 We tried first a constant c alone instead of c \Theta N , then linked it to N . 77 was chosen for c
more features a dataset has (in other words, the larger N is), the harder the
problem of feature selection (parity-5 is more difficult than parity-2, e.g.), and
hence more tries are needed. When LVF loops MAX TRIES times, it stops.
An alternative to this stopping criterion is to let LVF run forever to take full
advantage of its "anytime algorithm" nature (more in Section 6). The function
randomSet(seed) returns a set of features randomly. When the seed is changed
dynamically, a different set is generated. The function numOfFeatures(S) returns
the cardinality of set S. InconCheck(S; D) returns the inconsistency rate
of data D with selected features specified in S. printCurrentBest(S) prints
out subset S.
A more sophisticated version of LVF is like this: since we know the cardinality
of a better subset can only be smaller than C best - the cardinality of the
current best subset, we just need to randomly generate subsets whose cardinalities
are smaller than C best . For a new round of selection, we sample features
without replacement.
3.1 Measure of feature goodness
The inconsistency criterion (InconCheck(S; D) ! fl) is the key to LVF. The
criterion specifies to what extent the dimensionally reduced data is acceptable.
The inconsistency rate of the data described by the selected features is checked
against a pre-specified rate (fl). If it is smaller than fl, it means the dimensionally
reduced data is acceptable. The default value of fl is 0 unless specified. The
inconsistency rate of a dataset is calculated as follows: (1) two patterns are considered
inconsistent if they match all but their class labels; (2) the inconsistency
count is the number of all the matching patterns minus the largest number of
patterns of different class labels: for example, there are n matching patterns,
among them, c 1 patterns belong to label 1 , c 2 to label 2 , and c 3 to label 3 where
3 is the largest among the three, the inconsistency count
is and (3) the inconsistency rate is the sum of all the inconsistency
counts divided by the total number of patterns (P ). It can be easily shown
that if the inconsistency rate is 0 for both datasets with M and N features,
and FN is the original feature set and FM is the chosen feature subset, then
the conditional probability P(C exactly equals P(C j
for different possible classes, C, where f M and f N represent vectors of values of
respective feature vectors FM and FN .
The inconsistency criterion is a conservative way of achieving the "class
separability" which is commonly used in pattern recognition as the basic selection
criterion [6]. A limited version of this was first proposed by [2] as the
MIN-FEATURES bias on a binary domain. Instead of aiming to maximize the
class separability, our measure tries to maintain the original class separability of
the data. The inconsistency criterion is also in line with information-theoretic
in all the experiments in this paper. We tried not to use too large c so that for all (small and
large) datasets, we could use just one fixed MAX TRIES. The reader may do as we have done
in another version of LVF to link MAX TRIES to the percentage of the total search space
according to the desired quality of selected features.
considerations [28] which suggest that using a feature that is good for discrimination
provides compact descriptions of each of the two classes, and that these
descriptions are maximally distinct. Geometrically, this constraint can be interpreted
[17] to mean that (i) such a feature takes on nearly identical values
for all examples of the same class, and (ii) it takes on some different values for
all examples of the other class. The inconsistency criterion aims to retain the
discriminating power of the data for multiple classes after feature selection.
3.2 Theoretical analysis
Our analysis shows that LVF can give a good solution, or an optimal solution if
MAX TRIES is sufficiently large. With a good pseudo random number generator
[22], selecting an optimal subset of M features can be considered as sampling
without replacement. The probability of finding the optimal subset at the
(k+1)th experiment is 1
, and the probability of having to conduct (k+1) experiments
before finding the optimal subset is
\Theta ::: \Theta 1
where N is the number of original features. When N is large, MAX TRIES
2 N . Here we assume there is only one optimum. If there exist l optima as
in many applications, at the (k 1)th tossing, the probability of finding one
optimum is l
. Roughly, when the number of optima is doubled, the number
of run times can be halved.
Referring to the LVF algorithm, we notice that the inconsistency criterion is
checked only when C C best . Thus, when C best is reduced due to the random
search, the number of inconsistency checking is also reduced. As shown in
Section 5.1, for the real-world datasets, C best can be as few as one fifth of the
original number of features (Mushroom). In addition, the time complexity of
the checking is O(P ). Hence, LVF is expected to run fast. If the equivalently
good subsets are not required, the last two lines inside the for-loop of the LVF
algorithm can be removed, LVF can be made even faster.
3.3 Applying LVF to huge datasets (a practical case)
Feature selection is particularly useful when datasets are huge since many learning
algorithms may encounter difficulties. As mentioned earlier, feature selection
can help reduce the dimensionality of the datasets so that learning algorithms
can be used to induce rules. Hence, huge datasets are also an ultimate test for
a feature selection algorithm. LVF had an opportunity to undergo a real test of
huge datasets. In Section 5 (Empirical Study) below, the results of LVF on the
benchmark datasets are reported.
The datasets involved are related to the service industry. LVF was given
to a local institution 5 , which was in need of a method to reduce the number
of features before applying some machine learning algorithms to the datasets
due to the huge size of the datasets. Because the datasets are confidential, we
have no access to them. The users at the institution ran LVF independently
5 Japan-Singapore AI Centre, Singapore.
and without modification and provided the following account: One dataset (let
us call it HD1) has 65,000 patterns and 59 features; the other (HD2) has 5,909
patterns and 81 features. Both datasets are discrete, feature values range from
2 to 13. LVF found that 10 and 35 features were relevant for describing HD1
and HD2 respectively without sacrificing their discriminating power, after hours
of running LVF on a Sun Sparc workstation. Due to the long waiting time, they
did another experiment in which only 10,000 patterns of HD1 were used, it took
LVF about 5 minutes to complete its run and obtained the same results. The
results are summarized in the table below. The stark difference between hours
and minutes inspired us to extend the work of LVF. In short, their findings
manifest two points: (1) LVF significantly reduced the number of features; and
(2) reducing the number of patterns significantly reduced the run time. It is the
second finding that leads us to incremental feature selection.
Data #Features #Patterns #Selected Time
hours
hours
The largeness of a dataset can be differentiated into two types: (1) horizontal
largeness - the number of features, and (2) vertical largeness - the number
of patterns. In our implementation of LVF, we have considered overcoming
the horizontal largeness by applying a Las Vegas algorithm in order to avoid
exhaustive search and attack the vertical largeness by using a hash mechanism
in order to speed up. However, the above practical case shows that more can
be done in overcoming the vertical largeness. Hence, in the following, when we
mention largeness, we mean the vertical one (P ).
4 Incremental Probabilistic Feature Selection
Although LVF can generate optimal/suboptimal solutions (see the experimental
results below), when datasets are huge, as shown in Section 3.2, checking
whether a dataset is consistent still takes time due to its O(P ) complexity. It is
only natural to think about an incremental version of LVF that can significantly
reduce the number of inconsistency checkings. Studying the LVF algorithm, we
notice that if we reduce the data, we can decrease the number of checkings.
However, features selected from the reduced data may not be suitable for the
whole data. The following algorithm is designed to achieve that features selected
from the reduced data will not generate more inconsistencies than those
from the whole data. Furthermore, this is done without sacrificing the quality
of feature subsets which is measured by the number of features and by their
relevance.
LVI algorithm
percentage of the data used for feature selection,
dataset of N features, fl - allowed inconsistency rate;
Output: sets of M features satisfying the inconsistency criterion
of D randomly chosen ;
the rest data */
loop
if (checkIncon(subset, D 1 , inconData) !=
return subset;
else
remove(inconData,
loop
In LVI, checkIncon() is similar to InconCheck() in LVF. In addition, it
saves the inconsistent patterns of D 1 to inconData. The experiments below are
designed to demonstrate the claims made above on LVI. The incremental algorithm
(LVI) starts with a portion of data (p%) and an acceptable inconsistency
rate (fl) which is usually set to 0 if there is no prior knowledge, or the minimum
value of fl can be obtained from applying InconCheck(F; D) in LVF where F is
the set of N features. LVI splits the data D into D 0 and D 1 where D 0 is p% of
D and D 1 is the remaining. LVI uses a subset of features (subset) for D 0 found
by LVF to check subset on D 1 . The actual inconsistency found in D 0 is fi fl.
If the inconsistency rate on D 1 does not exceed stops. Otherwise, it
appends those patterns (inconData) of D 1 , which cause the additional incon-
sistency, to D 0 , and deletes inconData from D 1 . The selection process repeats
until a solution is found. If no subset is found, the whole set is returned as a
solution.
5 Empirical Study
The error probability plays the most important role in the feature selection
algorithms. Ultimately, it is always used as a meta-selection criterion [25]. That
is, regardless of different feature selection algorithms, the subset with the lowest
estimated error will always be selected for classification tasks. An error is caused
by a wrongly classified pattern. The number of errors divided by the number
of total patterns in the set gives us the error rate. Each dataset is split into
two sets (training vs. testing). Error rates are obtained for both sets. It is the
error rate of the testing set that estimates the performance of a classification
algorithm. In order to check error rates before and after features selection,
both artificial and real-world datasets are used in the study of the effectiveness
of LVF and LVI. These datasets are either commonly used in comparison or
having known relevant features. All but two (CorrAL and Parity5+5) datasets
can be obtained from the UCI Repository [18].
Artificial
ffl CorrAL The data was designed in [8]. There are six binary features,
I is irrelevant, feature C is correlated
to the class label 75% of the time. The Boolean target concept is
chose feature C as the root. This is an example of
datasets in which if a feature like C is removed, a more accurate tree will
result.
ffl Monk1, Monk2, Monk3 The datasets were taken from [27]. They
have six features. The training datasets provided were used for feature
selection. Monk1 and Monk3 only need three features to describe the
target concepts, but Monk2 requires all the six. The training data of
Monk3 contains some noise. These datasets are used to show that relevant
features should always be selected.
ffl Led17 This data is generated artificially by a program at the UCI data
mining repository. It generates 24 features among which the first 7 are
used to display a value between 0 - 9 in the seven segment display system.
The remaining 17 features are generated randomly. All the values are
binary except the class which takes a value between 0 and 9 (representable
in seven segments). The number of patterns to be generated is determined
by the user. 20000 patterns were generated for our experiments.
ffl Parity5+5 The target concept is the parity of five bits. The dataset
contains of them are uniformly random (irrelevant). The
training set contains 100 patterns randomly selected from all 1024 pat-
terns. Another independent 100 patterns are drawn to form the testing
set. Most heuristic feature selectors will fail on this sort of problems since
an individual feature does not mean anything.
Real-World
ffl LungCan The Lung Cancer data describes 3 types pathological lung cancers
found in UCI repository. This data contains only patterns and 56
features taking the values 0-3.
ffl SoybeanL In the UCI machine learning repository, we found training
and testing datasets in two separate files containing 307 and 376 patterns
respectively. It contains 35 features describing symptoms of 19 different
diseases in soybean plant.
ffl Vote This dataset includes votes from the U.S. House of Representatives
Congress-persons on the 16 key votes identified by the Congressional
Quarterly Almanac Volume XL. The dataset consists of 16 features, 300
training patterns and 135 test patterns.
Table
1: Notations: C - the number of distinct classes, N - the number of
features, S - the size of the dataset, S d - the size of the training data, S t - the
size of the testing data. Training and testing datasets are split randomly if not
specified.
Dataset C N S S d S t
LungCan 3 56
Mushroom 2 22 8125 7125 1000
ffl Mushroom The dataset has a total of 8124 patterns, of which 1000 patterns
are randomly selected for testing, the rest are used for training. The
data has 22 discrete features. Each feature can have 2 to 10 values.
ffl Krvskp This is the data for Chess End-Game - King+Rook versus King+Pawn
on a7. The Pawn on a7 means its one square away from queening. Its the
King+Rook's side (white) to move. The data contains 3196 patterns and
36 features. The class value 1 indicates white can win, which means white
can check the black pawn not to advance and vice versa. Each pattern is
a board description for this chess end-game. The first 36 features describe
board and the last one is the classification.
The major measurements of these datasets are summarized in Table 1. Since
most datasets in the first group do not have a large number of patterns, we
choose Vote and Mushroom plus ParityMix, Led17 and Krvskp to form the
second group of datasets to show the effectiveness of LVI in relation to the
size of datasets (small, medium, large). ParityMix is composed by having two
Parity5+5's side by side so that there are 20 features in total.
5.1 Effectiveness of LVF
For the artificial datasets, the evaluation of LVF is simple since the relevant
features are known. However, for the real-world datasets, it is not clear what
the relevant features are. Therefore, whether the selected features are relevant
or not can be only determined indirectly. One way is to see the effect of fea-
Table
2: Results of 100 runs of LVF on the datasets with one example of the
minimum set of features for each dataset. N - number of original features, M -
number of selected features, F - frequency.
Vote
Mushroom 22 4 (57), 5 (43) A4, A5, A12, A22
6 Allowing 5% inconsistency. If not, four features are selected: the above chosen 3 plus A1.
ture selection through a learning algorithm. Among many choices, we chose
C4.5 [24] and NBC [29] in our experiments because (1) C4.5 is a decision tree
induction algorithm that works well on most datasets as reported by many re-
searchers; and (2) it employs a heuristic to find the simplest tree structures.
(Naive Bayesian Classifier) employs the Bayes rule by assuming features
are independent of each other and is an approximation of Bayesian Classifiers -
the optimal classifier. NBC is chosen because it works in a different way from
that of C4.5. LVF is run 100 times on each training dataset. The numbers of
selected features and frequencies are reported in Table 2 under the condition
that the inconsistency criterion be satisfied. Also reported is a sample of these
selected features for each dataset which can directly be used by readers in their
experiments.
For the artificial datasets, the relevant features are always selected, albeit a
few of irrelevant ones are also chosen sometimes. For the problem like Parity5+5,
LVF correctly identifies the correct features all the time, plus one irrelevant feature
sometimes. For the real-world datasets, the number of features is reduced
at least by half to less than one fifth of the original. Table 2 shows that those
features in the last column are necessary in order to satisfy the inconsistency
criterion (the inconsistency rate is 0 except for Monk3). These features are
used by C4.5 and NBC to test if its performance improves compared to using
all features. Ten-fold cross validation is usually recommended, t-test is used
instead of Z-test in calculation of P-values since we need to take into account
the small sample effect (10 data in each sample for 10-fold cross validation).
The default settings of C4.5 are used in the experiments. For the experiments
"after" feature selection, only the features shown in the last column of Table 2
are used. Given in Tables 3 and 4 are the average accuracy rates of C4.5 before
and after applying feature selection to the datasets. The same applies to NBC:
instead of reporting the tree size, we report the table size.
Table
3: 10-fold cross validation results on Tree Size and Error Rates of NBC
before and after applying LVF to the datasets. P-val stands for P-value of t-test;
and "-" means that the pooled variances of "before" and "after" are zero.
Table
Size
Dataset Before After P-val Before After P-val
Monk2 36.0 36.0 - 37.4 37.4 1
Monk3 36.0 22.0 - 3.63 3.63 1
LungCan 722.8 95.4 0.0 56.66 63.33 .6685
Vote 98.0 62.0 - 9.9 9.9 1
Mushroom 236.0 74.0 - 0.33 1.16 .0004
In cases indicated by "-", the comparison between "before" and "after" is
obvious. Tables 3 shows that results are consistent with the known fact that
there are no bad features from the standpoint of Bayesian decision rules [26].
In all the datasets tested using NBC, only table sizes are all reduced (except
Monk2) due to feature selection; error rates are not significantly changed in
seven out of nine datasets. For the two datasets (SoybeanL and Mushroom),
the latter's error rate increases a little in absolute percentage, but SoybeanL's
error rate is much worse after feature selection. This is because the training
dataset has fewer patterns than the test dataset (recall that the division is done
by the data contributor and because features were selected based on the training
data, then both datasets were put together to run 10-fold cross validation). To
verify this conjecture, we did another experiment in which features were selected
using both data sets (training and testing). Fifteen (instead of fourteen) features
were chosen (they are A
and results of 10-fold cross validation on NBC and C4.5 are 14.4% (7.0% before)
and 9.7% (7.3%) respectively. Thus, error rates are lower with all data used for
feature selection. The only improvement of NBC's performance is on Parity5+5,
but it is not statistically significant.
Results in Table 4 suggests that the performance of C4.5 improves in general.
That is, the tree size is getting smaller and the error rate lower. For the artificial
datasets, this experiment further shows with the relevant features, C4.5 does
better than that with the full set of features. For the real-world datasets, C4.5 is
also doing better with the selected features. This indicates that LVF has selected
relevant features for these datasets. In particular, C4.5 did poorly on Parity5+5
Table
4: 10-fold cross validation results on Tree Size and Error Rates of C4.5
before and after applying LVF to the datasets. P-val stands for P-value of t-test;
and "-" means that the pooled variances of "before" and "after" are zero.
Tree Size
Dataset Before After P-val Before After P-val
Monk1 41.9 41.0 .0782 1.3 0.0 .1937
Monk2 14.3 14.3 1 35.4 35.4 1
Monk3 19.0 19.0 - 1.1 1.1 1
LungCan 18.3 16.6 .03 56.7 57.5 .2627
7.3 15.2 .0001
Vote 14.5 6.1 .0001 5.3 5.5 .8357
before feature selection. Nevertheless, C4.5 did as well on Mushroom with 22
features as with 4 features. This demonstrates that C4.5 does select relevant
features for some datasets, though not for all. The only serious deterioration
of C4.5's performance is seen in the results for SoybeanL. The reason is given
above in explaining NBC's poor performance on the dataset.
The gain from feature selection differs for NBC and C4.5. The difference
is due to the way in which features are used to induce a classifier. C4.5 is a
selective induction algorithm that selects the best feature at each test for tree
branching. NBC uses all features' conditional probabilities in determining a
pattern's class. Since NBC assumes that features are conditionally independent
given the class, the conditional probabilities of an irrelevant feature given the
class will be approximately the same, so it is not a good discriminant.
5.2 Effectiveness of LVI
For this set of experiments, we want to verify four claims: (1) LVI may not be
suitable for small datasets; (2) LVI can run faster than LVF on large datasets;
(3) LVI does not sacrifice the quality of the selected features; and (4) if no
solution can be found by LVF in the earlier runs, neither it can in later runs
(earlier runs start with less data). Five datasets in the second group are chosen
for experiments. They are (1) Vote, (2) Mushroom, (3) ParityMix, (4) Krvskp,
and (5) Led17.
The experiments are conducted as follows. For each dataset, starting with
10% of the data (D 0 ) for feature selection, we run LVI 10 times, recording the
number of features, features, and selection time in each run. Subsequently, we
do the same experiments with 20%, 30%, ., 90%, and 100% of the data. The
average time and number of features are computed for each experiment. Using
100% of data as the reference, we calculate the P-values for each sized D 0 . A low
P-value (e.g., ! 5%) suggests that the NULL hypothesis that the two averages
are the same be rejected. Refer to Figures 1 and 2 for varied P-values shaded
differently.
We summarize the findings from the experiments as follows.
ffl The effectiveness of LVI becomes more obvious when the data size is larger.
LVI performs well on all the three datasets. If the data size is small
(around a few hundred) as in Vote even the time saving for the best D 0
is not much. However, the saving is significant in the case of ParityMix
and a clear trend can be observed. This is due to the overheads required
by incremental feature selection. Since our inconsistency checking is fast
(O(P )), if P is not sufficiently large, the time saving will not be apparent,
it may even be negative if P is too small. This is why LVI is more suitable
for large sized datasets.
ffl Another issue is the number of patterns with which LVI should start for a
dataset. Having either too few or too many patterns affects the LVI's per-
formance. If too few patterns are used (D 0 is too small), LVF could select
few features that cannot pass the inconsistency check on the remaining
data (D 1 ), that is, inconData can be large. The worst case is that after
the first loop, D 0 becomes D (the whole dataset). One case of too small a
D 0 can be observed in Figure 1 for Vote when 10% of the data was used;
it took longer time than that using 20% - 50% of D. If too many patterns
are used (D 0 is large), the overheads (i.e., the time spent on those steps
inside the loop of the LVI algorithm after the LVF call) plus the time on
LVF may exceed the time of simply running LVF on D. In the cases of
Vote and Mushroom, the difference in times is not statistically significant
when 70% or more of D is used.
ffl The incremental algorithm does not have to sacrifice the quality of feature
selection. The time saving is mainly due to (1) small D 0 which is usually
a portion (say 10%) of D, and (2) by remembering inconsistent patterns,
LVI can avoid checking wrong guesses twice. The quality is measured here
in two dimensions. One is the number of features, and the other is the relevance
of the features. As shown in Figure 2, if there is some statistically
significant difference in the number of features selected between various
sized D 0 's against D, the numbers of features are lower than using 100% of
the data; otherwise, there is no statistically significant difference according
to t-test. For ParityMix, the relevant 5 features are always selected
plus 1 or 2 irrelevant/redundant ones. For the Vote and Mushroom, the
relevance test is done through a learning algorithm (C4.5 and NBC here).
If their performances do not deteriorate or even improve, we conclude that
these features are relevant. Experimental results shown in Tables 3 and
4 have verified the sets of features for the two datasets via 10-fold cross
validations. When the quality of the selected features can be warranted,
the reduction can simplify data analysis, rule induction, as well as data
collection in future.
ffl LVI can scale up. Time complexity of a feature selection algorithm can
be described along two dimensions: number of features (N ) and number
of patterns (P ). By approximating MAX TRIES of LVF with 77\ThetaN
(reduced from 2 N ), the time complexity of LVF is mainly determined by
P since N is relatively very small. The incremental version, LVI, makes
it possible to start with a fixed small number of patterns (e.g., a few
thousand for D 0 ), no matter how large the original dataset is. The experimental
results show that the time saving by so doing is statistically
significant when P is large.
ffl For data Krvskp, no feature can be removed from 10% data to 100% data
used. It indicates the other side of incremental feature selection by LVF: if
LVF cannot reduce features based on a smaller portion of data, then more
or all data cannot help reduce features either, other things being equal. It
will help if we extend the run time, for example, linking MAX TRIES to
the percentage of the total search space.
6 Discussion and Conclusion
The time performance of LVF is not reported for the first set of data because (1)
LVF completes its run fast (in a few seconds of elapsed time); (2) there is not
much to compare with among the small datasets; and (3) the time measurements
of LVI for the large datasets indicate the time performance of both LVF and
LVI. Both algorithms are simple to implement and fast to obtain results. By
predefining fl according to prior knowledge, LVI and LVF can handle noisy
data, as shown in the case of Monk3. Both can deal with multiple class values.
Another feature of LVF is related to so-called anytime algorithms [3] that are
algorithms whose quality of results improves gradually as computational time
increases. LVF prints out a possible solution whenever it is found; afterwards
LVF reports either a better subset or equally good ones. This is a really nice
feature because while it works hard to find the optimal solution, it provides
near optimal solutions. There is no need for a user to wait for results until
the end of search for optimal/suboptimal solutions as other types of search
do. The longer LVF runs, the better the solutions it produces. One salient
feature of LVI is its scaling capability without losing the quality of selected
features. The suggested modification - sampling subsets of features without
replacement of selected features and constraining subset generation by the newly
found minimum number of features should allow LVF to work faster.
In order to verify that a filter feature selector can easily be turned to a wrapper
one, LVW is built to prove the case [14]. If a favorite induction algorithm is
available, LVF can be easily transformed into LVW. The experimental results
show that LVW is much slower than LVF. This finding is consistent with the
results reported in [10].
There may be a problem with using inconsistency as a feature selection
criterion when one feature alone (such as social security number) can guarantee
that there is no inconsistency in the data. Obviously, this feature is irrelevant
for rule induction. The problem can be solved by leaving this feature out of the
feature selection process. If there is no prior knowledge, it will just take one
run of LVF to locate this kind of features 7 . Another run of LVF with the other
features will identify the right set of features.
LVF only works on discrete features since it relies on the inconsistency cal-
culation. One way is to apply a discretization algorithm (e.g., Chi2 [13]) to
discretize the continuous features first before one runs LVF. Other possibilities
are (1) to simply treat a continuous feature as a discrete one in some cases;
and (2) to apply LVF only to the discrete features when the number of features
is large. More work is needed. The search for new criteria in addition to the
inconsistency continues.
LVI and LVF can find other uses as well. As mentioned earlier that Las
Vegas algorithms may not be as fast as some domain specific heuristic methods,
LVI can still play a role as a reference in design of a domain specific heuristic
method. This is because it is not an easy task to verify a heuristic method,
especially when datasets involved are huge. LVI can be most helpful in this
case to validate feature subsets found by the heuristic method. Another feature
is that LVF may produce a number of equally good solutions for one dataset
based on the inconsistency criterion. One solution can be chosen according to its
predictive accuracy of a learning algorithm. That is, we choose a solution that
generates the best accuracy. This suggests a straightforward extension of this
work, i.e., a combined filter and wrapper model of this incremental probabilistic
algorithm. The significant advantage of LVI is that we move one step further
in handling the large sized datasets. It is a necessary addition to the present
repertoire 8 . So far, all algorithms are of automated feature selection. We have
not mentioned another important practical issue - using domain knowledge in
feature selection. Domain knowledge or expert's understanding of the data can
help tremendously in feature selection. For instance, domain knowledge can be
used to verify the finding of automated feature selection; domain knowledge can
be used to remove some obviously irrelevant or redundant features; and domain
knowledge can also help in designing heuristics. When expertise is available,
one should always start feature selection from what is known first, and apply
automated selection algorithms next.
Acknowledgments
The authors would like to thank H.Y. Lee for the suggestions on an earlier
version of this paper and H.L. Ong and A. Pang for providing the results on their
applying LVF to huge datasets at Japan - Singapore AI Center. Thanks also go
to Manoranjan Dash and Farhad Hussain for conducting some experiments using
7 Recall that one run of LVF has MAX TRIES loops.
8 Both LVI and LVF are available for research purposes upon request.
LVF and LVI, and Jian Shu for implementing NBC used in the experiments.
The suggestions by anonymous referees have also significantly helped improve
the paper.
--R
Tolerating noisy
Learning boolean concepts in the presence of many irrelevant features.
Deliberation scheduling for problem solving in time-constrained environments
Fundamentals of Algorithms.
Feature selection methods for classifications.
Pattern Recognition: A Statistical Approach.
Feature selection: Evaluation
Irrelevant feature and the subset selection problem.
The feature selection problem: Traditional methods and a new algorithm.
Wrappers for performance enhancement and oblivious decision graphs.
Toward optimal feature selection.
Selection of relevant features in machine learning.
Chi2: Feature selection and discretization of numeric attributes.
Feature selection and classification - a probabilistic wrapper approach
A probabilistic approach to feature selection - a filter solution
Concept learning through feature selection.
Principled constructive induction.
UCI repository of machine learning databases.
Feature selection using rough sets theory.
A branch and bound algorithm for feature subset selection.
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--TR
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590876 | A Neural Network Diagnosis Approach for Analog Circuits. | This paper presents a neural network system for the diagnosis of analog circuits and shows how the performance of such a system can be affected by the choice of different techniques used by its submodules. In particular we discuss the influence of feature extraction techniques such as Fourier Transforms, Wavelets and Principal Component Analysis. The system uses several different power supplies and as many neural networks in parallel. Two different algorithms that can be used to combine the candidate sets produced by each network are also presented. The system is capable of diagnosing multiple faults even if trained on single ones. | Introduction
During the past years, the authors have been involved
in several projects on analog circuit diagnosis
and quality control of electrical components.
The aim of this paper is to present the diagnostic
system developed by them and to show how
the performance of such a system can be affected
by the choice of different techniques used by its
submodules.
The system is based on neural networks , and
is used for off-line diagnosis of analog circuits affected
by catastrophic multiple faults . It may handle
linear and nonlinear circuits in transient or
steady state behavior.
1.1. Analog circuit faults
Fault diagnosis of analog circuits is a complex
problem. Classical solutions require either a huge
amount of calculation if parameter identification
methods are used, or a great number of simulations
of faulty conditions if fault dictionary methods
are used [23, 24].
The faults in analog circuits may be catastrophic
faults, that cause a large and sudden variation of
the circuit parameter values, and deviation faults,
associated to slight variations of the circuit parameter
values from their nominal values [2]. Since
statistics have shown that 80-90% of analog circuit
faults are catastrophic [19], we chose to study
faults of this kind, such as short circuits and open
circuits between two terminals of a component.
In some applications (regulation systems, nuclear
plants, etc.) a prompt fault detection is
necessary to avoid damaging the controlled process
any further. A diagnostic system capable of
detecting a fault during its occurrence performs
what is called on-line diagnosis.
In many other applications (quality control
of circuits, post-mortem diagnosis of electronic
boards, etc.) the diagnostic procedure may be
applied in an off-line fashion, in the sense that
the diagnosed device need not be operative. In
these cases there is no strict time constraint and
even computationally intensive diagnostic sys-
tems, such as those based on parameter identification
or fault dictionary methods [23, 24], qualitative
reasoning [11, 13] model-based and rule-based
expert systems [7, 28] etc., may be used. In the
case of electric circuits, off-line diagnosis offers an
additional advantage: suitable voltage supply configurations
may be chosen in order to maximize
the observability of the faults [10].
1.2. Diagnosis as pattern recognition
Classically, a pattern recognition system is composed
of three modules [12]. A transducer acquires
data on a physical device and passes them
to a feature extractor whose purpose is to reduce
the data by computing certain features (or prop-
erties). These features will be used by a classifier
to make a final decision on the state of the device.
A circuit diagnostic system is a particular pattern
recognition system, in which the physical device
is an analog circuit, and the state that must
be recognized is the set of faulty components. In
particular, in the diagnostic system we have de-
veloped, the classifier is a neural network.
This type of diagnostic system offers some advantage
over other classical diagnostic methods.
Rule-based systems. These diagnostic systems
use compiled sets of rules to associate a symptom
to its cause. On the contrary, a neural
network automatically derives the symptom-
cause correspondence during the training, and
does not require an explicit formalization. It
is well known that this formalization is the
bottleneck of rule-based system technology.
Note that there is a small price to pay for this.
A rule-based system has a symbolic-heuristic
approach to diagnosis and is generally able to
justify its deduction from the rules used to
compute the diagnosis. A neural network, on
the contrary, has a numerical-algorithmic approach
and the knowledge is implicitly memorized
in the weights of its synapses. Thus, to
justify its deduction a neural network requires
additional rule extraction techniques [9].
Model-based systems. These diagnostic systems
usually require the complete knowledge
of the circuit scheme and a model of its be-
havior. Using neural networks it is possible
to avoid the problems connected with the calculation
of circuit parameters and in general
to the modeling.
Fault dictionary method. This method can be
used to identify only those faults whose signature
has been previously computed and added
to the dictionary. Neural networks on the
contrary - as reported in several works -
may be able to recognize fault configurations
not explicitly included in the training set. In
[14, 21, 31] neural networks trained to recognize
single faults are successfully used to diagnose
multiple faults. In [36] neural networks
accurately classify previously unseen fault signatures
belonging to a deviation fault class
known by a few samples.
There have been several works where neural
networks have been compared with other pattern
classifiers in diagnosis applications. In the domain
of single fault diagnosis of circuits, a comparison
with Gaussian maximum likelihood and K-nearest
neighbors is presented in [26] where neural net-
works, once trained, are shown to significantly
reduce the time of the diagnosis, although they
do not offer improvements in the diagnostic accu-
racy. The same result was independently reported
in [36], comparing neural networks and K-nearest
neighbors classifiers in the diagnosis of deviation
faults.
In the diagnostic system we present, the transducer
is an acquisition board that measures the
voltage values in a given set of test points . Other
choices are possible as we will discuss in Section 2.
As an example, Spence et al. [34, 35] have used
nonintrusive circuit measurements (such as infrared
images or magnetic field images); however,
nonintrusive measurements have been proved to
be very ineffective, in the sense that they can only
be used to recognize a limited number of faults.
Kirkland and Dean [22] obtained good results using
current measurements; however, current mea-
Neural Diagnosis for Analog Circuits 3
surements are often impractical, since they would
require the opening of the circuit, and this is
clearly not possible on printed circuits.
We have investigated several feature extraction
techniques and have studied their influence on
the performance of the diagnostic system. In
this paper we compare Fourier Transforms [14],
Wavelets [8], Principal Components Analysis [15],
and Sampling. In [16] Mean and Root-Mean-
Square Values of the test point voltages were used
as features, but due to the large amount of lost
information they could only recognize a limited
number of faults.
We also observed that the performance of the
diagnostic system heavily depends on the choice
of power supplies. In particular, it is often the
case that a given supply can only lead to the detection
of a particular subset of all possible faults.
A suitable set of different supplies may be used to
build a diagnostic system that combines different
diagnoses (one for each supply) dramatically improving
the performance of the diagnostic system.
In the paper we also present two algorithms that
can be used to combine these different diagnoses.
The paper is structured as follows. In Section 2
we recall relevant work on the use of neural networks
for circuit diagnosis. In Section 3 we describe
the architecture of the proposed diagnostic
system and discuss the important issue of simulation
versus acquisition. In Section 4 we discuss
the choice of power supplies and how this affect
the diagnosis. In Section 5 we describe different
techniques that can be used by the features extraction
module to compactly represent the behavior
of the circuit. In Section 6 we present the structure
of neural network classifier and show how
it is trained. In Section 7 we present two algorithms
that can be used to combine the diagnosis
computed by different networks. In Section 8 we
present statistics of the system performance when
diagnosing two different circuits: a board part of
a DC motor drive, and an oscillator.
2. Relevant work
Other approaches to the use of neural networks
for circuit diagnosis have recently been published.
Keagle et al. [21] discuss how networks trained
to recognize single faults may be used to detect
multiple faults. Tests are performed on a digital
circuit consisting of nine logical gates affected by
stuck-at 1 or stuck-at 0. The paper also presents
results on the performance of the diagnostic system
as a function of the network architecture.
Meador et al. [26] compare feedforward neural
network performance with other classifiers: gaussian
maximum likelihood and K-nearest neigh-
bors. In each experiment a single parameter deviation
fault on an operational amplifier circuit is
considered. The classifiers must separate the input
patterns corresponding to the correct behavior
and to the faulty one.
Parten et al. [29] propose using neural networks
as part of a model-based expert system for diagnosing
lumped parameter devices. The purpose of
the net would be that of solving the equations ruling
the behavior of the diagnosed device, modeled
as a set of interconnected components.
Thompson et al. [38] consider the problem of
diagnosing an IC board with approximately
components, both analog and digital. They use
a backpropagation neural network with a modular
structure, i.e., each part of the net recognizes
a particular fault.
Totton and Limb [39] use neural networks to diagnose
a circuit board part of a digital telephone
exchange. They observed from historical data that
failures on four types of components account for
more than 85% of all faults. This led them to
construct a network whose four outputs signal the
presence of a faulty component of a given type,
i.e., the network does not pinpoint the faulty component
but simply detects what type of component
is faulty.
Spence et al. [34] use a different approach to
the single fault diagnosis of printed circuit boards
(PCB). The difference between the malfunctioning
infrared image and the image of a correctly
functioning PCB is interpreted by an artificial
neural network to diagnose some types of
faulty components. In a subsequent work Spence
[35] presents a different test method based on the
interpretation of the magnetic field close to the
PCB. Although these methods can only recognize
a limited number of faults, they have the advantage
of requiring nonintrusive measurements.
Rutkowski [31] was the first to suggest the use of
neural networks for the diagnosis of multiple faults
on analog DC circuits. In this introductory work,
4 Fanni, Giua, Marchesi and Montisci
the main focus is on testing the capability of the
network to generalize from single to double fault
diagnosis. In the application example presented
in the paper, only a limited number of faults are
considered.
Bernieri et al. [3] use a neural net-work
for on-line analysis of dynamic discrete-time
systems whose input/output behavior is
ruled by equations of the form: y
f(y . The network
at the k\Gammath instant receives as inputs the value
of y and is trained to
estimate the value of given parameters that rule
the behavior of the system. Parameter deviations
over a given threshold are symptoms of faults.
Kirkland and Dean [22] have reported using input
current measurements as circuit images.
Gu et al. [17] combine neural networks and expert
systems into a single diagnostic system. To
each component is associated a neural network
trained to recognize the component's fault. The
expert system acts as a coordinator between the
different neural networks, supplying suitable inputs
to the networks and deriving a diagnosis from
the analysis of the networks' output.
Spina and Upadhyaya [36] have considered the
problem of diagnosing deviation faults in linear
circuits. A white noise source is used to automatically
generate test patterns. Fault signatures
are generated associating to a single component a
value equal to the nominal value plus 50%. The
network can correctly classify previously unseen
patterns corresponding to deviation faults of different
magnitude.
All these works highlight the prominence in a
neural diagnostic system of the aspects related to
feature extraction and circuit supplies, thus leading
us to a systematic exploration of these issues.
The present paper summarizes the results that its
authors have obtained throughout a long period of
time and that have only partially been presented
in the papers referenced in the rest of this section.
In [16] is discussed how networks trained to recognize
single faults on analog circuits in dynamic
behavior may be used to detect multiple faults.
The neural network identifies the faulty components
from the mean values of the voltage measurements
in a given set of test points. In general
it was observed that the network is able to diagnose
multiple faults on two and three components,
although less sharply than in the single fault case,
due to the presence of false alarms. The set of multiple
faults was chosen among those single faults
well recognized by the network.
In [14] Fourier transforms are used as features
of the circuit image, and multiple neural networks
were used in parallel by the diagnostic system.
This improved the performance of the diagnostic
system with respect to the previous one.
In [8] Wavelet transforms are used as features.
Wavelets proved to be a good data compression
technique when the circuit is studied during a
transient. In fact, one can increase the number
of wavelets only in particular time intervals depending
on the degree of approximation required.
In [15] Principal Component Analysis is used in
the feature extraction phase. The main advantage
of such a technique lies in the fact that it gives a
simple automatic procedure to compress the data.
3. Architecture of the diagnostic system
The architecture of the proposed diagnostic system
is shown in Figure 1.
Testing procedure (horizontal path)
Given a circuit to diagnose, we apply a suitable
power supply and acquire the voltage signals
at a given set of test points, constructing the circuit
image. We extract significant features, as discussed
in Section 5, from the image and use them
as inputs to a neural network that has been previously
trained to recognize single faults on that
circuit. The neural network will generate the candidate
set , i.e., the set of components recognized
as faulty.
In Section 4, we will show that to increase the
number of detectable faults it is necessary to use
different supplies. Consequently, we will have several
neural networks, one for each supply consid-
ered. Repeating the procedure described above
for all supplies, we obtain several candidate sets.
These sets will be combined to derive a single diagnosis
using suitable algorithms, as described in
Section 7.
Training procedure (vertical path)
The diagnostic system is built training the neural
networks that will be used in the testing procedure
Neural Diagnosis for Analog Circuits 5
Each neural network is trained using a set
of patterns corresponding to all possible single-
faults, as detailed in Section 5 and 6. The training
patterns are constructed from the faulty circuit
images using the same feature extraction technique
that will be used in the testing.
It may be possible to obtain each faulty circuit
image using an acquisition board. One has to pro-
duce, one by one, all single faults on the circuit
and then has to acquire the corresponding faulty
circuit image. This procedure is not practical in
many cases. Thus we resorted to PSpice simulation
of the circuit behaviour in faulty conditions.
On the contrary, when constructing the circuit
image in fault-free condition, both real acquisitions
and PSpice simulations are possible. As we
will later discuss, several real acquisitions will be
used to estimate the magnitude of the measurement
noise.
Our results showed that if the circuit PSpice
model is accurate enough, there is no difference
between a network trained with "simulated" patterns
and a network trained with "acquired" pat-
terns. In fact, the distance between a simulated
and an acquired pattern has the same order of
magnitude of the distance (due to measurement
noise and component parameter tolerance) between
two patterns acquired during the same fault
condition.
4. Power supplies
One of the main problems in the diagnosis of circuits
is the presence of undistinguishable and undetectable
faults.
Consider two (or more) components, say k and
k 0 in parallel as in Figure 2.(a). Clearly the behavior
of the circuit is the same whenever component
k or component k 0 is short circuited. The
same problem appears when we consider open circuit
faults of series components as in Figure 2.(b).
Faults of this kind are called undistinguishable, in
the sense that they produce the same voltage configuration
at the available test points.
A similar problem may arise when a fault is un-
detectable. In this case, the measured behavior of
the fault-free circuit is the same as the measured
behavior of the faulty circuit.
The presence of undistinguishable and undetectable
faults may have different causes.
ffl Topology of the circuit , as in the examples discussed
above.
ffl Limited number of test points , that may not
allow detection of an abnormal behavior of the
circuit.
ffl Components whose measured behavior is the
same when faulty or correctly functioning. We
recall some of the possible causes.
Operating point of the component. Consider
the diode in Figure 2.(c). It is reverse
biased and thus for all practical purposes
its behavior is the same when the diode is
functioning well or when it is affected by
an open circuit fault.
Frequency content of the supplies. Some
frequency components may not be suitable
for highlighting a given fault. In DC
steady state, for instance, capacitors behave
as open circuits and inductors behave
as short circuits, as shown in Figure 2.(d)
and
Figure
2.(e), respectively.
Protection subcircuits. The behavior of
the protection components is not supposed
to affect the overall behavior unless other
faults are present.
There is little we can do to resolve the ambiguity
due to the topology of the circuit or due to the
choice of test points. However, we may try to
resolve the ambiguity due to the behavior of the
circuit by an appropriate choice of power supplies.
As an example, a different choice of supply, such
a high frequency square voltage, force the diode in
Figure
2.(c) to alternatively switch from reverse to
forward bias, and the capacitor and inductance in
Figure
2.(d),(e) to work in AC.
This problem has also been discussed by Dague
et al. [10]. These authors add an external stimulation
in suitable points so as to disturb the circuit
operating conditions.
We will train different networks to process the
data collected for each different supply configura-
tion. Thus, our diagnostic system is composed of
several neural networks, each one specialized in
detecting a given set of faults. When the system
is used to diagnose a circuit, each network will
produce a set of candidates, i.e., of possibly faulty
6 Fanni, Giua, Marchesi and Montisci
combination of
candidate sets
power
supplies
circuit
under test
feature
extractors
neural nets
PSpice model
of circuit
under test
acquisition
board diagnosis
feature
extractor
single faults
simulations
training
Fig. 1. The proposed diagnostic system architecture.
components. The overall diagnosis can be computed
by means of different algorithms, given in
Section 7.
5. Feature extraction techniques
We assume that the information on the circuit be-
havior, i.e., the circuit image, is given by the voltage
measurements in a set of available test points.
These points are usually given by the circuit board
manufacturer and cannot be arbitrarily chosen.
(a) (b)
(c) (d) (e)
Fig. 2. Examples of undistinguishable and undetectable
faults.
Since the voltage signal at each test point is a
function of time, we need to extract significant features
to compactly represent the circuit behavior.
Extensive experimental studies showed the influence
of the particular feature chosen. The feature
used in [16] was the mean value (MV). The diagnostic
system performances improved when root-
mean-square values (RMSV) or a combination of
MV and RMSV were used. When MV or RMSV
are used, all the information on the dynamic behavior
is lost. Thus other feature extraction techniques
are required. We discuss here four different
techniques: Fourier Transforms , Wavelets , Principal
Components Analysis , and Sampling.
During the training , the goal of the feature extraction
procedure is to construct an (s\Thetar) matrix
X . Each row of this matrix represents the circuit
behavior during one of the s acquisitions and each
column represents the value of a particular fea-
ture. Each row of X is use as a training pattern
input for the neural network, hence there will be
r nodes in the network input layer, and s training
patterns, as discussed in Section 6.
During the testing of a circuit, the same feature
extraction procedure is used to derive the inputs
that will be given to the neural network.
In this section, we mainly discuss the feature
extraction module as used during the training.
Consider a circuit with n components and a
given set of m test points.
Neural Diagnosis for Analog Circuits 7
The voltage of all test points is measured on a
real circuit by an acquisition board during p acquisitions
in the absence of faults. These measurements
will be used to estimate the magnitude of
measurement noise.
On the contrary, the faulty circuit images, i.e.,
the voltage of all test points in presence of a fault,
are constructed via PSpice simulation. We consider
two single faults for each bipolar component:
open circuit and short circuit. We also considered
faults on components with more than two
terminals. As an example, in the circuit shown
in
Figure
4, there are trimmers and operational
amplifiers. We considered two possible faults on a
trimmer (cursor stuck up and cursor stuck down)
and just one single fault on an operational amplifier
(it was made inoperative by feeding with
exceedingly high voltage).
In general, let s 0 be the number of the single
faults taken into account; then one needs to collect
images.
5.1. Fourier transforms
A simple technique for compacting the information
given by the circuit image without losing the
dynamics of the system is given by the Fourier
analysis that converts the signals into frequency
components [37].
We compute the Fast Fourier Transform
of the sampled voltage signal measured at each
test point. If we have t voltage samples, we obtain
- for each test point -
components and we take the amplitude of each
component.
We are now ready to construct the input pattern
matrix . The matrix has initially s rows, one
for each acquisition, and m \Delta q columns, one for
each feature, i.e., frequency component computed
at each test point. Thus the input pattern matrix
takes the form X
qg. The first p rows of X 0 are associated to the
fault-free acquisitions.
Matrix X 0 is still unusable because of its high di-
mensionality. Domain dependent knowledge may
be used to further reduce its number of columns
[37].
The data reduction algorithm we propose, requires
two phases.
1. Remove features that give no information. We
compute for each column j the difference \Delta j
between its maximum and minimum element.
We also compute the difference ffi j between the
maximum and minimum element in the first p
rows of the column: this is an index of the numerical
uncertainty associated to the value of
feature j during the p different fault-free ac-
quisitions. Fix a threshold ' ? 1. If \Delta j 'ffi j
then the variation of the feature j has the
same order of magnitude of the numerical uncertainty
and column j will be removed. We
used a value of
2. Scale the inputs. To improve separability between
patterns we scale the columns of the
input pattern matrix in the interval [\Gamma1; 1].
3. Select a subset of significant features. The
idea is to keep only those columns that are
necessary to distinguish between different pat-
terns. Fix a threshold oe ! 1. If j x 0
oe then the variation of feature j is not large
enough to distinguish pattern i from pattern
We used a value of
We proceed as follows.
begin
let the initial set of significant features be S :=
for i := 2; s (* compare each row i of X 0
with all previous ones *)
begin
ffl let S i;i 0
oeg be the set
of those features, i.e., columns, that may
be used to distinguish between patterns i
and
is not empty then
add to S the most significant feature, i.e.,
feature j such that j x 0
We thus obtain a new matrix X of order (s \Theta r)
with r m \Delta q.
The data reduction algorithm we use with FFT
falls into the category of unsupervised feature extraction
methods [4], i.e., methods that do not use
information on the target data. Note, however,
that the data reduction is performed opportunis-
tically, by projecting the features onto a subspace
8 Fanni, Giua, Marchesi and Montisci
that still contains all information required to separate
the input patterns.
5.2. Wavelets
The origins of Wavelets date back to 1909, when
Haar proposed them as a viable solution to function
decomposition problems. In fact Fourier se-
ries, as stated in its original formulation, show a
non-uniform convergence even for particular continuous
functions. Wavelets approach is more
suitable than Fourier one, especially when signals
are non-stationary. Both "time-frequency" and
"time-scale" wavelets are suited to signal analysis
ranging from "quasi-stationary" to fractal structure
type. Mathematicians speak of "atomic de-
composition" of signals, where wavelets are the
elementary constituents. The various wavelets
are obtained from a single wavelet by scaling and
shifting operations.
There are several definition of wavelets. One
possible is the following [27]: a wavelet is a function
y(x) in L 2 (IR) such that 2 j=2 y(2 is an
orthonormal basis for L 2 (IR). The most frequently
used wavelets are the Grossmann-Morlet wavelets,
that are also similar to Daubechies wavelets and
to Gabor-Malvar wavelets. The last algorithm is
of time-frequency type, while the former is a time-scale
algorithm.
In the wavelet theory [30, 25] any signal of finite
energy can be represented as a linear combination
of wavelets whose coefficients represent the features
we want to extract, and indicate how close
the signal is to a particular basis function.
Discrete wavelet transform (DWT) is a relatively
recent method whose biggest potential has
been found to be signal compression. The two major
advantages of the wavelet transformation are
that it can zoom in time discontinuity and that
it is possible to construct an orthonormal basis,
localized in time and frequency.
An important issue of wavelet analysis is the
choice of the proper type of wavelet and of
the methodology to use, i.e., time-scale, time-frequency
or a combination of the two.
In our diagnostic system, Haar wavelets are chosen
to realize data compression of circuit-image
information. Decomposition proposed by Haar results
as follows:
R 1f(t) h i (t) dt , and s n (t) is the
n\Gammath order summation which uniformly converges
to the signal f(t), and Haar wavelets are defined
as:
Here, the scaling factor is a power of 2, and
k defines the time shift with respect to the basic
wavelet H, that is the unit square window func-
tion. The various wavelets (n ? are obtained
starting from the basic wavelet
ing, scaling and shifting operations. It is important
to note that the time range has to be limited
in the [0; 1] interval. This is not limiting because
real signals always have a finite time length and
this will become the new time unit. It is also possible
to realize a suitable time windowing of the
signal.
Thus, it is possible to project the time signal
onto a set of mutually orthonormal wavelets. The
number of the wavelets may be arbitrary, depending
on the required approximation in reconstruction
or, as in the present case, on the amount of
information to extract from the signal.
Because a circuit image results from a set of
digital acquisition, signals are not continuous in
time, but discrete due to sampling. Hence, a discrete
transform has to be used and particular care
is required to compute the inner products.
The construction of the input matrix X using
wavelets follows the same procedure presented in
Section 5.1 for Fourier transforms and will not be
repeated here.
5.3. Principal components analysis
Principal Component Analysis (PCA) is another
unsupervised feature extraction method. Compression
by means of PCA is accomplished by projecting
each data vector along the directions of the
individual orthonormal eigenvectors of the covariance
matrix of data. As the first few eigenvalues
of the covariance matrix contain most of the signal
energy, the dimensionality of the data can be
Neural Diagnosis for Analog Circuits 9
greatly reduced without losing much information
on the input data.
It may happen that the information associated
to the discarded PC subspace is important for the
subsequent classification phase [4] and in this case
PCA is not suitable. However, PCA is a potentially
useful method because it works in many ap-
plications. In [1] PCA is used for terrain classifi-
cation, and it is shown that it can lead to a significant
improvement in the classifier performance.
In [6] there is a comparison between Gabor filters
and PCA as feature extraction methodologies
applied to SAR images segmentation with neural
networks.
Let s be the number of the circuit behaviors
taken into account, and t be the number of samples
for each test point voltage. Each circuit image
is represented by m \Delta t values. We have a (s \Theta m \Delta t)
data matrix X 0 which could be used as input for
the neural network.
As previously mentioned, preprocessing is necessary
to extract from these data the salient fea-
tures. We would like to reduce the number of
columns of this matrix from m \Delta t to r m \Delta t,
with acceptable loss of information. Using PCA
[20] this compression is accomplished projecting
the s circuit images along the directions of the
principal eigenvectors of the covariance matrix of
Given the data matrix (X th column
represents a circuit im-
age, the covariance matrix of these data is the
s
The eigenvectors of this matrix form an orthonormal
basis, and any vector ~x i can be represented
with respect to this basis by means of a coefficient
vector with m \Delta t elements.
To reduce the data dimension, it is possible to
consider only those eigenvectors associated to the
dominant eigenvalues of C. Fix a threshold c 2
be the ordered set
of eigenvalues of C, i.e., j j+1 . We say that
there are r dominant eigenvalues if
are the eigenvectors associated to the
dominant eigenvalues, we may use as compressed
representation of a vector ~x i the coefficient vector:
We used a value of
Thus, the data matrix X 0 is reduced to a (s \Theta
r) matrix X . The same compression technique
will be used on subsequent circuit images acquired
during the test phase.
5.4. Sampling
Given the circuit image (i.e., the sampled voltage
signals at all test points) one may compact the
data retaining just a limited number q of the t
samples.
Experimental results [16] showed that this is not
a viable technique if the circuit is in AC steady-state
or if there are many test points. In fact, this
leads to a neural network with too many nodes
in the input layer, i.e., too many features. This
may reduce the performance of the classification
system and leads to a higher computational cost
of the training.
However, this technique was effective when
studying short transients on circuits with a limited
number of test points. The choice of the samples
to retain must be opportunistic, and depends on
the signal variation pattern.
6. Neural model
As proposed in most of the literature discussed in
Section 2, we use a three level neural network with
sigmoid activation functions and backpropagation
learning with generalized delta rule.
6.1. Fault coding
The network has r input nodes, i.e., as many as
there are columns in the input pattern matrix X
derived with any of the different feature extraction
procedures previously described. The output
nodes of the network are as many as the number
of circuit components n.
We construct the s input-output patterns that
will be used to train the neural network for the
diagnosis of the circuit as follows. Each pattern is
given by a pair (~x i ; ~y i ). The vector ~x i is the th
row of matrix X while the associated vector ~y i is
defined as follows:
component k is not faulty
during the th acquisition;
component k is faulty
during the th acquisition.
This general scheme must be altered to take into
account undistinguishable faults.
Topologically undistinguishable faults are easy
to deal with. From an inspection of the circuit
a list of all sets of parallel components is made.
Then, a single short circuit fault acquisition for
each set C i of parallel components is considered.
There will be a single training pattern (~x
such a fault. The vector ~y i is such that y i
for all k 2 C i , while all other components have a 0
value. A dual procedure takes care of sets of series
components.
Two faults i and i 0 are behaviorally undistinguishable
if
where oe is the threshold introduced in section 5.1.
A fault i is behaviorally undetectable if the condition
k1 oe is satisfied for all input
vectors ~x 0 obtained in the faulty free condition.
We combine the patterns of behaviorally undistinguishable
faults (as we did for topologically undistinguishable
faults) and remove from the training
set the patterns associated to undetectable faults.
The fault coding here described is different from
the one presented in [16], that defined the vector
~y i as follows:
component k is short circuited
during the th acquisition;
0:5 if component k is not faulty
during the th acquisition;
component k is open circuited
during the th acquisition.
The new coding gives sharper identification of the
faulty component and is more robust when diagnosing
multiple faults because the values of interest
(0 and 1) are obtained by "pushing" the sigmoid
function toward saturation. Note also that
there is a difference with the coding in [31] where
each output node is associated to a catastrophic
fault and not to a component.
Once the net has been trained, it may be used to
perform the diagnosis of the circuit. The net must
be given the features extracted from the measured
test point voltages as input vector ~x. The net
will produce an output vector ~y; a value of y(k)
close to 1 will pinpoint a fault of component k; a
value close to 0 will denote that the component is
correctly functioning.
Although the net has been trained with the results
of single fault acquisitions, it is potentially
able to diagnose multiple faults. In this case, two
or more elements of ~y will be close to 1.
6.2. Network structure and training
The basic architecture we used consists of a three
layers backpropagation network. Since the input
patterns have been preprocessed to eliminate
undistinguishable faults, and thus they are sepa-
rable, we are sure that eventually there will be a
network capable of correctly learning all patterns.
We use early-stopping [4] to avoid overfitting.
This consists in measuring, during the training,
the error with respect to an independent set of
patterns, called validation set , and in stopping the
training when this error reaches a minimum.
Caruana [5] has shown that if early-stopping
is used the number of nodes in the hidden layer
may vary without appreciably affecting the performance
of a neural network, provided it is sufficiently
large. The results of our simulations, not
reported in this paper, seem to confirm this general
rule.
The validation set used for the stopping is independent
from the training set. We construct it by
performing a new set of PSpice simulations (one
for each fault) randomly changing the parameter
values of the components within their tolerance
range and by adding to the voltage signals of the
test points a noise whose magnitude is equivalent
to the measurement noise observed during the p
fault-free acquisitions.
7. Combining different diagnosis
In the diagnosis of circuits, we have underlined the
importance of using more than one power supply.
In fact, it is often the case that a given supply can
only lead to the detection of a particular subset
Neural Diagnosis for Analog Circuits 11
of all possible faults. The use of different supplies
leads to the use of several neural networks N i , each
of which produces its own candidate set A i . The
final diagnosis must be computed combining these
sets of candidates.
The combination of neural networks is a problem
that has been discussed in the literature and
is reviewed in [32]. In particular, since we use neural
networks that are all trained on the same task,
our approach falls into the ensemble (or commit-
It is clear that the "union" of two sets of candidates
magnifies the influence of false alarms, while
the "intersection" can be used to filter false alarms
at the risk of removing some faulty components
from the diagnosis. Keeping this in mind, we propose
two different ensemble algorithms.
Let us first give the following definitions. For
each candidate k let v(k) be the number of votes
it receives, i.e., the number of nets that consider k
malfunctioning, and let
v(k). We consider
all non-empty intersections of v candidate
sets; assume there are ff of such intersections and
denote them R u , with We also define
u the index of the intersection R u with the
smallest cardinality (should there be more than
one such intersection we randomly pick one).
Algorithm 1
The first algorithm considers as faulty all those
candidates that have received the highest number
of votes. The corresponding diagnosis is:
ff
R u
Algorithm 2
The second algorithm considers as faulty all those
candidates that have received the highest number
of votes and that belong to the intersection with
the smallest cardinality. By considering only the
smallest intersection we hope to filter out some
false alarms. The corresponding diagnosis is:
An example is shown in Figure 3. Here D
Note that these algorithms do not give different
weight to the candidate sets of each network,
but simply perform boolean operations on these
sets. We are currently investigating the possibility
of associating to each candidate set a different
weight, depending on how the network has learned
to recognize the single fault on each component
that belongs to the candidate set.
8. Experimental results
We discuss the results obtained by the different
diagnostic strategies presented in this paper. Two
circuits are studied: a DC motor drive board, and
an astable multivibrator.
Training As discussed above, we use early stop-
ping, hence we need both a training and a validation
set of patterns.
The training patterns corresponding to each single
fault condition are constructed using a PSpice
model of the circuit. This choice gives patterns
corresponding to a circuit where the component
parameters have nominal values and the voltage
signals in each test point are noise free.
The validation set is constructed by performing
a new set of PSpice simulations where parameter
tolerance and measurement noise are introduced.
Testing During the test phase, we consider a real
circuit and the different faults are implemented by
manually shortcircuiting or opening each component
terminals. The circuit measurements are col-
A 3
Fig. 3. Example of diagnosis combination.
lected through a National Instrument Corporation
acquisition board.
Thus the test patterns are determined independently
of the training patterns. Furthermore, the
test patterns are affected by measurement noise
and by the error due to the parameter tolerance
of the circuit components.
When diagnosing a circuit, we observe the net-work
output corresponding to the input pattern
derived from the measurements. Let us recall that
the network output layer has as many nodes as
there are components. During the training phase
we have coded a fault on component k assigning
a value 1 to the corresponding output node, while
a value 0 was assigned to the output node of a
fault-free component.
In general, during the test phase the value of
each output node may take any value between 0
and 1. A value close to 0 (1) of an output node
will be interpreted as the absence (presence) of a
fault on the corresponding component. Threshold
values need to be set to discriminate between these
two cases.
Let vmax be the maximum value of all output
nodes. If vmax ! 0:2 we consider the circuit
as fault-free and the candidate set will be
empty. If vmax 0:2 we consider the circuit as
faulty, and the candidate set will contains all components
whose corresponding output node has a
value greater than 0:5v max .
8.1. DC motor drive board
We present the results obtained diagnosing the
circuit in Figure 4, part of a DC motor drive.
The same circuit has also been diagnosed in
[8, 13, 14, 15, 16]. In the figure, the
test points are marked by numbers within circles,
while the are labeled by numbers
in square brackets.
Training There are 70 single faults to consider
on this circuit. In fact, the circuit is composed of
36 components but only one fault is considered for
each of the two operational amplifiers. Thus, the
overall training set should consist of 76 training
patterns - the additional six being obtained by
acquisitions of the circuit behavior in absence of
fault.
The following sets contain topologically undistinguishable
faults:
28s, 29sg,
17og, 30/31o =f30o, 31og. Here 10s represents
a short circuit fault on component 10, 16o represents
an open circuit fault on component 16, etc.
Thus, the training set is reduced to 62+6 patterns
by combining the conflicting patterns as discussed
in Section 6.1.
We have used three different voltage supplies
and thus three different networks.
1. The first network N 1 is trained with patterns
acquired when the circuit has close to nominal
voltage supplies: V 1
\Gamma12 (V).
Fourier The number of significant frequency
components is This gives rise to
columns in the input matrix X 0 ,
that are reduced to r = 14 in the matrix
X .
The set of behaviorally undetectable faults
for this net is: f3o, 6o, 11o,12s, 15o, 23o,
24o, 25s, 27o, 29o, 31s, 32s, 33o, 34og.
The sets of behaviorally undistinguishable
faults are: f4o, 6sg, f21o, 27/28/29s, 30sg,
f34/35s, 36og, f 35o, 36sg.
Wavelets The number of significant wavelets
is 8. This gives rise to m \Delta
columns in the input matrix X 0 , that are
reduced to r = 15 in the matrix X .
The set of behaviorally undetectable faults
for this net is: f3o, 6o, 11o, 12s, 15o, 23o,
24o, 25s, 27o, 29o, 31s, 32s, 33o, 34og.
The sets of behaviorally undistinguishable
faults are: f21o, 27/28/29s, 30sg, f34/35s,
36og, f35o, 36sg.
PCA Assuming a threshold 0:999, the
number of dominant eigenvalues (i.e., the
number of columns of the X matrix) is
25.
The set of behaviorally undetectable faults
for this net is: f3o, 6o, 11o, 12s, 15o, 23o,
24o, 25s, 27o, 29o, 31s, 32s, 33o, 34og.
Neural Diagnosis for Analog Circuits 13
Fig. 4. DC motor drive board.
The sets of behaviorally undistinguishable
faults are: f4o, 6sg, f21o, 27/28/29s, 30sg,
f34/35s, 36og, f35o, 36sg.
2. The second network N 2 is trained with
patterns acquired when the circuit has
far from nominal periodic voltage supplies:
are zero-mean square
waves with 160 Hz frequency, 4 V peak-to-
Fourier The number of significant frequency
components is 8. This gives rise to
columns in the input matrix X 0 ,
that are reduced to r = 15 in the matrix
X .
The set of behaviorally undetectable faults
for this net is: f10o, 11o, 12s, 21o, 22o,
23o, 24o, 25s, 27/28/29s, 28o, 29o, 30s,
31s, 32sg.
The sets of behaviorally undistinguishable
faults are: f1o, 3sg, f4o, 6sg, f34/35s,
36og.
Wavelets The number of significant wavelets
is 9. This gives rise to m \Delta
columns in the input matrix X 0 , that are
reduced to in the matrix X .
The set of behaviorally undetectable faults
for this net is: f8 up , 10o, 11o, 12s, 14o, 17s,
21o, 22o, 23o, 24o, 25s, 27/28/29s, 28o,
29o, 30s, 31s, 32sg.
The sets of behaviorally undistinguishable
faults are: f4o, 6sg, f14/15s, 16sg, f18o,
19og, f34/35s, 36og.
PCA Assuming a threshold 0:999, the
number of dominant eigenvalues is
The set of behaviorally undetectable faults
for this net is: f10o, 11o, 12s, 21o, 22o,
23o, 24o, 25s, 27/28/29s, 28o, 29o, 30s,
31s, 32s, 34/35s, 36og.
The set of behaviorally undistinguishable
faults is: f1o, 3sg.
3. The third network N 3 is trained with patterns
acquired when the circuit has step voltage
supplies:
(V).
Fourier The number of significant frequency
components is 8. This gives rise to
columns in the input matrix X 0 ,
14 Fanni, Giua, Marchesi and Montisci
that are reduced to r = 15 in the matrix
X .
The set of behaviorally undetectable faults
for this net is: f10o, 12s, 14o, 21o, 23o, 25s,
27o, 28o, 29o, 31s, 32s, 34og.
The sets of behaviorally undistinguishable
faults are: f1o, 3sg, f4o, 6sg, f22o,
27/28/29s, 30sg.
Wavelets The number of significant wavelets
is This gives rise to m \Delta
columns in the input matrix X 0 , that are
reduced to r = 17 in the matrix X .
The set of behaviorally undetectable faults
for this net is:f10o, 12s, 21o, 23o, 25s, 27o,
28o, 29o, 31s, 32s, 34og.
The sets of behaviorally undistinguishable
faults are: f1o, 3sg, f4o, 6sg, f22o,
27/28/29s, 30sg.
PCA Assuming a threshold 0:999, the
number of dominant eigenvalues is
The set of behaviorally undetectable faults
for this net is: f10o, 12s, 21o, 23o, 25s, 27o,
28o, 29o, 31s, 32sg.
The sets of behaviorally undistinguishable
faults are: f1o, 3sg, f4o, 6sg, f22o,
27/28/29s, 30sg.
Testing We are now ready to study the performance
of the neural diagnostic systems previously
constructed.
In the initial phase, we test the systems on a
fault-free circuit. We observed that when diagnosing
a real circuit in absence of faults, all networks
correctly identify this behavior, in the sense
that all output nodes have a value less than the
assigned threshold of 0:2 and thus the candidate
set is always empty.
In a second phase, we consider faulty circuits.
Table
1 compares the performance (in percent) of
the different systems. The first columns of the table
shows the diagnosis of N 1 , N 2 , and N 3 and
the diagnosis obtained combining the candidate
sets of the three nets with Algorithm 1 and Algorithm
2, using Fourier, Wavelets, and PCA, re-
spectively. The last two columns show the results
obtained combining the candidate sets of the nine
nets (three for each feature extraction technique)
with Algorithm 1 and Algorithm 2.
We consider a fault correctly diagnosed if the
candidate set of the net contains a subset of
the components associated to this fault, taking
into account topologically undistinguishable fault
classes. Let us consider some examples in the circuit
of Figure 4. The fault 16o belongs to the
topologically indistinguishable fault class 16/17o;
we say that it is correctly identified if the candidate
set is either f16g or f17g or f16, 17g. The
fault 16s is correctly identified if the candidate set
is f16g.
Single faults
The first row block of Table 1 shows the diagnosis
of the 62 possible single faults. There are
three different classes of diagnosis.
Class A 1 Faults correctly diagnosed.
Undistinguishable faults: these are the
faults that we have classified as behaviorally
undistinguishable during the training. As an
example, in net N 1 with Fourier, we have identified
9 undistinguishable faults, i.e., 14% of
the total 62 faults.
Class C 1 Undetected faults: these are the faults
that we have classified as behaviorally undetectable
during the training.
Double faults
The second row block of Table 1 shows the performance
of the different systems when diagnosing
double faults. Each double fault consists in the
simultaneous presence of two faulty components.
Note that not all possible pairs of single faults
constitute a double fault: e.g., a bipolar component
cannot be simultaneously open- and short-
circuited. We have considered a sample of 168
different double faults randomly chosen from the
total population. This sample was large enough
to satisfy the 2 test for the six different classes
of diagnosis.
These are the classes considered.
Class A 2 Both faults correctly diagnosed.
Only one fault correctly diagnosed.
Class C 2 At least one fault correctly diagnosed
with one or two false alarms.
Class D 2 At least one fault correctly diagnosed
with more than two false alarms.
Only false alarms.
Neural Diagnosis for Analog Circuits 15
Triple faults
The third row block of Table 1 shows the performance
of the different systems when diagnosing
triple faults. We have considered a sample of 181
different faults randomly chosen out of the total
population.
These are the classes considered.
Class A 3 All three faults correctly diagnosed.
Only one or two faults correctly diagnosed
Class C 3 At least one fault correctly diagnosed
with one or two false alarms.
Class D 3 At least one fault correctly diagnosed
with more than two false alarms.
Class F 3 Only false alarms.
Discussion
In the case of multiple faults, we consider correct
all diagnoses in class A and in class B. In fact,
starting from class B we may use an incremental
repair procedure, substituting the faulty components
one by one. Diagnosis in class C may also
be useful.
From the table it can be seen that the use of
several networks improves the system performance
provided that a good procedure is used to combine
the results of the networks. In particular, Algorithm
1 and Algorithm 2 give the same results
when diagnosing: (a) single faults; (b) multiple
faults using a system composed of many nets in
parallel. When diagnosis multiple faults, if the
system is composed by a small number of neural
nets Algorithm 2 performs better because it exalts
the filtering effect of the intersection operator, reducing
the number of diagnoses in class A but
increasing the total number of diagnoses in class
A+B.
All three feature extraction techniques give
comparable results. PCA performs better than
the other two when diagnosing single and double
faults, but seems to be less robust when diagnosing
three simultaneous faults. Unlike Fourier and
Wavelets, PCA requires less data preprocessing in
the feature extraction phase, as discussed in Section
5.
8.2. Astable multivibrator
Dague et al. in [10] remarked that oscillators are
difficult to diagnose because most faults cause
the same type of symptoms. This is exactly the
case in which a proper choice of the power supplies
can improve the diagnosability of the circuit.
They proposed using an external "stimulation"
and showed the results obtained using their diagnostic
expert system on the astable multivibrator
shown in Figure 5. In this section we present results
obtained using our diagnostic system on the
same circuit.
Training We chose test-points in the
nodes labeled 1 and 2 in the figure. The number of
components is single faults have been
considered. In fact, we consider 6 faults for each
transistor: short circuit between base and emitter
(QBEs), base and collector (QBCs),collector and
emitter (QCEs); open circuit on the base (QBo),
collector (QCo), and emitter (QEo). The circuit
does not contain topologically undistinguishable
faults.
To be able to compare the results of our diagnostic
system with the system developed by
Dague, we used the same voltage supply proposed
in [10]. It consists of the superposition of the
nominal supply PS (a continuous voltage signal of
+5V) and of an external stimulation EP (a voltage
pulse of 10V amplitude, applied in
and lasting 1s). Since we consider a unique sup-
ply, we use a single neural network for each feature
extraction.
We used a PSpice model of the oscillator to collect
the training and validation patterns for all
faulty conditions, as previously described.
The circuit has been studied in transient behaviour
and the voltage signals in each test point
have been collected in the interval 0:9 \Xi 40s with
a sampling interval of t 0:1s. This gives rise
to a circuit image before feature extraction composed
of for each test point.
We have used all different feature extraction
techniques described in Section 5.
Fourier The number of significant frequency
components is This gives rise
to m \Delta columns in the input matrix
Table
1. Diagnosis of the circuit in Figure 4 (in percent).
Fourier Wavelets PCA F+W+P
Single faults (62 fault cases)
Double faults (168 fault cases)
Triple faults (181 fault cases)
A3
28 19 21 62 74 23 21 23
are reduced to in the matrix
X .
There are no behaviorally undetectable faults.
The sets of behaviorally undistinguishable
faults are: fRC2s, Q2CEsg, fRB2s, Q2BEsg.
Wavelets The number of significant wavelets is
This gives rise to m \Delta columns
in the input matrix X 0 , that are reduced to
in the matrix X .
There are no behaviorally undetectable faults.
The sets of behaviorally undistinguishable
PS
RI
Fig. 5. Astable multivibrator. PS is the nominal power
generates the stimulation voltage pulse.
faults are: fC1o, RC2s, RB2s, Q2CEs,
Q2BEsg.
PCA Assuming a threshold 0:999, the number
of dominant eigenvalues (i.e., the number
of columns of the X matrix) is
There are no behaviorally undetectable faults.
The sets of behaviorally undistinguishable
faults are: fRC2s, Q2CEsg, fRB2s, Q2BEsg.
Sampling The Sampling feature extraction retains
samples (out of a total of
for each test point spaced with an hyperbolic
law so as to have more samples during the
initial phase of the transient and only a few
as the circuit reaches the steady state. For
Table
2. Diagnosis of the circuit in Figure 5 (in percent).
Fourier Wavelets PCA Sampling
Single faults (24 fault cases)
Double faults (140 fault cases)
28 8 23 41
Neural Diagnosis for Analog Circuits 17
Table
3. A comparison between the diagnosis of the circuit in Figure 5 done with Dague's Expert System and with Neural
Network with Sampling.
Defect Expert System Neural Network
(Dague et al.) with Sampling
RC1s RC1, Q1, RC1
fC2g \Theta fPS, CX, C1, EP, Q2, RB1, RB2, RC2, RIg
RB1s (*) Q1 RB1
double candidates C1
Q1CEs Q1, C2, Q1
fRC1g \Theta fCX, C1, EP, Q2, RB2, RC2, RIg
Q1BEs Q1, C1, RB1 Q1
fC2g \Theta fPS, EP, Q2, RB2, RC2, RIg
RC1o RC1, Q1, C2 RC1
RB1o RB1, Q1, C1 RB1
C1o C1, Q2, RC2 fC1, RC2g
Q1Eo Q1 Q1
Q1Bo Q1 Q1
Q1Co Q1, C1, RB1, C2, RC1 Q1
(*) Note that a short-circuit on RB1 induces destruction of Q1.
Then the ae-th retained sample is the ae -th
sample, as shown in Figure 6.
There are no behaviorally undetectable faults.
The sets of behaviorally undistinguishable
faults are: fC1o, RC2og, fRC2s, Q2CEsg,
fRB2s, Q2BEsg.
Testing In a first phase, we test the systems on a
fault-free circuit. We observed that when diagnos-
tr
r
Fig. 6. Sampling of the total measurements.
ing a real circuit in absence of faults, all networks
correctly identify this behavior.
In a second phase, we consider faulty circuits.
Table
2 compares the performances (in percent)
of the different neural diagnostic systems. The
classes of diagnosis are the same defined in the
previous example.
The first row block of Table 2 shows the diagnosis
of the 24 single faults considered.
The second row block of Table 2 shows the diagnosis
of the 140 double faults considered. Note
that in this case the total number of possible double
faults is 240.
We can see that in this particular case Sampling
appears to be the most effective feature extraction
technique (and it is also the easiest to implement).
As remarked before, however, it is a viable solution
only because we have a small number of test
points (two in this example) and the circuit has a
short transient.
Comparison with Dague's Expert System
In
Table
3 we compare the results obtained using
our diagnostic system with sampling feature
extraction (Neural Network with Sampling) with
the results obtained by Dague's expert system.
Note that although only 12 faults are considered
by Dague, our diagnostic system has been trained
to recognize all 24 single faults.
We observe that the neural network has been
able to correctly classify 11 faults in class A and
only 1 fault (C1o) in class B. The expert system,
on the contrary, can very rarely correctly identify
the faulty component (only 3 diagnosis in class A
including the diagnosis of RB1s).
The results show that the neural network performs
much better than the expert system. This
is a consequence of its ability to exploit the information
on each single-fault behavior of that particular
circuit and to generalize. This information
is not taken into account by the expert system,
that reasons on more abstract principles.
9. Conclusions
We have shown how a neural network, trained to
recognize catastrophic single faults, may be used
to diagnose multiple faults on analog circuits.
In general we observe that the network is almost
always able to learn and recall the single fault
patterns presented during the training. Multiple
faults on two and three components may also be
diagnosed, although less sharply than in the single
fault case, due to the presence of false alarms. In
most cases, however, the network is able to detect
at least one of the malfunctioning components.
Thus one may use an incremental repair proce-
dure, substituting the faulty components one by
one.
We consider several different power supplies in
order to detect those faults that do not modify
the circuit behavior under nominal supplies. We
use several neural networks "in parallel", one for
each different supply configuration. Each network
is specialized in detecting a given set of faults.
Thus, it is not necessary to force a network to
recognize a fault that is more easily detected by
another one.
The use of different networks, leads to the problem
of composing different sets of candidates into
a single diagnosis. We showed that a suitable
choice of the composition algorithm may dramatically
improve the system performance, especially
when diagnosing multiple faults.
We compared the results obtained by our system
when using different feature extraction tech-
niques. In fact, the performance of the diagnostic
system is noticeably affected by the choice of
features that we consider as representative of the
device behavior.
Although we have only presented two simple examples
of diagnosis, extensive experiments convinced
us that this approach is fairly general and
that it gives better results than other diagnostic
systems, such as expert systems, whenever it can
be applied.
--R
Ghaloum S.
Salama A.
A Neural Network Approach for Identification and Fault Diagnosis of Dynamic Systems.
Neural Networks for Pattern Recogni- tion
Extra capacity rarely hurts generalization if yo u use early stopping.
SAR Image Segmentation Using Textural Information and Neural Classifiers.
A Spectrum of Logical Definitions of Model-Based Diagnosis
Wavelet Analysis for Diagnostic Problems.
Extracting Comprehensible Models from Trained Neural Networks.
Luciani P.
Diagnosing Multiple Faults.
Hart P.
Qualitative Dynamic Diagnosis of Circuits.
A Multiple Neural Network Diagnostic System for Analog Circuits Based on Fourier Transforms.
Diagnosis of Electrical Circuits Using Neural Networks and Principal Components Analysis.
Neural Networks for Multiple Fault Diagnosis in Analog Circuits.
Yang Y.
Introduction to the Theory of Neural Computation.
A DC Approach for Analogue Fault Dictionary Determination.
Fundamentals of Digital Image Processing.
Murphy J.
Dean J.
Selected Papers on Analog Fault Diagno- sis
Testing and Diagnosis of Analog Circuits and Systems.
A theory for multiresolution signal de- composition: the wavelet representation
Stategy for Diagnosis.
Saeks R.
IEEE SP Magazine
A Neural Network Approach to Fault Location in Non Linear DC Circuits.
On combining Artificial Neural Nets.
Artificial Neural Systems.
Burris D.
Printed Circuit Board Diagnosis Using Artificial Neural Networks and Circuit Magnetic Fields.
Linear Circuit Fault Diagnosis Using Neuromorphic Analyzers.
A Guide to Neural Computing Appli- cations
Sutton J.
Limb P.
--TR
--CTR
Francesca Cau , Alessandra Fanni , Augusto Montisci , Pietro Testoni , Mariangela Usai, A signal-processing tool for non-destructive testing of inaccessible pipes, Engineering Applications of Artificial Intelligence, v.19 n.7, p.753-760, October, 2006
Barbara Cannas , Francesca Cau , Alessandra Fanni , Augusto Montisci , Pietro Testoni , Mariangela Usai, Neural NDT by means of reflected longitudinal and torsional waves modes in long and inaccessible pipes, Proceedings of the 5th WSEAS/IASME International Conference on Systems Theory and Scientific Computation, p.94-102, September 15-17, 2005, Malta | multiple fault diagnosis;analog circuits;neural networks |
590909 | Knowledge Extraction from Transducer Neural Networks. | Previously neural networks have shown interesting performance results for tasks such as classification, but they still suffer from an insufficient focus on the structure of the knowledge represented therein. In this paper, we analyze various knowledge extraction techniques in detail and we develop new transducer extraction techniques for the interpretation of recurrent neural network learning. First, we provide an overview of different possibilities to express structured knowledge using neural networks. Then, we analyze a type of recurrent network rigorously, applying a broad range of different techniques. We argue that analysis techniques, such as weight analysis using Hinton diagrams, hierarchical cluster analysis, and principal component analysis may be useful for providing certain views on the underlying knowledge. However, we demonstrate that these techniques are too static and too low-level for interpreting recurrent network classifications. The contribution of this paper is a particularly broad analysis of knowledge extraction techniques. Furthermore, we propose dynamic learning analysis and transducer extraction as two new dynamic interpretation techniques. Dynamic learning analysis provides a better understanding of how the network learns, while transducer extraction provides a better understanding of what the network represents. | Introduction
There has been a lot of interest lately in knowledge
structures and their representation in articial
neural networks [Holldobler, 1990, Kurfe, 1991,
Sperduti et al., 1995, Wermter, 1995, Hallam,
1995, Medsker, 1995, Sun, 1995, Wermter et al.,
1996, Elman et al., 1996, Craven, 1996, Wermter,
1999]. Articial neural networks (or connectionist
networks) have already demonstrated interesting
learning results for various classication tasks.
However, it continues to be very di-cult to understand
the underlying representations within the
connectionist networks which lead to this perfor-
mance. A better understanding of the connectionist
representations learned is not only important
for improving the credibility of a computational
technique, but also for improving the net-work
performance and the integration possibilities
with symbolic representations.
Several attempts have been made to interpret
connectionist networks, focusing on feedforward
networks in particular [Andrews and Diederich,
1996, Abe et al., 1993, Shavlik, 1994]. For in-
stance, visualizations of internal activations or
weight strengths can be used to get an impression
of the internal knowledge [Hinton, 1986, Gorman
and Sejnowski, 1988]. Some eort has also
been made to reduce the network size in order to
simplify the knowledge expressed therein by elimi-
28 Wermter
nating very small weights. Furthermore, groups of
similar weights can be replaced with their average
strength [Shavlik, 1994]. In addition, techniques
such as hierarchical cluster analysis have been
used to interpret connectionist networks. Never-
theless, often the interpretation of the dynamics
of the learning process and the underlying knowledge
has been neglected, especially in the case of
dynamic recurrent neural networks.
The interpretation of recurrent networks is more
di-cult than that of non-recurrent feedforward
networks, since the previous context in recurrent
networks has an important dynamic in
uence
within these networks. The internal states in recurrent
networks do not only depend on the input
but also on the internal state of the local memory
based on previous inputs [Elman, 1995, Giles
and Omlin, 1993, Omlin and Giles, 1996]. For
this reason, to date the focus has been primarily
on smaller recurrent networks and articially generated
data. For instance, an interesting current
approach interprets the training of a SRN network
that has two input, two output and two internal
elements in learning the sequence a n b n [Wiles and
Elman, 1996]. It has been discovered that the net-work
behaved like a spiral which moved to and
from a x point. Whereas this seems to be a plausible
interpretation of the behavior of recurrent
networks trained for the learning of the sequences
a n b n , dierent interpretations are required when
we move to dierent tasks and data sets closer to
real-world scenarios.
In the past, we have developed a large \real-
world" system for spoken language analysis which
makes extensive use of SRN networks [Wermter
and Weber, 1997, Wermter and Meurer, 1997].
The spoken input is recognized by a speech recognizer
and analyzed at the syntactic, semantic
and dialog levels based on an incremental analy-
sis, parallel syntactic and semantic interpretation,
and robust processing of errors. To date, however
it is not yet possible to focus on the interpretation
of the learning process and the interpretation
of the connectionist knowledge. In this
paper, we are primarily concerned with a detailed
interpretation of the learning behavior as well as a
symbolic interpretation of the learned knowledge
after training. In order to carry out such a detailed
analysis we will concentrate on a syntactic
transformation task as a representative task for
our large-scale speech/language system. The task
for the recurrent network is to process sentences
and associate their syntactic classes at the phrasal
level, e.g. noun phrase, prepositional phrase etc.
Using this task, we analyze a recurrent neural
network using many dierent techniques. We have
structured the paper as follows. First, we introduce
our representative syntactic transformation
task. Then, we dene and illustrate a) dynamic
learning analysis, b) weight analysis, c) hierarchical
activation analysis, d) component activation
analysis, and e) transducer extraction. We rigorously
compare these techniques on the same net-work
and the same data set and argue that these
dierent techniques provide mutually complementary
interpretations. The contribution of this paper
is a particularly broad and concrete analysis
of the knowledge extraction process which has not
been done before. Furthermore, we propose dynamic
learning analysis and transducer extraction
as two new interpretation techniques. Dynamic
learning analysis provides a better understanding
of how the network learns while transducer extraction
provides a better understanding of what the
network represents.
2. Extracting structured knowledge using
syntactic analysis task
In order to examine a number of dierent techniques
for extracting structured knowledge from
connectionist networks in a rigorous manner, we
will focus on a particular task. In our spoken
language environment [Wermter and Lochel,
1996, Wermter and Weber, 1997, Wermter and
Meurer, 1997], we have trained many variations
of SRN networks [Elman, 1991] with many sentences
using various corpora of several thousand
words each.
Based on a corpus of sentences from the domain
of scheduling appointments (2355 words), table 1
summarizes the accuracy of label assignment on
the unknown test set. The related experiments
and results have been reported elsewhere in detail
[Wermter and Lochel, 1996, Wermter and Weber,
1997, Wermter and Meurer, 1997, Wermter, 1998].
Here we just want to illustrate the real-world net-work
performance in table 1. The focus, how-
Knowledge Extraction from Transducer Neural Networks 29
ever, is on an analysis of the process of extracting
explicit knowledge from implicitly learned knowl-
edge. In this paper, we concentrate on syntactic
phrasal assignment (marked by *) in table 1.
Table
1. Performance of some networks on the test set of
the appointment scheduling corpus
Task Accuracy on test set
Basic syntactic disambiguation 89%
Basic semantic disambiguation 86%
Syntactic phrasal assignment* 84%
Semantic phrasal assignment 83%
Dialog act assignment 79%
Word repair detection 94%
Phrase repair detection 98%
To demonstrate this process of knowledge ex-
traction, we will here use 15 of these sentences
(containing 76 words) from the domain of appointment
scheduling. For illustration purposes, we
concentrate on the learning of a syntactic phrasal
assignment task where a sequence of basic categories
of words is associated with a sequence of abstract
syntactic categories. The actually occurring
syntactic basic categories are noun (n), verb (v),
adverb (a), adjective (j), preposition (r), determiner
(d) and pronoun (u). The abstract phrasal
categories are noun group (ng), verb group (vg),
and prepositional group (pg). The task of the recurrent
network is to learn to assign phrasal categories
on the basis of basic syntactic categories
in order to support a robust
at understanding
of spontaneously spoken language. Below, we
show some example utterances from the corpus,
together with the syntactic categories at the basic
and the phrasal level.
1. I (u ! ng) thought (v ! vg) in (r ! pg) the
2. That (u ! ng) is (v ! vg) the (d ! ng)
Thursday (n ! ng) after (r ! pg) Easter (n
Based on these seven basic syntactic and three
phrasal syntactic categories, we use an SRN net-work
with seven input units, three internal units
and three output units (the networks in the actual
system contain more categories and have been
trained with several thousand words, but for illustration
purposes we restrict ourselves to this
smaller network). The learning rate was 0.05 and
momentum 0.9. The weight updates were performed
incrementally after each training pattern.
Each training pattern consisted of the basic syntactic
category at the input layer and the abstract
phrasal category at the output layer.
Figure
1 shows a simplied example of such a
recurrent network for the task of syntactic phrase
assignment.
Output
Input
Context-
layer
AAAA
AAAA
AAAA
Noun
group
Prepositional
group
Verb
group
Pronoun Noun
Adjective
Verb
Adverb
Preposition
Determiner
Fig. 1. Recurrent network for knowledge extraction for
syntactic phrase assignment
The activation of an output element O j (t) at
time t in SRN networks is computed on the basis
of the weighted activation H i (t) of all incoming
connections limited by the logistic function f .
O
The activation of an element on the internal
layer H l (t) is computed in a similar manner. Here
the activation of the input layer I k (t) at time
t is used as the activation of the internal layer
at the previous time step t 1.
Wermter
3. Dynamic learning analysis: knowledge
structuring during lazy learning
In the past, most work on knowledge structures
and connectionist networks has focused on static
connectionist network representations. However,
important insights can be gained by examining
how certain knowledge structures emerge and develop
time before a certain task is learned
completely.
Frequently, the interpretation of the learning
behavior is just demonstrated by means of the
learning curve of the overall error reduction over
time. However, the learning curve is just the rst
step in a more detailed analysis and can only provide
preliminary hints about the performance of
a network over the training time. Figure 2 shows
the learning curve with the overall sum squared
error over time.
patterns 50000 100000 150000 2000000.20.6Error
Fig. 2. Learning curve for syntactic phrasal assignment
The learning curve shows that the speed of
learning diers substantially over time. Further-
more, we can see dierent stages during the learning
process. In the beginning, learning proceeds
fast, but later learning is slower and it takes longer
to make signicant improvements. For instance,
between 70000 and 140000 it seems that learning
is about to nish before there is a nal signicant
improvement.
We will now examine how the network reaches
its performance. We start the analysis directly after
the random initialization of the weights. This
is the state before learning starts. We want to
give an overview of the overall performance for all
input patterns at dierent time steps. To this ef-
fect, we show the error for each of the 76 patterns
of the demonstration set at dierent time steps.
Figure
3 shows the individual error for each of the
76 patterns before training.Individual patterns
Fig. 3. Performance for individual patterns before learn-
ing
Based on the random initialization, all patterns
show a relatively high error. At this point, it
is to be expected that the values of each output
element dier from the desired value 0 or
1 by 0:5. Therefore the expected error for an
individual pattern for three output elements is
expected error
value is conrmed in this gure.
As shown in gure 2, the error decreases quickly
at the start of the training. The state after 100
patterns of the training set is shown in gure 4.
First, we can observe that after 100 training pat-
terns, the error for some of the 76 patterns shown
could be reduced signicantly. Other patterns still
show a high error. Obviously, the network has
started to learn patterns selectively.
Knowledge Extraction from Transducer Neural Networks 31
Individual patterns
patterns
other patterns
Fig. 4. Performance for individual patterns after 100
training patterns
A more detailed analysis revealed that the patterns
with a lower error are exactly those patterns
which belong to the noun group NG. After only
100 patterns, the network has recognized that the
global error can be minimized signicantly by focusing
on the NG patterns, since these patterns
occur more frequently than, for instance, prepositional
groups or verb groups. Therefore, at rst
the network has learned a constant mapping of all
patterns to the noun group, since this reduces the
overall error most at this stage. This explains why
certain patterns in gure 4 still exhibit a high error
and others a low error. The patterns with a
low error are exactly the patterns which have been
classied correctly as noun groups.
Figure
5 shows the detailed performance after
patterns. After the network has learned a constant
mapping to NG, we can observe that the
performance for the NG patterns has improved
even further. However, we also observe that V G
patterns have been learned. A more detailed analysis
of the output preferences reveals that at this
stage, in addition to all NG, all V G patterns have
also been learned correctly. This is also demonstrated
in gure 5. All the remaining error patterns
at this stage are those patterns which should
belong to a prepositional group PG but which are
still categorized as noun groups NG. All NG patterns
and all V G patterns are classied correctly.
After the network has learned the most frequent
NG patterns, the second most frequent V G patterns
are learned. Thus, one could state that the
network pursues a conservative lazy learning strategy
and learns frequently occurring and simple
regularities rst.
Individual patterns
patterns
patterns
patterns
Fig. 5. Performance for individual patterns after 600
training patterns
Afterwards, the network attempts to improve
all patterns, especially the remaining patterns for
prepositional groups PG. The occurring nouns,
pronouns, determiners, and adjectives can either
be part of PG patterns or NG patterns. In order
to resolve this potential for ambiguity, previous
context must be used to learn the correct class
assignment. Again, we have an example of the
conservative lazy learning strategy of the network,
Individual patterns
3 exceptions of PG patterns
patterns
NG patterns, VG patterns
Fig. 6. Performance for individual patterns after 3000
training patterns
Wermter
since at rst the network has learned patterns
which do not need previous context knowledge for
the category assignment. Only after the simple
non-context-dependent category assignments have
been learned, are those patterns learned which require
the context of previous pattern assignments.
The state of the network after 3000 patterns is
shown in gure 6. All patterns are classied correctly
with the exception of three. Comparing g-
ures 5 and 6, the remaining error for the individual
patterns could be reduced signicantly. For the
learning of the PG patterns, it was necessary for
the network to integrate the local preceding con-
text. After 150000 patterns all regularities have
been learned as shown in gure 7. In comparison
with gure 6 we point out the smaller scaling of
the vertical axis. At this stage all patterns have
been learned, even though there are dierences between
the error rates of individual patterns. In order
to reach this 100% correctness on the training
set, it may be necessary to give up a reasonably
good state at a certain stage in order to reach an
even better stage later. This is also re
ected in
the global learning curve in gure 2.
Individual patterns
all NG patterns,VG patterns,
PG patterns correct
Fig. 7. Performance for individual patterns after 150000
training patterns
In general, the network pursues a conservative
lazy learning strategy. First, simple and frequently
occurring generalizations of one category
are learned. Only when the network cannot minimize
its error signicantly any more, are other
frequently occurring categories integrated. Fur-
thermore, only when all those patterns have been
learned that do not require previous local context,
are those patterns learned that require context for
the correct category assignment of otherwise ambiguous
input. Finally, any remaining exceptions
are learned. During this conservative learning process
it may be possible that the overall error increases
brie
y in order to reach a better overall
state later.
4. Weight analysis for knowledge extrac-
tion
Visualizations of internal weight strengths can be
used to get an impression of the internal knowl-
edge. In our experiments, the training set was
learned correctly after 150000 patterns and this is
where we start our analysis. We start with such
a weight analysis since weights provide the lowest
level of interpretation of a connectionist represen-
tation. Figure 8 shows the weights of the network
for three dierent time steps. It is illustrated how
the weights change over time during learning.
In this gure the identiers of the source connectionist
elements are shown horizontally and the
identiers of the goal connectionist elements are
shown vertically. We start with the horizontal
axis. From left to right, we can see the weights
from the threshold element (S), from the input
connectionist elements for the syntactic basic categories
(n, j, v, a, r, u, d), from the three internal
elements and from the three context
elements c3). In the vertical axis from
top to bottom, we see the weights to the three
internal elements and to the output
elements representing the abstract syntactic categories
(VG, NG, PG).
Knowledge Extraction from Transducer Neural Networks 33
after 100 patterns
after 600 patterns
after 150000 patterns
Fig. 8. Weight analysis at the beginning of training (100 patterns), during training (600 patterns) and after training
(150000 patterns)
White boxes represent positive weights, black
boxes negative weights. The size of the boxes
corresponds to the size of the weights. The copy
connections from the internal layer to the context
layer are not changed. Therefore, they are not
shown since they are always equal to 1.
We start with the analysis of the rst third of
gure 8. After random initialization, this rst
third shows all weights of the network after 100
patterns. At this point, all NG patterns can be
classied correctly, but no other patterns have
been learned yet. The network has learned a constant
output in order to reduce the overall error
as much as possible. We can see in gure 8 why
the network produced this constant NG class.
34 Wermter
We can see that the weights from the input elements
of the syntactic basic categories (n, j, v, a,
u, d) to the internal elements are
relatively small and similar. The same holds for
the weights from the context elements (c1, c2, c3)
to the internal elements. This is due to the random
initialization at the beginning of the train-
ing. The weights from the internal elements to
the output elements of the abstract syntactic categories
are negative for V G and
PG; those from the internal elements to NG are
close to 0. This is the reason why the network produces
constantly the NG category at this stage.
Now we focus on the state of the network after
presenting 600 patterns, also shown in gure 8.
At this point, all NG and all V G patterns are
assigned correctly. This is also re
ected in the
weights. We observe positive weights from n,
and d to the internal elements and positive weights
from the internal elements to NG. However, we
see negative weights from v to the internal elements
and from the internal elements to V G. The
PG patterns are not categorized correctly at this
point. One reason for this is that the PG patterns
depend signicantly on the previous con-
text. However, at this point, the network has
just learned the obvious preferences and is only
just starting to change the weights of the context
layer.
The network state after 150000 patterns is
shown at the bottom of the gure. In the internal
layer, a distributed representation has developed.
Therefore, a direct interpretation is not easily pos-
sible. However, it is observed that the rst internal
element is primarily important for PG detec-
tion, the second internal element plays an important
role in V G assignment and the third internal
element is important for NG. Nevertheless, this
is a distributed rather than a local representation
and there is additional in
uence from other ele-
ments. Furthermore, the weights of the context
layer (from c to h) have changed. This is necessary
in order to learn the PG group assignment.
Generally speaking, we can explain certain phenomena
using this type of weight analysis at the
lowest interpretation level of a network. However,
it is di-cult to extract explicit knowledge and a
deeper understanding of the behavior of the net-work
directly from the weights. Reasons for this
di-culty include (1) the static representation of
the weights which does not show the dynamics of a
recurrent network, (2) the distribution of weights
and activation, and (3) the number of weights,
especially in the case of larger networks. There-
fore, some eort could be made to reduce the size
of the network by eliminating very small weights.
Furthermore, groups of similar weights could be
replaced with their average strength. Nonethe-
less, weight analysis is still too detailed for larger
networks.
5. Component activation analysis for
knowledge extraction
Weight analysis focuses on the weights and provides
a very low-level analysis. One way to address
this problem is to move towards activation analysis
where the activations of internal elements are
analyzed. Since internal elements receive activation
from a number of weighted connections, the
activation of an internal element integrates several
weighted connections and provides a higher
abstraction level of analysis.
In order to demonstrate how such an analysis is
performed, we will use the same SRN network we
have introduced in the previous section and store
all vector representations of the internal layer for
each pattern. These vector representations constitute
the input to a cluster algorithm which provides
a hierarchical representation in the form of
a dendrogram. Vectors with similar vector representations
will end up in the same cluster.
Figure
9 shows the initial part of patterns as
they were clustered according to their internal
activations. It can be clearly observed that the
internal representations re
ect the classication
according to the three classes
PG. That is, based on the weights, the internal
layer has learned representations which particularly
support this classication. A single word can
appear in dierent contexts and can lead to different
internal representations. For instance, the
word \the" is shown with two dierent represen-
tations. One representation is its use as part of
the NG class, and the other as part of the PG
class. Therefore, we nd both representations at
dierent positions within the dendrogram.
Knowledge Extraction from Transducer Neural Networks 35
ALL$U$NG
I$U$NG
I$U$NG
US$U$NG
WE$U$NG
WEDNESDAY$N$NG
MORNING$N$NG
IN$R$PG
WEEK$N$PG
IS$V$VG
COME$V$VG
MUST$V$VG
IS$V$VG
Fig. 9. Hierarchical cluster analysis of internal classication representations (for visibility purposes, only a portion is shown
6. Principal component analysis for
knowledge extraction
Another kind of analysis which can be used for
interpreting the internal representations of clas-
sications is principal component analysis. Figure
shows the result of this analysis for our
current task. All vectors from the internal layer
and the corresponding identiers provide the input
for the principal component analysis. Vectors
which dier substantially from each other are
depicted in the gure with a large distance. It
can also be observed that the internal representations
re
ect the preference mappings learned for
the three category classes. NG, V G, and PG patterns
are distributed across dierent areas. Thus,
the classication of the internal representations
can be clearly seen. This shows that the network
has actually learned the classication task well.
After learning has been completed, the internal
representation characterizes the preference map-
ping. According to cluster analysis or principal
component analysis, similar internal vector representations
are responsible for the representation
of similar preference assignments to equal cate-
gories. However, the interpretation of the weights
by means of Hinton diagrams and of the activations
via cluster analysis and principal component
analysis only provides a limited form of structuring
to the extracted knowledge.
36 Wermter
I$U$NG
IN$R$PG
THURSDAY$N$NG*
MORNING$N$NG
WE$U$NG
UNS$U$NG
ON$A$PG
THE$D$NG*
IS$V$VG
WEDNESDAY$N$NG
I$U$NG*
COME$V$VG MUST$V$VG
I$U$NG*
THAT$U$NG*
IS$V$VG
ALL$U$NG*
TUESDAY
COULD$V$VG
US$U$NG
LET$V$VG*
US$U$NG
DAS$U$NG*
IS$V$VG
MAKE$V$VG*
TILL$R$PG*
WE$U$NG
IN$R$PG
MARCH$N$PG
OTHER$J$NG
THAT$U$NG*
IS$V$VG
Fig. 10. Principal component analysis of internal classication representations
7. Transducer extraction
Words and sequences of words can be represented
as syntactic, semantic, and pragmatic category
preferences. Then they can be input, for instance,
to SRN networks. Each input representing a sequence
of category preferences is associated with
a sequence of corresponding output preferences.
This simple description of sequence analysis is
similar to the function of synchronous sequential
machines [Booth, 1967, Kohavi, 1970, Shields,
1987], although preferences and learning are not
yet considered in such machines. Therefore, we
shall focus on extensions of synchronous sequential
machines for representing sequential knowl-
edge, especially synchronous Moore machines. We
start with the basic denition of a synchronous sequential
machine which is also called a transducer:
Denition of a Synchronous Sequential Ma-
chine, Transducer
A synchronous sequential machine M is a tuple
1. I , O nite, nonempty sets of input and output
2. S nonempty set of states
3. The function f s : I S ! S is state transition
function
4. The function f o is an output function. If the
output depends on the state and the input,
the machine is a so-called Mealy machine with
the output function f O. If the
output only depends on the state the machine,
the later is a so-called Moore machine with
the output function f These synchronous
sequential machines are sometimes
called transducers.
A sequential machine assigns an output and a
new state to an input and an old state. This can
be done for a whole sequence of inputs and states
in discrete time. The set S is not necessarily -
nite [Booth, 1967], although this is assumed in
the case of nite machines. Whereas automata
Knowledge Extraction from Transducer Neural Networks 37
or acceptors of languages decide whether a certain
input belongs to the corresponding grammar,
these sequential machines are transducers which
change their internal states dynamically, depending
on the inputs and the previous states, while
also providing an output for each input.
Mealy and Moore machines are slightly dier-
ent from each other. Moore machines determine
the state rst and afterwards this state is used to
provide the output. In contrast, the output in a
Mealy machine depends also directly on the current
input. However, it can be shown that for each
Moore machine there is an equivalent Mealy machine
and vice versa [Booth, 1967, Hopcroft and
Ullman, 1979].
In our case, we concentrate on Moore machines
since the output in certain neural networks is
based on the internal state. This holds, for in-
stance, for feedforward networks or SRN networks.
Whereas sometimes [Sun, 1995] a sequential machine
has been used to model a single element of
a neural network, we want to use a sequential machine
as a description for a whole network. This
is also motivated by the fact that real neuron systems
can be seen as physical entities which perform
state transitions [Churchland and Sejnowski,
1992].
Now we can specify language knowledge by describing
Moore machines and their state transition
function f s and output function f o . We can also
integrate f s and f o to a function f : IS ! OS.
Then f corresponds for instance to the transformation
within a SRN network. The specication
of a Moore machine could be performed by using
state tables. A potential entry for the task of
assigning syntactic phrasal categories to syntactic
basic categories could be:
If verb and current state = prep. group
then new state = verbal group and output = verbal
group
It may not be possible to assign a direct interpretation
to a state. For this reason, simple
identiers may be used:
If verb and current state = 4
then new state = 5 and output = verbal group
It is possible to dene state transition tables
which assign each combination of input and current
state an output and a new state. In this
way, a symbolic synchronous sequential machine
is specied. If clear regularities are known beforehand
and the number is limited, such tables can
be composed manually. However, the number of
input and state combinations quickly gets so large
that automatic procedures become necessary.
The above-mentioned state transition tables are
discrete symbolic. Therefore, they do not support
gradual representations. For instance, the input
or the state could be ambiguous and dierent
gradual preferences could exist for dierent inter-
pretations. For instance \meeting" could have a
stronger preference for its syntactic interpretation
as a noun and a smaller preference for a verb form.
Consequently, we want to use preferences for the
input, output, and states of such machines. Preferences
of this type should be able to take values
from [0; 1] m so that multiple preferences can be
represented and integrated.
If we extend a single category (as in: if
verb) to an n-dimensional preference for the input
and an m-dimensional preference for the output
then we obtain a new synchronous machine
which we will call a preference Moore machine.
Now we want to describe such a synchronous sequential
preference Moore machine which transforms
sequential input preferences to sequential
output preferences. We will see that simple recurrent
networks or feedforward networks can be
interpreted as neural preference Moore machines.
Furthermore, we will show how symbolic and neural
knowledge can be integrated quite naturally
using preference Moore machines.
38 Wermter
Denition of a Preference Moore Machine
A preference Moore machine PM is a synchronous
sequential machine which is characterized
by a 4-tuple
and S being non-empty sets of inputs, outputs and
states. O S is the sequential preference
mapping and contains the state transition
function f s and the output function f o . Here I ,
O and S are n-, m- and l-dimensional preferences
with values from [0; 1] n , [0; 1] m and [0; 1] l , respectively
A generalized version of a preference Moore machine
is shown in gure 11 on the left. The preference
Moore machine realizes a sequential preference
mapping, which uses the current state preference
S and the input preference I to assign an
output preference O and a new state preference.
Preference mapping
States
AAAA
AAAA
Output
Input
Fig. 11. Neural preference Moore machine and its relationship to a SRN network
Now we describe a new technique of extracting
the knowledge within a recurrent network in the
form of a transducer. A symbolic transducer can
be extracted from our recurrent network which
assigns to each input vector of basic syntactic
categories a new output vector of phrasal categories
depending on the previous context. In our
network, the internal state and the context were
represented by a three-dimensional vector. For
simplicity, each strict symbolic interpretation of
a three-dimensional vector can take 2 3 , that is 8
states. In order to acquire a symbolic interpretation
of the network, we presented all patterns
from the training set and stored the internal state
vectors at the hidden layer of the network. For
each output vector and for each state vector the
next corner preference was determined using the
Euclidean distance metric. Thus the Euclidean
distance metric assigned one of three symbolic abstract
syntactic phrase categories to each output
vector and one of eight state number identiers to
each state vector.
Knowledge Extraction from Transducer Neural Networks 39000 100010 110011
n:ng
r:pg
v:vg
d:ng
d:ng
n:ng
r:pg
u:ng
d:ng
n:ng
n:ng
d:pg
j:pg
n:pg
v:vg
d:ng
a:ng
a:pg
j:ng
n:ng
v:vg
v:vg
r:pg v:vg
u:ng
r:pg
n:ng
n:ng
d:ng
u:ng
n:ng
r:pg
a:pg
v:vg
v:vg
u:ng
Fig. 12. Transducer extraction from a recurrent network for the example sentence \That (u:ng) is (v:vg) the (d:ng)
Thursday (n:ng) after (r:pg) Easter (n:ng)".
Figure
12 shows the knowledge learned by the
network as an extracted symbolic transducer. The
corner nodes represent the eight strict states, the
center node represents the start state of the trans-
ducer. At the edges we nd the symbols for the
single transductions. Input and output categories
are separated by a colon, e.g. d : ng means that -
starting from the source state of this edge - a determiner
preference d is assigned to a noun group
preference ng and the transduction is made to the
end state of this edge. In the extracted transducer
we can see some clear regularities at certain
states. For instance, the transductions to state
100 are primarily responsible for the assignments
to the prepositional group pg. Other examples
are the transductions to state 010 and to state
000, which are primarily responsible for the verbal
group (vg) assignment. Furthermore, gure 12
shows the example transductions for the sentence
\That is the Thursday after Easter". Beginning
with the start state at the center, we see the transduction
ng for the word \That" which assigns
the noun group ng to the pronoun u. Then,
assigns a verb group vg to the verb \is". Then the
transductions ng ng assign the noun group
ng to \the Thursday". Finally the transductions
assign the prepositional group pg
to the sequence \after Easter". Dierent abstract
syntactic categories (ng, pg) can be assigned to
the same category (n) depending on the learned
previous context.
n:ng
r:pg
v:vg
d:ng
d:ng
n:ng
r:pg
u:ng
d:ng
n:ng
n:ng
d:pg
j:pg
n:pg
v:vg
d:ng
a:ng
a:pg
j:ng
n:ng
v:vg
v:vg
r:pg
v:vg
u:ng
r:pg
n:ng
n:ng
d:ng
u:ng
n:ng
r:pg
a:pg
v:vg
v:vg
u:ng
Fig. 13. Transducer extraction from a recurrent network for the example sentence \I (u:ng) thought (v:vg) in (r:pg) the
(d:pg) next (j:pg) week (n:pg)".
More detailed (less detailed) transducers can
be obtained if the state and output vectors are
mapped to more (fewer) nodes. Thus, the general
abstraction level of such a symbolic transducer
can be quite variable. The symbolic transducer
represents an abstraction of the detailed network
knowledge but this abstraction also hides some of
the numerical complexity and allows a direct symbolic
interpretation which provides a summary of
the network behavior.
To give an example, gure 13 shows the transductions
for the example sentence \I thought in
the next week". Beginning with the start state
at the center, we see the transduction ng
for the word \I", which assigns the noun group
ng to the pronoun u. Then, assigns a
verb group vg to the verb \thought". Finally
the transductions r : pg d :
assign the prepositional group \pg" to the word
sequence \in the next week". One advantage of
this transducer extraction is the higher abstraction
level used for the representations of the recurrent
network which leads to a better understanding
of its function. The original network contains
more detailed knowledge in the numerical weights
and activations, but it is not possible to see the
declarative sequential symbolic knowledge which
this network represents. The extraction of a symbolic
transducer allows a better understanding of
the learned sequential knowledge which is represented
in a more explicit manner.
Knowledge Extraction from Transducer Neural Networks 41
8. Discussion and Analysis
8.1. Comparison of knowledge extraction technique
There has been some previous work on using individual
techniques in isolation for interpreting neural
networks and extracting structural knowledge
from them. In this paper, we have analyzed ve
such dierent techniques using the same trained
network in order to interpret the network knowl-
edge. Such extensive comparisons of detailed net-work
knowledge are needed in order to gain a
better understanding of the knowledge extraction
represented in neural networks.
We have also introduced two new techniques
here: dynamic learning analysis and transducer
extraction. Dynamic learning analysis examines
the formation and development of categories over
time during learning. Thus, it provides a much
deeper understanding of how the neural network
arrives at its learned representation. Transducer
extraction was developed to represent the sequential
processing in a recurrent network at a higher
level of abstraction.
In general, we found that dierent interpretation
techniques provide dierent views of the
knowledge contained in a neural network. Thus,
there is not a single best technique for all dier-
ent aspects of knowledge extraction. The use of
a particular technique depends rather on the requirements
of the interpretation. In table 2, we
illustrate and summarize the general properties of
the ve dierent techniques.
Dynamic Learning Analysis (DLA) is based on
the output representations and provides a high
level of understanding based on these known output
representations. This technique is easy to
interpret and can be used with other network
types. On the other hand, it does not particularly
support recurrent networks, symbolic integration,
and
exible knowledge structuring. Furthermore,
structural relationships cannot be extracted.
Transducer extraction (TE) is a new technique
which uses output representations as well as internal
activations. The main advantages of this
technique are the high level of understanding in
the form of an extracted symbolic transducer, the
specic support for the sequentiality of recurrent
networks and the possibility for extracting structural
relationships. Such an extracted transducer
can be integrated with other symbolic knowledge,
e.g. other coded symbolic transducers. Further-
more, dierent transducers can be generated with
exibility, based on the number of states used in
the internal activation layer. This leads to a relatively
straightforward interpretation of the net-work
involved compared to the other techniques,
but it also requires the additional eort of extracting
this symbolic transducer from the internal activations
and the output representations.
If we compare DLA and
and CAA, we can see that DLA and are techniques
that specically provide high level interpretations
for dynamic learning and processing. We
argue that WA, HAA, and CAA are techniques
with a tendency towards a general, detailed, but
low-level interpretation. DLA and TE, however,
are techniques for specialized, high-level, dynamic
interpretation. Focusing on output interpretations
and the dynamics of recurrent networks provides
a new level of understanding. Whereas a lot
of previous work has focused on low-levels of in-
terpretation, we believe that in the future, higher
levels of interpretation and knowledge extraction
will be required.
8.2. Related work on transducer extraction and
related work
Finite state automata and transducers have been
widely used in various forms within traditional
e.g. [Hopcroft and Ullman,
1979]. Basically, automata and transducers are always
in a certain context state and they analyze a
certain word (symbol). Then they move to a new
state and potentially generate a new word (sym-
bol). By using changing states, it is possible to
encode the sequential context.
Although nite automata or regular languages
are not su-cient to describe all possible constructions
of natural language completely (see e.g.
[Winograd, 1983]), automata still constitute a central
minimal requirement for the representation
of natural language. Thus, they occupy the lowest
level in the Chomsky hierarchy of languages
[Hopcroft and Ullman, 1979]. Furthermore, it is
possible to design e-cient realizations of nite automata
for dierent domains [Kaplan, 1995], e.g.
42 Wermter
Table
2. Comparison of dierent knowledge extraction techniques: Dynamic Learning Analysis (DLA), Weight Analysis
(WA), Hierarchical Activation Analysis (HAA), Component Activation Analysis (CAA), Transducer Extraction (TE).
Further abbreviations: Activations/Weights/Outputs and Low/Medium/High.
Network representations used O W A A AO
General level of understanding H L M M H
Specic support for recurrent networks L L L L H
Degree of structural relationships L L M M H
Integration with symbolic knowledge L L M M H
Flexibility in level of knowledge structuring L L M M H
Computational eort L L M M M
Easiness of interpretation H L M M H
Generality and portability to other networks H H H H M
for morphology, lexicon access, information extraction
from sentences, syntactic tagging, etc.
Recurrent networks have the potential to learn
a sequential preference mapping f
automatically, based on input and output examples
(see gure 11), whereas traditional Moore
machines or Fuzzy-Sequential-Functions [Santos,
1973] involve manual encoding. It has been recently
illustrated how SRN networks can emulate
each symbolic Moore machine and each nite automaton
[Kremer, 1995, Kremer, 1996]. It has
also been shown however [Goudreau and Giles,
1995, Goudreau et al., 1994] that a recurrent net-work
with only a single input layer, one context
layer, and one output layer, the so-called Single-
layer-rst-order-network, is not su-cient for the
realization of arbitrary nite automata.
In natural language processing, representations
have to be at least as powerful as nite au-
tomata. Consequently, Single-layer-rst-order-
networks are not appropriate, which is why we
have used SRN networks here. These recurrent
networks contain nite transducers as a special
case, but also support much more powerful properties
based on their gradual m-dimensional preference
representations. For instance, it could be
shown that SRN networks can emulate certain restricted
properties of a pushdown automaton, in
particular the recursive representation of structures
with a limited depth [Elman, 1991, Wiles
and Elman, 1996].
Apart from traditional symbolic regular rep-
resentations, gradual and learned representations
can also be represented. Furthermore, the number
of input, state, and output preferences is not necessarily
nite. Therefore, neural preference Moore
machines are more powerful than nite transduc-
ers. Our recurrent neural networks can be seen as
learning augmenting
a simple nite symbolic transducer with
respect to learning within a gradual preference
space. From this perspective, symbolic knowledge
is a special abstract region in a neural preference
space.
An important line of research on automata and
recurrent networks has been reported in [Giles
et al., 1992, Goudreau and Giles, 1995, Tino et al.,
1995]. Giles and colleagues studied both nite
state automata and neural networks, but there are
substantial dierences with our research. They
started often with a known nite state automaton,
which was used to generate sequences for it. Then
these sequences were used for training a second-order
neural network. Using a partition algorithm,
a nite state automaton was extracted from the
network activations, minimized and compared to
the original known nite state automaton. In this
way, Giles and colleagues could study the computational
properties of the extraction particularly
well, but the nite state automata also frequently
relied on relatively simple 1/0 sequences.
Knowledge Extraction from Transducer Neural Networks 43
Our motivation and methodology is dierent
from theirs in several respects. We assume that
the initial nite state automaton or transducer is
not known. Especially for real-world problems,
the interesting case is the one where such an automaton
is not known in advance. Whereas it
is interesting for comparison and sequence gen-
eration, generating sequences with a nite state
automaton already introduces certain regularities
into the training set. Thus, sequence generation
has an important in
uence on the learning behav-
ior, something which we want to rule out. In
fact, we are more interested in situations where
we do not know the machine which has to be ex-
tracted. Especially with noisy real-world learning
data, the underlying regularities may be quite disparate
from regularly generated sequences.
Furthermore, the task of our networks is quite
dierent. The second-order networks employed by
Giles and colleagues are trained for recognition.
The output layer represents state representations
which can be fed back to the input layer at the
next step. Our recurrent networks perform an assignment
task, where a sequence of inputs is associated
with a sequence of outputs. We are not determining
whether a certain sequence belongs to a
certain automaton, but what the simple
at structure
of this sequence is. That is, we are interested
in transducer extraction rather than recognizer ex-
traction. In general, there are no designated nal
states in our networks, since the network - and the
extracted symbolic transducer - produce output
as long as input is provided. This transducer behavior
is therefore quite dierent from the recognition
performance reported in [Giles and Omlin,
1993], which is based on acceptors for articial
languages.
9. Conclusion
The main contribution of this paper is a particularly
broad analysis of knowledge extraction
for recurrent networks. In addition, we propose
dynamic learning analysis and transducer extraction
as two new dynamic interpretation tech-
niques. Dynamic learning analysis provides a
better understanding of how the network learns,
while transducer extraction provides a better understanding
of what the network represents. After
learning, a conservative \lazy learning" strategy
leads to connectionist representations which
can be described as symbolic transducers. These
transducers allow for a much better interpretation
of the sequential network knowledge compared to
the standard analysis using hierarchical clustering
or Hinton diagrams. Weight analysis, cluster
analysis, and principal component analysis are detailed
but static. In contrast, our new method
for extracting symbolic transducers can describe
the learned classication performance much bet-
ter, since transducer extraction considers the sequential
character of the learned representations
in a recurrent network and allows a better symbolic
inspection. Possibilities for direct integration
with symbolic classiers can be explored in
future work. We conclude that dynamic learning
analysis and transducer extraction have a lot
of potential for improved knowledge structuring
based on recurrent networks.
--R
Extracting algorithms from pattern classi
Rules and Networks.
Sequential Machines and Automata Theory.
The Computational Brain.
Extracting Comprehensible Models from Trained Neural Networks.
Distributed repre- sentations
Language as a dynamical system.
Learning and extracted
On recurrent neural networks and representing
Hybrid Prob- lems
Learning distributed representations of concepts.
Introduction to Automata Theory
Finite state technology.
Switching and Finite Automata Theory.
On the computational power of Elman-style recurrent networks
A theory of grammatical induction in the connectionist paradigm.
Hybrid Intelligent Systems.
Extraction of rules from discrete-time recurrent neural networks
Fuzzy sequential func- tions
A framework for combining symbolic and neural learning.
An Introduction to Automata Theory.
Learning distributed representations for the classi
Finite state machines and recurrent neural networks.
Hybrid Connectionist Natural Language Processing.
The hybrid approach to arti
Preference moore machines for neural fuzzy integration.
Building lexical representations dynamically using arti
SCREEN: Learning a at syntactic and semantic spoken language analysis using arti
Learning to count without a counter: A case study of dynamics and activation landscapes in recurrent net- works
Language as a Cognitive Process.
--TR | symbolic interpretation;SRN networks;analysis of connectionist learning;knowledge extraction;neural network learning |
590930 | Two-Loop Real-Coded Genetic Algorithms with Adaptive Control of Mutation Step Sizes. | Genetic algorithms are adaptive methods based on natural evolution that may be used for search and optimization problems. They process a population of search space solutions with three operations: selection, crossover, and mutation. Under their initial formulation, the search space solutions are coded using the binary alphabet, however other coding types have been taken into account for the representation issue, such as real coding. The real-coding approach seems particularly natural when tackling optimization problems of parameters with variables in continuous domains.A problem in the use of genetic algorithms is premature convergence, a premature stagnation of the search caused by the lack of population diversity. The mutation operator is the one responsible for the generation of diversity and therefore may be considered to be an important element in solving this problem. For the case of working under real coding, a solution involves the control, throughout the run, of the strength in which real genes are mutated, i.e., the step size.This paper presents TRAMSS, a Two-loop Real-coded genetic algorithm with Adaptive control of Mutation Step Sizes. It adjusts the step size of a mutation operator applied during the inner loop, for producing efficient local tuning. It also controls the step size of a mutation operator used by a restart operator performed in the outer loop, for reinitializing the population in order to ensure that different promising search zones are focused by the inner loop throughout the run. Experimental results show that the proposal consistently outperforms other mechanisms presented for controlling mutation step sizes, offering two main advantages simultaneously, better reliability and accuracy. | INTRODUCTION
.
Genetic algorithms (GAs) are general purpose search algorithms which use principles inspired by
natural genetic populations to evolve solutions for problems ([Goldberg (1989a), Holland (1992)]).
The basic idea is to maintain a population of chromosomes, which represent candidate solutions
for the specific problem, that evolves over time through a process of competition and controlled
variation. The following bibliography may be examined for a more detailed discussion about GAs:
[B-ack (1996), B-ack et al. (1997), Goldberg (1989a), Holland (1992), Michalewicz (1992)].
Under their initial formulation, the search space solutions are coded using the binary alpha-
bet. However, other coding types have been considered for the representation issue, such as real
coding, which would seem particularly natural when tackling optimization problems of parameters
with variables on continuous domains. Then a chromosome is a vector of floating point numbers,
the size of which is kept the same as the length of the vector, which is the solution to the prob-
lem. GAs with this type of coding are called real-coded GAs (RCGAs) (see [Herrera et al. (1998),
Surry et al. (1996)]). There are other types of Evolutionary Algorithms (EAs), i.e., implementing
the idea of evolution ([B-ack (1996)]), which are based on real coding as well. These are Evolution
Strategies ([Schwefel (1995)]) and Evolutionary Programming ([Fogel (1995)]). This paper deals with
RCGAs.
Population diversity is crucial to a GA's ability to continue the fruitful exploration of the search
space ([Li et al. (1992)]). If the lack of population diversity takes place too early, a premature
stagnation of the search is caused. Under these circumstances, the search is likely to be trapped in
a local optimum before the global optimum is found. This problem, called premature convergence,
has long been recognized as a serious failure mode for GAs ([Eshelman et al. (1991)]).
The mutation operator may be considered to be an important element for solving the premature
convergence problem, since it serves to create random diversity in the population ([Spears (1993)]).
Different techniques have been suggested for the control, during the GA's run, of parameters associated
with this operator, depending on either the current state of the search or other GA related
parameters ([Angeline (1995), Herrera et al. (1996b), Hinterding et al. (1997)]). They try to offer
suitable diversity levels for avoiding premature convergence and improving the results. In the case
of working with real coding, a topic of major importance involves the control of the proportion or
strength in which real-coded genes are mutated, i.e., the step size ([B-ack et al. (1996a)]).
The objective of this paper is to formulate a mechanism for the control of mutation step sizes for
RCGAs, which should handle and maintain population diversity that in some way helps produce good
chromosomes, i.e., useful diversity ([Mahfoud (1995)]). We present TRAMSS, a Two-loop RCGA
model with Adaptive control of Mutation Step Sizes that attempts to do this. It is made up by two
loops, an inner loop and an outer one:
Inner loop. It is designed for processing useful diversity in order to lead the population toward
the most promising search areas, producing an effective refinement on them. So, its principal
mission is to obtain the best possible accuracy levels.
The inner loop performs the selection process and fires the crossover and mutation operators.
Furthermore, for achieving its objective, it controls the step size of the mutation operator.
ffl Outer loop. It introduces new population diversity, after the inner loop reaches a stationary
point where there are no improvements, that helps the next one to reach better solutions.
Therefore, it attempts to induce reliability in the search process.
The outer loop iteratively performs the inner one, and later, it applies a restart operator that
reinitializes the population by mutating all genes, using a step size that is adapted as well,
throughout the runs for this loop.
For doing this, the paper is set up as follows: in Section 2, we analyze two mutation issues, the
ways in which the control of mutation step sizes may be made and the idea of the restart operator; in
Section 3, we present TRAMSS, in Section 4, we describe the experiments carried out for determining
the efficacy of the proposal; and finally, some conclusions are dealt with in Section 5.
In this Section, we explain two issues that will be included as important components in the conceptual
foundation of TRAMSS, mutation step size control (Subsection 2.1) and the restart operator
(Subsection 2.2).
2.1 Mutation Step Size Control
In general, the mechanisms presented for controlling parameters associated with EAs may be
assigned to the following three categories ([Hinterding et al. (1997)]):
ffl Deterministic Control. It takes place if the values of the parameters to be controlled are
altered by some deterministic rule, without using any feedback from the GA. Usually, a time-varying
schedule is used.
Adaptive Control. It takes place if there is some form of feedback from the GA that is used
to determine the direction and/or magnitude of the change to the parameters to be controlled.
The rules for updating parameters that are used by this type of control and, by the previous
one, are termed absolute adaptive heuristics ([Angeline (1995)]) and, ideally, capture some
lawful operation of the dynamics of the EA over a broad range of problems.
ffl Self-adaptive Control. The parameters to be controlled are encoded onto the chromosomes
of the individual and undergo mutation and recombination.
Next, we describe mechanisms for the control of mutation step sizes that belong to each one of
these categories.
2.1.1 Deterministic Step Size Control
In [Michalewicz (1992)], a mutation operator for RCGAs, called non-uniform mutation, was pre-
sented, which is based on the absolute adaptive heuristic "to protect the exploration in the initial
stages and the exploitation later". It implements this idea by decreasing the step size as the GA's
execution advances. Let us suppose that this operator is applied on a real-coded gene, x 2 [a; b]
(a; b 2 !), at generation t, and that T is the maximum number of generations, then it generates a
gene, x 0 , as follows:
ae
with - being a random number that may have a value of zero or one, and
where r is a random number from the interval [0; 1] and b is a parameter chosen by the user. This
function gives a value in the range [0; y] such that the probability of returning a number close to
zero increases as the algorithm advances. The size of the gene generation interval shall be smaller
with the passing of generations. This property causes this operator to make an uniform search in
the initial space when t is small, and very locally at a later stage, favoring local tuning.
The non-uniform mutation operator has been widely used, reporting good results ([Herrera et
al. (1996a), P'eriaux et al. (1995), Sefrioui et al. (1996)]). It is considered to be one of the most
suitable mutation operators for RCGAs ([Herrera et al. (1998)]).
2.1.2 Adaptive Step Size Control
The (1+1)-Evolution Strategy ((1+1)-ES) ([Schwefel (1995)]) is an EA that uses adaptive step
size control. It attempts to adapt its mutation step size to the problem according to the absolute
adaptive heuristic: "expand the step size when making progress, shrink it when stuck". This heuristic
will be denoted as E/S heuristic.
(1+1)-ES works using a continuous representation and a mutation operator based on normally
distributed modifications with expectation zero and given variance, oe, as the step size. It operates
on a vector of variables by applying mutation with identical oe to each variable, so generating a
descendant. The better of ancestor and descendant is considered as the new starting point. (1+1)-
ES applies the E/S heuristic for adapting oe by means of the 1/5 success rule. This rule uses the
results obtained by mutation in the last few generations: if more than one fifth of the mutation have
been successful, the step size is increased, otherwise it is decreased.
In [De La Maza et al. (1994)], a dynamic hill climbing algorithm is presented, which uses the
E/S heuristic as well. We would like to point out that the model proposed in this paper, TRAMSS,
uses important ideas that are present in this algorithm.
2.1.3 Self-Adaptive Step Size Control
In [Schwefel (1995)], an EA model, called (-Evolution Strategy ((-ES), is developed that
uses a mechanism for the self-adaptive step size control.
In (-ES, - parents create - offspring by means of recombination and mutation, and the best
offspring individuals are deterministically selected to replace the parents. Therefore, - should
be greater than -. The main quality of the algorithm is its ability to incorporate the standard
deviations (step sizes) and the correlation coefficients of normally distributed mutations into the
search process, such that adaptation not only takes place in the object variables, but also in these
parameters according to the current local topology of the search space. This property is called
self-adaptation ([B-ack (1996), Schwefel (1995)]). Self-adaptation exploits the indirect link between
favorable parameter values and fitness function values, being capable of adapting the parameters
implicitly, according to the topology of the objective function ([B-ack et al. (1996b)]).
Therefore, each population individual consists of three vectors, representing the
object variable, the standard deviation and the rotation angle values, respectively. The vector ~x has
dimensions, equal to the number of problem variables. The n oe dimensions of a vector ~oe can be
up to n (in this case, each object variable x different step size oe i associated to
it), and n ff can be up to (2\Deltan\Gamman oe )\Delta(n oe \Gamma1). n ff may be set to zero, indicating that the rotation angles
are not considered, as is assumed in this paper.
For more information about (-ES refer to [B-ack (1996), Schwefel (1995)]. Other mechanisms
for the self-adaptive step size control are to be found in [Fogel (1995), Hinterding (1995),
Ostermeier et al. (1994)].
2.2 Restart Operator
Premature convergence causes a drop in the GA's efficiency; the genetic operators do not produce
the feasible diversity to tackle new search space zones and thus the algorithm reiterates over the
known zones producing a slowing-down in the search process. Under these circumstances, resources
may be wasted by the GA searching an area not containing a solution of sufficient quality, where
any possible improvement in the solution quality is not justified by the resources used. Therefore,
resources would be better utilized in restarting the search in a new area, with a new population
et al. (1995)]). This is carried out by means of a restart operator. Next, we review some
different approaches to this operator.
ffl In [Goldberg (1989b)], it was suggested restarting GAs that have substantially converged, by
reinitializing the population using both randomly generated individuals and the best individual
from the converged population.
ffl In [Eshelman (1991)], upon convergence, the population is reinitialized by using the best individual
found so far as a template for creating a new population. Each individual is created
by flipping a fixed proportion (35%) of the bits of the template chosen at random without
replacement. If several successive reinitializations fail to yield an improvement, the population
is completely (100%) randomly reinitialized.
ffl In [Maresky et al. (1995)], a selectively destructive restart is proposed that does not completely
destroy the converged population; a percentage of the converged genes will survive untouched
to begin the next convergence stage. A probability of gene reinitialization, p r , is used: the
higher the rate, the more genes are initialized. Experiments carried out with some p r values
showed that different problems have different optimal reinitialization probabilities. This model
seems to provide an improved method for renewing genetic diversity in GA search. Intuitively,
the complete reinitialization of the population forgets the previous solutions, therefore it cannot
make use of previously discovered building blocks.
ffl In [Grefenstette (1992)], a similar mechanism, called partial hypermutation model, was in-
troduced, which replaces, at each generation, a percentage of the population by randomly
generated individuals. The percentage is called replacement rate. The intended effect is similar
to the one of the previous approach: to maintain a continuous level of exploration of the
search space, while trying to minimize disruption for the ongoing search.
Other important GA models based on the restart operator are ARGOT ([Shaefer (1987)]), Dynamic
parameter encoding ([Schraudolph et al. (1992)]) and Delta coding ([Whitley et al. (1991)]).
OF STEP SIZES.
In this Section, we present TRAMSS. It uses:
ffl An instance of the absolute adaptive E/S heuristic, presented in Subsection 2.1.2, for the
adaptive step size control of the mutation operator applied in the inner loop, and
ffl An instance of its opposite version, denoted here as S/E heuristic, for the adaptive step size
control of the mutation operator used by the restart operator that is executed by the outer
loop.
Next, in Subsection 3.1, we examine the application of the E/S and S/E heuristics for step size
control in RCGAs, and, in Subsections 3.2 and 3.3, we present the TRAMSS inner and outer loops,
respectively.
3.1 The E/S and S/E Heuristics
Let's suppose that an RCGA is applying a mutation operator with ffi being its step size. If a
stationary state is detected (the fitness of the best individual or the average fitness have not been
improved during the previous generations), there are two possible causes concerning ffi:
1. It is too high, which does not allow the convergence to be produced for obtaining better
individuals, or
2. It is too low, which induces a premature convergence, with the search process being trapped
in a local optimum.
On the one hand, if we decided to include an adaptive control of ffi based on the instance of the
E/S heuristic "increase ffi when making progress, decrease it when stuck", a stationary state caused
by (1) would be suitably tackled, since ffi would become lower, so introducing more convergence.
However, this heuristic would not be adequate if the stationary state is caused by (2), because it
would complicate the problem even more.
Precisely, this last circumstance will occur as the number of iterations increases. Since the RCGA
will find more difficulties for making progress, the natural trend of the instance of the E/S heuristic
will be to lead ffi to lower values, so producing more convergence. The possibility of this problem has
been claimed by some authors. For example, in [B-ack et al. (1995)], the following was stated about
the 1/5 success rule:
". the 1/5 success rule may cause premature stagnation of the search due to the deterministic
decrease of the step size whenever the topological situation does not lead to a
sufficiently large success rate".
For complex problems, this effect will probably become a premature convergence. This explains
the following claim, again about the 1/5 success rule ([Angeline (1995)]):
". this heuristic is especially useful in smooth multimodal environments of the type well
studied by the ES community but would be less applicable in discontinuous or extremely
rough environments".
On the other hand, if we are inclined to use the instance of the S/E heuristic "decrease ffi when
progress is made, increase it when there are no improvements", a stationary state produced by (2)
will be adequately attacked, since ffi would be greater and so, more diversity is introduced with the
possibility of escaping from the local optimum. However, an important problem may occur: as no
improvements are made by the RCGA, higher ffi values are tried, so introducing too much diversity
and not considering the possibility that convergence may be suffice for improving results.
So, all these facts show that serious problems may arise when the E/S and S/E heuristics are
applied separately. However, we think that a mechanism applying both of these heuristics would
handle the population diversity suitably to avoid the premature convergence problem and improve
the behavior of the search process.
The adaptive RCGA model proposed, TRAMSS, includes this idea: it uses the E/S heuristic for
adapting the step size of a mutation operator applied in the inner loop and the S/E heuristic for
adapting the step size of a mutation operator used by a restart operator performed in the outer loop.
3.2 TRAMSS Inner Loop
The inner loop performs the usual process (selection, crossover and mutation) over a number of
G, called time-interval between observations. Then, depending on the progress of the
population mean fitness found throughout these generations, it adjusts the step size of the mutation
operator, and calculates a new value for G. Next, we fully describe these steps where a minimization
problem is assumed.
Selection, Crossover and Mutation (Step 2.2). Over the time-interval between observations, G,
the following selection mechanism and crossover and mutation operators are applied.
ffl The selection probability calculation follows linear ranking ([Baker (1985)]), with
and the sampling algorithm is the stochastic universal sampling ([Baker (1987)]).
The elitist strategy ([De Jong (1975)]) is considered as well. It involves making sure that the
best performing chromosome always survives intact from one generation to the next. This
is necessary since it is possible that the best chromosome disappears, due to crossover or
mutation.
ffl We have tried different crossover operators, which are presented in Subsection 4.2.
ffl The mutation operator used is denoted as Mutation(ffi), where ffi is the step size (0 - ffi - 1).
This operator is defined as follows: If x 2 [a; b] is a gene to be mutated, then the gene resulting
from the application of this operator, x 0 , will be a random (uniform) number chosen from
Clearly, the higher ffi is, the greater changes on x are produced.
Adaptive Control of ffi (Step 2.3). After G generations, the ffi parameter used by the mutation
operator is adapted following a particular instance of the E/S heuristic: "increase ffi when observing
progress on -
f (population mean fitness), decrease it when stuck". ffi is kept in the interval [ffi min ; \Delta],
where \Delta is a parameter calculated by the outer loop, as described in Subsection 3.3, and ffi min
is the minimum threshold defined by the user (in experiments we assume a value of
1.0e-100).
The inner loop ends when ffi reaches the ffi min value. By finishing with a fine grained search
with small step sizes, we are sure that a local optimum, or the global one, will be located precisely
([De La Maza et al. (1994)]). It will stop as well, when a maximum number of generations is reached.
The update rates for ffi depend on the number of previous successive observations that were
successful or not successful. Two variables, yes and no, are used for recording these occurrences,
respectively. If progress is made during many successive previous observations
fOld -
fOld being
the population mean fitness of the previous iteration), then the increasing rate for ffi is very high
(in particular, ffi is multiplied by 2 yes ), whereas if these observations were not successful, then the
decreasing rate is high (ffi is divided by 2 no ). In this way, when the search process is located in a
local optimum and improvements are still not surely expected by reducing ffi, the inner loop duration
will not be too long.
Time-interval Calculation (Step 2.4). The time-interval between observations, G, is calculated
depending on the current values of ffi with regard to \Delta. If ffi is similar to \Delta, then the time-interval
is high (G 0
100 in the experiments), and if it is lower, the time-interval will become like Gmin
in the experiments). This allows ffi values similar to \Delta to be used for a long time (\Delta is
considered a good starting point for ffi, because it is adapted in the outer loop on the basis thereof,
as we will explain in Subsection 3.3). Furthermore, we need to point out that the initial \Delta, \Delta 0
, was
assigned to 1 in the experiments, in order to favor exploration during the initial stages of the first
inner loop's run.
Figure
1 shows the pseudocode algorithm for the whole TRAMSS inner loop. In short, the
objective of this loop is to find and refine local optima (or the global one), in an efficient way.
TRAMSS INNER LOOP
1.
2. while (ffi - ffi min ) and (not termination-condition) do
2.1. -
fOld := -
2.2. perform Selection, Crossover and Mutation(ffi) over G generations;
2.3. if ( -
fOld -
f) then
else
2.4. G := G0 \Delta ffi=\Delta; if
Fig. 1. TRAMSS Inner Loop Structure
3.3 TRAMSS Outer Loop
The outer loop randomly initializes the population that will be handled throughout the TRAMSS
run. It fires the inner loop, and when this one returns, it applies a restart operator based on a step
size that is adaptatively controlled throughout its execution. Now, we explain, in depth, the main
issues related to this loop.
Restart Operator Application (Step 3.4). The outer loop applies a restart operator, called
applies Mutation(\Delta) to all the genes in the chromosomes stored
in the population. The objective of this operator is similar to the one of the partial restart operators
for binary-coded GAs, described in Section 2.2, i.e., to maintain a continuous level of exploration of
the search space, while trying to use the promising zones located as a kind of sketch. It attempts
to ensure that new and promising genetic material is available in the population for being handled
and treated by the next inner loop.
Adaptive Control of \Delta (Step 3.3). The outer loop adapts the \Delta parameter, using information
obtained after each inner loop run, by means of an instance of the S/E heuristic: "decrease \Delta when
observing progress on fBest (fitness of the best element found so far), otherwise increase it. This is
implemented by dividing the previous \Delta value by 2 or multiplying it by 2, respectively. The new \Delta
value will be the first value for the ffi parameter used in the next inner loop.
The pseudocode algorithm for the outer loop is depicted in Figure 2. To sum up, the outer loop
attempts to introduce adequate diversity levels for allowing the subsequent inner loop processing to
be capable of finding, better local optima, or the global one, every time. For this reason, it uses
the f best for the adaptive step size control. When no better local optima are found after the last
inner loop runs, the outer loop produces more diversity in order to increase the probability of having
access to a better one, which will be refined by the next inner loop. On the other hand, if better
solutions are being found by previous inner loop's runs, \Delta becomes low, so avoiding, for the moment,
great destructive effects of the restart operator.
TRAMSS OUTNER LOOP
1.
2. run Initialize;
3. while (not Termination-condition) do
3.1. fOldBest := fBest ;
3.2. run Inner Loop;
3.3. if (fOldBest - fBest ) then
else
3.4. run Restart(\Delta);
Fig. 2. TRAMSS Outer Loop Structure
Minimization experiments on the test suite, described in Subsection 4.1, were carried out in order
to study the behavior of the TRAMSS model. In Subsection 4.2, we describe the algorithms built
in order to do this and, in Subsection 4.3, we show the results and discuss some conclusions about
them.
fSph fRos
fSph
fSch fGri
fSch
d
fSch
fRas ef10
Fig. 3. Test functions
4.1 Test Suite
For the experiments, we have considered six test functions used in the GA literature: Sphere
model (f Sph ) ([De Jong (1975), Schwefel (1981)]), Generalized Rosenbrock's function (f Ros ) ([De
Jong (1975)]), Schwefel's Problem 1.2 (f Sch ) ([Schwefel (1981)]), Griewangk's function ([Griewangk
(1981)]), Generalized Rastringin's function (f Ras ) ([B-ack (1992), T-orn et al. (1989)]), and Expansion
([Whitley et al. (1995)]). Figure 3 shows their formulation. The dimension of the
search space is 25.
fSph is a continuous, strictly convex and unimodal function.
fRos is a continuous and unimodal function, with the optimum located in a steep parabolic
valley with a flat bottom. This feature will probably cause slow progress in many algorithms since
they must permanently change their search direction to reach the optimum. This function has been
considered by some authors to be a real challenge for any continuous function optimization program
([Schlierkamp-Voosen et al. (1994)]). A great part of its difficulty lies in the fact that there are
nonlinear interactions between the variables, i.e., it is nonseparable ([Whitley et al. (1996)]).
fSch is a continuous and unimodal function. Its difficulty lies in the fact that searching along
the coordinate axes only gives a poor rate of convergence, since the gradient of fSch is not oriented
along the axes. It presents similar difficulties to f Ros , but its valley is much narrower.
fRas is a scalable, continuous, separable and multimodal function which is produced from fSph
by modulating it with a
fGri is a continuous and multimodal function. This function is difficult to optimize because it
is nonseparable ([M-uhlenbein et al. (1991)]) and the search algorithm has to climb a hill to reach
the next valley. Nevertheless, one undesirable property exhibited is that it becomes easier as the
dimensionality is increased ([Whitley et al. (1996)]).
is a function that has nonlinear interactions between two variables. Its expanded version,
built in such a way that it induces nonlinear interaction across multiple variables. It is
nonseparable as well.
4.2 Algorithms
We have built five different TRAMSS versions that apply the following crossover operators: Arithmetical
([Michalewicz (1992), Wright (1991)]), Max-min-arithmetical ([Herrera et al. (1997)]), Discrete
([M-uhlenbein et al. (1993)]), Fuzzy recombination ([Voigt et al. (1995)]), and BLX-ff ([Eshel-
man et al. (1993)]). The TRAMSS versions are called TRAMSS-AR, TRAMSS-MMA, TRAMSS-DI,
TRAMSS-FR and TRAMSS-BLX, respectively.
We have implemented five classical RCGAs based on these crossover operators that apply the
non-uniform mutation operator (Subsection 2.1.1.) and, the same selection strategy as the one used
by the TRAMSS inner loop. They are called RCGA-AR, RCGA-MMA, RCGA-DI, RCGA-FR and
RCGA-BLX, respectively.
The crossover probability in these RCGAs and in the TRAMSS versions is 0:6, the mutation
probability 0:005, and the population size 60 chromosomes.
Now, we present the definition of the crossover operators. Let us assume that
are two real-code chromosomes that have been selected to apply the crossover
operator to them. Below, the effects of the five crossover operators are shown.
Arithmetical crossover. An offspring,
We have considered
Max-min-arithmetical crossover. It generates the following four offspring:
In particular, we have considered 0:25. The resulting descendents are the two
best of the four aforesaid offspring.
Discrete crossover. z i is a randomly (uniformly) chosen value from the set fx g.
Fuzzy recombination. The probability that the i-th gene in the offspring has the value z i is given
by the distribution p(z i
g, where OE x i
and OE y i
are triangular probability distributions with
the following features (d - 0:5):
Triangular Prob. Dist. Minimum Value Modal Value Maximum Value
d have been set to 0.5 in the experiments.
BLX-ff crossover. z i is a randomly (uniformly) chosen number from the interval [Min \Gamma I \Delta
We have
assumed
All these crossover operators may be ordered with regard to the way randomness is used for
generating the genes of the offspring: 1) Arithmetical and Max-min-arithmetical crossovers do not
use it, 2) Discrete crossover considers discrete probability distributions, where there are only two
possibilities introduces uniform continuous probability distributions, and
Fuzzy recombination applies triangular continuous probability distributions, and therefore, it may
be considered as a hybrid between discrete crossover and BLX-ff.
Probability distributions used by BLX-ff and Fuzzy recombination are calculated according to
the distance between the genes in the parents (x i and y i ), and the ff and d values, respectively. So,
they fit their action range depending on the diversity of the population using specific information
held by the parents ([Eshelman et al. (1993)]).
We have also implemented two (-ESs, with 100. The first one is denoted as
(15; 100)-ES1 and uses n the other, denoted as (15; 100)-ESn, uses n i.e., the number
of variables. n for both of them. They apply discrete recombination (random exchanges
between parents) on object variables and local intermediate recombination (arithmetic averaging)
on standard deviations (see [B-ack (1996)] for a more detailed description of these operators). The
number of potential parents involved in the recombination of the object variable is 15, and that for
the standard deviations is 15, as well. The standard deviations are initialized to value of 3.0. This
parameterization is very usual for (-ES ([B-ack et al. (1995)]).
All TRAMSS versions and RCGAs were executed 15 times, each one with 10,000 generations,
except TRAMSS-MMA and RCGA-MMA, that performed 5,000 generations, since each Max-min-
arithmetical crossover application needs four evaluations. (15; 100)-ES1 and (15; 100)-ESn were executed
times, each one with 4,000 generations. In this way, the number of fitness function
evaluations required by all algorithms are similar.
4.3 Results
For each function, we introduce the average of the best fitness function found at the end of each
run (A), the best of these values (B), and the percentage of success with respect to the thresholds
shown in Table 1 (S). Table 2 shows the results obtained.
Test Thresholds
fSph 1.0e-150
fRos 1.0
fSch 1.0e-3
fRas
Table
1. Thresholds for the test functions
First, we consider the behavior of the TRAMSS algorithms compared with their corresponding
RCGAs. Then, we comment on the results obtained by (-ES1 and (-ESn.
4.3.1 RCGAs vs. TRAMSS
In general, TRAMSS-AR, -MMA, -DI, -FR and -BLX outperform their corresponding RCGAs,
RCGA-AR, -MMA, -DI, -FR and -BLX, respectively, for all functions, with regard to all performance
measures. To explain this ability, first we focus on the study of the results on the unimodal functions,
fSph , f Ros and fSch , and then, on the multimodal ones, f Ras , fGri and f ef10 .
Analysis for unimodal functions.
ffl We have observed that during runs of the TRAMSS instances on all unimodal functions, the
outer loop performed the inner loop only once. This means that the inner loop has been
controlling ffi to properly suit the local nature of the landscape in these functions. In this way,
improvements on -
f (population mean fitness) predominated, so obtaining very accurate results.
Thus, we may say that the implementation of the E/S heuristic used for controlling ffi is highly
suitable for dealing with unimodal functions. In fact, we should point out that this heuristic has
already been used for designing efficient local search procedures ([De La Maza et al. (1994)]).
fSph fRos fSch
Algorithms A B S A B S A B S
TRAMSS-FR 4.5e-153 2.1e-163 100.0 1.6e+01 2.7e-03 20.0 2.7e-04 2.7e-05 100. 0
TRAMSS-BLX 2.2e-176 2.7e-188 100.0 1.3e+01 4.9e-01 13.3 7.4e-08 2.2e-09 100 .0
(15, 100)-ESn 0.0e+00 0.0e+00 100.0 3.4e+00 7.9e-03 13.3 7.8e+03 3.4e+03 0.0
fRas fGri ef10
Algorithms A B S A B S A B S
TRAMSS-MMA 2.3e-14 0.0e+00 33.3 2.7e-02 0.0e+00 13.3 6.9e-01 0.0e+00 66.7
Table
2. Results of experiments
ffl For fSph , the A and B measures of the TRAMSS algorithms are very accurate to a higher
degree than the ones for the respective RCGAs based on non-uniform mutation (see the very
good results of TRAMSS-FR and TRAMSS-BLX for this function).
ffl For f Ros and fSch , more complex functions, these measures are better as well. The joint effects
of the E/S heuristic and the application of high ffi values during many initial generations allow
the inner loop to be capable of tackling the difficulties associated with the localization of the
optimum in these functions. For example, we may highlight the good B results obtained by
TRAMSS-FR for f Ros , 2.7e-03, and TRAMSS-BLX for fSch , 2.2e-9.
These results indicate that the inner loop has been generating useful diversity throughout the
runs for the unimodal functions. On the other hand, the non-uniform mutation does not take
into account whether the diversity being generated is useful or not. It only decreases the step size
depending on the time without observing if improvements are made, or not. For the case of f Ros ,
this fact does not allow a good search direction to be established for reaching the optimum, and so,
no good final results are obtained (see the results of RCGA-AR, RCGA-FR and RCGA-BLX for this
Analysis for multimodal functions.
ffl The participation of the restart operator in the outer loop allowed reliability to be improved
on the multimodal functions, with regard to the RCGAs based on non-uniform mutation. For
f Ras and fGri , all TRAMSS implementations reached the global optimum, at least once (see B
measure).
ffl Other examples of improvements on reliability, for the case of fGri , involve increasing in the
S measure (expressed with regard to the fitness of the global optimum) of TRAMSS-AR (60%)
in front of RCGA-AR (0%), and TRAMSS-BLX (80%) as contrasted with RCGA-BLX (26.7%).
ffl For the case of
, the union of the effective local tuning of the inner loop and the introduction
of a spread search by means of the restart operator allows very good results to be obtained.
For example, the A measure of TRAMSS-FR for this function is the best of all the algorithms
executed, 1.4e-14, and the B measure of TRAMSS-BLX is very exact, 1.5e-44.
On the other hand, the decreasing of the step size performed by the non-uniform mutation does
not allow the search direction to escape from a possible stagnation in a local optimum, when working
on multimodal functions. In particular, for the case of using arithmetical and discrete operators, we
may observe how this disadvantage induced very poor performance measure values for this type of
functions.
TRAMSS and Crossover Operators.
ffl An important issue that should be highlighted is that the improvements of the TRAMSS
implementations on their corresponding RCGAs based on non-uniform mutation operator are
more notable for most test functions when using Fuzzy recombination and BLX-ff.
The adaptive ffi control performed in the inner loop depends on the changes produced on
f , which are determined by the joint effects of the selection process, and the mutation and
crossover operators. Let us consider only the interactions between the last ones. The mutation
operator generates diversity and the crossover operator would have to use it for creating better
individuals. If the crossover operator achieves this task, then the mutation operator would be
generating "useful diversity", and so evolution is introduced. Therefore, only if the mutation
and the crossover operators are being suitably coupled, the success of the inner loop may be
accomplished. Results have shown that in the case of using Fuzzy recombination and BLX-ff,
this circumstance is held. In particular, we highlight the very good A, B and S measures of
TRAMSS-FR and TRAMSS-BLX for all functions.
These facts lead us to think that the associated property of these crossover operators (to fit
their action range depending on the diversity of the population) is the main one responsible
for making this union so profitable. It would allow these operators to exploit the diversity
generated by the mutation operator.
ffl We should underline the good results of the TRAMSS version based on the Max-min-arithme-
tical crossover as well, TRAMSS-MMA, for the unimodal fSph and the multimodal
. It
reached the global optimum of the first one for 100% of the runs (see A measure) and was the
only algorithm that found the global optimum of the second one (see B measure). The process
of selecting the two best offspring from a set of four ones with different properties along with
the effects of the TRAMSS model allow the best search regions to be located and refined.
4.3.2 (-ESs vs. TRAMSS
performance measures for (15, 100)-ES1 show good behavior. The adaptation mechanism
concerning this algorithm is similar to the one in the TRAMSS model to the fact that all genes
belonging to the same chromosome are mutated using the same step size. However, all TRAMSS
implementations have outperformed this algorithm.
On the other hand, (15, 100)-ESn shows a similar behavior for fSch and f Ras than (15, 100)-ES1.
Furthermore, it is outperformed by TRAMSS-FR and TRAMSS-BLX in the complex f ef10 . However,
its performance for the unimodal functions fSph and f Ros and for the multimodal one fGri was very
good. For these functions, (15, 100)-ESn certainly benefited from the greater degree of freedom by
working with n different self-adaptive step sizes per individual in contrast to a single one in the case
of the TRAMSS algorithms.
5 CONCLUDING REMARKS.
In this paper, we have presented TRAMSS, a two-loop RCGA model that adjusts the step size
of a mutation operator applied during the inner loop, for producing an efficient local tuning, and
controls the step size of a mutation operator used by the outer loop, for reinitializing the population
in order to ensure that different promising search zones are focused by the inner loop throughout
the run. An instance of the E/S heuristic was used for implementing the adaptive mechanism in the
inner loop whereas an instance of its opposite, the S/E heuristic, was considered for the outer loop.
Five TRAMSS algorithms were built using five crossover operators for RCGAs, Arithmetical,
Max-min-arithmetical, Discrete, Fuzzy recombination and BLX-ff, which represent different ways in
which randomness may be used for generating real-coded genes. The principal conclusions derived
from the results of experiments carried out are the following:
ffl The TRAMSS model allows the control of useful population diversity to be accomplished for
improving accuracy in the case of unimodal functions, and, both reliability and accuracy for
the multimodal ones, with regard to RCGAs based on the non-uniform mutation operator.
ffl The adaptive control of step size performed by TRAMSS couples suitably with Fuzzy recombination
and BLX-ff. Their interactions allow TRAMSS to manage useful diversity, so inducing
an effective behavior on all test functions. We suggested that this occurs thanks to the fact
that these crossovers adjust the intervals for the generation of genes depending on the current
population diversity.
ffl With the TRAMSS model, the performance of the strategy of selecting the two best offspring
from a set of four with different properties (Max-min-arithmetical crossover) is enhanced.
Finally, we should point out that TRAMSS extensions may be followed in two ways: 1) control the
parameter associated with the Fuzzy recombination and BLX-ff, respectively, in order to exploit,
even more, the profitable combination between TRAMSS and these crossover operators, and 2)
study the possible application of dynamic crossover operators, such as the Dynamic FCB-crossovers
([Herrera et al. (1996a)]) and Dynamic Heuristic FCB-crossovers ([Herrera et al. (1996c)]), which
are based on the same absolute adaptive heuristics as the non-uniform mutation operator.
--R
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--TR
--CTR
Manuel Lozano , Francisco Herrera , Natalio Krasnogor , Daniel Molina, Real-coded memetic algorithms with crossover hill-climbing, Evolutionary Computation, v.12 n.3, p.273-302, September 2004 | mutation operator;premature convergence;real-coded genetic algorithms |
590937 | Probabilistic Pattern Matching and the Evolution of Stochastic Regular Expressions. | The use of genetic programming for probabilistic pattern matching is investigated. A stochastic regular expression language is used. The language features a statistically sound semantics, as well as a syntax that promotes efficient manipulation by genetic programming operators. An algorithm for efficient string recognition based on approaches in conventional regular language recognition is used. When attempting to recognize a particular test string, the recognition algorithm computes the probabilities of generating that string and all its prefixes with the given stochastic regular expression. To promote efficiency, intermediate computed probabilities that exceed a given cut-off value will pre-empt particular interpretation paths, and hence prune unconstructive interpretation. A few experiments in recognizing stochastic regular languages are discussed. Application of the technology in bioinformatics is in progress. | INTRODUCTION
Language inference is a classical problem in machine learning, and continues to be an
important and active research topic. The basic problem is, given a set of example behaviours
or strings, automatically infer a corresponding language (grammar, automata,
expression,.) which generates or recognizes those examples. Genetic algorithms (GA) and
genetic programming (GP) have been applied towards languages inference, with varying
degrees of success. Although successful inference is possible, the generic inference problem
is not entirely well-suited for solution by evolutionary search. There are a number
of reasons for this. For example, some genome encodings do not preserve useful language
characteristics during crossover. Even small local changes to such genomes can be catas-
trophic, which does not lend itself well to genetic reproduction and evolutionary search.
An even more acute weakness is that "all or nothing" problems such as the language
inference problem are not entirely natural for GP. An acceptable language inference
minimally requires that the solution language correctly recognize all positive test cases, and
reject all negative ones. This essential criteria may also be supplemented by efficiency con-
cerns, such as a relatively small number of states or grammar rules. The resulting search
space is a difficult one to navigate with evolutionary techniques, due to these stringent
requirements for language correctness and completeness. On the other hand, it is generally
recognized in the GP community that problems which require an "acceptably close"
solution are typically the best candidates for successful solution with GP. Pragmatically
speaking, giving the fitness function a larger degree of freedom for evaluating a successful
solution will substantially increase the chances of the discovery of acceptable solutions.
This research addresses the inference of stochastic regular languages using genetic
programming. Stochastic languages are formal languages with probability distribution
associated with the language set. The stochastic language inference problem is similar to
the classical inference problem, with the additional requirement that the distribution of
strings recognized by the stochastic language conform to some desired target distribution.
At first, this may seem intuitively more complex than non-stochastic language inference,
since it is unclear what impact the determination of probability distributions has on the
It turns out, however, that the inclusion of string distributions can simplify the
inference problem. Hypothesized languages are now allowed to generate erroneous strings
so long as they fall within an acceptably small probability of occurence. In other words, the
use of language distributions introduces a more generous degree of freedom for generated
solutions. This is ideal in a GP setting, as it simplifies the search space substantially for
evolutionary search.
The target language used here is a probabilistic regular expression language, henceforth
called Stochastic Regular Expressions (or SRE). Although theoretically weaker than
stochastic context-free languages studied elsewhere, it was nevertheless chosen due to both
its amenability to concise GP representation, and its ability to naturally solve the substantial
number of problems in the "regular language domain". The stochastic regular
expression language is closely related to stochastic regular grammars and stochastic finite
automata, the latter commonly referred to as Hidden Markov Models in the literature.
Some SRE language implementation issues had to be addressed before GP could
be successfully applied to stochastic regular expression problems. Firstly, an efficient implementation
of SRE interpretation was necessary. Interpretation of an SRE expression
requires that the probability of recognizing a given string is generated. Since intermediate
probabilities would be computed during the interpretation of a string, these values can
be used to terminate or prune unproductive interpretation paths whose probabilities are
smaller than some supplied cut-off probability. Given the extensive testing that is necessary
during fitness evaluation, such pruning greatly increases the speed of GP runs. The
SRE language is implemented in a grammatical GP system, which permitted the use of
syntactic language constraints to further enhance evolution efficiency.
Two example experiments proved that probabilistic language inference is indeed
possible with SRE and GP. The more complex of these experiments indicated that the complex
search space often resulted in premature convergence. A minor language enhancement
to this experiment resulted in failed inferences by the GP system. From this experience,
it can be deduced that the fitness evaluation strategy used here is not a general purpose
solution to all stochastic language problems, but rather, is suitable to a class of stochastic
regular languages whose members are structurally related to one another.
An outline of the paper is as follows. Related work is reviewed in section 2. Section
3 defines the syntax and semantics of the stochastic regular expression language, and
discusses the algorithm for processing SRE expressions. Section 4 outlines the genetic programming
system used. Two example experiments are discussed in section 5. A discussion
and future directions conclude the paper in section 6.
Formal language induction has a long history as a fundamental problem in machine learning
and Booth 1975a, Fu and Booth 1975b, Angluin 1992, Sakakibara 1997). The specialized
topic of stochastic languages has also been studied for some time (Fu and Huang 1972).
A stochastic grammar differs from a conventional grammar in that each grammar rule is
marked with a probability associated with its use, and the set of probabilities for a grammar
encode a probability distribution for the resulting derived language. (Fu 1982) has
an extensive treatment of stochastic grammars, their derivation, and their application in
pattern recognition. Stochastic grammars are also more complex than their non-stochastic
kin, as the distributions inherent with the language introduce a new dimension of membership
criteria. For example, all context free languages are also stochastic context free
languages (all probabilities are 1); however, there may be many stochastic context free
languages having essentially the same membership set, but vastly different distributions
over that set. Language equivalence issues are therefore more discriminating than in a
non-stochastic setting. Stochastic context free languages enjoy both expressitivity and
tractable properties, for example, the existence of useful inference algorithms (Lari and
Young 1990). They have also found practical use in language processing (Charniak 1993).
Stochastic regular languages, albeit descriptively weaker than stochastic context-free
languages, have also found their practical niche in applications. Regular languages
are definable by finite automata, regular grammars, and regular expressions (Hopcroft and
Ullman 1979). Similarly, stochastic regular languages are defined by stochastic versions
of these three representations. Examples of work in stochastic grammar inference is in
(Maryanski and Booth 1977, van der Mude and Walker 1978, Carrasco and Forcada 1996,
Carrasco and Oncina 1998). Stochastic finite automata are defined in terms of Hidden
Markov Models (HMM) (Rabiner and Juang 1986). An HMM is a finite automaton with
probabilities marking the transition links between nodes. Each node is connected to all
other nodes, and so the network itself is maximally connected. When particular transitions
are not required, the probabilities associated with those nodes are set to zero. HMMs have
found extensive use in language and speech processing (Rabiner 1989, Charniak 1993).
Strangely enough, stochastic regular expressions have not been extensively studied; one
example paper is (Garg et al. 1996).
Language inference has been successfully done using genetic algorithms (GA) and
genetic programming (GP). The distinguishing difference between GA and GP approaches
is one of denotation: a pure GA uses a binary encoding for the genome, while a GP uses a
variable-sized parse tree. Some of the following use encodings with characteristics of both
approaches.
With respect to regular languages, an early work in evolving finite automata is in
(Zhou and Grefenstette 1986). They used a GA with a binary encoding of the automata
as a set of state transitions, capped at a size of 8 states. A weakness of this encoding is
that the represented automata are susceptible to destructive effects during crossover and
mutation. Their unspecified fitness function scores language performance (ability to accept
positive strings and reject negative examples) and automata size.
(Dunay et al. 1994)'s approach is similar to (Zhou and Grefenstette 1986), except
that finite automata are denoted in GP-style nested S-expression notation.
(Dupont 1994) uses an automata-theoretic partition representation for regular
languages. This has the advantage of preserving language properties of chromosomes
during GA reproduction, unlike the more fragile FA represention in (Zhou and Grefenstette
1986). His fitness function scores both language performance and automata size. He
successfully evolved a large set of regular languages, including the benchmark Tomita
languages (Tomita 1982).
(Brave 1997) uses an abstract "cellular encoding" representation for deterministic
FA's, which builds the network structure of a FA during interpretation. The intention of
this denotation is to preserve structural properties of a language during evolutionary re-
production. His automata are embellished with boolean operators which permit automata
composition. The fitness function tallied the number of correctly classified sentences. All
but one of the Tomita languages were successfully inferred using this technique.
(Longshaw 1997) uses a straight-forward state-transition representation for au-
tomata. However, his GA uses a population seeded with correct but overly general au-
tomata. Specialized reproduction operators manipulate automata by duplicating or refining
states. The overall intention is to refine the general automata into a more specific one
for the language in question. His fitness function scores example classification performance
and automata size.
(Svingen 1998) uses a GP on regular expressions. Regular expressions are directly
encoded as program trees, and fitness is based on correct example classification. He
successfully evolved the Tomita languages.
Context-free languages have also been studied. (Wyard 1991) uses a GA to evolve
context-free grammars. Chromosomes takes the form of lists of production rules, which
guarantees correctness at all times. The fitness function scores example classification per-
formance. Two simple CFG's were successfully evolved.
(Lankhorst 1994) uses a vector encoding to represent productions. His
fitness function is more involved than most others, as it scores example classificaton perfor-
mance, the length of substrings of examples correctly classified, the degree of determinism
of grammars, and the ability of the grammar to generate correct strings not included in
the example set. These additional evaluation considerations give the GA more information
with which to drive evolution. He applied the GA to a number of CFG and regular
languages.
(Lucas 1994) uses a binary-encoded normal form for CFG productions, which
preserves language properties during reproduction, and promotes convergence. His fitness
strategy scores example classification and grammar size.
(Sen and Janakiraman 1992) applies a GA towards inferring deterministic push-down
automata, which is an alternative to the grammar representation for CFG's. Fitness
scores example recognition performance, and whether the PDA attempts to erroneously
'pop' an empty stack. (Lankhorst 1995) extends this idea towards nondeterministic push-down
automata. His fitness additionally considers prefix sizes and the stack size after a
string has been consumed.
(Dunay and Petry 1995) use a Turing machine representation in their GA exper-
iments. Although this powerful notation can denote the entire set of languages in the
Chomsky, it does not necessary mean that search will be easy to accomplish, given the
inherent enormity of the search space in question. To solve some relatively simple examples
of regular, context-free and context-sensitive languages, they used a compositional
approach, in which the GA had access to TM building blocks evolved in earlier runs.
Finally, the evolution of stochastic languages has been studied. (Schwehm and
Ost 1995) uses a GA for evolving stochastic regular languages. Two different encodings are
studied - production rules with probabilities, and quotient automata. The fitness function
uses grammar complexity (number of productions), a modified - 2 test for distribution
conformance, and a measure of the grammar's ability to accept prefixes of the target
grammar. A few experiments were performed, and their GA performance compares well
with standard regular-language inference algorithms.
(Kammeyer and Belew 1997) uses a GA to evolve stochastic context-free gram-
mars. They use a liberal representation for grammars in which correct grammars are parsed
from the genome when evaluated; this permits intron or junk material to be included in
chromosomes. The fitness function evaluates the size of test example prefixes consumed
by a grammar, and uses cross-entropy to evaluate distribution conformance. They also use
a local search technique for finding production probabilities during evolution. A couple of
CFG's were successfully evolved.
3.1 Language Definition
The target language for the GP system is stochastic regular expressions, or SRE. The
language is very similar to one in (Garg et al. 1996), which is used for modeling the
qualitative behaviour of stochastic discrete event systems. Amongst other properties, they
prove that probabilistic regular language operations such as choice, concatenation, and
Kleene-closure forms a closed language, and hence an algebra. Although a few basic
properties will be illustrated here, the reader is referred to (Garg et al. 1996) for further
details. It is assumed the reader is familiar with basic concepts from formal language
theory (Hopcroft and Ullman 1979).
Two language variations, SRE and Guarded SRE (or gSRE), are used. We first
define SRE. Let ff range over alphabet
positive integers (0 - n - 1000), and f range over decimal values with a precision of 2
decimal places (0 - f ! 1:00). The syntax of SRE is recursively defined as:
Without loss of generality, the empty string ffl is not included in the alphabet.
The operators have the following meaning:
1. Atomic action ff : The action ff is generated.
2. Choice
This denotes a probabilistic choice of terms. Each choice expression
can be chosen with a probability:
For example, given the expression E (5), the term E 1 can be chosen with a
probability of 3=8 and E 2 with a probability of 5=8.
3. Concatenation "E followed by that of E 2 .
4. Kleene Closure can be repeatedly executed 0 or more times, and each
iteration occurs with a probability of f . The probability of E terminating execution
5. +Closure once, after which it repeatedly executes 0 or
more times using the same probability scheme as Kleene closure. +Closure is an
abbreviation for the following:
The Guarded SRE language is identical to SRE, except that a guarded choice
operator is used instead of the general choice in 2 above:
6. Guarded Choice
Here, each term in the choice expression is either prefixed with a unique atomic
action that is found nowhere else in the expression, or consists of a unique action
by itself. This makes guarded choice deterministic, unlike SRE's nondeterministic
choice.
Note that, even with guarded choice, gSRE is still a nondeterministic language, since the
closure operators are nondeterministic. The rest of the discussion in this section pertains
equally to both SRE and gSRE.
A derivation of a conventional regular expression E is the set of sentences, or
strings over the alphabet, derivable from it. This defines the language L(E) of E. This
notion of language derivation is similarly applicable to SRE, except that each string has
a probability value associated with it, and hence the language itself is associated with a
probability distribution of its members.
Alternatively, an intuitive way to consider SRE expressions is that every expression
defines a specific probability function over strings in \Sigma :
Using a denotational semantics style of representation (Stoy 1977), the probability function
for SRE expression E is denoted by [[E]], and its application to a particular string s is
denoted [[E]]s, which denotes the probability associated with string s in the language
L(E).
A probability function model of SRE is now given. Let
ffl Atomic actions:
(1)
ffl Choice (including guarded choice):
(2)
Since every term might recognize s, the overall probability for a choice expression is
the sum of all the term probabilities with respect to s.
ffl Concatenation:
In the first summation, s is decomposed into two substrings, each of which may
be consumed by a concatenated expression. Even though one term may recognize
its substring argument, if the other term does not recognize its respective substring,
then that term returns a probability of 0, and the overall probability for that instance
of decomposition is 0. The rest of the formula represents the cases when one entire
expression consumes s, while the other consumes ffl. If these other cases do not
succeed, then they return 0.
ffl Kleene closure:
The first formula accounts for empty strings, as the only way an iterated expression
should recognize an empty string is by not iterating. The other formula recursively
defines the general case. Here, one iteration of E will consume some portion of s,
and the rest of s is consumed by further iterations. The final term in this formula
represents when the first iteration consumes the entire string. It is assumed that an
iteration of a loop always consumes some non-empty string. Otherwise, the semantic
model would have to account for Kleene closure iterating indefinitely on an argument,
which is not useful behaviour.
ffl +Closure:
This is similar to the non-empty argument formula for Kleene closure, except that
the expression E will consume part of the string before iterations commence. This
can be seen by the lack of f value in the formula.
The nondeterministic nature of regular expressions is modeled in the above by
multiple argument decomposition in the concatenation and closure operators. Nondeterminism
can also arise in the (nonguarded) choice operator.
Membership in SRE is reflected by SRE expressions returning non-zero probabilities
for particular strings:
Definition 3.1 All probability functions pf must adhere to the following two characteristic
(Subrahmaniam 1979):
(i) for all x i in the sample space of the experiment:
(ii) For every event
(7)Consequently, if SRE expressions are to define well-formed probability functions,
all expressions must similarly respect these requirements.
Theorem 3.1 The SRE operators are well-formed probability functions.
Proof: The proof uses structural induction on SRE expressions. We show conditions (i)
and (ii) of Defn 3.1 hold for all operators. Let s 2 \Sigma .
(a) Atomic actions: trivially.
(b) Choice: i. From equations 2 and
By the induction hypothesis, X
Thus we have, X
ii. From equation 2, the greatest value for the sum
occurs when k. In this case, the sum reduces to
Equivalently, when all the summation is zero. And when any 0 !
the resulting summation is a fraction between 0 and 1. Hence it is a probability.
(c) Concatenation: i. From equation
Using equation 3, because s i ranges over all \Sigma , this becomes:
This translates to:
By the induction hypothesis, this simplifies to:
ii. Given a concatenation,
By the induction hypothesis, each of E 1 and
Hence their product must likewise be a probability.
(d) Kleene closure:
i. Starting with equation
Using equations 4, it translates as follows:
By the inductive hypothesis:
Doing some algebraic manipulation:
f
Note that the division by f \Gamma 1 is permitted because f ! 1 by definition.
ii. By induction on the length of a string s, it can be shown that
The base case is when ffl, in which case the probability is f from the first
equation in 4, and 1. For an arbitrary s 6= ffl, the probability from the second
equation in 4 is:
By incorporating the second term into the first term's summation, this is rewritten:
By the inductive hypothesis over s,
a probability. Furthermore, by the structural induction of expressions,
probability. Hence their product with f is a probability.
+Closure: Similar to (c) and (d) above.3.2 Implementation of an SRE Processor
Given a regular expression, determining whether particular strings are members of its language
is a tractable problem (Sipser 1996, Hopcroft and Ullman 1979). There are different
ways in which this may be performed. One technique is to convert the regular expression
into an equivalent nondeterministic finite automaton, which can be done in polynomial
time. Once this is done, a graph-searching algorithm reads a string character by charac-
ter, marking states of the FA that are still elligible as paths towards an acceptance state.
An advantage of the FA approach is that the nondeterministic FA can be polynomially-
time translated into a deterministic FA, which will then have more efficient recognition
characteristics during language recognition.
Alternatively, regular expressions can be symbolically interpreted directly. The
behaviour of each regular expression operator has a corresponding operational semantics,
which can be used to define a regular expression interpreter. This may be done from
the perspective of either language generation or language acceptance. One technical requirement
of the expression interpretation approach is that the interpreter must be able
to handle the nondeterministic nature of expression derivations, since regular expressions
are naturally nondeterministic in nature. The expression interpretation is similar to the
FA approach, in that there is a mapping between the states of a translated FA and the
derivation paths used by the interpreter when processing an expression.
Stochastic regular language recognition uses the same basic recognition schemes
as conventional regular languages, with the additional requirement that probabilities be
computed for strings. For example, if a FA is derived for a stochastic language, then the
links are marked with probabilities. The overall probability of accepting a given string is
then computed by computing the product of all the transition probabilities used from the
start state to the final accepting state. This probabilistic FA is known as a Hidden Markov
Model or HMM (Rabiner and Juang 1986). Therefore, given a stochastic regular language
as defined by SRE, the formulae of section 3.1 are incorporated into a translated FA or a
The SRE recognition system uses the expression interpretation approach described
above. The operational semantics use two relations. One relation, \Gamma! over E \Theta (\Sigma; p) \Theta E,
where p is a probability, represents single action transitions of expressions. This relation
is denoted,
The other relation, =) over E \Theta (\Sigma ; p) \Theta E, is the transitive closure of \Gamma! , and models
the generation of strings:
Figure
contains transitional rules for the relations, which define the structural
operational semantics of the SRE operators (Hennessy 1990). These inference rules define
an abstract interpreter for SRE expressions, and are the basis of an SRE recognizer. In
fact, with languages such as Prolog, these rules can be compiled into Prolog statements,
and then directly interpreted using Prolog's inference engine (Clocksin and Mellish 1994).
Furthermore, multiple solutions are obtained for nondeterministic SRE expressions using
Prolog's builtin backtracking mechanism.
The actual implementation of the SRE processor uses the above fundamental
ideas. The operational semantics implemented are a superset of the rules in Figure 1. The
Action ff
F (ff;p)
+Closure
Figure
1: Transitional semantics of SRE
implementation uses a logical grammar definition of SRE, which is part of the DCTG-GP
system (Ross 1999) (see Section 4). Prolog's backtracking is advantageously used to
investigate different paths of an expression's derivation. In addition, string recognition
is performed by pattern matching on an argument string and the generated string as
shown in the transitional semantics: when a match occurs, the current derivation path
is correct, while mismatches cause the current derivation to backtrack and test another
possible nondeterministic path. For example, one instance of backtracking may try different
terms in a Choice expression, while another may unwrap an iterative expression a varying
number of times. Such backtracking is assured of terminating because of the finite size of
input strings to be checked, as well as the assertion within the SRE semantics that empty
strings ffl can never be generated within the generative component of iterative operators
(they can only be generated when the iteration terminates).
One advantage of a stochastic language is that the computed probabilities of
strings can be used as an efficiency mechanism during expression recognition. The implementation
of the SRE recognizer is such that the probability of intermediate strings are
always known throughout the interpreter. When the current probability becomes smaller
than a user-supplied threshold, the current derivation path can be forced to terminate.
This prunes derivations of an expression which yield probabilities too small to be of conse-
quence. Of course, setting this threshold too large results in inaccurate probability values
for recognized strings, and may even erroneously reject legal strings. However, for many
experiments, especially with large strings to be recognized, this speeds up processing significantly
4.1 Grammatical SRE and gSRE
The GP system used for the SRE experiments is the DCTG-GP system (Ross 1999).
DCTG-GP performs grammar-based genetic programming, in which the target language
of the evolved program population is defined in terms of a context-free grammar (Lucas
1994, Whigham 1995, Wong and Leung 1995, Geyer-Shulz 1997). A major advantage
of grammatical GP systems is that the search space is syntactically constrained so that
evolution is given a helpful push towards program structures that are more sensible for
the problem at hand. The grammar used by DCTG-GP is a definite clause translation
grammar, or DCTG (Abramson and Dahl 1989). A DCTG is a logical version of a context-free
attribute grammar. Each DCTG production has a syntactic component, which defines
a context-free syntax rule. In addition, each production can have included with it one or
more semantic components. A semantic component defines some characteristic of syntactic
component to which it is attached. For example, one important SRE characteristic that is
defined in the DCTG grammar is the string recognition algorithm of Section 3.2. Since the
operational semantics of the SRE operators are very modular in nature, their recognition
behaviours can be encoded with the grammar rules that define the syntax of the operators
themselves. The overall result of this is a compact definition of the SRE language, in which
the syntax and semantics are conveniently unified together.
One syntactic constraint applied to both SRE and gSRE in the experiments in
section 5 is the following. Although not specified in the grammar of SRE (or gSRE), the
grammatical definition of SRE disallows iterative operators to be directly nested within
one another. In other words, expressions such as
are not allowed. The reason for this restriction is a pragmatic one. When GP was performed
without this restriction, many programs had multiply nested iterative expressions.
Such expressions are relatively expensive to interpret, due to the variety of nondeterministic
paths possible for interpreting them. In addition, nested iteration typically results in
strings with very low probabilities, since there is a probabilistic factor f associated with
executing every nested iterative expression. Moreover, the expense of nested iteration is
not justified by results, since any of these expressions can be replaced with a semantically
equivalent expression that uses only one iterative operator. This restriction does not imply
that an expression like
is illegal, since the concatenation operator means that the iteration operators are not
directly nested.
gSRE is also encoded as a syntactic constraint of SRE. Rather than permitting
any SRE expression as a term within a choice operator, only uniquely guarded terms are
permitted.
4.2 Other GP System Details
DCTG-GP uses standard GP strategies, such as tournament or roulette-wheel selection,
and steady-state or generational evolution. Relevant experimental parameters will be
illustrated in section 5. The system is implemented in Sicstus Prolog 3 on both Windows
and Silicon Graphics platforms.
5.1 General Strategy
The inference of a stochastic language can be considered to involve two different objec-
tives. Given a training set of positive (and possibly negative) examples, one task is to
infer a language which correctly classifies the training examples. This is equivalent to
non-stochastic language inference. An additional task required for stochastic language in-
ference, however, is to ascertain the stochastic distribution of the training examples. One
might naively presume that a statistical analysis of the training set could be performed,
and the results applied to the inferred language. Unfortunately, the situation is typically
more complicated than this, because the representation of the stochastic language as used
in the hypothesis will not likely permit a straight-forward application of the final string
distributions to its internal encoding. For example, if an HMM representation is used in
hypotheses, finding appropriate probability values for intermediate links in the network
that will correspond to the example set distribution is a challenging task. The significance
of the problem of determining distributions for HMM's and context-free languages has
spawned specialized training algorithms (Lari and Young 1990, Charniak 1993).
The inference strategy undertaken with the GP experiments is to let evolution
determine stochastic distributions in concert with example classification. Since SRE incorporates
probability values directly in expressions, treating numeric probability fields
is straight-forward in GP. It was found that this approach was sufficient for many experiments
undertaken. In fact, it was discovered that evolution using local search for
fine-tuning probability parameters lent no advantage over simple evolution of the parameters
The training sets used in the GP experiments consist of sets of positive examples
for the target language to be inferred. Each member of the set is a string, along with its
frequency with respect to the total number of strings in the set (typically 1000). Since the
format of the target languages is already known via a stochastic regular expression or gram-
mar, generating these sets is straight-forward. Unlike conventional language inference, the
probability distributions in training example sets permits stochastic languages to
forgo the need for negative examples. This is because the inference of a distribution that
matches that of the training set will automatically account for 'negative examples', which
have 0 probability in the distribution.
Stochastic language inference incorporates an implicit degree of error in any inferred
solution. This has ramifications on the GP fitness evaluation described below. It
also can be used to boost efficiency of computations performed during inference. As detailed
in section 3.2, string recognition can be pre-empted when intermediate probabilities
become smaller than some threshold limit set for the experiment. Similarly, the test set
can be pruned of strings whose frequency is below some limit set by the user. This limit
parameter should be set with the recognition threshold in mind. For example, if the threshold
is set to 0:001, then the test set limit could be likewise set to 1 for a test set of size
1000. Of course, there may be many nondeterministic derivations of an expression when
recognizing a string, and all the probababilities of these derivations will be summed to an
overall probability for that string. The less discriminating the recognition threshold and
test set limit, the more precise (albeit slower) the results.
Since GP experiments use a steady-state algorithm, there are not any discrete
generations. For convenience, however, a new generation is said to have occurred every
K reproductions, where K is the population size. Between generations, the test set is
regenerated. This prevents overfitting to one set of test data, and reflects the nature of
the stochastic languages, as each test set reflects a sampling from the actual distribution.
One disadvantage, however, is that a discrete test set is an approximation of the real
distribution of the language, and hence this introduces an unavoidable measure of noise.
This noise is compensated by the fact that multiple test sets are used during successive
generations, and their cumulative effect should reflect a more accurate model of the target
distribution. However, the population is not reevaluated for each newly generated test set,
and so the fitnesses of much of the population may be legacy values from earlier generations.
This is acceptable, because the test sets used for those generations are presumed to be as
statistically valid as those from any other generation.
The fitness evalution strategy used in the experiments is a modified - 2 test (Press
et al. 1992). The known distribution is taken to be the set T of test examples, and the
experimental set will be the results of the SRE recognition algorithm on each member
Each test set example string is given to the SRE processor, and an overall probility
that string is computed. Non-membership is reflected in a probability of 0. The
fitness formula is:
where d i is the frequency of example t i in test set T , is the
maximum prefix of t i recognized. The first term is the - 2 formula, and it is used when
the example string t i is completely recognized. The second formula is used when only
a prefix of t i is recognized. Its value is inversely proportional to the size of the prefix
recognized. Should none of t i be recognized, then this value becomes 2 \Delta d i (a normal
formula would use just d i ). This prefix scoring gives credit to expressions that recognize
portions of the examples, which helps drive evolution towards expressions that recognize
complete examples.
5.2 Experiment 1: Stochastic Iteration
The first experiment uses a simple stochastic regular language which can be naturally encoded
in SRE. The main intention of this experiment is to test the evolvability of stochastic
Kleene closure as modeled in SRE. The target language is a stochastic rendition of a regular
language suggested by Tomita 1 from his popular benchmarks for machine learning
(Tomita 1982). The target language written in SRE is:
This is a non-trivial language, especially in the stochastic domain, as the overall distribution
of each a and b term in all the strings should conform to the given probability of 0:5.
These terms may also generate empty strings, should iterations terminate immediately.
The parameters for the experiment are in Figure 1. Most are self-explanatory, and
the fitness function strategy was discussed earlier in Section 5.1. The initial population
is oversampled, and the running population is pruned from it using tournament selection.
Replacement is done using a reverse tournament selection (a sample of K members are
randomly selected, and the member with the lowest fitness is selected to be replaced).
His language is a b a b .
Table
1: Parameters
Parameter Value
Target language gSRE
Fitness function modified - 2
Generation type steady-state
Initial population size 750
Running population size 500
Unique population members yes
Maximum generations 50
Probability of crossover 0.90
Probability of mutation 0.10
Probability internal crossover 0.90
Probability terminal mutation 0.75
Probability numeric mutation 0.50
mutation range \Sigma0.1
Max reproduction attempts 3
Initial population shape ramped half&half
depth initial popn. 6, 12
Max depth offspring 24
Tournament size 5
Test set size 1000
test string size approx. 20
Min test example frequency 3
probability limit 0.0001
Mutation is performed on either terminal or nonterminals. If a nonterminal is to
be mutated, there is a 0:5 probability that it should be a numeric field. When a numeric
field is selected for mutation, its current value is perturbed \Sigma10% of the entire range for
that numeric type (a range of \Sigma100 for integers, and \Sigma0:1 for probabilities).
A test set is generated before every generation. Initially, 1000 strings are generated
for L 1 , and their frequencies are tallied. The maximum string size is approximately 20
(some may exceed this length). Should there be less than 3 instances of a given string, it
is pruned from the test set. This means that there are typically between 55 to
strings in the test set, each of which has its particular frequency for that particular sample
of the language. The number of unique strings in the test set is important for - 2 analyses,
as it is equivalent to bin size in the - 2 formula.
Table
2: Summary L 1
Total runs 15
# unique examples
Avg. test set - 2 142.22
(50 cases)
Fitness min 89.4 (- 2 =88.8)
Generation
Fitness
Average
Figure
2: Fitness curves (avg 15 runs)
Summary statistics for the best solutions from 15 runs are given in Figure 2. These
values are obtained using a common test set, since each run will have used a different test
set during its prior evolution. An average - 2 test of the test sets themselves is included, in
order to better evaluate the expression results. 50 pairs of random test sets were generated.
One of the pair was fixed as the independent variable, while the other was the dependent
variable. The sets were filtered for frequencies below the minimal test example frequency
in
Figure
1), and the - 2 was computed. The resulting 50 - 2 values were averaged.
A performance chart of the best and average population fitness averaged for 15
runs is in Figure 2. It can be seen that convergence to a local optimum has largely occured
by generation 10.
The best solution found
a
This is a nearly perfect solution, and the iterative probabilities within the range of what
might be expected given the stochastic error inherent with the random test sets. The
second best solution (- 2 =89.63) is:
a
The last term is interesting, in that the erroneous choice of a is not too acute a problem,
given the low probability of choosing it (0.11).
One of the poorer solutions (- 2 =132.85) is:
The inaccuracy occurs with the first term, which erroneously permits b to occur too fre-
quently, even though the low probability of 0:25 for the enclosing iteration helps reduce its
likelihood.
The worst solution obtained (- 2 =203.75) is:
(a
Note the repetition of particular numeric fields, such as 320 and 0.04, which is a sign of
population convergence. Simplifying this expression by removing iterative probabilities
less than 0.10 and expanding +Closure terms, it becomes:
a
which is obviously a suboptimal solution. This example shows the nature of introns within
expressions: virtually any expression can be intron code, so long as the associated
choice or iterative probability is low enough.
a A (0.4)
Figure
3: Target language
5.3 Experiment 2: Stochastic Regular Grammar
The second experiment evolves a more complex stochastic regular language. The target
language L 2 is taken from (Carrasco and Forcada 1996), and is defined by the stochastic
regular grammar in Figure 3. Each production has a probability on the right, which
denotes the probability that rule is selected with respect to the other productions for that
nonterminal.
Table
3: Summary L 2
Total runs 50
# unique examples 35
Avg. test set - 2 99.75
(50 cases)
Fitness min 66.39 (- 2 =65.06 )
The experimental parameters for these runs are identical to those in Figure 1. The
summary for 50 runs are in Figure 3. A performance plot for the best fitness and average
population fitness averaged for the 50 runs is in Figure 4.
The best solution (-
Simplifying by expanding +Closures and removing terms with probabilities less than 0.03,
Generation
Fitness
Average
Figure
4: Fitness curves (avg 50 runs)
this becomes:
It is difficult to see how this expression maps to the target grammar of Figure 3, and an
intuitive mapping may not even exist. However, its - 2 is impressive compared to the test
set average.
5.4 Limitations
Many language inference algorithms are easily thwarted by target languages having characteristics
antagonistic to the peculiarities of the algorithm in question. Often, these
languages are only subtlely different from ones that the algorithms have no problems inferring
The GP paradigm suffers a similar limitation. A variation of the language in
section 5.3 was tried,
which is just language L 2 with an additional string bbaaabab with probability 10%. 50
runs were performed using the same parameters as figure 1. None of the runs found an
acceptably close solution: the best solution had a fitness of 259 and -
40).
One reason that GP had problems evolving L 0
2 can be attributed to the linguistic
characteristics of SRE. Even though the above definition of L 0is a concise statement of
the language, the evolutionary process tries to unify the term bbaaabab and L 2 together in
a regular expression. This is difficult to do, because this string is an anomaly with respect
to the other strings in L 2 . Considering the stochastic regular grammar used to generate
is clear that strings are derived progressively and incrementally from one another,
and so strings of L 2 equal in length to bbaaabab are natural extensions of smaller strings
of the language. The anomalous string, however, is not derivable from L 2 , and hence a
natural model of the union of these languages in SRE cannot be inferred. This is especially
true given that bbaaabab has a 10% probability, which makes it a populous member. If it
had a smaller probability, it might be ignored as noise.
The above must be considered in light of the linguistic nature of all formal lan-
guages: some representations more naturally model particular languages than others. Even
though regular expressions, finite automata and regular grammars have the same expressive
power, some languages are more naturally and concisely denoted by regular expressions
than by finite automata, and vice versa. It could be the case that another representation
language, for example HMM's, may more naturally denote L 0
than SRE.
6 CONCLUSION
This paper presented a new means for evolving stochastic regular languages. Using a
probabilistic version of regular expressions as a language for evolution, genetic programming
is capable of evolving accurate expressions for stochastic regular languages. However,
some stochastic regular languages are more amenable to successful evolution than others.
It can be speculated that languages in which members have structural similarities with
one another are the most suitable for this paradigm. For more complex languages, more
sophisticated evolutionary techniques may be required.
It was found during experimentation that SRE had no evolutionary advantage
over gSRE with respect to the quality of solutions discovered. On the other hand, SRE
expressions were less efficient to process, and runs took much longer than the gSRE ones.
The use of SRE in a genetic programming context presents advantages over other
evolutionary experiments with stochastic languages. One advantage is that SRE is akin
to a programming language, with operators that have syntactic and semantic definitions
akin to conventional languages. Since GP is typically applied towards such languages as
Lisp, the encoding and processing of SRE within a GP environment is straight-forward.
More importantly, however, is that SRE has linguistic advantages over finite automata
and regular grammars: some stochastic languages are more naturally encoded in SRE
than these other representations. The L 1 experiment is a clear example of this point.
The linguistic clarity of L 2 is less apparent, although the solution is not overly complex
compared to the target grammar.
Like (Svingen 1998), this work uses a regular expression language directly for GP.
His work required fairly large populations and parallel populations in order to evolve the
Tomita languages. The fitness strategy used here is similar to that used in (Lankhorst
1994, Schwehm and Ost 1995), in that both language recognition performance and prefix
consumption are taken into consideration.
There are many directions for future work. The GP strategies used here were fairly
conventional, and more sophisticated approaches may be more applicable to stochastic
languages. In the experiments, the wide degree of qualitative variations between runs
indicates that evolution quickly gets stuck at suboptimal solutions. Parallel subpopulations
may help in this regard. Although it was found that local search using hill-climbing over
numeric fields was not advantageous to evolution, it is worth investigating the utility of
more sophisticated local search techniques akin to those used in stochastic context-free
languages (eg. the inside-outside algorithm).
Currently, the applicability of SRE in bioinformatics problems is being investi-
gated. A fundamental problem in DNA and protein sequencing is to determine a common
pattern shared amongst a family of sequences (Brazma et al. 1995), which can be used for
both search and analytical purposes. A number of techniques, such as HMM's and regular
pattern languages, are used for this purpose. SRE is a natural vehicle for this problem area,
since its regular expression basis conforms to the pattern languages commonly used (eg.
that used in the PROSITE database (Hofmann et al. 1999)), while its stochastic features
conveniently model the probabilistic characteristics of DNA sequences themselves.
Acknowledgement
Thanks to Tom Jenkyns for helpful discussions about probability. This
research is supported by NSERC Operating Grant 138467-1998.
--R
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590954 | Decision-Theoretic Planning for Autonomous Robotic Surveillance. | In this paper, we introduce a decision-theoretic strategy for surveillance as a first step towards automating the planning of the movement of an autonomous surveillance robot. In our opinion, this particular application is interesting in its own right, but it also provides a test-case for formalisms aimed at dealing both with (low-level) sensor, localisation, and navigation uncertainty and with uncertainty at a more abstract planning level. After a brief discussion of our view on surveillance, we describe a very simple formal model of an environment in which the surveillance task has to be performed. We use this model to illustrate our decision-theoretic strategy and to compare this strategy with other proposed strategies. We treat several simple examples and obtain some general results. | Introduction
In several projects, robots are employed to perform surveillance tasks. See, for example,
[5, 6, 7]. However, in most of these projects, the robots are typically used as a kind
of flexible sensor platform controlled by some external human operator who makes the
high-level decisions on where to go, and how to use the on-board sensors.
We are interested in automating this high-level decision making process to allow an
autonomous mobile robot plan itself how to perform the surveillance task. Little work
has been done in this area, with the the notable exception of [2, 3]. We believe this
autonomous robotic surveillance application is interesting in its own right, but it also
provides a test-case for formalisms aimed at dealing both with (low-level) sensor, local-
isation, and navigation uncertainty and with uncertainty at a more abstract planning
level.
In section 2, we give our view on what surveillance is. We then give in section
3 a simple formal model of an environment in which the surveillance task has to be
performed. In section 4 we describe several surveillance strategies that have been studied
in [3], and introduce our proposed decision-theoretic strategy. Finally, section 5 contains
some examples illustrating the various strategies and some preliminary results.
Surveillance
In dictionaries, surveillance is defined as a close watch kept over something or someone.
We feel one should add that the purpose of this close watch is to detect the occurrence
of some relevant events. Which events are considered relevant depends on the overall
reason for the surveillance.
Of course, this close watch is not limited to visual means. In principle, any kind of
sensor can be used. Also, the person or thing to be closely watched should be understood
to include areas, places, (parts of) aerospace, et cetera. Familiar examples of surveillance
tasks include the observation of suspected members of a criminal organization by the
police, the detection of airplanes in a no-fly zone by the military, and the look-out for
suspicious behaviour of people in a shop by a security-guard.
There are at least two disctinct difficulties involved in detecting relevant events.
1. The relevant events have to be in sensor range.
2. Once detected, the events have to be recognised as relevant.
For some applications, such as the observation of a place which is in plain view of a
(static) camera (see e.g. [1]), the first difficulty does not arise. The difficulty that
remains here is to recognise or classify particular events as belonging to the class of relevant
events. Such a classification may vary from rather straightforward (e.g. speeding
cars) to quite complex (e.g. suspect behaviour of people in a shop). Although the issue
of automatic classification of events is extremely important, and essential for artificial
intelligence, we concentrate our research on the first difficulty.
As soon as the relevant events are not guaranteed to occur within sensor range, it
becomes essential to get the available sensors in such a position that the relevant events
are (most likely) detected. Mobile robots seem to be excellently suited for automating
the process of moving sensors, since robot navigation and self-localization, although still
not trivial, are nowadays considered fairly standard tasks. Of course, the difficulty of
navigation and self-localization, and therefore of surveillance, depends on the type of
environment in which the tasks have to be performed. But some successes have been
obtained in a variety of environments, ranging from structured in-door environments,
like office buildings, to out-door environments, like aerospace.
The use of robots in surveillance tasks is not new, but in most of the existing projects
in this area, the robots are typically used as a kind of flexible sensor platform controlled
by some external human operator who makes the high-level decisions on where to go,
and how to use the on-board sensors. We are interested in automating this planning
task to make the surveillance robots more autonomous and less dependent on human
control.
Our main concern is to develop strategies or algorithms that make use of the available
recources (sensors, manoeuvring capabilities, et cetera) such that the probability
of relevant events remaining undetected is minimised. (This is only a preliminary description
of the goal of surveillance. See below.) The fact that detecting relevant events
does not guarantee them to be recognised as relevant, can (and should) be modelled by
introducing some uncertainty in the classification of events.
When evaluating surveillance strategies one should compare them with alternative
strategies, but also with alternatives to robotic surveillance, such as the use of security
cameras, smoke detectors, et cetera. Of course, one can also consider combining robotic
surveillance with simple static sensors and, for example, use a surveillance robot to check
out alarms raised by smoke detectors to decrease the amount of false alarms reported
to police and fire departments.
An important issue when evaluating surveillance strategies is to clarify the goal of
surveillance. We cannot simply say that the goal is to detect all relevant events, since
it is not possible to develop strategies that guarantee to achieve this goal in case the
relevant events are not necessarily in sensor range. We used as a preliminary description
that the probability of relevant events remaining undetected should be minimised. This
is similar to the description used in [6].
This description should at least be refined to take into account the fact that some
relevant events can have more serious consequences than others. near a storage
room of highly inflammable toxic material is more serious than a flooding of a rarely
used basement with no electric outlets.) This can be done by saying that not detecting
a relevant event involves some cost, which may vary depending on the event, and that
one should minimise the expected cost of not detecting relevant events.
Using such a decision-theoretic criterion has a number of advantages. For example,
at least from a theoretical point of view, it seems obvious how to further refine this
criterion to incorporate other types of cost, such as the danger of damage to the robot,
or the cost of false alarms (in case the recognition of relevant events is not completely
reliable). However, several serious problems remain.
The first problem is that the expected cost of the whole surveillance plan or decision
policy should be minimised. In general, such a policy cannot be reduced to a sequence of
simple decision problems, but involves looking (far) ahead at (many) possible outcomes
of (many) future actions. Therefore, in realistic applications one will probably need
heuristics and approximate methods to arrive at feasible solutions.
Another problem is that the application of decision theory requires a great amount
of data. The necessary probabilities of events and costs of possible outcomes of actions
are not always readily available. This is a good reason for preferring policies which are
robust, in the sense that they are not influenced much by small changes in the data.
Also, it can be argued that in realistic applications one should settle for satisfactory
policies rather than insist on optimal policies.
It should be mentioned that we explicitly adopt an asymmetrical view on surveillance,
in the sense that the goal can be described as detecting (or minimising the cost of not
detecting) relevant events. In [3], the goal of surveillance seems (at least sometimes) to
be understood as maintaining a maximally correct model of the state of the environment
with respect to both the presence and the absence of relevant events. We call this the
symmetrical view on surveillance.
We believe that for a surveillance robot with, for example, detecting fires or intruders
as primary goal, the asymmetrical view on surveillance is the most appropriate, and
the decision-theoretic formalisation of the goal is the most natural. In [4], the issues
mentioned in this section are discussed in more detail.
3 A Simple Environment Model
In this section we introduce a very simple, formal model of the environment for the
surveillance task.
Definition 3.1 An environment E is a tuple hX; A is a set
ng of mutually disjoint spatial areas, or locations, A 0 2 X is the start
location of the robot, A ' X \Theta X represents the relation of immediate accessibility (for
the robot) between locations, C is a function assigning to each location X i 2 X the cost c i
associated with not detecting a relevant event occurring at X i , P 0 is a function assigning
to each location X that at time 0 an event occurs at
is a set of transition probabilities. That is, for every
contains denoting the (prior) probability that at time t a relevant event
starts at denoting the (prior) probability that the relevant event at
stops at time t.
For the rest of this paper, we assume that for every
A. As a consequence, we can simplify the description of an environment
Further, we assume that P t (X
do not depend on the time t, and we drop the subscript t. We also assume that
the environment is connected, in the sense that for every there is a path
We write r t to denote the sensor range of the robot at time t, and we assume that
then we say that
is visited at time t. The decision strategies should decide which immediately accessible
location to visit next. For the moment, we do not take recognition uncertainty into
account and assume a relevant event to be detected whenever it is in the sensor range
of the robot.
The above definition provides a very abstract model of the decision problem. For
realistic applications, we have to take into account that a robot can have several sensors,
each with its own sensor range, that the sensor range is not necessarily an element of
X , that the actions of the robot may include changing its location, its orientation, and
possibly manipulating aspects of the environment, such as opening a door, and that the
exact state and dynamics of the environment, the exact position of the robot in the
environment, and the recognition of relevant events are uncertain, et cetera. See [4] for
a more detailed discussion of the simplifying assumptions of the given model.
In spite of the many simplifying assumptions, the model is sufficiently general to
capture the abstract environment used in [3] to experimentally compare different surveillance
strategies. In that experimental set-up, all locations are assumed to be immediately
accessible from each other, i.e., . The dynamics at each location is
modelled by a Markov process, and P (X are viewed as transition
probabilities between possible states of a location.
In our opinion, for many typical surveillance applications it is not appropriate to
model the dynamic behaviour of a location X i as a Markov process. For example,
although the start of a fire can be assumed to have an extraneous cause (nature or some
agent other than the surveillance agent), it is typically the case that the detection of a
fire influences the probability of the fire being extinguished. The detection of a relevant
event should trigger some appropriate response of the surveillance agent, such as raising
an alarm, or taking more direct countermeasures against the event.
Not all applications of surveillance are meant to trigger intervening responses to the
observed relevant event. For example, observations made in the context of a scientific
study are primary aimed at information gathering, not at intervening. However, when
interventions do play a role their effects should be incorporated in the model of the
surveillance problem. Since the particular actions triggered by a detection are them-
selves, strictly speaking, not part of the surveillance behaviour of the agent, we will
leave them out of our considerations.
In our examples, we assume that relevant events do not stop spontaneously, but that
they stop immediately when detected by the surveillance agent. Formally, P (X
0, and P t+1 (X It is possible to introduce some time
delay for the countermeasures to take effect, but this raises the problem of deciding how
important it is to monitor areas where relevant events are known or have been observed
to occur. It is also possible to allow P (X and to model the effect of the actions
triggered by observing a relevant event as an increase in P (X Our assumption
can be viewed as an extreme instance of this possibility.
Given our assuptions, it is possible to express P t (X in terms of P (X
and the amount of time that has passed since the last visit to X i .
Proposition 3.1 be an environment where P
implies that P t+1 (X
, where t 0 is the largest time point - t such that r t
4 Surveillance Strategies
In [3] a surveillance strategy is proposed based on the newly introduced notion of confi-
dence, which can be viewed as a second-order uncertainty measure. Whenever sensory
information about the state of a location becomes available, the probability of an event
occurring at that location at that time is updated, and one is assumed to be very confident
about this assessment of the state of the location. This confidence then drops
gradually over time during a period in which no fresh sensory information concerning
this particular location is obtained. The rate by which the confidence decreases depends
on the transition probabilities: the more likely the changes, the higher the decrease rate.
Specifically, the factor - p is used as confidence decrease rate, where p is the transition
probability leaving from the observed state and - is some unspecified parameter. The
actually used computation of confidence is slightly more complicated, due to the fact
that some time after the observation it is no longer clear which transition probability
should be used in the computation of the decrease rate.
In our model, the situation is simpler, since we assumed that when visiting a location
X i at time t, the robot either observes that no relevant event occurs at X i or the robot
immediately stops the relevant event. In both cases, the robot can be confident that no
relevant event is going on at X i after t. This confidence can decrease over time due to
the possibility that a relevant event starts after t. This rate of this decrease of course
depends on P (X 1). The transition probability P does not play a role.
Since the factor - P (X i !1) is meant to be a decrease rate, one can infer that
Lemma 4.1 Let my.
It follows that if one assumes that by visiting X i the confidence that no relevant
event is going on at X i becomes 1, then the location with the lowest confidence at time
t is the location X i such that P (X is the time of the
last visit to X i .
The policy proposed in [3] can be described as follows.
maximum confidence Choose the action that changes the sensor range to the neighbouring
location which has the lowest degree of confidence attached to it.
This policy is experimentally compared to the following policies.
random exploration Randomly choose a location as the next sensor range.
methodical exploration Choose all the locations, one after the other, and always in
the same order, as the sensor range at the next moment.
maximum likelihood Choose the action that changes the sensor range to the neighbouring
location with maximal uncertainty, where the uncertainty at location X i
is measured by min(P (X
Notice that both random and methodical exploration, as described above, allow
choosing non-neighbouring locations. Actually, in the experiments of [3] it is assumed
that all locations are directly accessible from each other X). This is only
realistic in case changing attention to a far removed location involves no or only a negligible
amount of cost or time. It is of course not difficult to restrict random exploration
to choosing randomly between neigbouring locations only, but it is not clear how to put
a similar restriction on methodical exploration.
One possible policy that can be considerd to be a local variant of methodical exploration
is the following.
interval Minimise the maximum time interval between visits of locations by
choosing the action that changes the sensor range to the neighbouring location
which has not been visited for the longest time.
We propose to use this minimax interval policy as a kind of reference strategy. Since
this strategy does not use information about the uncertainties, it can be used to clarify
how much other strategies which do use uncertainty information gain in efficiency.
It should be mentioned that in the case of the maximum likelihood policy many
uncertainty measures, including, for example, entropy, give rise to the same preferences
as min(P (X As we will see in section 5, the maximum likelihood
policy seems more appropriate for (symmetrical) surveillance understood as maintaining
a maximally correct model of the state of the environment with respect to both the
presence and the absence of relevant events, than for (asymmetrical) surveillance aimed
at detecting relevant events.
In [3], no explicit choice is made between such a symmetrical view on surveillance and
the asymmerical view we take. Several criteria are used to evaluate the performance of
the strategies in the experiments, including the (symmetrical) criterion of the percentage
of erroneous estimations of the state of each location and the (asymmetrical) criterion
of the percentage of non detected relevant events.
We propose a surveillance strategy based on decision-theoretic considerations. By
decision-theoretic surveillance we understand the kind of behaviour guided by the following
decision policy.
minimum expected cost Choose the action that minimises the expected cost.
This decision policy can be interpreted both globally and locally. Under the global
interpretation, the action that has to be chosen corresponds to the behaviour of the
surveillance agent from the start to the end of the surveillance task. There is not an
inherent end to a surveillance task, but in practice each particular task has a limited
duration (say, until the next morning when the employees return to the office building,
or until the batteries of the robot have to be recharged).
The (global) expected cost EC T until time T can be computed by the following
formula.
Notice that a choice to visit X i at t not only removes the term P t (X
the above sum, but it also has some indirect benefits, due to the fact that it reduces
t.
The behaviour of the surveillance agent from the start to the end of the surveillance
task can also be viewed as consisting of a sequence of simpler actions. One can apply the
above decision policy locally to choose at each time between the possible simple actions
by comparing the consequences of these simple actions, or perhaps by comparing the
(expected) consequences of small sequences of simple actions.
Let us say that an n-step policy compares the (expected) consequences of sequences
of n (simple) actions. Of course, the policy is more easily implemented for small n,
whereas, in general, it better approximates the global policy for large n. Since the goal
of surveillance should be interpreted as global minimisation of expected cost, it will
be interesting to study under what conditions the local policy is already (sufficiently)
equivalent to the global policy for small n.
None of the policies considered in [3] takes into account a notion of cost. If the
cost C(X i ) of not detecting a relevant event is the same for all X i , then minimising
the expected cost reduces to maximising the probability of detecting a relevant event.
Notice that this is still different from the maximum likelihood policy.
5 Examples and First Results
We have defined three decision policies for surveillance which make use of some kind of
uncertainty information, namely maximum confidence, maximum likelihood, and minimum
expected cost. The following proposition shows that these policies essentially agree
if there is no (relevant) uncertainty information to be used.
Proposition 5.1 Let Assume that for all
Then the maximum confidence policy
and the one step minimum expected cost policy both reduce to the minimax interval
policy. Also, for sufficiently small transition probabilities P 1), the maximum
likelihood policy will agree with the minimax interval policy.
It follows that we can expect a difference between the mentioned policies only in
the case of varying probabilities (or costs). Our first example illustrates the difference
between the various surveillance strategies introduced so far.
Example 5.1 Consider an environment is a set
consisting of two rooms,
0.8. The strategy based on maximum likelihood will always look
at room X 1 (where the uncertainty is maximal), and will never take a look at room X 2 .
The strategy based on methodical exploration goes back and forth between both rooms,
just as the maximum confidence policy does.
The strategy based on one step minimising expected cost is slightly more complicated.
again 0.8, since room X 2 was
visited at the immediately preceding time step. However, P only increased
to 0.75, which is not enough to get chosen. Only at time 3, P
increased above the 0.8 probability that a relevant event occurs in room 2. We thus obtain
a sequence where room X 1 is only chosen every third time step. See table 1, where for
the first six time steps the expected costs (of not visiting a room) are displayed. The one
step minimum expected cost policy chooses the room with the (maximal) expected cost
printed in boldface.
Table
1: The expected costs of example 5.1.
In this example, the maximum likelihood policy does not result in an exhaustive
exploration of the environment. Both maximum confidence and minimum expected cost
behave better in this respect. The problem with the maximum likelihood criterion is
that P (X not guaranteed to result in a preference to visit X i
rather than X In fact, if P (X
then the criterion prefers X j . Notice that not only in the artificial example above,
but also in practical applications, with sufficiently many locations and sufficiently high
transition probabilities, it can happen that P (X
Since the maximum likelihood criterion does not prefer locations where the chance
of detecting a relevant event is high, but is more interested in locations where the
occurrence of a relevant event is highly unknown, we conclude that the criterion is more
appropriate for symmetrical surveillance than for asymmetrical surveillance.
In example 5.1, the one step minimum expected cost policy results in a behaviour
which seems intuitively appealing, since it clearly reflects the fact that P (X
substantially lower than P confidence, just as methodical
exploration, treats both rooms the same. However, we will see below that this intuitive
appeal may be somewhat misleading.
The maximum confidence policy does also take the probabilities P
account, since the rate of confidence decrease is a function of these
probabilities. However, the decrease rate proposed in [3] does not result in a different
treatment of both rooms in the example. Thus one can view the example as an indication
that in the minimum expected cost policy the probabilistic information is used more
directly and taken more seriously than in the maximum confidence policy.
The maximum confidence policy does not consider at all the possibility that for some
areas it may be relatively more important to detect events. This is easily implemented in
the minimum expected cost policy by letting the cost C(X i ) of not detecting a relevant
event depend on the area X i . Such varying costs may cause a problem, since they may
prevent the one step mimimum expected cost policy to obtain an exhaustive exploration
of the environment.
It can be shown that if the cost of not detecting a relevant event is constant over the
different areas X i , then, in the long run, the one step mimimum expected cost policy
will result in an exhaustive exploration of the environment. More precisely, one can
show the following.
Proposition 5.2 Let Assume that for all
in the long run, every X i 2 X is visited when applying the
one step mimimum expected cost policy, and there is a finite upper bound N i on the
length of the time interval between visits of X i .
If relevant events will not stop spontaneously before they are detected, exhaustive
exploration implies that all relevant events will eventually be detected. But even among
policies that are 100% successful with respect to (eventually) detecting relevant events
there may be a difference in performance if, for example, early detection is considered
to be important.
Table
2: The expected costs of example 5.2.
The following simple modification of example 5.1 shows that, in general, proposition
5.2 is no longer valid (and the one step minimum expected cost policy is no longer
guaranteed to result in an exhausite exploration of the environment) if the costs are
allowed to vary.
Example 5.2 Consider the situation of example 5.1, but now assume C(X 1
3. Then the expected cost of not visiting room 1 has an upper bound of 1,
whereas the expected of not visiting room 2 is 2.4 even when it has been visited the
previous moment. Therefore, by the one step minimum expected cost policy, room 2 will
always be chosen. See table 2, where for the first four time steps the expected costs (of
not visiting a room) are displayed. The one step minimum expected cost policy chooses
the room with the (maximal) expected cost printed in boldface.
This effect of ignoring room 1 can be avoided by allowing the cost of not detecting
an event to grow as a function of the time passed since the event has started. However,
if the mentioned (expected) costs are correct, then this ignoring of room 1 may be
defensible. Intuitively, one should not require a surveillance agent to explore irrelevant
or unimportant areas of the environment. This is substantiated by the following.
Proposition 5.3 In the environment of example 5.2 the one step minimum expected
cost policy minimises the global expected cost.
Perhaps a more important problem than possibly preventing an exhaustive exploration
of the environment is that the varying cost can form an obstacle to obtaining
optimal behaviour using the local (one step) mimimum expected cost policy.
Example 5.3 Consider an environment is a set
consisting of three rooms,
As in example 5.2, the one step minimum expected cost policy will always choose
room 2. But now the (possibly justifiable) ignoring of room 1 will make it impossible to
visit room 0, and in the long run the expected cost of not visiting room 0 will be very
high.
Obviously, such problems can be solved theoretically by looking more than one step
ahead. Since looking ahead many steps is computationally expensive, it would be useful
to develop some methods for assessing the number of steps required to obtain satisfactory
behaviour.
Even for constant costs, the one step minimum expected cost policy is not guaranteed
to globally minimise the expected cost.
Proposition 5.4 In the environment of example 5.1 the one step minimum expected
cost policy does not minimise the global expected cost.
Actually, in example 5.1, the global expected cost of the one step minimum expected
cost policy is higher than that of the back and forth behaviour resulting from
the methodical exploration and the maximum likelihood policy. The two step minimum
expected cost policy already results in the same back and forth behaviour.
The cause of the problem with the one step minimum expected cost policy in our
model is that visiting a location at time t decreases the probability of a relevant event
occurring at that location after t. It is worth investigating whether one can take into
account such indirect benefits of visiting a location by adjusting the costs.
Proposition 5.5 Let t 0 be the time of the last visit to X i before t, and T be the time
of the next visit after t. Then the indirect benefits of a visit to X i at t are equal to the
following.
then the above expression provides a lower bound of the indirect benefits
of a visit to X i at t instead of later. Incorporating this amount of the indirect benefit into
the one step minimum expected cost policy is similar to employing a two step minimum
expected cost policy, and it result in the back and forth behaviour in the environment
of example 5.1.
Proposition 5.6 Let t 0 be the time of the last visit to X i before t. Then the indirect
benefits of a visit to X i at t have the following upper bound.
lim
This upper bound can be used to argue that it is not optimal to visit room 1 in the
environment of example 5.2.
6 Conclusions and Further Work
We do not claim that our discussion of decision-theoretic surveillance establishes the
minimum expected cost policy as the best policy for surveillance. To substantiate such
a claim, one needs to evaluate the performance of different possible strategies when
applied to much more realistic surveillance problems than those discussed thus far.
However, as we mentioned before, we believe that the decision-theoretic view on surveillance
is sufficiently general to, at least theoretically, incorporate many of the necessary
refinements to the model presented in this paper.
To properly evaluate surveillance strategies, one should clarify which kind of surveillance
task one is considering, since the term 'surveillance' is used to refer to quite
different problems. Actually, it may be impossible to find an unqualified best policy
for all surveillance problems. For example, the maximum likelihood policy seems to be
intimately connected with a symmetrical view on surveillance, whereas the minimum
expected cost policy is devised with the asymmetrical interpretation in mind. In gen-
eral, we expect the minimum expected cost policy to behave well in situations where
the probabilities and costs matter and where early detection is important.
We introduced a simple, formal model of an environment in which surveillance tasks
can be performed. The main purpose of this environment model is to clarify the distinctions
between different surveillance strategies. In the future, we plan to refine the
environment model presented in this paper to incorporate the uncertainties a real surveillance
robot has to face: sensor uncertainty, navigation uncertainty, et cetera. A simulation
program is being developed to experimentally evaluate different surveillance
strategies in varying environments. Also, some further theoretical work is needed to
clarify the situations under which local strategies can be used to obtain globally optimal
(or satisfactory) performance.
Acknowledgments
The investigations were carried out as part of the PIONIER-project Reasoning with
Uncertainty, subsidized by the Netherlands Organization of Scientific Research (NWO),
under grant pgs-22-262.
--R
Visual surveillance in a dynamic and uncertain world.
A new approach in temporal representation of belief for autonomous observation and surveillance systems.
Repr'esentation Dynamique de l'Incertain et Strat'egie de Perception pour un Syst'eme Autonome en Environnement ' Evolutif. Dissertation, L' ' Ecole Nationale Sup'erieure de l'A'eronautique et de l'Espace
Issues in surveillance.
On the lookout: The air mobile ground security and surveillance system (AMGSSS) has arrived.
SPAWAR Mobile Detection Assessment and Response System.
AUV survey design qpplied to oceanic deep convection.
--TR | autonomous robots;surveillance;decision-theoretic planning |
590960 | A Reusable Multi-Agent Architecture for Active Intelligent Websites. | In this paper a reusable multi-agent architecture for intelligent Websites is presented and illustrated for an electronic department store. The architecture has been designed and implemented using the compositional design method for multi-agent systems DESIRE. The agents within this architecture are based on a generic information broker agent model. It is shown how the architecture can be exploited to design an intelligent Website for insurance, developed in co-operation with the software company Ordina Utopics and an insurance company. | Introduction
Most current business Websites are mainly based on navigation across hyperlinks. A closer
analysis of such conventional Websites reveals some of their shortcomings. For example,
customer relationships experts may be disappointed about the unpersonal treatment of
customers at the Website; customers are wandering around anonymously in an unpersonal
virtual environment and do not feel supported by anyone. It is as if customers are visiting the
physical environment of a shop (that has been virtualised), without any serving personnel.
Marketing experts may also not be satisfied by the Website; they may be disappointed in
the lack of facilities to support one-to-one marketing. In a conventional Website only a
limited number of possibilities are provided to announce new products and special offers in
such a manner that all (and only) relevant customers learn about them. Moreover, often
Websites do not acquire information on the amounts of articles sold (sales statistics). It is
possible to build in monitoring facilities with respect to the amount of products sold over
time, but also the number of times a request is put forward on a product (demand statistics).
If for some articles a decreasing trend is observed, then the Website could even advice
employees to take these trends into account in the marketing strategy. If on these aspects a
more active role would be taken by the Website, the marketing qualities could be improved.
The analysis from the two perspectives (marketing and customer relationships) suggests
that Websites should become more active and personalised, just as in the traditional case
where contacts were based on humans. Intelligent agents provide the possibility to reflect at
least a number of aspects of the traditional situation in a simulated form, and, in addition,
enables to use new opportunities for, e.g., one-to-one marketing, integrated in the Website.
The generic agent-based architecture presented in this paper offers these possibilities.
This generic architecture for active intelligent Websites was first introduced for the
application domain of a department store, which has been analysed in co-operation with the
software company CMG (cf. [22]). It reuses the generic architecture of information broker
agents developed earlier (cf. [21]), which in turn was designed as a specialisation of the
generic agent model GAM introduced in [8]. As a second step the reusability of the generic
multi-agent architecture for active intelligent Websites has been tested by applying it in a
project on an intelligent Website for insurance in co-operation with the software company
Ordina Utopics and an insurance company (cf. [20]). The testbed chosen for this application
involves information and documents that need to be exchanged between insurance agents
and the insurance company main office. The goal of the intelligent Website is to provide
insurance agents with an accurate account of all relevant available documents and
information. The supporting software agents are able to provide a match (either strict or soft)
between demand and available information. They support pro-active information provision,
based on profiles of the insurance agents that are dynamically constructed. A prototype
system for this application is described in more detail in the second part of the paper.
In this paper in Section 2 the global design of a multi-agent architecture for an intelligent
Website is presented; the different types of agents participating in the Website are
distinguished. In Section 3 their characteristics and required properties are discussed. In
Section 4 the compositional generic information broker agent architecture is described and
applied to obtain the internal structure of the agents involved in the multi-agent architecture.
In Section 5 the insurance application domain is introduced. In Section 6 the application of
the architecture to insurance is discussed in more detail and illustrated by some example
behaviour patterns. Section 7 concludes the paper by a discussion.
Multi-Agent Architecture for Intelligent Websites
In this section a global multi-agent architecture, that can be used as a basis for an intelligent
Website, is introduced. Although the architecture is generic, for reasons of presentation some
of its aspects will be illustrated in the context of the insurance application.
The domain has been identified as a multi-agent domain. Therefore, it makes sense to
start with the agents as the highest process abstraction level within the system. Four classes
of agents are distinguished at the level of the multi-agent system (see Fig. 1):
. customers (human agents),
. Personal Assistant agents (software agents, denoted by PA),
. Website Agents (software agents, denoted by WA),
. employees (human agents).
In Fig. 1, the shaded area at the right hand side shows the agents related to the Website; the
shaded area at the left hand side shows the two agents at one of the customer sites. In this
figure, for shortness only two Website Agents, one employee, one Personal Assistant agent
and one customer (user of the Personal Assistant) are depicted. Moreover, for the sake of
simplicity, the Website itself is left out of the picture. The Website has the role of the
external world for the agents; note that is not considered an agent itself. All agents can have
interaction with this external world to perform observations. The Website agents and
employees can also perform actions in this world, e.g., to change the information on one of
the Webpages.
user
Website
Agent2
Website
Agent1
Personal
Assistant
employee
Fig. 1. The overall multi-agent architecture
Note that the Personal Assistant is involved as a mediating agent in all communication
between its own user and all Website Agents. From the user it can receive information about
his or her interests and profile, and it can provide him or her with information assumed
interesting. Moreover, it can receive information from any of the Website Agents, and it can
ask them for specific information. The Website Agents communicate not only with all
Personal Assistants, but also with each other and with employees. The customer only
communicates with his or her own Personal Assistant. This agent serves as an interface
agent for the customer. If a customer visits the Website for the first time this Personal
Assistant agent is instantiated and offered to the customer (during all visits).
The application domain to illustrate the architecture addresses the design of an active,
intelligent Website for a chain of department stores. The system should support customers
that order articles via the Internet. Each of the department stores sells articles according to
departments such as car accessories, audio and video, computer hardware and software,
food, clothing, books and magazines, music, household goods, and so on. Each of these
departments has autonomy to a large extent; the departments consider themselves small
shops (as part of a larger market). This suggests a multi-agent perspective based on the
separate departments and the customers. For each department in the department store a
Website Agent can be designed, and for each customer a Personal Assistant agent serves as
an interface agent.
3 Requirements for the Software Agents
The departments should relate to customers like small shops with personal relationships to
customers. The idea is that customers know at least somebody (a Website Agent) related to a
department, as a representative of the department and, moreover, this agent knows specific
information on the customer.
Website Agent - Interaction with the world
observation passive
observation active
its own part of the Website
product information
- presence of customers/Personal Assistants visiting the
Website
- economic information
- products and prices of competitors
- focusing on what a specific customer or Personal
Assistant does
- search for new products on the market
performing actions - making modifications in the Website (e.g., change
prices)
showing Web-pages to a customer and Personal
Assistant
creating (personal or general) special offers
- modification of assortment
Table
1. World interaction characteristics for a Website Agent
3.1 Characteristics and Requirements for the Website Agents
Viewed from outside the basic agent behaviours autonomy, responsiveness, pro-activeness
and social behaviour such as discussed, for example in [38] provide a means to characterise
the agents (see Table 3). Moreover, the following external agent concepts to define
interaction characteristics are used:
. interaction with the world (observation, action performance)
. communication with other agents
In
Tables
1 and 2 the interaction characteristics for the Website Agents have been specified
and illustrated for the case of the department store.
Website Agent - Communication
incoming from Personal Assistant:
- request for information
- request to buy an article
paying information
customer profile information
customer privacy constraints
from employee:
- requests for information on figures of sold articles
new product information
- proposals for special offers and price changes
- confirmation of proposed marketing actions
- confirmation of proposed assortment modifications
- proposals for marketing actions
- proposals for assortment modifications
from other Website Agent:
- info on assortment scopes
customer info
outgoing to Personal Assistant:
asking whether Website Agent can help
providing information on products
providing information on special offers
special (personal or general) offers
to employee:
- figures of articles sold (sales statistics)
- analyses of sales statistics
- numbers of requests for articles (demand statistics)
- proposals for special offers
- proposals for assortment modifications
to other Website Agent:
- info on assortment scopes
customer info
Table
2. Communication characteristics for a Website Agent
The following requirements have been imposed on the Website Agents:
. personal approach; informed behaviour with respect to customer
In the Website each department shall be represented by an agent with a name and face.
Furthermore, some of these agents (those who have been in contact with the customer) know
the customer and his or her characteristics, and remember what this customer bought
previous times.
. being helpful
Customers entering some area of the Website shall be contacted by the agent of the
department related to this area, and asked whether he or she wants some help. If the
customer explicitly indicates that he or she only wants to look around without getting help,
the customer shall be left alone. Otherwise, the agent takes responsibility to serve this
customer until the customer has no wishes anymore that relate to the agent's department. The
conventional Website can be used by the Website Agents to point at some of the articles that
are relevant (according to their dialogue) to the customer.
. refer customers to appropriate colleague Website Agents
A customer which is served at a department and was finished at that department can only be
left alone if he or she has explicitly indicated to have no further wishes within the context of
the entire department store. Otherwise the agent shall find out in which other department the
customer may have an interest and the customer shall be referred to the agent representing
this other department.
. be able to provide product and special offer information
For example, if a client communicates a need, then a product is offered fulfilling this need
(strictly or approximately), and, if available a special offer.
. dedicated announcement
As soon as available new products and special offers shall be announced to all relevant (on
the basis of their profiles) customers, (they shall be contacted by the store in case they do not
frequently contact the store).
Website Agent - Basic types of behaviour
Autonomy - functions autonomously, especially when no
employees are available (e.g., at night)
Responsiveness - responds to requests from Personal Assistants
- responds to input from employees
- triggers on decreasing trends in selling and demands
Pro-activeness - takes initiative to contact Personal Assistants
takes initiative to propose special offers to customers
- creates and initiates proposals for marketing actions
and assortment modifications
Social behaviour - co-operation with employees, Personal Assistants, and
other Website Agents
Table
3. Basic types of behaviour of a Website Agent
. analyses for marketing
The Website Agents shall monitor the amounts of articles sold (sales statistics),
communicate them to employees (e.g., every week) and warn if substantially decreasing
trends are observed. For example, if the figures of an article sold decrease during a period of
3 weeks, then marketing actions or assortment modifications shall be proposed.
. actions for marketing
Each Website Agent shall maintain the history of the transactions of each of the customers
within its department, and shall perform one to one marketing to customers, if requested.
The employees shall be able to communicate to the relevant Website Agents that they have
to perform a marketing campaign. The agent shall propose marketing actions to employees.
. privacy
No profile is maintained without explicit agreement with the customer. The customer has
access to the maintained profile.
Personal Assistant - Interaction characteristics
A. Interaction with the world
observation passive
observation active
observe changes and special offers at the Website
- observe the Website for articles within the customer
needs
performing actions
B. Communication with other agents
incoming from Website Agent:
product info
special (personal and general) offers
from customer:
customer needs and preferences
- agreement to buy
privacy constraints
outgoing to Website Agent:
customer needs
payment information
profile information
to customer:
product information
special offers
Table
4. Interaction characteristics for the Personal Assistant
3.2 Characteristics and Requirements for the Personal Assistants
For the Personal Assistants the interaction characteristics are given in Table 4, and their
basic types of behaviour in Table 5. The following requirements can be imposed on the
Personal Assistants:
. support communication on behalf of the customer
Each customer shall be supported by his or her own Personal Assistant agent, who serves
as an interface for the communication with the Website Agents.
. only provide information within scope of interest of customer
A customer shall not be bothered by information that is not within his or her scope of
interest. A special offer that has been communicated by a Website Agent leads to a
proposal to the customer, if it fits in the profile, and at the moment when the customer
wants such information
. sensitive profiling
Customers are relevant for a special offer if they have bought a related article in the past,
or if the offer fits in their profile as known to the Personal Assistant.
. providing customer information for Website Agents
every week the relevant parts of the profile of the customer is communicated to the
Website Agent, if the customer agrees.
. privacy
The Personal Assistant shall protect and respect the desired privacy of the customer.
Only parts of the profile information agreed upon are communicated.
Personal Assistant - Basic types of behaviour
Autonomy autonomous in dealing with Website Agents on behalf of
customer
Responsiveness responsive on needs communicated by customer
Pro-activeness initiative to find and present special offers to customer
Social behaviour with customer and Website Agents
Table
5. Basic types of behaviour for the Personal Assistant
4 The Internal Design of the Information Broker Agents
The agents in the multi-agent architecture for intelligent Websites presented in the previous
sections have been designed on the basis of a generic model for a broker agent. The process
of brokering as it often occurs as a mediating process in electronic commerce involves a
number of activities. For example, responding to customer requests for products with certain
properties, maintaining information on customers, building customer profiles on the basis of
such customer information, maintaining information on products, maintaining provider
profiles, matching customer requests and product information (in a strict or soft manner),
searching for information on the WWW, and responding to new offers of products by
informing customers for whom these offers fit their profile. In this section a generic broker
agent architecture is presented that supports such activities. This generic information broker
model has been used as a basis for both the Website Agents and the Personal Assistant
agents. As these architectures have been designed using the compositional design method for
multi-agent systems DESIRE, first a brief overview of DESIRE is presented (Section 4.1),
next the generic broker agent model is briefly discussed (Section 4.2), and finally the two
types of information broker agents that are used in the generic multi-agent architecture for
intelligent Websites are discussed: Website Agent (Section 4.3) and Personal Assistant
(Section 4.4).
4.1 Compositional Design of Multi-Agent Systems
The emphasis in DESIRE is on the conceptual and detailed design. The design of a multi-agent
system in DESIRE is supported by graphical design tools within the DESIRE software
environment. The software environment includes implementation generators with which
(formal) design specifications can be translated into executable code of a prototype system.
In DESIRE, a design consists of knowledge of the following three types: process
composition, knowledge composition, and the relation between process composition and
knowledge composition. These three types of knowledge are discussed in more detail below.
4.1.1 Process Composition
Process composition identifies the relevant processes at different levels of (process)
abstraction, and describes how a process can be defined in terms of (is composed of) lower
level processes.
of Processes at Different Levels of Abstraction
Processes can be described at different levels of abstraction; for example, the process of the
multi-agent system as a whole, processes defined by individual agents and the external
world, and processes defined by task-related components of individual agents. The identified
processes are modelled as components. For each process the input and output information
types are modelled. The identified levels of process abstraction are modelled as
abstraction/specialisation relations between components: components may be composed of
other components or they may be primitive. Primitive components may be either reasoning
components (i.e., based on a knowledge base), or, components capable of performing tasks
such as calculation, information retrieval, optimisation. These levels of process abstraction
provide process hiding at each level.
Composition of Processes
The way in which processes at one level of abstraction are composed of processes at the
adjacent lower abstraction level is called composition. This composition of processes is
described by a specification of the possibilities for information exchange between processes
(static view on the composition), and a specification of task control knowledge used to
control processes and information exchange (dynamic view on the composition).
4.1.2. Knowledge Composition
Knowledge composition identifies the knowledge structures at different levels of
(knowledge) abstraction, and describes how a knowledge structure can be defined in terms
of lower level knowledge structures. The knowledge abstraction levels may correspond to
the process abstraction levels, but this is often not the case.
of knowledge structures at different abstraction levels
The two main structures used as building blocks to model knowledge are: information types
and knowledge bases. Knowledge structures can be identified and described at different
levels of abstraction. At higher levels details can be hidden. An information type defines an
ontology (lexicon, vocabulary) to describe objects or terms, their sorts, and the relations or
functions that can be defined on these objects. Information types can logically be represented
in order-sorted predicate logic. A knowledge base defines a part of the knowledge that is
used in one or more of the processes. Knowledge is represented by formulae in order-sorted
predicate logic, which can be normalised by a standard transformation into rules.
Composition of Knowledge Structures
Information types can be composed of more specific information types, following the
principle of compositionality discussed above. Similarly, knowledge bases can be composed
of more specific knowledge bases. The compositional structure is based on the different
levels of knowledge abstraction distinguished, and results in information and knowledge
hiding.
4.1.3 Relation between Process Composition and Knowledge Composition
Each process in a process composition uses knowledge structures. Which knowledge
structures are used for which processes is defined by the relation between process
composition and knowledge composition.
4.2 A Generic Information Broker Agent Architecture
The generic information broker agent architecture was designed as a refinement of the
generic agent model GAM (cf. [8]), supporting the weak agency notion (cf. [38]). First we
will briefly describe the generic model GAM and next we discuss how this model was
refined to the generic information broker model.
4.2.1 The generic agent model GAM
At the highest process abstraction level within the compositional generic agent model GAM
introduced in [8], a number of processes are distinguished that support interaction with the
other agents. First, a process that manages communication with other agents, modelled by
the component agent interaction management in Fig. 2. This component analyses incoming
information and determines which other processes within the agent need the communicated
information. Moreover, outgoing communication is prepared. Communication is modelled in
a first-order logic approach, comparable, for example, to KIF. Communication from agent A
to B takes place in the following manner:
. the agent A generates at its output interface a statement of the form:
. the information is transferred to B; thereby it is translated into
If needed, it is not difficult to replace this format by more extensive formats used in KQML
or FIPA-ACL.
Next, the agent needs to maintain information on the other agents with which it co-
operates: maintenance of agent information. The component maintenance of world information is included to
store the world information (e.g., information on attributes of products). The process own
process control defines different characteristics of the agent and determines foci for behaviour.
The component world interaction management is included to model interaction with the world (with
the World Wide Web world, in the example application): initiating observations and
receiving observation results.
The agent processes discussed above are generic agent processes. Many agents perform
these processes. In addition, often agent-specific processes are needed: to perform tasks
specific to one agent, for example directly related to a specific domain of application. This is
the purpose of the component Agent Specific Task. Fig. 2 depicts how the generic agent is
composed of its components.
communicated
observation
results
to wim
observed
agent
communicated
agent
Agent task control
Own
Process
Control
Maintenance
of Agent
Information
Agent
Task
Maintenance
of World
Information
Agent
Interaction
Management
World
Interaction
Management
own process info to wim
own process info to aim
own
process
info to
own
process
info to
mwi info to be communicated
communicated
info to ast
communicated world info
observations and actions
observed
info to ast
observed
world info
action and observation info from ast
communication info from ast
agent info to opc
world info to opc
agent info to wim
agent info to aim
world info to aim
world info to wim
Fig. 2. Composition within the generic information broker agent model
4.2.2 Refinement of GAM to the generic information broker agent model
The refinement of a generic model may involve both specialisation of the process
composition and instantiation of the knowledge composition. The specific refinement
discussed here only involves instantiation of the knowledge composition. Part of the
exchange of information within the generic broker agent model can be described as follows.
The broker agent needs input about scopes of interests put forward by agents and
information about attributes of available products that are communicated by information
providing agents. It produces output for other agents about proposed products and the
attributes of these products. Moreover, it produces output for information providers about
interests. In the information types that express communication information, the subject
information of the communication and the agent from or to whom the communication is
directed are expressed. This means that communication information consists of statements
about the subject statements that are communicated.
Within the broker agent, the component own process control uses as input belief info, i.e.,
information on the world and other agents, and generates focus information: to focus on a
scope of interest to be given a preferential treatment, i.e., pro-active behaviour will be shown
with respect to this focus. The component agent interaction management has the same input
information as the agent (incoming communication), extended with belief info and focus info. The
output generated includes part of the output for the agent as a whole (outgoing communication),
extended with maintenance info (information on the world and other agents that is to be stored
within the agent), which is used to prepare the storage of communicated world and agent
information.
Information on attributes of products is stored in the component maintenance of world
information. In the same manner, the beliefs of the agent with respect to other agents' profiles
(provider attribute info and interests) are stored in maintenance of agent information. The component agent
specific task uses information on product attributes and agent interests as input to generate
proposals as output. For reasons of space limitation the generic and domain-specific
information types within the agent model are not presented; for more details; see [21]. The
information broker agent may have to determine proposals for other agents. In this process,
information on available products (communicated by information providing agents and kept
in the component maintenance of world information), and about the scopes of interests of agents
(kept in the component maintenance of agent information), is combined to determine which agents
might be interested in which products.
4.3 The Website Agent: Internal Design
The broker agent architecture provides an appropriate means to establish the internal design
of the two types of agents involved.
For the Website Agent, the internal storage and updating of information on the world and
on other agents (the beliefs of the agent) is performed by the two components maintenance of
world information and maintenance of agent information. In Table 6 it is specified which types of
information are used in these components. Profile information on customers is obtained from
Personal Assistants, and maintained with the customer's permission. Also identified
behaviour instances of the Personal Assistants can give input to the profile. Profile
information can be abstracted from specific demands; how this is performed may depend on
the application that is made.
Website Agent - Maintenance of Information
world information - info on products within the Website Agent's
assortment
- info on special offers
agent information - info on customer profiles
- info on customer privacy constraints
- info on customer preferences in communication
- info on which products belong to which other Website
Agent's assortments
- info on providers of products
Table
6. Maintenance information for the Website Agent
The component agent interaction management identifies the information in incoming
communication and generates outgoing communication on the basis of internal information.
For example, if a Personal Assistant agent communicates its interests, then this information
is identified as new agent interest information that is believed and has to be stored, so that it
can be recalled later.
In the component agent specific task specific knowledge is used such as, for example:
. if the selling numbers for an article decrease for 3 weeks, then make a special offer
with lower price, taking into account the right season
. if a customer asks for a particular cheap product, and there is a special offer, then this
is proposed
. if an article is not sold enough over a longer period, then take it out of the assortment
Within this component non-strict (or soft) matching techniques can be employed to relate
demands and offers.
4.4 The Personal Assistant: Internal Design
In this section some of the components of the Personal Assistant are briefly discussed.
For the Personal Assistant, as for the Website Agent, the internal storage and updating of
information on the world and on other agents is performed by the two components maintenance
of world information and maintenance of agent information. In Table 7 it is specified which types of
information are used in these components.
Personal Assistant - Maintenance of Information
world information - product information
special offers
agent information - customer needs and profile
customer privacy constraints
offers personal to the customer
- Website Agents assortment scopes
Table
7. Maintenance information for a Personal Assistant
As in the previous section, the component agent interaction management identifies the
information in incoming communication and generates outgoing communication on the basis
of internal information. For example, if a Website Agent communicates a special offer, then
this information is identified as new agent information that is believed and has to be stored,
so that it can be recalled later. Moreover, in the same communication process, information
about the product to which the special offer refers can be included; this is identified and
stored as world information.
4.5 Profile modelling approaches that can be used within the agents
Within the generic architecture for Website Agents and Personal Assistants no commitment
has been made to specific approaches to user profiling. In this section a number of these
approaches are briefly discussed (for a more detailed treatment, see [11]). The profile of a
user can be used to determine how interesting an information item is to that user. It can be
used to select and prioritise information items in a personalised manner. The structure and
properties of profiling approaches may vary with the application area in which they are used.
For example, in multi-attribute decision systems (see [3],[23],[37]) the user profile or
preference for an item is defined in terms of values of various attributes of the item and the
preferences of the user towards those attributes (i.e., the importance of those attributes). On
the other hand, in the area of recommendation systems the profile may as well be defined in
terms of statistical correlation between users and their rated items.
The preferences of a user towards a set of items can be defined in terms of the content of
the items (content information) or the preference of the items by a society of users
(collaborative or social information). In the content-based approach a user is defined to have
preference for an item if the item is similar in attribute values to other items that are
preferred by the user. Also ratings for the (relative) relevance of attributes for a user are
often included.
In the collaborative-based approach a user is defined to have preference for an item if the
user is similar (in preferences of other items) to other users who have preference for the
item. Both the content information as well as the collaborative information can be used to
construct user profiles.
The construction of a profile can be a time consuming matter. For example, in the
content-based approach the user may have to express his or her preferences towards various
(combinations of) attributes and attribute values in extensive forms. Some systems (e.g., see
[12]) derive the preferences of a user by suggesting an item to the user and ask her to correct
this suggestion. The user corrects the system's suggestion by indicating why the suggested
item does not match his or her needs. Based on these corrections, profiles of users are
constructed or updated. In other, collaborative-based applications such as recommendation
systems, a user may be asked to rate several, sometimes hundreds, of (other) items before an
item can be recommended.
A number of systems employ methods to induce the user profile by observing the
behaviour of that user over time (e.g., see [16],[26],[29],[31]). These methods are usually not
intended to fully model user profiles, but to model the more frequent and predictable user
preferences. Applications that require huge efforts from their users may become ineffective
(e.g., see [27],[28]). To model user profiles in an application, a balance is to be found
between the amount of interaction with the user and the effectiveness of the constructed user
profile.
Modelling user profiles on the basis of content or collaborative information can be
considered as a learning problem where the aim is to learn the so-called preference function
for a certain user. The preference function for a user maps items from a certain domain to
some values that express the importance of those items for that user. Various types of
preference functions may exist. The type of a preference function characterises the structure
of profile (e.g., see [23],[37]). Another profile learning approach, based on Inductive Logic
Programming, can be found in [5], [6], [11].
Several collaborative-based recommendation systems have been introduced in which the
preferences of users are modelled automatically. Examples of online recommendation
systems that employ a collaborative approach are MovieFinder [39] and FireFly [13]. The
preferences of a user are modelled automatically by observing the behaviour of that user and
applying statistical methods to the observed behaviour (e.g., see [4],[16],[17],[33]).
In contrast to the collaborative-based approach, the content-based approach can be
applied only when items are described in terms of properties and attribute values. The
content-based profiling approaches have been used in online recommendation systems such
as BargainFinder [1] and Jango [18]. Unlike collaborative-based preference models, the
content-based preference models are also used in applications such as integrative negotiation
where the utility function is defined in terms of user preferences towards various attribute
values (e.g., see [3],[15],[23],[29],[37]). The collaborative-based and content-based
approaches do not exclude each other; in fact they can be combined into an integrated
approach to model user profiles (see [2]).
The effectivity of collaborative-based and content-based approaches to profiling may
depend on the application. For example, collaborative-based profiling approaches may be
more effective in applications where it is unrealistic to collect a large amount of information
about the preferences of an individual user, or where the number of users is too large. Using
collaborative-based profiling models is also effective for applications where the content of
the items neither is available nor can be analysed automatically by a machine (e.g. items like
a picture, video, sound). However, the collaborative-based profiling approaches are less
effective for applications like integrative negotiation (e.g., see [3],[14]) in retail Electronic
Commerce where negotiation is considered to be a decision making process over items that
are described as multiple interdependent attributes.
5 Reuse of the Generic Architecture in the Insurance Domain
The reusability of the designed generic multi-agent architecture was tested in a new domain:
insurance. In this section this domain is briefly introduced. One of the largest insurance
companies in the Netherlands is organised on the basis of (human) mediating insurance
agents. To support these agents a Website was created with information about products
offered by the company, forms to support administrative actions, and other related
information. The Website is structured around four main sections: Store, Desk, Newsstand
and Office.
The store provides information about the insurance products offered. The various
insurance policies can be found here, as well as request forms for more information,
brochures, and personalised proposals . From the store a couple of useful programs can be
downloaded as well: spreadsheets, an anti-virus toolkit, and an insurance dictionary.
When the insurance agent is faced with a problem, he or she can turn to the desk. Apart
from a Frequently Asked Questions page also a form is available for specific questions. The
desk further contains the editorials that address certain problems in depth. Finally, an address
book is available, in which the various departments and teams operating within the company
can be found.
At the news-stand the visitors of the site can find the most recent information.
Newsletters can be found, and a calendar can be checked for upcoming events. Furthermore,
various links to other interesting sites and assorted articles are offered here. Whenever new
interesting sites or articles are added, the visitor can be notified of this by email.
At the office, the sale of insurance products is supported. Here resources to improve the
insurance agent's job can be found: telemarketing scripts, newsletter articles, advertisements
that only need further filling out and sales letters. Furthermore, the agent can find its
personal production figures for the company's products.
The Website consists of a collection of variable information sources: images, programs,
documents, addresses, phonebooks and personal data. New information is added daily.
Keeping up to date with the most recent relevant information, is time-consuming. The multi-agent
system has been developed to support the human agent in this task.
The aim of the multi-agent system integrated in the Website is to improve the use of the
resources offered by the Website. From the visitors point of view, more interesting
information can be obtained. The agent, with its knowledge of the user improves the
customer experience. Application forms can be offered, already (partially) filled out by the
software agent. The employees maintaining the Website can use information collected by the
multi-agent system to improve marketing. The appropriate visitor can be contacted about
new (possibly personalised) products or offers that are relevant to him or her.
6 Instantiation of the Generic Architecture
The generic multi-agent architecture has been instantiated for the new domain of insurance
described in Section 5. Application-specific information types and knowledge bases were
specified and included in the model. The system is explained for two cases: behaviour
initiated by an information request of a user (user initiated), and behaviour initiated by
update or addition of information to the Website (Website initiated). In both cases, after
initiation a reactivity chain is triggered. In the first case the main reactivity chain follows the
path
user-PA-WA-PA-user
The first half of this path deals with the queries, and the second half (back) with answers on
these queries. In the second case the main reactivity chain follows the path
The first half deals with voluntarily offered information (one-to-one marketing), and the
second half (back) with feedback on usefulness of the offered information (in order to update
profile information). In the explanation of these behavioural traces, it is shown which
knowledge bases were used to instantiate the generic architecture.
6.1 Information used in the system
This system is only a prototype; as such it does not work with the actual information on the
Website. Instead a sample of the information objects on the Website was selected and a
description of each of these was made.
In cooperation with employees from the insurance company the following attributes were
selected to describe the information:
. Title: The title of the information object.
. Author: The department or person that created the information object.
. Subject: Subject of the information object.
. First Relation: The first related subject.
. Second Relation: The second related subject.
. Date: Date of creation/availability.
. Language: The language used in the information.
. Persistency: An indication of how soon the information will be outdated.
. Kind: The form of the information object (mailform, text, audio, etc.
. Type: The type of information in the information object (e.g., FAQ,
newsletter, personal information).
. URL: The hyperlink to the actual information object
Fig. 3. User interface for asking questions and stating user interest
6.2 Behaviour initiated by a user
When a user asks a question, the Personal Assistant agent performs a number of actions. The
question is analysed to find similarities to previous questions and if these exist, new interests
are created within the user profile. Furthermore, the agent attempts to respond to the
information request using information available within the Personal Assistant itself and by
contacting the appropriate Website Agents.
First it is described how an answer to a question is found. Next, the process of updating
the user profile is discussed.
Handling a question. The behaviour of the system is first described from the user's point of
view. Subsequently, the processes that are invisible for the user a described in more details.
The user interaction
A trace is described in which a user needs information about car insurance. As a first step the
user communicates this question to the Personal Assistant using the interface (Fig. 3): the
user selects the subject 'car insurance' in the scrollable list under the heading `Subjects'. The
Personal Assistant will start to acquire useful information on behalf of its user.
Fig. 4. Display for the answers to questions and offers made (by a Personal Assistant)
The Personal Assistant inspects all information it has in store and it contacts appropriate
Website Agents for more information. All the gathered relevant information is
communicated to the user, using the display in Fig. 4; each title is a link to a description of
the information. The user can indicate whether or not he or she evaluates the information as
interesting.
if query(Q:QUERY_ID, scope(subject, S:SUBJECT))
and object_scope(O:OBJECT_ID, scope(related_subject, S:SUBJECT))
then possible_answer_to_query(O:OBJECT_ID, Q:QUERY_ID);
if query(Q:QUERY_ID, scope(A:ATTRIBUTE, V1:VALUE))
and object_scope(O:OBJECT_ID,scope(A:ATTRIBUTE, V2:VALUE))
and not subject
and not
then rejected_answer_for_query(O:OBJECT_ID, Q:QUERY_ID);
if possible_answer_to_query(O:OBJECT_ID, Q:QUERY_ID)
and not rejected_answer_for_query(O:OBJECT_ID, Q:QUERY_ID)
then selected_answer_to_query(O:OBJECT_ID, Q:QUERY_ID);
Table
Knowledge involved in user-initiated behaviour
The processes within the multi-agent system
When Personal Assistant agent receives a question from the user, it identifies the
communication as a question in the component agent interaction management. The question is
further processed in the task specific component determine proposals of the Personal
Assistant. That component matches the request to the information objects available in the
memory of the agent (component maintenance of world information). Two types of matching are
covered: strict matching and soft matching. For strict matching, attributes need to have
exactly the same value, or an overlapping value range. For soft matching, it can be specified
when values of attributes are considered close (but not necessarily equal) to each other. This
closeness relation may be based on various techniques. In the current prototype the closeness
relation for the subject attribute is taken as a point of departure, abstracting from the manner
in which it is determined. One of the matching rules is rule r1 in Table 8.
The subject of the query is matched with the related subject of the object under
consideration. If the rule succeeds, the object is selected as a possible answer. A criterion for
this possible answer to become a definite answer is that the object does not differ on other
attributes (see rule r2). Rule r3 is used to derive the final answer to the question.
Simultaneously, in the same component determine proposals, the relevant Website Agents
the are selected. This is done in three steps. First, the Personal Assistant agent looks for a
Website Agent that is known to provide information about the subject occurring in the query;
see rule r4. Rule 4 makes use of the agent model for the Website Agent that is stored by
thePersonal Assistantwithin component maintenance of agent information. Information about the
subjects that a Website Agent can provide is expressed by the statement
webagent_subject(W:WA, S:SUBJECT) .
Rule 4 will not succeed, however, when the question does not contain a subject term or
when the Personal Agent does not know a relevant Website Agent. In this case the Personal
Assistant agent uses a second method to determine an appropriate Website Agent, by
considering another part of the agent models it maintains of Website Agents; see rule r5.
Finally as a fail-safe, each Personal Assistant has a default Website Agent it can contact. The
name of this default Website Agent is stored in the component own process control and is also
available in the component determine proposals. The final selection of the Website Agent is
performed by the knowledge specified in rules r6 to r9.
if query(Q:QUERY_ID, scope(subject, S:SUBJECT))
and webagent_subject(W:WA, S:SUBJECT)
then main_wa_for_answer(W:WA, Q:QUERY_ID)
and found_wa_for_query(Q:QUERY_ID);
if query(Q:QUERY_ID, S:SCOPE)
and can_provide_scope(W:WA, S:SCOPE)
then secondary_wa_for_answer(W:WA, Q:QUERY_ID)
and found_wa_for_query(Q:QUERY_ID);
if main_wa_for_answer(W:WA, Q:QUERY_ID)
then selected_wa_for_answer(W:WA, Q:QUERY_ID);
if secondary_wa_for_answer(W:WA, Q:QUERY_ID)
then selected_wa_for_answer(W:WA, Q:QUERY_ID);
if not found_wa_for_query(Q:QUERY_ID)
and default_wa(W:WA)
then selected_W:WA_for_answer(W:WA, Q:QUERY_ID);
Table
9 Knowledge involved in selection of Website Agents
Next the selected_wa_for_answer and selected_object_for_query information is transferred to the
component agent interaction management where communication to the selected Website
Agent(s) is actually initiated (see Table 9.
Website Agents handle questions in the same way as the Personal Assistant. The
component determine proposals of a Website Agent attempts to find a match with the known
information objects. The matches are communicated back to the Personal Assistant. The
component agent interaction management of the Personal Assistant passes the received answers
on to its user.
if communicated_by(query_answer(Q:QUERY_ID, object_scope(O:OBJECT_ID, S:SCOPE)),
pos, W:WA)
then to_be_communicated_to(query_answer(Q:QUERY_ID, object_scope(O:OBJECT_ID,
S:SCOPE)), pos, user);
Table
9 Knowledge involved in communication to user
The information contained in received answers is also stored by the Personal Assistant: in
the future it can supply this information by itself.
Update of user profile. The focus of the current prototype lies on the agent interaction and
document selection. Profile management had a lower priority. Therefore the mechanisms for
profile management used are simple. As stated earlier, the Personal Assistant compares
questions to each other. When similarities are found in three questions, these similarities are
added to the user profile. This is performed by the (composed) component interest creator.
A new question is first compared to all previous questions. A simple method has been
chosen to create these candidates: whenever three different questions match on one or more
attribute values, these attribute-value pairs are selected as a candidate interest specification;
see rule r11 in Table 10. The three query id's are combined to create a temporary candidate
if asked(query(Q1:QUERY_ID, scope(A:ATTRIBUTE, V:VALUE)))
and asked(query(Q2:QUERY_ID, scope(A:ATTRIBUTE, V:VALUE)))
and asked(query(Q3:QUERY_ID, scope(A:ATTRIBUTE, V:VALUE)))
and not
and not
and not
then candidate_for_interest(candidate_id(Q1:QUERY_ID, Q2:QUERY_ID,
Q3:QUERY_ID), scope(A:ATTRIBUTE, V:VALUE));
if candidate_for_interest(C:CANDIDATE_ID, scope(A:ATTRIBUTE, V1:VALUE))
and belief(interest(I:INTEREST_ID, scope(A:ATTRIBUTE, V2:VALUE))
and not
then different(C:CANDIDATE_ID, I:INTEREST_ID);
if new_interest_id(I:INTEREST_ID)
and approved_candidate(C:CANDIDATE_ID, S:SCOPE)
then to_be_created(interest(I:INTEREST_ID, S:SCOPE));
Table
Knowledge involved in profile update: user-initiated case.
6.3 Behaviour initiated by the Website
The second type of behaviour discussed here is initiated by the Website. First the behaviour
to directly serve the user is discussed, and subsequently the behaviour to update the user
profile is described in more detail.
Offering the user new information. First the behaviour shown to the user is described.
Next a more detailed description is given of the processes within the multi-agent system
itself.
The user interaction
The Personal Assistant takes the initiative to notify its user when relevant information has
been found, using the display depicted in Fig. 4. Again, the user can click on a title to get
more information about the proposal (Fig. 5). Furthermore, the user can choose to accept the
proposed information or to reject it.
The processes within the multi-agent system
When new information becomes available at a Website, the Website Agent identifies
possible interested parties. The Website Agent has built a profile of the Personal Assistants it
has been in contact with. In the component determine proposals the Website Agent uses this
information to match the new information to the Personal Assistants interests; see rule r13 in
Table
11.
if new_object_scope(O:OBJECT_ID, S:SCOPE)
and interest(P:PA, I:INTEREST_ID, S:SCOPE)
then partly_matched_new_object(O:OBJECT_ID, P:PA, I:INTEREST_ID);
if offered_object_scope(O:OBJECT_ID, S:SCOPE)
and interest(I:INTEREST_ID, S:SCOPE)
then partly_matched_offer(O:OBJECT_ID, I:INTEREST_ID);
if partly_matched_offer(O:OBJECT_ID, I:INTEREST_ID)
and not rejected_offer(O:OBJECT_ID, I:INTEREST_ID)
then accepted_offer(O:OBJECT_ID, I:INTEREST_ID);
Table
Knowledge involved in Website-initiated behaviour
For each scope in the new object a comparison to the existing interests in the profile is made.
When they match, the object is partly selected. However, on another scope, the interest and
the new object may differ. Only if all of the scopes of the object match is the object selected.
The offer is made by the component agent interaction management. The Personal Assistant
receives this offer and compares it to the interests in its user profile. This is performed in the
Personal Assistant's determine proposals, as it is done in the Website Agent; see rule r14.
Again, when no conflicting scopes can be found between the interest and the offered object,
it is selected using rule r15. The selected offer is communicated to the user, who can reply to
the offer.
Update of user profile. After the user has communicated to the Personal Assistant whether
he or she rates the offer interesting or not, a profile update process is initiated, if necessary,
by removing those interests repeatedly receiving negative feedback. This feedback is used in
the component interest remover to select interests for removal. Similar to the creation of new
interests, a simple mechanism is used to select interests for removal. A circular list is kept of
the last three responses to offers based on an interest. This list has three objects; when all
three objects show a negative response, the interest is marked for removal; see rule r16 in
Table
12.
if last3_suggestions_response(last_id1, rejected, I:INTEREST_ID)
and last3_suggestions_response(last_id2, rejected, I:INTEREST_ID)
and last3_suggestions_response(last_id3, rejected, I:INTEREST_ID)
then to_be_confirmed(remove(I:INTEREST_ID))
if removal_response(I:INTEREST_ID, confirmed)
and believe(interest(I:INTEREST_ID, S:SCOPE))
then to_be_removed(interest(I:INTEREST_ID, S:SCOPE));
Table
Knowledge involved in profile update: removal in Website-initiated case
An interest marked for removal is not automatically removed. Before actual removal, the
user has to give his or her approval. When the user disapproves of the removal, the three last
responses to that interest are reset; thus again three rejections in a row must be received
before the agent considers the interest for removal. When the user approves, the removal is
performed; see rule r17. As for the interest creator, this component reasons about changes in
interests and is therefore at a meta-level compared to the component maintenance of agent
information. The interest is actually removed by an information link, similar to how interests
are created.
In this paper a generic, reusable multi-agent architecture for active intelligent Websites is
presented. This generic architecture for active intelligent Websites was first designed for one
application domain: a department store (cf. [22]). This application reuses the generic
architecture of information broker agents developed earlier (cf. [21]), which in turn was
designed as a specialisation of the generic agent model GAM introduced in [8]. The model
has been designed in such a way that the generic, reusable structures are separated from the
application-specific aspects in a transparent manner.
The reusability of the generic multi-agent architecture for active intelligent Websites has
been tested in a second application: a project on an intelligent Website for insurance in co-operation
with the software company Ordina Utopics and an insurance company (cf. [20]).
The outcome of this test was clearly positive. With not much effort (an investment of only a
few person months) a prototype multi-agent system for an intelligent Website in insurance
has been designed and implemented, based on the generic architecture. The actual work
concentrated mainly on the specification of the domain concepts and application-specific
knowlege bases.
A Website, supported by the architecture introduced has a more personal look and feel
than the usual Websites. Within the architecture, also negotiation facilities (e.g., as in [38])
can be incorporated.
In the agent literature, a number of architectures for (information) broker agents can be
found; e.g., [9], [10], [25], [30], [32], [36],. The design of most of these architectures is not
formally specified in detail; usually they are only available in the form of an
implementation, and at the conceptual level some informal pictures and natural language
explanations. In general, the aim for the development of these architectures in the first place
is to have a working piece of software for a specific type of application. The design of the
generic architecture for intelligent Websites introduced in this paper has a different aim. The
generic model was meant as a unified design model, formally specified in an
implementation- and domain-independent manner at a high level of abstraction. The (multi)
agent architecture described here was designed and implemented in a principled manner,
using the compositional design method for multi-agent systems DESIRE [7]. Due to its
compositional structure it supports reuse and maintenance; a flexible, easily adaptable
architecture results. A success criterion for this aim is the possibility to specialise and
instantiate the model to obtain conceptual, formal specifications of design models for
different applications. The positive experience in the insurance domain, discussed above,
shows that the aim was achieved.
Applications of broker agents (addressed in, e.g., [9], [10], [24], [25], [30], [32], [34],
[36]), often are not implemented in such a principled manner: without an explicit design at a
conceptual level. Compared to, for example, systems designed using CORBA, or other
object-based methods, a main difference is that in our approach functionality can be
specified at the level of design in an explicit declarative manner (in the form of ontologies
and knowledge bases). Especially for applications in knowledge-intensive domains this
provides appropriate means to specify a design. The RETSINA approach (cf., [34], [35]) is
more comparable to the design method DESIRE as such, and not to the generic architecture
for the specific application type of intelligent Websites proposed here. A difference is that
DESIRE is based on a formal specification language for design models. The same difference
applies to the work on SIMS, described in [24]. However, in [24], also the problem of
information integration is addressed (i.e., integration of information expressed in different
ontologies), which has not (yet) been addressed in the architecture proposed here. A next
step is to refine our model with possibilities for information integration, for example adopted
from SIMS.
The question whether the approach scales up has not been explicitly investigated in the
research reported, by performing experiments. Since an essentially distributed approach has
been chosen, the Personal Assistant agents can all be implemented on an own server. Also it
is possible to implement different Website Agents on different servers, thus avoiding too
much interaction overload of one server.
For the particular application in insurance the generic broker agent model has been
instantiated with domain ontologies and domain knowledge. In the prototype some of these
instantiations have been done in an ad hoc manner, without the intention to propose these
instantiations as a generic approach for more domains. Current research addresses more
principled manners to use dynamic taxonomies in profile creation and techniques from
inductive logic programming to induce profiles from examples. In [11] (see also Section 4.5
above) an overview is given of a number of these profiling approaches and it is shown how
they can be incorparated). As an example, in further research a component-based generic
agent architecture for multi-attribute (integrative) brokering and negotiation has been
developed in co-operation with, among others, Dutch Telecom KPN. The agent architecture
was designed as a refinement of the compositional generic agent model GAM. Within the
component Maintenance of Agent Information (MAI) within this agent architecture, a profile
of the human user of the agent is maintained, which includes
. evaluation functions per attribute assigning to each attribute value an evaluation
value between 0 and 1,
. importance factors (between 0 and 1) for the different attributes.
Within such a more sophisticated content-based profile model (which, for example, is also
used in [3], it can be expressed, for example, that a car with colour blue is evaluated as 0.9,
whereas a yellow colour is evaluated as 0.1, and a CD player of high quality is rated 0.8
whereas a CD player with low quality as 0.2. Moreover, the attribute 'colour' can be
assigned, e.g., importance 0.6, whereas the attribute 'CD player' can be assigned importance
0.8.
Acknowledgements
Pascal van Eck (Vrije Universiteit) supported other experiments with some variants of the
broker agent model. Working on a design for the department store application with various
employees of the software company CMG provided constructive feedback on the
architecture introduced. The authors are also grateful for discussions with employees of
Ordina Utopics (in particular, Richard Schut) and an (anonymous) insurance company.
--R
Recommendation as Classification: Using Social and Content-Based Information in Recommendation
Enabling Integrative Negotiations by Adaptive Software Agents.
Springer Verlag.
Learning Collaborative Information Filters.
Lookahead and discretization in ILP.
Scaling up inductive logic programming by learning from interpretations.
Formal specification of Multi-Agent Systems: a real-world case
Compositional Design and Reuse of a Generic Agent Model.
An Agent Marketplace for Buying and Selling goods.
Modeling User Preferences and Mediating Agents in Electronic Commerce
Freuder and Richard J.
Latent Class Models for Collaborative Filtering.
Design of Collaborative Information Agents.
A Multi-Agent Architecture for an Intelligent Website in Insurance
Springer Verlag
Compositional design and maintenance of broker agents.
Information Broker Agents in Intelligent Websites.
Decisions with Multiple Objectives: Preferences and Value Trade-offs
Agents for Information Gathering.
On Using KQML for Matchmaking.
Learning to filter news.
Machine Learning for Adaptive User Interfaces.
User Modeling in adaptive interfaces.
An Agent that Assists Web Browsing.
Information Brokering in an Agent Architecture.
The identification of interesting web sites.
Issues in Automated Negotiation and Electronic Commerce: Extending the Contract Network.
"Word of Mouth"
Designing behaviors for information agents.
IEEE Expert 11
Toward a Virtual Marketplace: Architectures and Strategies.
Multicriteria Decision-aid
--TR | information agent;intelligent website |
591005 | Multi-Dimensional Modal Logic as a Framework for Spatio-Temporal Reasoning. | In this paper we advocate the use of multi-dimensional modal logics as a framework for knowledge representation and, in particular, for representing spatio-temporal information. We construct a two-dimensional logic capable of describing topological relationships that change over time. This logic, called (Propositional Spatio-Temporal Logic) is the Cartesian product of the well-known temporal logic PTL and the modal logic S4u, which is the Lewis system S4 augmented with the universal modality. Although it is an open problem whether the full PSTL is decidable, we show that it contains decidable fragments into which various temporal extensions (both point-based and interval based) of the spatial logic RCC-8 can be embedded. We consider known decidability and complexity results that are relevant to computation with multi-dimensional formalisms and discuss possible directions for further research. | Introduction
It is widely accepted that many kinds of AI applications require high-level
reasoning involving spatial and temporal concepts (see e.g. (Hayes,
1979; Hobbs and Moore, 1985; Davis, 1990)). Several approaches have
been applied to representing these concepts: some researchers have developed
specialised computation-oriented representations such as solid
geometry (Requicha, 1980), constraints techniques (Allen, 1981) and
spatio-temporal database architectures (Guting et al., 2000); others
have employed techniques of formal logics. Typically, logical representations
have made use of the expressive power of rst-order languages
(e.g. (McCarthy and Hayes, 1969; Clarke, 1981; Allen, 1984; Hayes,
1985; Kowalski and Sergot, 1986; Randell et al., 1992b)); but, since
Supported by the EPSRC under grants GR/K65041 and GR/M56807
c
2000 Kluwer Academic Publishers. Printed in the Netherlands.
rst-order logic is undecidable, such languages do not provide an eec-
tive reasoning algorithm unless supplemented by some special purpose
inference mechanism.
An alternative to rst-order logic is the framework of propositional
modal logic, which extends classical propositional calculus with one
or more 'modal' operators, interpreted on some relational or algebraic
structures. Appropriate structures may be chosen to correspond to an
aspect of reality, such as time or space, which one wishes to describe
within a logical representation.
For example, one may think of time as a sequence of points represented
by the natural numbers hN; <i on which temporal operators |
'some time in the future', `since' etc. | can be dened. For instance
is true at a time point t 2 N (in symbols t
only if t 0 t. The resulting propositional temporal
logic, PTL, is decidable in PSPACE (Sistla and Clarke, 1985) and by
Kamp's theorem (Kamp, 1968) it is as expressive (on hN; <i) as the
rst-order language with unary predicates P i (x), < and =. Temporal
logics of this sort are widely used in computer science (Manna and
Pnueli, 1992; Stirling, 1992).
Modal representations of space are relatively new to computer sci-
ence. However, the possibility of giving topological interpretations of intuitionistic
propositional logic has been known since the thirties (Stone,
1937{8; Tarski, 1938). Tang (1938) and later McKinsey and Tarski
(1948) interpreted the necessity operator of the Lewis modal system
S4 as the interior operator of topological spaces and proved that S4
is sound and complete with respect to this interpretation (for details
see e.g. Chagrov and Zakharyaschev (1997)). Since S4 is decidable in
PSPACE, this suggests that it might provide a computationally viable
language for describing topological information. That S4 really can be
used for qualitative spatial representation and reasoning was observed
by Bennett (1996, 1998), who embedded the spatial language RCC-8
into extended with the universal modality 8 (for all
points in the space). This encoding, which will be explained in Section 2,
provides a decision procedure for a spatial language that can express
a large class of signicant topological relations including all those in
RCC-8.
As we have seen, propositional modal and temporal logics may be
decidable and hence amenable to automated reasoning (Demri, 1994;
Giunchiglia and Sebastiani, 1996; Horrocks, 1998; Dixon et al., 1998;
Voronkov, 1999; De Nivelle et al., 2000). This of course means that
they are less expressive than rst-order logic. Moreover it is clear that
modalities of single type (say, only temporal or only spatial) are not
comprehensive enough for many real applications, which require rea-
Time
Space
Space
Figure
1. Motion modelled in terms of a Cartesian product of space and time
soning with a range of dierent concepts. What we actually need is
systems combining several types of modal operator. Such multi-modal
logics have been the subject of much recent research | e.g. (Kracht
and Wolter, 1991; Spaan, 1993; Marx and Venema, 1997; Gabbay,
1998; Wolter, 2000; Gabbay et al., 2000). Central problems in this eld
concern how modalities can be combined, whether interplay between
operators preserves decidability and other meta-logical properties, and
what is the computational complexity of the resulting systems.
Whereas modalities can be successfully combined as logically independent
operators (Kracht and Wolter, 1991; Fine and Schurz, 1996),
a key feature of most of the more interesting systems is the interaction
between dierent modalities. This applies in particular to modal operators
relevant to describing spatial and temporal concepts. Consider, for
instance, a moving region (the black disk) depicted in Fig. 1. If we can
model space as an S4 u -frame (i.e., a quasi-order) F and if the
ow of
time is represented by a linear order G, then the whole spatio-temporal
'universe' can be viewed as the Cartesian product F G, in which the
act 'horizontally' to talk about spatial regions, while
the temporal operators act 'vertically' taking care of their movements
in time.
Cartesian products of Kripke frames are typical examples of multi-dimensional
structures that serve as models of multi-dimensional modal
logics . The idea of a multi-dimensional modal logic was introduced by
presented an axiomatisation for a 2-dimensional
extension of S5. Products of modal logics were investigated in (Sheht-
man, 1978; Gabbay and Shehtman, 1998; Marx, 1999). In computer science
and AI, multi-dimensional modal logics are used e.g. for constructing
temporal epistemic logics for multi-agent and distributed systems
et al., 1995), temporal logics of parallel processes (Reynolds,
1997), temporal, epistemic, and dynamic description logics (Baader and
Ohlbach, 1995; Wolter and Zakharyaschev, 1998, 1999, 2000c). Further
details and development of multi-dimensional modal logic can be found
in (Gabbay et al., 2000).
Although the idea of constructing multi-dimensional modal logics
may appear natural and simple, the resulting 'hybrids' often turn out
to be very complex or undecidable, even if the one-dimensional components
are 'almost tractable'. As with other frameworks for knowledge
representation there is a delicate balance between expressive power and
computational complexity. Let us consider, for instance, the following
nave example of a 'compass' spatial logic (Venema, 1990).
In our everyday practice, we often connect spatial structures with
this or that system of coordinates. For instance, in geographical maps
we have four compass directions: North, South, West, and East. Using
these we can say e.g. that Moscow is to the North-East of London.
Spatial relations of this sort can be expressed in a modal language with
four operators N ('somewhere to the ('somewhere to
the South') interpreted in the Cartesian product of two linear orders,
say, hR; <i hR; <i in the standard Kripke-style manner:
etc.
That 'Moscow is to the North-East of London' can be represented in
this language by the formula London It is well known
from modal logic that the satisability problem for a language with
a single operator is NP-complete for the frames hR; <i or hN; <i,
(Ono and Nakamura, 1980). However, the satisability problem for a bi-modal
language (with N and E ) in hR; <ihR;<i or hN; <ihN;<i
turns out to be undecidable, even 1
and Reynolds, 1999; Reynolds and Zakharyaschev, 2001). The interval
temporal logic of Halpern and Shoham (1986) can be interpreted in
the f(x; y) 2 N N : x yg part of hN; <i hN; <i (see (Venema,
1990)), and this logic is also not recursively enumerable. There are
many other examples of undecidable multi-dimensional modal logics,
e.g. S5S5S5 (Maddux, 1980) or even any logic between KKK
and S5S5S5 (Hirsch et al., 2000).
Thus, we see that multi-dimensional modal logics are not easy to deal
with, and we need to be careful in constructing eective and expressive
spatio-temporal formalisms. For example, the straightforward attack
on the problem by means of using the Cartesian products of frames
for S4 and the
ow of time hN; <i (or any other innite linear
has not brought any result yet: whether the logic of such 2-dimensional
frames is decidable remains one of the challenging open problems in
the eld.
In the rest of this paper, we discuss possible uses of multi-dimensional
modal logics for spatio-temporal representation and reasoning, giving
initial results already obtained in this direction. Thereby we would like
both to attract the attention of the knowledge representation community
to this novel and promising approach and also to give incentive
to logicians to investigate modal languages that may turn out to have
practical utility.
2. Region Connection Calculus and a Modal Interpretation
The Region Connection Calculus (RCC) is a rst-order theory proposed
by Randell et al. (1992a) for qualitative spatial representation
and reasoning. 1 The basic language of RCC contains only one primitive
predicate C(X; Y ), read as 'region X is connected with region Y '. 2
Many other spatial relations can be dened in terms of the C primi-
tive. Of particular signicance are the eight relations depicted in Fig. 2.
These form a pairwise disjoint and exhaustive set of relations known as
RCC-8. Essentially the same set (but applied to the more restricted
class of connected regions) has been independently identied as useful
in the context of Geographical Information Systems (Egenhofer and
Franzosa, 1991). In English, the relations can be described as: Dis-
Connection, External Connection, Partial Overlap, Tangential Proper
Part, Non-Tangential Proper Part and Equality. Formal notations for
these relations are given under the diagrams. The part relations are
asymmetric, so each has an inverse, su-xed by 'i'.
The RCC formalism was originally presented as a nave theory in
the spirit of Hayes (1979), so no specic model was assumed. However,
it has been found that the theory can be interpreted in classical point-set
topology (Gotts 1996a; Bennett 1997,1998). Thus, we can take as
models of RCC, topological spaces, U is a non-empty
set, the universe of the space, and Ian interior operator on U . 3
comprehensive account of this theory and its applications can be found in
(Cohn et al., 1997).
Randell et al. (1992a) also considered an extended language incorporating an
additional primitive function conv(r), which denotes the convex-hull of region r. We
will comment brie
y on convexity later (Section 6).
3 This means that Imust satisfy the axioms I(X) X,
and I(X \ Y
a
a
a
a
a b
a
a
a b
Figure
2. Basic Relations in the RCC Theory
Individual variables of RCC range over non-empty regular closed sets of
T, i.e., an assignment in T is a map a associating with every variable X
a set a(X) U such that
is the closure operator on U dual to I. The connection relation C(X; Y )
is then interpreted as meaning that the point sets denoted by X and
Y share at least one point:
The full rst-order theory of RCC is too expressive to be computationally
useful and is in fact undecidable (this follows from (Grzegor-
czyk, 1951); results applying more specically to RCC can be found in
(Gotts, 1996b) and (Dornheim, 1998)). Fortunately, there are various
decidable (and even tractable) fragments of RCC.
As was mentioned above, for certain applications one can limit the
relations employed to the set RCC-8. Moreover, according to the experiments
reported by Knau et al. (1997) the eight predicates turn out
to be 'cognitively adequate' in the sense that people indeed distinguish
between those relations. Formally, the language of RCC-8 consists of a
set of individual variables X called region variables, the eight
binary predicates DC, EC, PO, EQ, TPP, TPPi, NTPP, NTPPi, and the
Boolean connectives (:, out of which we can construct
spatial formulas.
RCC-8 is interpreted in topological spaces
the region variables range over non-empty regular closed sets in T, and
the eight predicates are dened in the following way:
For example, according to this denition, EC(X; Y ) means that X and
Y share at least one point but do not share any interior point (i.e. they
only share boundary points).
The main reasoning task for RCC-8 can be formulated as follows:
given a nite set of spatial formulas, decide whether is satis-
able in a topological space, i.e., whether there exists a topological
space T and an assignment a in it such that T j= a .
That this satisability problem is decidable was observed by Bennett
(1994), who exploited the results of Tarski (1938) to encode RCC-8
into intuitionistic propositional logic. Subsequently, Bennett (1996) gave
another embedding of RCC-8 into the more expressive logic containing
Lewis's S4 modality and an additional universal modality, whose
meaning will be explained below. In the current paper we shall refer
to this bi-modal logic as S4 u . The embedding is based on the result of
McKinsey and Tarski (1948) according to which S4 is characterised by
the class of topological spaces in the following sense:
Given a topological space
interpret the propositional variables in ' as subsets of U , the Boolean
connectives as the corresponding set-theoretic operations, the necessity
operator I as I, and the possibility operator C as C (we denote the
usual box and diamond of S4 by I and C to emphasize their
topological meaning). Now, if the value of ' is the whole space U , no
matter what sets are assigned to its variables and what topological
space is taken, then ' is provable in S4, and vice versa.
According to (Goranko and Passy, 1992), this completeness theorem
still holds if we extend S4 with the universal modalities 8 and
9 interpreted in T as 'for all points in T' and `there is a point in T',
respectively. That is the value of 8' (under a certain interpretation) is
U if the value of ' (under this interpretation) is U , otherwise it is ;;
:8:'.
It is easy to see that the language of the resulting logic S4 u is expressive
enough to encode the topological meaning of the RCC-8 formulas
as dened above. Given such a formula ', we replace in it occurrences
of RCC-8 predicates with the corresponding modal formulas, e.g.
etc.
are propositional variables) and add to the result the conjunct
for every region variable X i in ' to ensure that the propositional variables
are interpreted by non-empty regular closed sets of topological
spaces. The resulting modal formula is denoted by ' y .
Now, if we recall that some (in particular, all nite) topological
spaces are determined by Kripke frames hW; Ri for S4 | every such
frame gives rise to the topological space hW; Ii, where
for any X W | and that S4 u has the nite model property (Goranko
and Passy, 1992), then we immediately obtain
Theorem 1. For every RCC-8 formula ', the following conditions
are equivalent:
(i) ' is satisable in a topological space,
(ii) ' y is satisable in a topological space,
(iii) ' y is satisable in some nite Kripke frame for S4.
Thus we reduce the satisability problem for RCC-8 formulas to
the satisability problem for propositional bimodal formulas in Kripke
frames for S4 u , which is decidable (Goranko and Passy, 1992).
Moreover, one can show that every satisable formula of the form
' y can be satised in a partially ordered Kripke frame each point in
which has at most two (incomparable) successors, and the number of
worlds in this frame is linear in the number of symbols in 'y (and
consequently also linear in the length of the RCC-8 formula '). This
result was obtained by Renz (1998); a somewhat more general theorem
is proved in (Wolter and Zakharyaschev, 2000a). It follows in particular
that the satisability problem for RCC-8 formulas is NP-complete;
see (Renz and Nebel, 1997; Renz and Nebel, 1999) where maximal
tractable fragments of RCC-8 are also described. It also follows that
all satisable RCC-8 formulas can be satised in R n for any n 1.
(Note however that S4 u is not complete with respect to R n .)
3. Point-based temporal RCC-8
One approach to constructing spatio-temporal logics is to combine
RCC-8 with point-based temporal logics, for instance, the well-known
propositional temporal logic PTL interpreted on the
ow of time hN; <i
and having the temporal operators S (Since) and U (Until). Other
standard operators can be dened:
('at the next moment "),
(p _ :p) U ' (`at some time in the future '')
in the future ''),
and similarly for their past counterparts f , and .
The spatial regions occupied by the objects under consideration may
change with time passing by, but the topological space in which they
are moving always remains the same. This nave picture is formalised
by the following concept of topological temporal model.
Denition 1. A topological temporal model (or tt-model, for short)
based on a topological space is a triple of the form
a, an assignment in T, associates with every region
variable X and every moment of time n 2 N a non-empty regular
closed subset of U . For each n, we take a n to be the function dened
by a
There are several dierent ways of introducing a temporal dimension
into the syntax of RCC-8. The most obvious is to allow applications of
the operators S and U (along with the Booleans) to spatial formulas.
We call the resulting spatio-temporal language ST 0 .
Denition 2. For a tt-model ai, an ST 0 -formula ', and
dene the truth-relation (M; n) meaning `' holds in M at
moment n,' by induction on the construction of ':
if ' contains no temporal operators, then (M; n) an ';
(M; n) there is k > n such that
every l such that n < l < k;
(M; n) there is k < n such that
every l such that k < l < n.
allows us to say many things about changes of spatial relationships
over time. For example, means that
'Kosovo will not always be part of Yugoslavia.' It is also expressive
enough to constrain movements to be continuous, in so far as one can
describe possible continuous transitions among the RCC-8 relations
that can hold between two regions (such transitions were identied by
Randell et al. (1992a) and the importance has been emphasised more
recently by Muller (1998)):
etc.
However, the expressive power of ST 0 is limited in that one can only
employ the RCC-8 relations to compare region variables at the same
point in time; so it does not allow one to describe spatial relations
between the extensions of a region variable at dierent times. For ex-
ample, there is no way to say something like 'The extension of the EU
is a part of what its extension will be next year (i.e. at the next time
To overcome this limitation we introduce the logic ST 1 , which extends
ST 0 by allowing applications of the next-time operator f not only
to formulas but also to region variables. Thus, arguments of the RCC-
8 predicates are now region terms, which consist of a region variable
that may be prexed by an arbitrarily long sequence of f operators.
For instance f f EU could denote the region occupied by the EU in
two years time and one can write formulas such as P(EU ; f f EU ).
The semantics of ST 1 is just that of ST 0 extended by the following
clause:
Using ST 1 we can now talk about the changing extensions of individual
region variables. For instance that 'the
EU will never shrink'. The new construct may also be used to rene the
continuity assumption by requiring that
i.e., 'regions X and f X either coincide or overlap.' We can also express
the condition that the extension of a region variable X is xed for the
future: or that it has at most two distinct states, one
on even days, another on odd ones: X). Note, that the
only in models based on
innite topological spaces, whereas for formulas of ST 0 (and of course
nite topological spaces always su-ce.
It may appear that ST 1 can compare regions that are separated
in time only by limited sequences of time points. However, using an
auxiliary variable, whose extension is constrained to be constant over
time, we can write, for instance,
which is satisable i 'someday in the future the present territory
of Russia will be part of the EU.' This contrasts with the formula
meaning that there will be a day when Russia (its
territory on that day | perhaps without Chechnya but with Byellorus-
sia) becomes part of the EU.
Imagine now that we want to say that every location in Europe will
pass through the Euro-zone, but only the land currently occupied by
Germany will use the Euro forever. Unfortunately, we do not know
which countries will form within Europe in the future, so we can't
simply write down all formulas of the form What
we need is the possibility of constructing regions containing all the
points that will belong to region X at some time in the future and
containing just those points included in all future states of X . Then we
can write: P(Europe;
Similarly, the formula P(Russia; that all points of the
present territory of Russia will belong to the EU at some time in the
future (but perhaps at dierent moments of time).
To permit fully general application of temporal operators to region
variables we dene the language ST 2 in which region variables may be
prexed by arbitrary strings of the operators f , . The semantics
of the new operators are given by:
k>n a(t; k)),
k>n a(t; k)).
Finally, we can make our languages ST i , for even more
expressive by allowing applications of the Boolean operations to region
terms. Their semantical meaning is dened as follows:
i be the resulting family of languages. In these languages we
can write formulas such as EQ(UK,Great Britain _ Northern Ireland),
meaning that the extension of the UK is the sum of the extensions of
Great Britain and Northern Ireland.
4. Modal Encoding of ST +The decidability and complexity results for the ST i languages were
proved in (Wolter and Zakharyaschev, 2000b) by embedding them into
the two-dimensional propositional modal logic S4 u PTL, the Cartesian
product of the logics S4 u and PTL. We call this logic Propositional
Spatio-Temporal Logic or PSTL. Its connectives are: the Booleans,
the necessity and possibility operators I and C of S4, the universal
necessity and possibility operators 8 and 9, and the temporal operators
S and U . It is not known whether the full PSTL (or even S4PTL) is
decidable. However, its fragments corresponding to the ST i languages
are quite manageable.
The modal translation from ST
into PSTL is dened by extending
the y transformation specied in Section 2 to handle region terms containing
Boolean and temporal operators. The region terms are already
ostensibly formulas of PSTL; however, we must bear in mind that
these terms are intended to denote only regular closed subsets of the
space. The easiest way to take this into account is to prex CI to every
subterm occurring in an ST +-formula.
What is the intended semantics of PSTL? As we have seen, when
encoding pure RCC-8 into S4 u we can specify the semantics with
either topological models or Kripke models and these are equivalent
in terms of validity. However, the addition of the temporal component
makes the situation more complicated in that the topological and
Kripke semantics do not perfectly agree on the class of valid formulas.
Let us dene these two types of model structures:
Denition 3. A Kripke PSTL-model is a triple
is a quasi-order (a frame for S4) and V, a valuation, is a
map associating with every propositional variable p and every n 2 N a
subset V(p; n) W . For each spatial point u 2 W and each time point
N, the truth-relation (u; n)
(u; n)
(u; n)
(u; n) such that uRv,
(u; n) there is k > n such that
plus the standard clauses for S and the Booleans. A PSTL-formula '
is satised in K if (u; n)
Denition 4. A topological PSTL-model is a structure
in which is a topological space and U is a map associating
with every propositional variable p and every n 2 N a set U(p; n) U . U
is then extended to arbitrary PSTL-formulas in the following way:
there is a k > n such that x 2 U(;
m) for all m; n < m < k.
Consequently the dened temporal operators are interpreted by
A PSTL-formula ' is satised in N if U('; n) 6= ; for some n 2 N.
The sets of PSTL-formulas satisable in Kripke models and topological
models turn out to be dierent. Of course, every Kripke model
is equivalent to some topological model. But the converse does not
hold. A good example is provided by the formula
This is valid in every Kripke PSTL-model because from any space-time
point in such a structure any given other point is reachable by
forward transition along the time line followed by transition along the
accessibility relation of F just in case it is reachable by rst moving
along the accessibility relation and then forward along the time dimension
(since the accessibility relation remains constant for all time
points). However, in certain innite topological spaces (e.g. R) one can
construct innite sequences X n of closed sets such that S
closed. In a topological model based on such a space need not
have the same denotation as C + p. This subtlety concerning innite
unions is not accounted for by the Kripke approach.
The divergence between topological and Kripke PSTL-models is
problematic because whereas the topological models correspond to the
desired spatial interpretation, most currently known methods of determining
decidability and complexity are based on Kripke models.
Fortunately, Wolter and Zakharyaschev (2000b) have been able to show
that the two semantics agree on the satisability of ST +-formulas as
long as we adopt a reasonably natural Finite State Assumption (FSA).
The FSA requires that over the innite sequence of time points each
region that one can refer to can have only nitely many distinct extensions
(but it may change its extension innitely often). Although this
restriction rules out many mathematically interesting possibilities, it
is perfectly satisfactory for a wide range of practical applications (for
example planning tasks where we want to get from an initial to a nal
situation | in a nite number of steps). To formalise the FSA we dene
the following restricted class of tt-models:
Denition 5. Say that a tt-model or is an
FSA-tt-model, if for every region term t there are nitely many regular
closed sets A U such that fa(t; n) g.
It can be shown that an ST
2 -formula is satisable in an FSA-tt-
model i it is satisable in an FSA-model based on a nite topological
space.
Because of the very restricted combinations of operators that result
from translating RCC-8 predicates and their region term arguments,
the modal translations of ST
-formulas form a rather special fragment
of the modal language PSTL. As was mentioned above, Renz (1998)
showed that an RCC-8 formula ' is satisable i ' y is satisable in
a Kripke model based on an S4-frame of depth 1 and width 2
(which means that it contains no chains of more than 2 distinct points,
and no point has more than 2 distinct successors); and it turns out that
this result can be generalised to formulas of ST +and ST +:
Theorem 2.
a) An ST
-formula ' is satisable in a FSA-tt-model i ' y is sat-
isable in a Kripke PSTL-model (which also satises FSA) whose
underlying S4-frame is of depth 1 and width 2. 4
-formula ' is satisable in a tt-model i ' y is satisable in
a Kripke PSTL-model of depth 1 and width 2.
This result makes it possible to use the method of quasi-models
(Wolter and Zakharyaschev, 1999) to prove that the satisability problem
for all the languages ST
in tt-models is decidable.
Given the rather weak interaction between time and space in ST 0 it
is not hard to show that the satisability problem for ST 0 formulas
in tt-models is PSPACE-complete (the same as that of PTL). The
satisability problem for ST 1 formulas in tt-models is decidable in
EXPSPACE (and becomes NP-complete for the sublanguage of ST 1
with only the temporal operator f ). If we restrict the admissible models
to FSA-models then the satisability problem for ST 2 formulas
is decidable in EXPSPACE. Moreover, all the complexity results just
given remain valid after replacing ST i with ST
4 Actually, since occurrences of + within the region terms of ST
are always
regularised by adding the prex CI, one might conjecture that the FSA is not
necessary. However, at present we cannot envisage how to prove this.
The topological temporal models we were considering above are
based on the discrete
ow of time N. By replacing N with Q, R, or
any other strict linear order we can extend our semantics to cover
dierent
ows of time. But then the question arises as to whether the
decidability and complexity results proved for N can be extended to,
say, the logic determined by the class of arbitrary strict linear orders,
by the reals R, or by the rationals Q. It turns out that for the language
of ST 0 we can easily extend the decidability proof for N by using the
fact that the propositional temporal logics based on those
ows of time
are decidable (Gabbay et al., 1994). As concerns ST 1 , observe that the
operator f is meaningless for dense linear orders|thus for Q and R this
language reduces to ST 0 . Decidability of ST 1 interpreted in arbitrary
linear orders is an open problem. And nothing is known about the
decidability of ST 2 interpreted in
ows of time dierent from N. We
conjecture, however, that the methods developed in (Hodkinson et al.,
2000) can be used to prove the decidability of ST 2 based on any of the
ows of time mentioned above (under FSA).
5. RCC-8 and Interval Temporal Logic
We have seen how the RCC-8 theory can be temporalised by combining
it with a point-based temporal logic. However, since the region-based
approach to spatial reasoning was inspired by and closely mirrors
the interval-based approach to temporal reasoning (Allen, 1981; Allen,
1984) (they both take extended entities, rather than points as primi-
tives) it would seem far more natural to temporalise RCC-8 by combining
it with an interval-based logic. In this section we show that this
can indeed be done; and moreover, that the resulting system can in fact
be embedded by suitable syntactic denitions within the point-based
logic PSTL dened above.
We remind the reader that Allen's logic has thirteen basic relations
between time intervals | Before(i; j), Meets(i; j), Overlaps(i; j),
The set of formulas constructed using these predicates and the Booleans
can be regarded as a temporal 'twin' of RCC-8.
Following Allen (1984), we write HOLDS(R; i) to say that the relation
R holds during some time interval i. Thus HOLDS(PO(X; Y ); i)
means that during interval i regions X and Y partially overlap. Let
us call an ARCC-8 formula any Boolean combination of basic temporal
predicates and formulas of the form HOLDS('; i) where ' is an
RCC-8 formula. Here is a simple example of a valid entailment in this
unsophisticated language:
HOLDS(TPP(Hong Kong
HOLDS(DC(Hong Kong ; UK); j);
There are dierent views on how intervals should be modelled in
dierent time
ows. A common interpretation is that the intervals
are treated as ordered pairs of distinct points of the domain Q or R.
Within our framework we can in fact adopt any of these models; but,
for simplicity of presentation it is convenient to employ a semantics
where intervals are arbitrary convex non-empty subsets of the time
points of an arbitrary time
ow. Thus we give the following semantics
for ARCC-8:
Denition 6. An it-model (interval topological model) is the triple
is a strict linear order (modelling the
intended
ow of time); is a topological space; and assignment
a associates with every interval variable i a non-empty convex subset
of W and with every region variable X and every moment of time u it
associates a regular closed set a(X; u) in T.
The truth-relation is dened inductively as follows (clauses for the
Booleans are standard):
and not M j= a Before(i; j);
The other interval relations are dened similarly;
M j= a HOLDS('; i) i for every point u 2 a(i) we have T
We now show how by using some ideas of Blackburn (1992) the
language ARCC-8 can be directly embedded into PSTL. For convenience
we dene undirected universal and existential modalities over
time: is true at every moment) and
We extend the translation function y dened above to
encode the interval relations and HOLDS predicate. For example, we
replace in '
and the other Allen relations can be encoded in similar fashion. Here
the variables t i are just ordinary propositional variables; however, we
need to constrain their interpretation so that they can be employed
to stand for intervals. Therefore, we add to the resulting formula the
conjuncts
for every interval variable i occurring in ', thus obtaining ' y . The rst
conjunct ensures that t i is non-empty; 5 the second that it is a convex
interval of the time series; and the nal conjunct means that the value
of t i is constant relative to the spatial dimension.
It can be shown that an ARCC-8 formula ' is satisable in an
it-model just in case its modal translation ' y is satisable in a Kripke
PSTL-model. Moreover, the satisability problem for ARCC-8 formulas
in it-models is NP-complete. Thus ARCC-8 has the same computational
complexity as both RCC-8 and the constraint language of
Allen's temporal interval predicates (Vilain et al., 1986).
6. How Far Can Multi-Dimensional Modal Logic Take Us?
We have seen how the 2-dimensional language PSTL provides an expressive
formalism for representing spatio-temporal information, which
encompasses both topological constraints and linear temporal logic. We
now consider what other concepts one might want to represent and note
some di-culties that arise due to known complexity results.
A concept that is very useful for describing real situations is that
of spatial convexity. The addition to the rst-order RCC formalism
of a function giving the 'convex-hull' of a region proposed in (Randell
et al., 1992a) (such a function and a convexity predicate true of convex
regions are interdenable). First-order languages including topological
relations and a convexity predicate have been found to be highly
expressive (Pratt, 1999). Although satisability of a combination of
5 If we replace this conjunct by out 'intervals'
consisting of a single time point.
RCC-8 relations and convexity predicates is known to be decidable
is has also been shown to be as hard as solving systems of non-linear
constraints over R (Davis et al., 1999). It is unclear whether modal
logics can contribute to reasoning about convexity. The obvious models
of multi-dimensional space within which convexity constraints could
be specied involve cross products of linearly ordered innite frames
corresponding to coordinate axes of the space. However, it is known
that all modal logics of such products are undecidable (Reynolds and
Zakharyaschev, 2001). Balbiani et al. (1997) have given a modal logic
of incidence geometry within which one can express collinearity and
betweenness (and hence convexity) but it is not known whether this
logic is decidable.
Another concept crucial to real situation descriptions is that of connectedness
of regions. The objects we normally think and talk about
are connected in the sense that any two points within an object can be
joined by a path that lies entirely within the object. One of the most
important open problems in spatial reasoning is whether there is a decision
procedure for testing satisability of sets of topological relations
holding among connected planar regions. A lower exponential complexity
bound follows from the results of (Kratochvl, 1991; Kratochvl and
Matousek, 1991) concerning an analogous problem for planar graphs;
and this result applies even in the case where we deal with only the
basic non-disjunctive relations of RCC-8. We speculate that a modal
analysis might shed some light on this decision problem.
One might also want to introduce more expressive power in the
temporal dimension. In order to describe continuous changes more adequately
one may want to employ a logic based on a dense model of
time (Barringer et al., 1986). Within such a time
ow, we must consider
whether spatial relationships hold over open or closed intervals.
This problem has be examined by Galton (1997,2000), who argues that
whether a spatial relation is true at the bounding point of an interval
over which it holds depends on the nature of the relation in question.
This analysis may allow for more detailed description of changing spatial
relationships than is possible within Allen's (1984) treatment. Also,
rather than treating time as a linear ordering one might like to model
alternative possible histories in terms of some branching structure,
such as is described by the logic CTL (Emerson and Halpern, 1986).
Semantically it is straightforward to combine the spatial interpretation
of S4 u with any reasonable temporal logic; but so far nothing is known
about the complexity of combinations with non-linear time-
ows.
In considering decidability and complexity of multi-modal systems,
logicians have almost always looked at the problem for arbitrary formulas
of the resulting modal language. However, as we saw in the case
of PSTL, sub-languages with severely restricted syntax (e.g. ST (+)
may be able to express a signicant vocabulary su-cient for many
applications. Thus we suggest that looking for such sub-languages is a
research area of great potential.
In order to really demonstrate the utility of multi-dimensional logics
one would need to develop and implement reasoning algorithms capable
of carrying out useful reasoning tasks. Automated reasoning with multi-modal
logics is still in its infancy; but recently there have been some
successes in developing proof methods (Dixon et al., 1998; Hustadt
et al., 2000). A tableau calculus for the local cubic logic LC 2 , which is
closely related to S5S5, has been implemented by Marx et al. (1999).
7. Conclusion
We have outlined the general structure of a knowledge representation
formalism, based on multi-dimensional modal logics. This framework
seems to be well suited for representing a large vocabulary of useful
high-level spatio-temporal relations. Our decidability and complexity
results for the languages ST (+)
i and ARCC-8 show that this approach
enables one to construct very expressive yet eective spatio-temporal
languages. We hope that the reader will take from this paper not only
the particular details of the language PSTL but a wider appreciation
of the possibilities of applying multi-dimensional modal logics to the
development of Articial Intelligence. We would also like to suggest that
spatio-temporal reasoning is an area within which cooperation between
pure logicians and researchers tackling specic reasoning problems arising
in applications can lead to both interesting theorems and powerful
practical algorithms.
--R
Modal Logic.
'A calculus of individuals based on 'connection
Representations of Commonsense Knowledge.
Principles of Knowledge Representation and Reasoning:
International Journal of Geographical Information Systems 5(2)
Reasoning About Knowledge.
Temporal Logic: mathematical foundations and computational aspects
Fibring Logics.
Formal Theories of the Commonsense World.
Journal of Combinatorial Theory
The temporal logic of reactive and concurrent systems.
Principles of Knowledge Representation and Reasoning:
Journal of Logic and Computation
Casopis pro p
Revised version in (Weld and De Kleer
Readings in Qualitative Reasoning About Physical Systems.
The decision problem for combined modal logics.
Fundamenta Informaticae
Frontiers of Combining Systems 2.
--TR
--CTR
Alfredo Burrieza , Manuel Ojeda-Aciego, A Multimodal Logic Approach to Order of Magnitude Qualitative Reasoning with Comparability and Negligibility Relations, Fundamenta Informaticae, v.68 n.1-2, p.21-46, January 2005
Andrzej Skowron , Piotr Synak, Complex Patterns, Fundamenta Informaticae, v.60 n.1-4, p.351-366, January 2004
Norihiro Kamide, Linear and affine logics with temporal, spatial and epistemic operators, Theoretical Computer Science, v.353 n.1, p.165-207, 14 March 2006
Andreas Schfer, Axiomatisation and decidability of multi-dimensional Duration Calculus, Information and Computation, v.205 n.1, p.25-64, January, 2007
Stphane Demri , Deepak D'Souza, An automata-theoretic approach to constraint LTL, Information and Computation, v.205 n.3, p.380-415, March, 2007
Frank Wolter , Michael Zakharyaschev, Qualitative spatiotemporal representation and reasoning: a computational perspective, Exploring artificial intelligence in the new millennium, Morgan Kaufmann Publishers Inc., San Francisco, CA, | modal logic;multi-dimensional logic;spatio-temporal reasoning |
591458 | Slipping and Tripping Reflexes for Bipedal Robots. | Many robot applications require legged robots to traverse rough or unmodeled terrain. This paper explores strategies that would enable legged robots to respond to two common types of surface contact error: slipping and tripping. Because of the rapid response required and the difficulty of sensing uneven terrain, we propose a set of reflexes that would permit the robot to react without modeling or analyzing the error condition in detail. These reflexive responses allow robust recovery from a variety of contact errors. We present simulation trials for single-slip tasks with varying coefficients of friction and single-trip tasks with varying obstacle heights. | Introduction
R OUGH terrain occurs not only in natural environments
but also in environments that have been constructed
or modified for human use. Currently, most legged
robots lack the control techniques that would allow them
to behave robustly on such relatively simple rough terrain
as stairs, curbs, grass, and slopes. Even smooth terrain
becomes difficult to traverse if it includes small obstacles,
loose particles, and slippery areas. Many control systems
for bipedal robots have assumed steady-state running over
smooth surfaces, but some have explored control techniques
for rough terrain. Statically stable robots, which always
maintain their balance over at least three legs, have used
controllers with foot-placement algorithms to insure viable
footholds. However, for dynamically stable robots,
which run with a ballistic flight phase, constraints on timing
and foot placement increase the difficulty of designing
controllers that can anticipate rough terrain or react to er-
rors. This paper demonstrates the effectiveness of preprogrammed
high-level responses to errors during locomotion
in a complex dynamic environment. A suite of responses
allows a simulated, three-dimensional, bipedal robot to recover
from slipping on low friction surfaces and tripping
over small obstacles (Figure 1).
Many ground contact errors would be avoided if the control
system could guide the robot around slippery areas
and obstacles. However, the approximate nature of sensor
information obtained at a distance means that it is not
always possible to sense the surface properties of terrain
before making contact. For example, small holes, bumps,
debris, and sticky or slippery areas are difficult to detect
from a distance with current technology. If the robot cannot
detect and avoid or prepare for surface features in ad-
vance, then robust locomotion on rough terrain requires
that the robot respond to unexpected features after the
contact error has occured and before the robot crashes. For
College of Computing, Georgia Institute of Technology, Atlanta,
GA 30332-0280. [gboonejjkh]@cc.gatech.edu.
Submitted to Autonomous Robots.
Slip Sequence
Trip Sequence
Fig. 1. Examples of a Slip and Trip. Without the addition of
reflexes for recovering from slips and trips, the simulated robot does
not respond successfully to slippery areas or contact with an obstacle.
Foot Foot
3 degree of freedom
hip joint for each leg
1 degree of freedom
telescoping leg joint
for each leg
Rearward Leg
Body
Direction of Travel
Forward Leg
Fig. 2. Biped Structure. The simulated bipedal robot consists of a
body and two telescoping legs. Each leg has three degrees of freedom
at the hip and a fourth degree of freedom for the length of the leg.
dynamically stable robots, the time available for modeling
the surface and planning an appropriate reaction is severely
limited. In the case of the dynamically stable bipedal robot
shown in Figure 2, the controller may have less than a few
hundredths of a second in which to choose or plan an appropriate
recovery.
We define reflexes as responses with limited sensing and
no explicit modeling. That is, the robot can detect a slip
or a trip, but makes no attempt to estimate the properties
of the surface or obstacle or to calculate a corresponding
recovery plan. Instead, the slipping and tripping sensors
trigger fixed responses. These reflexes are defined at a high
level, such as reconfigurations of the leg positions, and at
a low level, such as modifications of servo gains. Just as
animal motor programs can be considered both open-loop
and closed-loop[1], several low-level feedback control laws
operate during the primarily open-loop reflex responses.
For example, a reflex may reconfigure the leg position, but
sensing is used to determine transitions in the leg controller
state machine during the recovery step.
Fig. 3. Physical Biped Slip. Planar two-legged robot running
across an oily spot on the floor. Footage from the MIT Leg Labora-
tory. [Frames: 0, 35, 70, 91, 105, 140]
During experimentation with a physical, planar biped,
the robot sometimes slipped on hydraulic oil or tripped on
cables in its path. Because the robot had no responses
customized for these error conditions, it almost always immediately
crashed. This paper reports a set of fixed reflexes
that enable robust recoveries for a simulated three-dimensional
robot in tasks involving a single slip or trip.
In the next section, we describe previous approaches to
legged locomotion in rough terrain. In Section III, we consider
biological reflexes. Section IV describes the simulated
bipedal robot and its control system. The slipping prob-
lem, slipping reflexes, and simulation results are presented
in Section V, followed by the tripping problem, tripping re-
flexes, and results in Section VI. The reflex approach and
results are discussed in Section VII.
II. Locomotion on Rough Terrain
A suitable foothold is one that allows a legged system
to maintain balance and continue walking or running. For
statically stable locomotion, the difficulty is not in placing
the robot's feet on footholds, but in deciding which
locations on the terrain provide suitable footholds. Successful
locomotion on rough terrain was demonstrated by
the Adaptive Suspension Vehicle[2] and by the Ambler[3],
[4], [5]. These large, statically stable machines traversed
grassy slopes, muddy cornfields, and surfaces that included
railroad ties and large rocks. Static stability allowed these
robots to emphasize detection at a distance and avoidance
of obstacles and uncertain footholds.
Klein and Kittivatcharapong[6] proposed algorithms for
insuring that foot forces remain within the friction cone
and identifying situations in which these constraints, or the
desired body forces and torques, could not be achieved.
Their work addressed prevention of slipping and did not
consider sensor noise or responses to unmodeled surfaces.
For dynamically stable robots, the control of step length
for locomotion on rough terrain interacts with the control
of balance. Hodgins and Raibert[7] implemented three
methods for controlling step length of a running bipedal
robot. Each method adjusted one parameter of the run-0.5hip
altitude
forward
speed
(m/s)
pitch
-2020leg
angle
foot
position
time
Fig. 4. Slipping Data of the Physical Robot. The physical
planar robot slipped on oil during a laboratory experiment at the
point indicated by the vertical dotted lines. The top three graphs
show the height, forward speed, and orientation of the body. The
bottom two graphs show the angles of each leg and the position of
each foot on the ground. For each step but the last, the foot is
stationary while it is on the ground.
ning cycle: forward running speed, running height, or duration
of ground contact. In laboratory demonstrations, a
biped running machine used these methods for adjusting
step length to place its feet on targets, leap over obstacles,
and run up and down a short flight of stairs. However,
unlike the tasks described below, the size and location of
the objects were known to the controller in advance.
developed algorithms for running on terrain that
was known to be slippery. By running slowly, the robot
generated nearly vertical foot forces. His controller used
a priori knowledge or estimation of friction coefficients to
prevent slipping by confining control forces and torques to
slip-free regions.
Kajita and Tani[9] used an ultrasonics sensor to construct
a ground profile of terrain that consisted of horizontal
surfaces at varying heights. Yamaguchi et al have
built a bipedal robot that uses feet to sense ground inclinations
and plan appropriately[10], although it was not able
to react to slips or trips.
III. Reflexive Responses to Errors
Biological systems use many different reflexes in locomotion
and manipulation. Reflexes help to restore balance
when perturbations occur during walking or stand-
ing[11], [12], [13]. The role of reflexes in walking is com-
plex: the same stimulus elicits a different response in the
stance phase than in the swing phase[14], [15], [16]. Touching
the foot of a cat or human during a swing phase, for
example, will cause the leg to flex, raising the foot. If an
obstacle caused the stimulus, this response might lift the
foot over the obstacle and allow walking to continue. During
the stance phase, a stimulus delivered to the foot causes
the leg to push down harder, resulting in a shorter stance
phase. Although these actions are opposite, both facilitate
the continuation of locomotion.
Robotics has adopted the term "reflex" from the biological
literature, but in both biology and robotics the precise
definition of the term varies from study to study. Most
researchers in robotics use the term to mean a quick response
initiated by sensory input. Some require reflexes to
be open-loop and to proceed independently of subsequent
sensory input[17], [18]; others apply the term more loosely
to describe actions that are performed with feedback until
a terminating sensory event occurs[19]. In some cases, reflexes
refer to general purpose actions[20], [21] and in others
only to actions taken to correct errors or to compensate for
disturbances[19] or transitions[22].
Brooks's subsumption architecture[21] combined several
simple reflex-like actions to produce complex behaviors
such as six-legged walking. A global gait generator specified
the order of leg use while inhibitory connections between
the legs prevented conflicting reflexes from acting
simultaneously. Other hexapod robot researchers have designed
subsumption controllers for rough terrain[23] and
have integrated reactive leg control with gait planning for
rough terrain[24].
Hirose[20] built and controlled a statically stable
quadruped that used a reflexive probing action to climb
over objects and to walk up and down steps without visual
input or a map of the terrain.
Wong and Orin[19] implemented two reflex responses for
a prototype leg of the Adaptive Suspension Vehicle. Using
velocity and hydraulic pressure information from sensors
at the joints, they were able to detect foot contact and
slippage. A foot contact reflex reduced the peak forces at
touchdown. A foot slippage reflex was used to detect and
halt slipping.
Reflex responses have also been used in manipulation.
Tomovic and Boni[17] used a reflex response to implement
grasping for the Belgrade prosthetic hand. Bekey and To-
movic[18] continued the exploration of prosthetic control
systems with a rule-based technique that relied on sensory
data and fixed response patterns.
IV. Dynamic Bipedal Robots
The simulated robot used in our research is based on
a planar bipedal robot constructed by Raibert and col-
leagues[25], [26]. The simulated robot is three-dimensional
and has three controlled degrees of freedom at each hip and
one for the length of each leg (Figure 2). In the physical
robot, the leg contains a hydraulic actuator in series with
an air spring. The simulation models the spring and actuator
as a linear spring with a controllable rest length. In
Fig. 5. Simulated Biped Slip. The dark circle represents an area
of the floor with a reduced coefficient of friction. Without slipping
reflexes, the simulated robot is unable to complete a step on a slippery
surface. The first leg slips, almost immediately becoming airborne
as it accelerates forward. As the body falls, the second leg hits the
surface and also slips. The second leg continues to accelerate forward.
[Friction coefficient: 0.04. Times (s): 0.0, 0.06, 0.09, 0.11, 0.12, 0.13]
experiments with the physical robots, hydraulic fluid occasionally
created slippery spots that caused the robot to fall
Figures
3 and 4). A simulation of a similar fall is plotted
in
Figures
5 and 6. The physical robot was also able to
climb stairs and jump over boxes[26]; however, the positions
of the obstacles were known in advance. The current
research extends the controller to handle unexpected slips
and unanticipated collisions with a box.
The controller achieves dynamically stable, steady-state
running by decomposing the control problem into three
largely decoupled subtasks: hopping height, forward ve-
locity, and body attitude. Hopping height is maintained
by adding enough energy to the spring in the leg during
stance to account for the system's dissipative losses.
Forward velocity is maintained by choosing a leg angle at
touchdown that provides symmetric deceleration and acceleration
as the leg compresses and extends. The attitude
of the body (pitch, roll, and yaw) is maintained with
proportional-derivative servos that apply torques between
the body and the leg while the foot is on the ground.
The robot control system is implemented as a state machine
that sequences through the flight and stance phases
for each leg, applying the control laws that are appropriate
for each state. As shown in Figure 7, flight is followed
by a stance phase of four states. During loading , the
foot makes contact with the ground and begins to bear the
weight of the robot. During compression, the leg spring
is compressed by the downward velocity of the robot. After
the spring has stopped the vertical deceleration of the
body, the body begins to rebound during thrust . As the leg
reaches maximum extension during unloading , it ceases to
bear weight. After liftoff, the roles of the legs are reversed
and the second leg is positioned forward in anticipation
of touchdown. For further details on the control system,
see [26] and [25].
The control system's state machine depends on measurements
of leg length to determine state transitions dur-0.20.61.0
hip
altitude
forward
speed
(m/s)
-0.3
-0.2
-0.0pitch
(rad)
leg
angle
(rad)
foot
position
time
Fig. 6. Slipping Data of the Simulated Robot. After taking
five steps on a surface with a friction coefficient of 1.0, the simulated
robot steps on a region with a coefficient of 0.20 and slips. Because
no slipping recovery strategies are active, the robot falls. The top
three graphs show the height, forward speed, and orientation of the
body. The bottom two graphs show the leg angles of both legs and
the position of each foot on the ground. When the foot slips (vertical
dotted line), it leaves the ground and the other foot soon impacts.
Loading Compression Thrust Unloading
Direction of Travel
Flight
Fig. 7. Control States. Running is achieved by dividing each step
into several states and applying the appropriate control laws during
each part of the running step.
ing steps. Slips may interfere with control by altering leg
lengths unexpectedly. The transition from loading to com-
pression, for example, occurs when the leg has shortened
by a small amount. After a slip, the leg may lengthen. Not
only must slipping reactions prevent these errors, but they
must minimize interference with normal control, such as
the adjustment of body attitude.
V. Slipping
The impact of the foot on the ground, the weight of the
robot, and the forces and torques generated by the hip and
leg servos create a force on the ground during a step (Fig-
F
Direction
of Travel
Fig. 8. Foot Forces and the Friction Cone. During a step, the
foot produces forces on the ground, F , with horizontal and vertical
components,F h and Fv . Slipping occurs when the angle of the impact
force is outside the friction cone.
ure 8). Slipping occurs when the horizontal component of
the force of the foot on the ground, F h , exceeds the maximum
force of static friction generated by the ground. A
simple model of this interaction is that the maximum force
of static friction is directly proportional to the normal force
of the ground on the foot, F v . Under this model, slipping
will occur when the horizontal component of F exceeds the
vertical component times the coefficient of static friction:
where - s is the coefficient of static friction. When slipping
occurs, the horizontal force returned by the ground is given
by
where - d is the coefficient of dynamic friction and the sign
of F h should remain unchanged. These relationships define
a friction cone, illustrated in Figure 8. When the force of
the foot on the ground lies within the friction cone, the
foot does not slip. The angle of the cone is given by
Note that this cone is defined for foot forces, not leg angles.
The motion of the leg prior to impact affects the direction
of the foot's force on the ground, as do the control torques
applied to the hip joint and the leg spring. Foot forces are
most likely to exceed the friction cone at the beginning or
end of a step, when the angle of the force vector is greatest.
Slips at the beginning of a step are more likely than slips
at liftoff because the foot is moving with respect to the
ground at touchdown. In contrast, the foot is stationary at
liftoff. Slips during liftoff are often less critical because the
step is nearly complete; the controller has already executed
corrections during the step.
Our simulations assumed minimal sensory information:
the properties of the surface and the extent of the slipping
area were not available to the control system. The
controller could not adjust the leg configuration prior to
touchdown or try to position the foot outside the slippery
area to find a secure foothold. Neither the forces on the feet
nor the coefficients of friction were available to the control
system. However, the control system could detect slips.
In the simulation, slips were detected when a foot moved
while in contact with the ground. The control system of a
Increase Leg Force
Increase Hip Torque
Direction of Travel
Fig. 9. Same-Step Reactions. When a slip has been detected, a
torque can be applied at the hip to reduce the horizontal force on the
ground or the leg can be extended to increase the vertical force.
physical robot can detect slips indirectly by measuring joint
angles and velocities or structural forces. Direct methods
include encoder wheels and micro-slip detectors.
When the control system has detected a slip, it can attempt
to continue the step or abandon that step and pull
the leg off the ground. In the first case, hip torques or leg
forces can be applied to increase the vertical component of
the foot force while decreasing the horizontal component,
thus returning the force vector to within the friction cone.
If the step is abandoned, one of the legs can be positioned
during the next flight phase so that the leg angle at the
next touchdown will be near vertical or both legs can be
moved to a triangular configuration. In the simulations
described here, we defined a response to be successful if
the robot was able to continue running after slipping and
taking a recovery step in the slippery region, then taking
subsequent steps on a non-slippery surface. Changes in velocity
or hopping height were not considered failures provided
that the control system was able to maintain balance
and return to steady-state running.
A. Same-Step Response Strategies
Reacting to a slip requires careful management of the
horizontal and vertical components of the forces generated
by the impact of the foot on the ground. Initial responses
to a slip can attempt to alter the force vector immediately
by generating a torque at the hip or a force axial to the leg
Figure
9).
The first reaction responds to a slip by increasing the
hip torque by a fixed amount. In most cases, this action
increases the vertical component of the foot's force
on the ground. After the foot stops slipping, the hip controller
reverts to its normal task of correcting pitch errors.
This strategy may have undesirable consequences because
a torque applied at the hip also increases the forward velocity
of the body thus increasing the likelihood of a slip
on a subsequent step. Applying a torque at the hip also interferes
with the correction of body attitude during stance
and tends to increase the pitch of the body.
The second reaction responds to a slip by compressing
the leg spring a fixed amount to increase the vertical force
at the foot and regain a foothold. In a normal running
step, the leg spring stores energy during the stance phase
and causes the body mass to have approximately equal and
opposite vertical velocities at liftoff and touchdown. To
maintain the duration of flight, the control system length-
l d
l
d
l
l
Ground
Contact
Maximum
Compression
Detected
Maximum
Compression
Fig. 10. Forcing the Foot into the Ground. In a normal step
(top), energy is added into the leg spring at the moment of maximum
compression. In the forced step (bottom), the loading on the leg is
increased just after touchdown, forcing the foot into the ground and
shortening the step duration. l d is the desired leg length. \Deltal is the
change in desired leg length that returns the robot to the desired
hopping height.
Rebound
Compress
Detect Slip Set Front Leg
Front Foot Reposition
Detect Slip Rebound
Compress
Set Rear Leg
Rear Foot Reposition
Detect Slip Rebound
Compress
Triangle
Stable Triangle
Fig. 11. Repositioning Strategies. After a slip has been detected,
the initial step is abandoned and one or both legs are repositioned for
the next step. The leg angle at touchdown on the next step will be
closer to vertical, keeping the impact force vector within the friction
cone.
ens the leg to add energy equivalent to that lost due to
internal mechanical losses and to the impact of the un-
sprung mass of the lower leg with the ground. In a normal
step, thrust occurs at the moment of maximum compression
of the spring (Figure 10). In responding to a slip, the
control system may alter this sequence by extending the
leg as soon as the slip is detected. If the leg is close to
vertical, this extension increases the vertical component of
the foot's force on the ground and may stop the slip. The
extension also adds energy into the leg spring. The extra
energy is removed later in the step by lengthening the leg
spring when the leg is vertical, leaving the hopping height
unchanged (Figure 10).
One effect of this reaction is to slow the robot, a desirable
6effect when the surface is slippery. However, the
foot forcing reflex may lead to a crash if the leg geometry
and velocity is such that extending the leg increases the
horizontal forces on the foot more than the vertical forces.
Thus, the foot forcing reflex may not be sufficient in itself
to recover from slips.
The foot forcing reaction shortens the period of time
during which the spring is passively compressed, leading to
a shorter stance phase and a style of running that utilizes
quick hops rather than long strides. We have observed that
this quick-stepping behavior is a useful method for running
briefly on slippery surfaces because the leg angle at touchdown
is near vertical. However, the shorter stance phase
also reduces the available time for correcting the body attitude
and makes steady-state running difficult to achieve.
B. Repositioning Strategies
The step on which the initial slip occured may be abandoned
by immediately lifting the foot; the resulting flight
phase provides a brief opportunity to prepare for another
landing on the slippery surface. By reconfiguring the legs
during the flight phase following the initial slip, the control
system can attempt to keep the foot forces within the friction
cone. Because the coefficient of friction is not known,
the size of the friction cone is unknown. Therefore, the
best place for the foot at the next touchdown is directly
under the body, making the leg vertical at touchdown.
Figure
11 diagrams the strategies that reposition the legs.
Figures
12, 13, and 14, contain sequences showing the repositioning
strategies involved in recovering from a slip.
After a slip has been detected, both legs may be used in
the recovery by configuring them in a narrow fixed triangle
vertically centered under the body. The control system
attempts to hold this triangle throughout the subsequent
step and does not apply the normal pitch, roll, and yaw
adjustments. Instead, the robot bounces, letting the geometric
configuration provide stability rather than using
active control. The leg angles in normal running are nearly
symmetric during the flight phase of steady-state running;
the control system only has to equalize the leg lengths to
create a symmetric triangle. Because the extent of the friction
cone is unknown, the triangle is narrowed so the legs
are close to vertical. When both feet contact the ground,
foot forcing is applied to each to reduce the time of stance.
After both feet have lifted off the ground, the control returns
to a normal flight state.
C. Slipping Results
The slipping strategies were tested in simulation by varying
the initial velocity of the robot and the coefficient of
friction to produce multiple runs. For each trial, a circular
slippery area was simulated at the location of the first
footfall. During successful runs, the robot stepped once in
the slippery area and then five additional times on a non-
slippery surface. The initial velocity was 2.5 \Sigma 0.25 m/s.
The size of the slippery area for each reaction strategy was
large enough to prevent a foot from sliding to the edge, a
situation that allowed an easy recovery. The slippery area
was small enough that subsequent footfalls were located
outside of it. Twenty friction coefficients between 0.025
and 0.5 were used. Both static and dynamic coefficients
were set to the same value for each trial of 20 simulations
with different initial velocities. The robot was judged able
to recover from a slip at a given coefficient of friction if at
least half of the trials were completed successfully.
For the successful trials, we computed a measure of the
error at touchdown of the step after the recovery step that
followed the slip. The error measure was the summed absolute
values of differences between the actual and desired
angles for the body yaw, ff, pitch, fi, and roll, fl:
The error calculation was designed to measure how well
the slip recovery strategy had positioned the robot after
the slip step, the recovery step, and the subsequent ballistic
flight. The errors for the successful trials were averaged
to compute the data shown in Figure 15. This graph illustrates
the tradeoff between the two types of strategies.
With no active reflexes, the controller is able to negotiate
friction coefficients as low as 0.28. Upon contact, the foot
slides; as it is loaded, the vertical and horizontal forces
increase, pushing the foot back under the body. Eventually
the forces on the foot reenter the friction cone, slipping
ceases, and a normal step ensues. The foot forcing strategy
causes the foot to slide further out from under the body,
leading to fewer recoveries at lower coefficients of friction
than the steady-state control system. We observed this
effect for several running speeds and heights. However, it
may be a consequence of the geometry of the robot design;
foot forcing may be useful for slow moving robots or those
with other gait patterns. The hip torque reflex succeeds at
pulling the leg back and enables recoveries as low as 0.22.
Note that hip torque does indeed increase the body pitch,
producing increased errors shown in the graph.
The repositioning strategies delay error correction while
the legs are reconfigured. As a result, the repositioning
strategies produce larger errors upon return to normal running
than the foot forcing and hip torque reflexes. However,
the repositioning strategies are able to recover from slips on
surfaces with smaller coefficients of friction. By lifting the
leg and repositioning it within the friction cone, the front
and rear repositioning reflexes are able to recover from surfaces
with coefficients as low as 0.07 and 0.15, respectively.
The front repositioning strategy is more successful than the
rear repositioning strategy because it more effectively reduces
the relative speed of the foot over the ground before
impact. Because the robot is moving forward while the
foot is airborne, bringing the rear leg forward increases the
relative speed between the foot and the ground. The front
repositioning strategy brings the front leg back, reducing
the relative speed. On impact, the foot with the lower relative
speed is subjected to smaller horizontal forces and is
less likely to slip.
The robot experiences increased slipping as the coefficient
of friction decreases, but it often recovers because
Fig. 12. Front Leg Repositioning. The front leg is lifted and repositioned for a more vertical impact. [Friction coefficient: 0.20. Times
Fig. 13. Rear Leg Repositioning. The rear leg is brought under the slipping robot to arrest the fall. The newly planted leg slips upon
takeoff, but the step is successful because the body attitude is not disturbed significantly. The robot is able to continue running. [Friction
coefficient: 0.20. Time (s): 0.0,
Fig. 14. Stable Triangle Recovery. After detecting a slip, the robot forms a stable triangle. Although the legs slip just prior to liftoff,
the control system is able to recover because the slip is symmetric and occurs at the end of the step. [Friction coefficient: 0.02. Times (s):
Friction Coefficient0.10.30.50.70.9Error
None
Foot Force
Hip Torque
Front Reposition
Rear Reposition
Stable Triangle
Fig. 15. Touchdown Errors. If the robot recovers from a slip,
it starts the next step with some error. This graph illustrates the
tradeoff between smooth running and slip recovery. Lower curves
indicate smaller errors in body and leg angle. Longer curves indicate
that a greater range of friction coefficients can be tolerated.
the slips occur at the end of the recovery step. Figure 13
shows a normal ground contact and rebound followed by
a slip upon takeoff. Because the hopping height, forward
speed, and body attitude control algorithms have already
been applied, the slip has little effect on the configuration
of the robot. Figure 14 shows slipping upon takeoff for
the stable triangle strategy, which applies no attitude correction
during the recovery step. However, as Figure 14
shows, both legs slip symmetrically, cancelling the effect of
their torque on the body. Thus, the stable triangle reflex
is capable of recovering from surfaces with coefficients as
low as 0.025.
VI. Tripping
For steady-state running, the control system detects expected
events, such as foot contact or initial leg spring
compression, and uses these signals to transition between
control states. During each state, it applies the appropriate
collection of control laws. Tripping occurs when the robot
feet or legs encounter unexpected obstacles, causing the
controller to execute inappropriate servo commands (Fig-
ure 16).
To explore reflexive responses to tripping, we considered
the task of returning the robot to steady-state running after
Fig. 16. Simulated Trip. The front foot contacts the vertical face
of a box and slides down the surface. With no response, the robot is
unable to continue running and crashes. [Times (s): 0.0, 0.05, 0.09,
a collision with a box. The existing controller allowed the
robot to continue running for some unexpected contacts.
For example, foot contacts on the top surfaces of boxes,
though premature in the flight phase, allowed a normal step
to occur. Oblique contacts, such as brushing the side of the
box, also did not usually prevent running from continuing.
Other contacts, such as a foot or leg contacting the vertical
face of a box, resulted in crashes.
A. Tripping Responses
As in the case of slipping, the sensing requirements were
minimal. The controller detected only that a contact with
a foot or leg had occurred. It did not detect where on
the leg the contact had occurred. These conditions could
be determined on a physical robot with contact sensors on
the legs or via the existing joint angle sensors.
When a leg or foot hits the front surface of a box, a foot
must be repositioned to find a foothold on or beyond the
box. If the forward foot hits the box, either the forward or
the rear foot can be retracted and repositioned to contact
the top surface of the box, where good footholds are avail-
able. We call these strategies the "front lift" and "rear lift"
reflexes, depending on which leg is lifted to the top surface
of the box. If the rear leg hits a box, the leg can be pulled
back, allowing it to pass over the box without contact. We
refer to this strategy as "rear pull." These reflexes are diagrammed
in Figure 17 and shown in Figures 18, 19 and 20.
B. Tripping Results
To test the tripping reactions, boxes of varying heights
were placed in the path of a robot running in steady state.
For the front lift and rear lift reflexes, the vertical face
of each box was divided into 20 impact heights and the
robot was released with the front foot 2 cm from the box
at each height. For the rear pull reflex, the robot was
placed straddling boxes of varying heights with the forward
foot making an initial ground contact in a normal running
step. As the box height increased, the rear leg eventually
contacted the box as it swung forward. In all simulations,
Compress Rebound
Detect Trip Set Front Leg
Front Lift Trip Response
Rebound
Detect Trip Compress
Set Rear Leg
Rear Lift Trip Response
Rebound
Detect Trip Compress
Pull Rear Leg
Rear Pull Trip Response
Fig. 17. Trip Recovery Strategies. After a trip has been detected,
one of the legs is repositioned in an attempt to contact the top surface
of the obstacle or avoid it entirely.
the initial forward speed of the robot was varied by a small
random factor.
With no reflex responses, the robot was unable to continue
running following a trip. The front lift and rear lift
response curves show that as the box heights increase, the
tripping reflexes are less likely to produce a recovery (Fig-
ures 21 and 22). The number of crashes increases as the
box height increases. This increase in crashes is due to the
increasing distances to the box top as the height increases.
If the foot hits the box near the top, there may be sufficient
time to lift it to the top of the box. However, as the box
height increases, fewer potential contact points are near the
top edge of the box.
To measure the disturbance to normal running, we computed
the same error measure as was used in the slipping
trials. The error measure was the sum of the absolute values
of the errors between actual and desired yaw, ff, pitch,
fi, and roll, fl:
The bottom graphs in Figures 21 and 22 show that if the
robot is able to recover, it does so with approximately the
same error independent of box height.
The front lift reflex causes less touchdown error than
does the rear lift reflex. To recover with the front foot, the
foot must lift over the box edge, whereas a recovery with
the rear foot must move the rear foot from its position behind
the robot to the box. The rear lift reflex accumulates
more errors during the additional flight time.
With no reflex responses, the robot is unable to recover
when the rear leg hits a box of any height. However, Figure
23 shows that pulling the leg back after the initial con-
Fig. 18. Front Lift Trip Response. The front leg is lifted and repositioned to achieve a better foothold. [Times (s): 0.0,
Fig. 19. Rear Lift Trip Response. The rear leg is lifted and repositioned to achieve a better foothold. [Time (s): 0.0, 0.07, 0.09, 0.11,
Fig. 20. Rear Pull Trip Response. When a leg hits an obstacle while swinging forward, it is pulled back to allow it to clear the obstacle.
[Times
0.25515Number
of
Crashes
Front Lift Response Error Curves
at
Touchdown
(rad.)
Fig. 21. Front Lift Results. The top graph shows the number
of crashes as the obstacle height increases. The bottom graph shows
the average error in body attitude at the start of the next step after
recovering from a trip. As the box height increases, trips more often
lead to crashes. Note however, that the errors remain relatively
constant for those trials where the robot is able to recover and continue
running. There were 20 runs per box height. Box heights below
5 cm did not cause trips; box heights above 28:75 cm did not allow
recovery.
tact allows the robot to pass the leg over boxes as high
as 23 cm without crashes. For boxes between 23 cm and
cm, the leg, though pulled back, hits the box again, but
may still be able to recover. Above 25 cm, the boxes are
too high for the retracted leg to pass over, increasing the
number of crashes.
VII. Discussion and Conclusions
We have considered the problem of creating reflexes for
slipping and tripping given only the information that a
slip or a trip has occurred. We evaluated two kinds of responses
to slipping, one-step strategies and two-step strate-
gies, depending on whether the correction was applied in
the slip step or in the following step. Responses that continue
the slipping step produce smoother recoveries but
only for higher friction coefficients. Responses that abandon
the slipping step are capable of negotiating surfaces
with a larger range of friction coefficients but accumulate
larger errors.
Our slipping simulations focused on traversing a patch
in which one footfall slipped; however, some observations
can be made regarding running on a slippery surface. For
higher coefficients of friction, the strategy with the smallest
errors, the hip torque reaction, is most likely to succeed.
The repositioning strategies are limited because continual
0.25515Number
of
Crashes
Rear Lift Response Error Curves
at
Touchdown
(rad.)
Fig. 22. Rear Lift Results. Taller boxes are more likely to cause a
crash. However, if the robot does recover, it does so with a relatively
constant error. The rear lift reflex recovers about as often as the front
lift reflex (Figure 21), but with higher resulting errors. There were
runs per box height.
0.25515Number
of
Crashes
Rear Pull Response Error Curves
at
Touchdown
(rad.)
Fig. 23. Rear Pull Results. Pulling the tripping foot back so it
passes over the box allows the robot to continue running, but with
some additional attitude error. For box heights below 13:75 cm, the
rear foot passes over the box without tripping due to the retraction
of the leg during running. There were 20 runs per box height with
variation in the initial velocity of the robot.
slipping would cause them to abandon every other step.
However, all of the reflexive strategies except the hip torque
strategy reduce the forward velocity during slip recovery,
thus making the foot forces more vertical on subsequent
steps. Preliminary results indicate that only a few slipping
reactions may be required to achieve steady running on a
slippery surface without slipping.
If the foot is moving with respect to the ground at touch-
down, the horizontal force on the ground is increased in the
direction of motion, thereby increasing the danger of slip-
ping. Strategies for running on slippery surfaces should
try to reduce the relative motion of the foot between the
ground prior to impact. This principle, commonly called
ground-speed matching, is useful in slip prevention. It also
reduces the impact of ground contact and is used by animals
and human runners.
We evaluated several reflexes that repositioned the foot
after a trip to find a viable foothold or to avoid the box. For
trips in which the forward foot struck the vertical face of the
lifting either the front or rear foot allowed recoveries.
However, lifting the front foot produced the smallest errors
at the start of the subsequent step. For trips in which the
rear leg hit the box, pulling the leg back to let it pass over
the box allowed the robot to continue running, but with
some additional error in body attitude.
The slipping and tripping reflexes have been validated
for single slip or trip tasks. The next task is to integrate
the reflexes to enable running through general rough terrain
with arbitrary obstacles and slippery areas. Additional
controllers may be used to select among the applicable reflexes
based on sensing or modeling of the environment.
Finally, within the time constraints of the rapidly evolving
dynamic system, limited replanning may be used to aid
recovery.
These slipping and tripping reflexes are robust despite
their minimal sensing requirements. Without determining
friction or obstacle properties, without modeling the
surface, and without online planning, the reflexes enable
the robot to continue running under many circumstances.
Even if more sensing and computational resources are available
for foot placement, surface modeling, and replanning,
reflexes such as these will remain necessary due to sensing
and modeling errors.
Slipping and tripping reflexes are fundamental to many
rough terrain problems. Slopes, uneven surfaces, and small
obstacles create oblique impact angles that can cause slips
and trips. Reflexive responses will facilitate the successful
traversal of these terrains in combination with other
reflexive strategies for foothold errors such as adhesions,
bounces, and loss of firm footing.
VIII.
Acknowledgments
This project was supported in part by NSF Grant No.
IRI-9309189 and funding from the Advanced Research
Projects Agency.
--R
Motor Control and Learning
"The adaptive suspension vehicle"
"Configuration of autonomous walkers for extreme terrain"
"Terrain mapping for a walking planetary rover"
"Perception, planning, and control for autonomous walking with the ambler planetary rover"
"Optimal force distribution for the legs of a walking machine with friction cone con- 1straints"
"Adjusting step length for rough terrain locomotion"
"Realistic animation of legged running on rough ter- rain"
"Adaptive gait control of a biped robot based on realtime sensing of the ground profile"
"De- velopment of a dynamic biped walking system for humanoid: Development of a biped walking robot adapting to the humans' living floor"
"Adapting reflexes controlling the human pos- ture"
"Fixed patterns of rapid postural responses among leg muscles during stance"
"Balance adjustments of humans perturbed while walking"
"Stumbling correct reaction: A phase-dependent compensatory reaction during locomotion"
"Phasic control of reflexes during locomotion in vertebrates"
"Corrective responses to perturbation applied during walking in humans"
"An adaptive artificial hand"
"Robot control by reflex ac- tions"
"Reflex control of the prototype leg during contact and slippage"
"A study of design and control of a quadrupedwalking vehicle"
"A robot that walks: Emergent behaviors from a carefully evolved network"
"Robot impact control inspired by human reflex"
"Control of a six-legged robot walking on abrupt terrain"
"Developing planning and reactive control for a hexapodrobot"
Legged Robots That Balance
"Running experiments with a planar biped"
--TR
--CTR
Christophe Sabourin , Olivier Bruneau , Gabriel Buche, Control Strategy for the Robust Dynamic Walk of a Biped Robot, International Journal of Robotics Research, v.25 n.9, p.843-860, September 2006
Tao Geng , Bernd Porr , Bernd Florentinwrgtter, A Reflexive Neural Network for Dynamic Biped Walking Control, Neural Computation, v.18 n.5, p.1156-1196, May 2006 | rough terrain;tripping;slipping;reactive control;reflexes;biped locomotion |
591464 | Globally Consistent Range Scan Alignment for Environment Mapping. | A robot exploring an unknown environment may need to build a world model from sensor measurements. In order to integrate all the frames of sensor data, it is essential to align the data properly. An incremental approach has been typically used in the past, in which each local frame of data is aligned to a cumulative global model, and then merged to the model. Because different parts of the model are updated independently while there are errors in the registration, such an approach may result in an inconsistent model.In this paper, we study the problem of consistent registration of multiple frames of measurements (range scans), together with the related issues of representation and manipulation of spatial uncertainties. Our approach is to maintain all the local frames of data as well as the relative spatial relationships between local frames. These spatial relationships are modeled as random variables and are derived from matching pairwise scans or from odometry. Then we formulate a procedure based on the maximum likelihood criterion to optimally combine all the spatial relations. Consistency is achieved by using all the spatial relations as constraints to solve for the data frame poses simultaneously. Experiments with both simulated and real data will be presented. | Introduction
1.1 Problem Definition
The general problem we want to solve is to let a mobile robot explore an unknown environment
using range sensing and build a map of the environment from sensor data. In this paper, we
address the issue of consistent alignment of data frames so that they can be integrated to form
a world model. However, the issue of building a high-level model from registered sensor data is
beyond the scope of this paper.
A horizontal range scan is a collection of range measurements taken from a single robot position.
In previous robot navigation systems, range scans have often been used for robot self-localization
in known environments [3]. Using range measurements (sonar or laser) for modeling an unknown
environment has also been studied in the past [11, 4, 8]. A range scan represents a partial view of
the world. By merging many such scans taken at different locations, a more complete description
of the world can be obtained. Figure 1 gives an example of a single range scan and a world model
consisting of many scans.
a b
Figure
1: Building world model from range scans. (a) One range scan in a simulated world; (b)
model consisting of many scans. The small circles show the poses at which the scans are taken.
The essential issue here is to align the scans properly so that they can be merged. But the
difficulty is that odometry information alone is usually inadequate for determining the relative
scan poses (because of odometry errors that accumulate). On the other hand, we are unable to
use pre-mapped external landmarks to correct pose errors because the environment is unknown.
A generally employed approach of building a world model is to incrementally integrate new data
to the model. When each frame of sensor data is obtained, it is aligned to a previous frame or to
a cumulative global model. Then the new frame of data is integrated into the global model by
averaging the data or using a Kalman filter [1, 10, 11, 4, 8]. A major problem with this approach
is that the resulting world model may eventually become inconsistent as different parts of the
model are updated independently. Moreover, it may be difficult to resolve such inconsistency if
the data frames have already been permanently integrated.
To be able to resolve inconsistency once it is detected at a later stage, we need to maintain
the local frames of data together with their estimated poses. In addition, we need a systematic
method to propagate pose corrections to all related frames.
Consider an example as shown in Fig. 2(a). The robot starts at P 1 and returns to a nearby
location P n after visiting along the way. By registering the scan taken at P n against
scan P n\Gamma1 , the pose of P n can be estimated. However since P n is close to P 1 , it is also possible to
derive pose P n based on P 1 by matching these two scans. Because of errors, the two estimates of
could be conflicting. If a weighted average of the two is used as the estimate of P n , the pose
of P should also be updated as otherwise the relation P will be inconsistent with its
previous estimate. This inconsistency could be significant if the looped path is long. Similarly,
other poses along the path should also be updated. In general, the result of matching pairwise
scans is a complex, and possibly conflicting, network of pose relations. We need a uniform
framework to integrate all these relations and resolve the conflicts.
In this paper, we present such a framework for consistently registering multiple range scans. The
idea of our approach is to maintain all the local frames of data as well as a network of spatial
relations among the local frames. Here each local frame is defined as the collection of sensor data
measured from a single robot pose. The robot pose, in some global reference frame, is also used
to define the local coordinate system of the data frame. Spatial relations between local frames are
derived from matching pairs of scans or from odometry measurements. We treat the history of
robot poses in a global coordinate system (which define all the local frame positions) as variables.
Our goal is to estimate all these pose variables using the network of constraints, and register the
scans based on the solved poses. Consistency among the local frames is ensured as all the spatial
relations are taken into account simultaneously.
Figure
2 shows an example of consistently aligning a set of simulated scans. Part (a) shows the
original scans badly misaligned due to accumulated pose errors. Part (b) shows the result of
aligning these scans based on a network of relative pose constraints (with edges indicated by line
segments).
a
Pn
Figure
2: An example of consistently aligning a set of simulated scans. (a) Original scans badly
misaligned due to accumulated pose errors; (b) the result of aligning these scans based on a
network of relative pose constraints. The constraints are indicated by line segments connecting
pairs of poses. Two types of constraints are used: those derived from aligning a pair of scans
(marked by both solid and dotted lines), and those from odometry measurements (marked by
solid lines).
1.2 Related Work
The first project that systematically studied the consistency issue in dynamic world modeling
is the HILARE project [2]. In this system, range signals are segmented into objects which are
associated with local object frames. Each local frame is referenced in an absolute global frame
along with the uncertainty on the robot's pose at which the object frame is constructed. New
sensor data are matched to the current model of individual object frames. If some object which
has been discovered earlier is observed again, its object frame pose is updated (by averaging).
In circumstances that the uncertainty of some object frame is less than the uncertainty of the
current robot pose, as it happens when the object frame is created earlier, and later the robot
sees the object again, the robot's pose may be corrected with respect to that object frame. After
correcting the current robot pose, the correction is propagated backwards with a "fading" effect
to correct the previous poses. Although the HILARE system addressed the issue of resolving
model inconsistency, its solution has the following potential problems. First of all, the system
associates local frames with "objects". But if the results of segmenting sensor data or matching the
data with model are imperfect, the "objects" and therefore the local frames may not be defined
or maintained consistently. When a previously recorded object is detected again, the system
only attempts to update the poses (and the associated frames) along the path between the two
instances of detecting this object, while the global consistency among all frames in the model
may not be maintained. HILARE uses a scalar random variable to represent the uncertainty of
a three-degree-of-freedom pose, therefore it can not model the confidences in the individual pose
components.
Moutarlier and Chatila presented a theoretical framework for fusing uncertain measurements for
environment modeling [14]. They first discussed two types of representations: relation-based
and location-based. In relation-based representation, an object is related to another by the
uncertain transform between their reference frames. A network of relationships is maintained as
the world model. When new observations are made, all the relationships need to be updated
to preserve consistency. In location-based representation, the global references of individual
object frames are maintained together with their uncertainties. When objects are re-observed,
these object frames and other related frames are updated with respect to the global reference
frame. After comparing these two approaches, Moutarlier and Chatila choose to use the location-based
approach. They treat the object and robot locations as state variables and maintain all
the object variance/covariance matrices as state information. A stochastic-based formulation
for fusing new measurements and updating the state variables is introduced. In addition to a
global updating approach, they also introduced a relocation-fusion approach which first updates
the robot position based on the new observations and then updates the object frames. The
relocation-fusion approach reduces the influence of sensor bias in the estimation, although the
algorithm is suboptimal.
In a series of work by Durrant-Whyte [5, 6, 7], the problem of maintaining consistency in a
network of spatial relations was studied thoroughly. In their formulation, the environment model
is represented by a set of spatial relations between objects. A probabilistic fusion algorithm similar
to the Kalman filter is employed to integrate new measurements to the a priori model. When
some relations are updated as a result of new observations, the consistency among all relations
are enforced by using explicit constraints on the loops of the network. The updating procedure is
formulated as constrained optimization and it allows new observations to be propagated through
the network while consistency between prior constraints and observed information is maintained.
In another similar approach, Tang and Lee [17] formulated a geometric feature relation graph for
consistent sensor data fusion. They proposed a two-step procedure for resolving inconsistency in
a network of measurements of relations. In the first step, a compromise between the conflicting
measurements of relations is achieved by the fusion of these measurements. Then in the second
step, corrections are propagated to other relations in the network.
The difficulty in maintaining model consistency in a relation-based representation is that the
relations are not independent variables. Therefore additional constraints are needed in formulating
an updating procedure. The constrained optimization approach seems very complicated and
difficult to apply in practice.
In view of the previous methods, we present a new approach which has the following distinctive
characteristics:
1. We use an unambiguous definition of an object frame as the collection of sensor measurements
observed from a single robot position. Thus we avoid the difficult task of segmenting and recognizing
objects (which the previous methods rely on in order to create and update object frames).
It is also important to note that we use a robot pose to define the reference for an object frame.
In a local frame, the relative object positions with respect to the robot pose are fixed (whose
uncertainty is no more than bounded sensing errors). During the estimation process, when the
robot position in the global reference frame is updated, effectively the global coordinates of all
objects in the current frame are updated accordingly. Therefore by maintaining the history of
robot poses, we also maintain the spatial relationships among the object frames.
2. Our approach uses a combination of relation-based and location-based representations. We
treat relations as primitives, but treat locations as free variables. This is different from the pure
relation-based approach in that we do not directly update the existing relations in the network
when new observations are made. We simply add new relations to the network. All the relations
are used as constraints to solve for the location variables which, in turn, define a set of updated and
consistent relations. On the other hand, our approach is different from the location-base approach
by Moutarlier and Chatila [14] in that we do not assume the covariance matrices between the
object frames as known. Our state information is the entire set of raw relations. We derive the
covariance matrices at the same time as we solve for the position variables.
3. We obtain direct spatial relations between object frames. Because our object frames are tied to
robot poses, odometry measurements directly provide spatial relations between the frames. More
importantly, we may align two overlapping frames of data (in our case range scans) to derive more
accurate relations between frames. In previous approaches, the robot typically relies on odometry
to first determine its new pose. Then the detection of objects allows the robot pose as well as
the object locations to be updated. Since the relations between object frames are updated rather
indirectly through the robot pose, biases in odometry measurements may lead to divergence in the
estimation of object positions, as reported in [14]. Moutarlier and Chatila propose an algorithm
that is supposed to address the divergence problem at the expense of a sub-optimal solution.
Our formulation does not have this problem, as we obtain direct spatial relations between object
frames by aligning the data, and therefore we are less sensitive to odometry biases.
2 Overview of Approach
We formulate our approach to multiple scan registration as one of estimating the global poses
of the scans, by using all the pose relations as constraints. Here the scan poses are considered
as variables. A pose relation is an estimated spatial relation between the two poses which can
be derived from matching two range scans. We also obtain pose relations from odometry mea-
surements. Finally, we estimate all the poses by solving an optimization problem. The issues
involved in this approach are discussed in the following subsections.
2.1 Deriving Pose Relations
Since we use a robot pose to define the local coordinate system of a scan, pose relations between
scans can be directly obtained from odometry which measures the relative movement of the robot.
In section 4.2, we will discuss the representation of odometry pose constraint and its uncertainty.
More accurate relations between scan poses are derived from aligning pairwise scans of points.
Here any pairwise scan matching algorithm can be used. One possible choice is the extension to
Cox's algorithm [3] where line segments are first fit to one scan and then points in another scan
are matched to the derived line segments. In our previous studies, we proposed another scan
matching algorithm which is based on direct point to point matching [12, 13]. In either case, the
scan matching algorithm takes two scans and a rough initial estimate of their relative pose (for
example from odometry information) as input. The output is a much improved estimate of the
relative pose.
After aligning two scans, we can record a set of corresponding points on the two scans. This
correspondence set will form a constraint between the two poses. In section 4.3, we will formulate
this type of constraint and its uncertainty as used in the estimation algorithm.
When we match two scans, we first project one scan to the local coordinate of the other scan,
and discard the points which are likely not visible from the second pose. The amount of overlap
between two scans is estimated empirically from the spatial extent of the matching parts between
the two scans. A pose relation is only derived when the overlap is significant enough (larger than
a given threshold).
2.2 Constructing a Network of Pose Relations
Given the pairwise pose relations, we can form a network. Formally, the network of constraints
is defined as a set of nodes and a set of links between pairs of nodes. A node of the network is
a pose of the robot on its trajectory at which a range scan is taken. Here a pose is defined as
a three dimensional vector (x; consisting of a 2D robot position and the home orientation
of the rotating range sensor. We then define two types of links between a pair of nodes. First,
if two poses are adjacent along the robot path, we say that there is a weak link between the two
nodes which is the odometry measurement of the relative pose. Second, if the range scans taken
at two poses have a sufficient overlap, we say that there is a strong link between the two nodes.
To decide whether there is sufficient overlap between scans, we use an empirical measure. The
spatial extent in the overlapping part of two scans should be larger than a fixed percentage of
the spatial extent covered by both scans.
For each strong link, a constraint on the relative pose is determined by the set of corresponding
points on the two scans given by the matching algorithm. It is possible to have multiple links
between two nodes. Figure 3 shows an environment and the constructed network of pose relations.
2.3 Combining Pose Relations in a Network
The pose relations in a network can be potentially inconsistent because they are not independent
variables (the number of relations may be more than the degrees of freedom in the network),
while the individually estimated relations are prone to errors. Our task is to combine all the
pose relations and resolve any inconsistency. This problem is formulated as one of optimally
Figure
3: Example of constructing a network of pose relations from matching pairwise scans. (a)
A simulated environment where the scan poses are labeled by circles; (b) the network of pose
relations constructed from matching overlapping scans.
estimating the global poses of nodes in the network. We do not deal with the relations directly.
Rather, we first solve for the nodes which constitute a set of free variables. Then a consistent set
of relations which represents a compromise of all a priori relations is defined by the poses on the
nodes.
An optimization problem is defined as follows. We construct an objective function from the
network with all the pose coordinates as variables (except one pose which defines our reference
coordinate system). Every link in the network is translated into a term in the objective function
which can be conceived as a spring connecting two nodes. The spring achieves minimum energy
when the relative pose between the two nodes equals the measured value (either from matching
two scans or from odometry). Then the objective function represents the total energy in the
network. We finally solve for all the pose variables at once by minimizing this total energy
function.
2.4 The Three-Node Example
Using the 3-node example, we illustrate the difference of our formulation from previous approaches
Assume that the network consists of three nodes: relations
. When there is new measurement -
the algorithm by
Durrant-Whyte [6] updates the three relations to T 0
3 based on an optimization criterion
which is subject to the constraint T 0
In our approach, we pool together all the relations T
T 1 to form an optimization
problem and solve for a new estimate for the nodes: P 0
3 . These node positions define a
consistent set of relations: T 0
1 . Note that the node positions are
so we do not need to solve a complex constrained system.
Moutarlier and Chatila [14] also treat the node positions as variables when updating the network
with new measurements. But they assume the knowledge of covariance matrices among the a
priori estimates of However, we only require the variances of individual measurement
errors on the relations T
are directly available from sensor models.
The rest of the paper is organized as follows. In section 3, we present the optimization criterion
by considering a generic optimal estimation problem. We derive a closed-form solution in a linear
special case. In section 4, we formulate the pose relations as well as the objective function in
the context of range scan registration. The closed-form solution derived in section 3 is applied to
solve for the scan poses. In section 5, we present experimental results.
Optimal Estimation from a Network of Relations
In this section, we formulate a generic optimal estimation algorithm which combines a set of
relations in a network. This algorithm will later be applied in section 4 in the context of robot
pose estimation and scan data registration.
3.1 Definition of the Estimation Problem
We consider the following generic optimal estimation problem. Assume that we are given a net-work
of uncertain measurements about n+1 nodes X Here each node X i represents
a d-dimensional position vector. A link D ij between two nodes X i and X j represents a measurable
difference of the two positions. Generally, D ij is a (possibly nonlinear) function of X i and
and we refer to this function as the measurement equation. Especially interesting to us is the
simple linear case where the measurement equation is
We model an observation of D ij as -
a random Gaussian error
with zero mean and known covariance matrix C ij . Given a set of measurements -
pairs of nodes and the covariance C ij , our goal is to derive the optimal estimate of the position
by combining all the measurements. Moreover, we want to derive the covariance matrices of
the estimated X i 's based on the covariance matrices of the measurements.
Our criterion of optimal estimation is based on the maximum likelihood or minimum variance
concept. The node position X i 's (and hence the position difference D ij 's) are determined in
such a way that the conditional joint probability of the derived D ij 's, given their observations
ij 's, is maximized. If we assume that all the observation errors are Gaussian and mutually
independent, the criterion is equivalent to minimizing the following Mahalanobis distance (where
the summation is over all the given measurements):
(D
Even if the observation errors are not independent, a similar distance function can still be formed.
However, it will involve the correlation matrices of the measurements. The assumption of independence
is actually not necessary in our formulation. The assumption makes practical sense as
the covariances of errors are difficult to estimate.
A typical application of the optimal estimation problem is in mobile robot navigation, where we
want to estimate the robot pose and its uncertainty in three degrees of freedom (x; '). The
observations are the relative robot poses from odometry, and also possible from matching sensor
measurements. We want to utilize all the available measurements to derive the optimal estimate
of the robot poses. Note that in this application, the measurement equation is non-linear because
of the ' component in the robot pose.
Our approach above differs from the one typically used within a Kalman filter formulation, in
which only the current pose is estimated, while the history of previous poses and associated
measurements is collapsed into the current state of the Kalman filter. Our objective, however, is
not simply getting from A to B safely and accurately, but also building a map of the environment.
It is, therefore, meaningful to use all the measurements obtained so far in the mapping process.
The old poses themselves are not particularly useful. But we are using the poses to define local
object frames. Thus maintaining the history of robot poses is equivalent to maintaining the
structure of the environment model. The advantage of using a pose to define a data frame is
that it is unambiguous and it avoids the difficult segmentation and object identification problem
present in other work.
Next, we study the case when the measurement equation is linear and we derive closed-form
solutions for the optimal estimates of the nodes and their covariances. Later, we will solve the non-linear
robot pose estimation problem by approximately forming linear measurement equations.
3.2 Solution of Optimal Linear Estimation
We consider the special case where the measurement equation has the simple linear
are the nodes in the network which are d-dimensional vectors
and the D ij 's are the links of the network. Without loss of generality, we assume that there is a
link D ij between every pair of nodes . For each D ij , we have an observation -
which is
assumed to have Gaussian distribution with mean value D ij and known covariance C ij . In case
the actual link D ij is missing, we can simply let the corresponding C
ij be 0. Then the criterion
for the optimal estimation is to minimize the following Mahalanobis distance:
0-i!j-n
Note that W is a function of all the position X i 's. Since we can only solve for relative positions
given the relative measurements, we choose one node X 0 as a reference and consider its coordinate
as constant. Without loss of generality, we let X
relative positions from X 0 .
We can express the measurement equations in a matrix form as
where X is the nd-dimensional vector which is the concatenation of is the
concatenation of all the position differences of the form D H is the incidence
matrix with all entries being 1, \Gamma1, or 0. Then the function W can be represented in matrix form
as:
D is the concatenation of all the observations -
ij for the corresponding D ij and C is the
covariance of -
D which is a square matrix consists of C ij 's as sub-matrices.
Then the solution for X which minimizes W is given by
The covariance of X is
If the measurement errors are independent, C will be block-diagonal and the solution can be
simplified. Denote the nd \Theta nd matrix H t C \Gamma1 H by G and expand the matrix multiplications.
We can obtain the d \Theta d sub-matrices of G as
Denote the nd-dimensional vector H t C
D by B. Its d-dimensional sub-vectors are the following
(let -
Then the position estimates and covariance can be written as
The above algorithm requires to be invertible. If the network is fully connected
and the individual error covariances are normally behaved, we believe it is possible to prove that
G is invertible. Note the dimension of G (number of free nodes) is less than or equal to the
dimension of C (number of edges) in a fully connected graph.
3.3 Special Networks
(b)
(a)
Figure
4: (a) Serial connection; (b) parallel connection.
We will apply the formula in Eq. 9 to two interesting special cases as in Figure 4. First, if the
network consists of two serially connected links, D 01 and D 12 , the derived estimate of X 2 and its
covariance matrix are
Another case to consider is the network which consists of two parallel links D 0 and D 00 between
two nodes X 0 and X 1 . If the covariance of the two links are C 0 and C 00 , the estimate of X 1 and
its covariance are given by
The solution is equivalent to the Kalman filter formulation. The above two cases correspond to
the compounding and merging operations given by Smith and Cheeseman [16], which are used to
reduce a complex network to a single relation. Smith and Cheeseman's algorithm has a limitation
that it only applies to networks formed by serial and parallel connections.
Figure
5: A Wheatstone bridge network.
Consider the network in the form of a Wheatstone bridge (Fig. 5). The estimate of X 3 can not
be obtained through compounding and merging operations. Therefore, the method by Smith
and Cheeseman can not be directly applied to simplify this network 1 , while in our method, the
variables can be solved from the linear system
G =B @
\GammaC \Gamma1-
The covariance matrix for the estimated position X 3 has a nice symmetric form (derived by
expanding G \Gamma1
1 It is possible to first convert a triangle in the network to an equivalent Y-shaped connection and then the
network becomes one with serial and parallel links. However, this Delta-to-Y conversion still can not turn every
network into a combination of serial and parallel connections.
4 Derivation of Pose Relations
In this section, we will apply the optimal estimation algorithm, as derived in section 3, to the
robot pose estimation and scan data registration problem. To do this, we need to derive linearized
measurement equations for the pose relations. In the following subsections, we study a constraint
on pose difference given by matched scans or odometry measurements. For each constraint, we
formulate a term in the form of Mahalanobis distance. For convenience in discussions of pose
measurements, we will first define a pose compounding operation.
4.1 Pose Compounding Operation
Assume that the robot starts at a pose its pose by
ending up at a new pose V a = (x a ; y a ; ' a ) t . Then we say that pose V a is
the compounding of V b and D. We denote it as:
The coordinates of the poses are related by:
y
This is the same compounding operation as defined by Smith and Cheeseman [16]. If we consider
that an absolute pose defines a coordinate system (the xy coordinates of the origin and the
direction of one axis), and a relative pose defines a change of coordinate system (a translation
followed by a rotation), then the compounding operation gives the pose which defines the new
coordinate system after the transformation. The compounding operation is not commutative,
but it is associative. We can thus define the compounding of a series of poses.
It is also useful to define the inverse of compounding which takes two poses and gives the relative
pose:
The coordinates are related by the following equations:
If D ab is the relative pose V a \Psi V b , the reversed relative pose D a can be obtained from
D ab by a unary operation:
We can verify that (\PsiD) \Phi
We also want to define a compounding operation between a full 3D pose
2D position vector . The result is another 2D vector u We still denote the
operation as
The coordinates for u 0 are given by the first two equations of the full 3D pose compounding
(Eq. 18,19). This 2D compounding operation is useful for transforming an non-oriented point
(typically from a range sensor) from its local sensor coordinate system to the global coordinate
system.
4.2 Pose Relations from Matched Scans
Let V a and V b be two nodes in the network and assume there is a strong link connecting the
two poses. From the pairwise scan matching algorithm, we get a set of pairs of corresponding
points: u a
where the 2D non-oriented points u a
k are from scan S a and S b ,
respectively. Each pair (u a
corresponds to the same physical point in the robot's environment
while they are represented in different local coordinate systems. If we ignore any sensing or
matching errors, two corresponding points are related by:
If we take the random observation errors into account, we can regard \DeltaZ k as a random variable
with zero mean and some unknown covariance C Z
k . From the correspondence pairs, we can form
a constraint on the pose difference by minimizing the following distance function:
F ab (V a
k(V a \Phi u a
If we notice that a pose change is a rigid transformation under which the squared Euclidean
distance k \Delta k 2 is invariant, we can rewrite the function in an equivalent form:
F ab (V a
k((V a \Psi V b ) \Phi u a
Thus F ab is a function of D . The solution of D 0 which minimizes F ab can be derived
in closed-form (see [12]). The relation D is the measurement equation.
In order to reduce F ab into the Mahalanobis distance form, we linearize each term \DeltaZ k . Let
close estimates of V a and V b . Denote \DeltaV
and \DeltaV
(the global coordinates of a pair of
matching points). Then for small \DeltaV a and \DeltaV b , we can derive from Taylor expansion:
V a \Phi u a
\DeltaV a \Gamma
\DeltaV b
V a \Phi u a
H a \DeltaV a \Gamma -
where
H a =B @
y a
We can rewrite Eq.
where
V a \Phi u a
H a \DeltaV a \Gamma -
Thus we can now regard D in Eq. 35 as the pose difference measurement equation to replace
. For the m correspondence pairs, we can form m equations as in Eq. 32. If we
concatenate the -
Z k 's to form a 2m \Theta 1 vector Z, and stack the M k 's to form a 2m \Theta 3 matrix M,
then F ab can be rewritten as a quadratic function of D:
F ab
We can then solve for the
D which minimizes F ab as
The criterion of minimizing F ab (D) constitutes a least-squares linear regression. In Eq. 32, M k is
known and -
Z k is observed with an error \DeltaZ k having zero mean and unknown covariance C Z
k . If
we assume that all the errors are independent variables having the same Gaussian distribution,
and further assume that the error covariance matrices have the form:
then the least squares solution -
D has the Gaussian distribution whose mean value is the true
underlying value and whose estimated covariance matrix is given by
is the unbiased estimate of oe
D)
D)
Moreover, we notice that Eq. 37 can be rewritten as
F ab (D) -
We can define the energy term W ab corresponding to the pose relation which is equivalent to a
Mahalanobis distance:
where
is the estimated covariance of -
D. Note that D (as given in Eq. 35) is the linearized pose difference
measurement equation.
In deriving the covariance matrix CD , we made assumptions that the matrix is diagonal and
the individual components of errors are zero mean Gaussian. It is probably difficult to justify
these assumption. However, we believe that they are reasonable ones in practice. If any other
estimates of the covariance matrices are available, they can certainly also be incorporated in our
global estimation formulation.
4.3 Pose Relations from Odometry
We also form an energy term in the objective function for each weak link. Suppose odometry
gives a measurement -
D 0 of the relative pose D 0 as the robot travels from pose V b to pose V a . The
measurement equation is:
We define the energy term in the objective function as follows:
where C 0 is the covariance of the odometry error in the measurement -
The covariance of measurement error is estimated as follows. Consider that a cycle of pose change
consists of: (1) the robot platform rotation by an angle ff to face towards the new target position;
(2) the robot translation by a distance L to arrive at the new position; (3) the sensor rotation
by a total cumulative angle fi (usually 360 ffi ) to take a scan of measurements while the platform
is stationary. We model the deviations oe ff , oe L , oe fi , of the errors in the variables ff, L, and fi as
proportional to their corresponding values, while the constant ratios are determined empirically.
The 3D pose change D derived as:
Then the covariance C 0 of the pose change D 0 can be approximated as:
fiC A J t (48)
where J is the Jacobian matrix consisting of the partial derivatives of (x; with respect to
\GammaL sin ff cos ff 0
We would also like to linearize and transform the measurement equation of D 0 to make the pose
difference representation for odometry measurements consistent with that for matched sensing
data. Consider the observation error \DeltaD
of odometry. Let -
close estimates of V a and V b . Denote \DeltaV
Then through Taylor expansion, the observation error \DeltaD 0 becomes:
V a \Psi -
b (\DeltaV a \Gamma -
H ab \DeltaV b ) (52)
where
sin -
H ab =B @
Notice that -
a
H a and -
H b are defined in Eq. 31. If we define a new observation
error
H a
then we can rewrite Eq. 52 as
H a \DeltaV a \Gamma -
where we denote
H a
V a \Psi -
H a \DeltaV a \Gamma -
Notice that now we are dealing with the measurement equation for D which is consistent with
that for matched sensing data. -
D can be considered as an observation of D. The covariance C
of -
D can be computed from the covariance C 0 of -
as:
H a
The energy term in the objective function now becomes:
ab -
4.4 Optimal Pose Estimation
Once we have uniformly formulated the two types of measurements, we can apply the estimate
algorithm in section 3 to solve for the pose variables. Denote the robot poses as
The total energy function from all the measurements is :
is the linearized pose difference between V j and
and -
ij is an observation of D ij ( -
is derived from the true observations, either range data or
odometry measurements). The covariance C ij is also known.
By regarding X
as the state vector corresponding to a node of the network as in
Section 3.2, we can directly apply the closed-form linear solution to solve for the X i 's as well as
their covariance C X
. The formulas are in Eq. 5 to Eq. 9. Then the pose V i and its covariance C i
can be updated as:
Note that the pose estimate V i and the covariance C i is given based on the assumption that the
reference pose is non-zero, the solution should be transformed
to
where
sin
4.5 Sequential Estimation
The estimation algorithm we previously discussed is a one-step procedure which solves for all
the pose variables at the same time. The algorithm is to be applied only after collecting all the
measurements. Yet it will be more practically useful if we have a sequential algorithm which
continuously provides estimates about the current or past pose variables after getting each new
measurement. Here we will describe such a sequential procedure.
Our sequential algorithm maintains the current best estimate about the poses of previously visited
places. At each step, a new location is visited and measurements about the new location as well
as the previous locations are gathered. By using these new measurements, the current pose can
be estimated while the previous poses can be updated.
be the pose vectors which we have previously estimated and let X n be the current
new pose which we are about to measure. Let X represent the concatenation of X
. Assume that we currently have an estimate X 0 of X whose inverse covariance matrix is C
Because we have no knowledge about X n yet, the X n component in X 0 contains an arbitrary
value and the matrix C
has all zeros in the last d rows and d columns, where
consider the addition of a set of new measurements relating X n to some of the past pose vari-
ables. Let the measurement equation, in matrix form, be is a constant matrix).
Assume that the set of measurements is -
D which is an unbiased observation of D whose error
has Gaussian distribution with covariance matrix CD . The updated estimate of X after using
the new measurements is the one which minimizes the following function, using the maximum
likelihood criterion, and assuming independent errors:
The solution can be derived as
D) (65)
and the new covariance of X is
A convenient way of updating X and CX is to maintain a matrix
(the summation is over different sets of measurements). Then at each step,
the updating algorithm is the following: First increase the dimensions of G and B to include the
new pose X n . Update G and B as
Then the new X and CX are given by
One potential problem with the above sequential updating procedure is that the state variable
expanding as it is augmented by a new state at each step. In case the robot path is
very long, the variable size may become too large, causing storage or performance problems. A
possible solution is to delete some of the old variables while adding the new ones.
We propose a strategy of reducing the number of state variables as follows. In order to choose
a pose to be deleted, we check all pairs of poses and find a pair (X the correlation
between the two poses is the strongest. We then force the relative pose between X i and X j to be
fixed as a constant. Then X i can be deleted from the state variables as it can be obtained from
When deleting the node X i from the network, we transform any link (X link
from X j to X k . Note that the covariance matrix CX contains all the pairwise covariance between
any two poses. A correlation ratio between two poses can be computed from the covariance and
variance.
By only fixing some relative poses, the basic structure in the network is still maintained. Thus
we are still able to consistently update all the pose variables once given new measurements. This
strategy is more flexible than the simple strategy of fixing selected absolute poses as constants.
Another approach to reducing the size of the system is to decompose the large network into smaller
components. The estimation algorithm is to be applied to each sub-network. The relative pose
between two nodes in different sub-networks can be obtained through pose compounding. If
there is a single link connecting two parts of a network, the poses in two parts can be estimated
separately and then combined with compounding, without loss of information. If, however, the
network is strongly connected that there are two or more links between any two nodes, then a
decomposition could give a sub-optimal estimation.
5 Implementation and Experiments
5.1 Implementation of Estimation Procedure
The implementation of the estimation algorithm is as follows. After building the network, we
obtain the initial pose estimates -
by compounding the odometry measurements. Then
for each link, we compute a measurement vector -
ij and a covariance matrix C ij according
to Eq. 38, 44 or Eq. 55, 57. Finally, we form a large linear system explained in
Section 3.2 and solve for the pose variables X.
The components needed to build G and B are C \Gamma1
ij . In the case of a strong link
(from matching a pair of scans), these components can be readily computed as C \Gamma1
which can be expanded into simple summations by noting the regularity in
the matrix M. In the case of a weak link (from odometry), these components can be computed by
multiplications of small matrices (3 \Theta 3). The most expensive operation in the estimation process
is to compute the inverse of a 3n \Theta 3n matrix G which gives the covariance of X.
The network is stored as a list of links and a list of nodes. Each link contains the following
information: type of link, labels of the two nodes, the computed measurement (relative pose),
and the covariance matrix of the measurement. Each node contains a range scan.
Note that we made linear approximations in the measurement equations in formulating the optimization
criterion. The first order approximation error is proportional to the error in the initial
pose estimate. Clearly, if we employ the newly derived pose estimate to formulate the linear
algorithm again, a even more accurate pose estimate can be obtained.
The iterative strategy based on this observation converges very fast. Typically, the first iteration
corrects over 90% of the total pose error correctable by iterating the process. It usually takes
four or five iterations to converge to the limit of machine accuracy.
5.2 Experiments with Simulated and Real Scan Data
We now present experiments of applying our algorithm to register simulated and real range scan
data. We first show an example using a simulated environment and measurements. This is useful
because ground truth is known. Then an example using real data is presented.
In the first example, we simulate a rectangular environment with a width of 10 units. The robot
travels around a central object and forms a loop in the path. There are 13 poses along the path at
a
Figure
Global registration of multiple scans using simulated scan data. (a) scans recorded in
a global coordinate system where the pose of each scan is obtained from compounding odometry
measurements. The scans align poorly because of accumulation of odometry error. (b) the result
of correcting pose errors. Both the dashed lines and solid lines show the constraints from matching
scan pairs. The solid lines also give the robot path and odometry constraints.
which simulated range scans are generated (with random measurement errors). We also simulate
a random odometry error (which is the difference between a pose change the robot thinks it made
and the actual pose change) at each leg of the trajectory. The magnitude of the accumulated
odometry error is typically in the range of 0.5 units.
We apply our iterative global pose estimation algorithm to correct the pose errors. In Fig. 6(a),
we show all the scans recorded in the initial coordinate system where the pose of each scan is
obtained by compounding odometry measurements. Due to the accumulation of odometry error
the scan data are aligned poorly. In Fig. 6(b), we show the result of correcting the pose errors
and realigning the scan data. Each line segment (either dashed or solid) in the figure represents
a strong link obtained from matching two scans. In addition, the solid lines show the robot
path which corresponds to the weak links. A plot of orientational and positional errors of the
poses along the path, both before and after the correction, is given in Fig. 7. Pose errors are
accumulated along the path while the corrected pose errors are bounded. For comparison, we
also apply a local registration procedure which matches one scan only to the previous scan. The
pose errors along the path after this local correction are also shown in Fig. 7. Although pose
errors are also significantly reduced after local corrections, they can still potentially grow without
bound. In this example, global registration produces more accurate results than local correction.0.010.030.050.070.090 2 4
Radian
Pose Number
Magnitude of Orientational Errors along the Path
Before correction
After local correction
After global correction0.10.30.50
Unit
Pose Number
Magnitude of Positional Errors along the Path
Before correction
After local correction
After global correction
a b
Figure
7: Pose errors along the path, before correction, after local correction, and after global
correction. (a) Orientational errors; (b) positional errors.
Then we present the experiment using real range scans and odometry data. The testing environment
is the cafeteria and nearby corridor in FAW, Ulm, Germany. The robot travels through
the environment following a given path. A sequence of 30 scans which were taken by the robot
with an interval of about 2 meters between scan poses were obtained. The laser sensor is a Ladar
2D IBEO Lasertechnik which is mounted on the AMOS robot. This laser sensor has a maximum
Figure
8: Consistent global registration of real range scans which are collected by a robot at
FAW, Ulm, Germany. (a) unregistered scans whose poses are subject to large odometry errors.
(b) registered scans after correcting the pose errors. The robot path estimated from odometry is
shown in dashed lines. The corrected path is shown in solid lines.
Figure
9: Mapping of a Hallway using the RWI Pioneer platform and a SICK laser range scanner
(a) Raw laser range scans (b) Aligned laser range scans.
viewing angle of 220 degrees. Thus having only the 2D positions of two poses close together
does not necessarily ensure a sufficient overlap between the scans taken at the two poses; we also
need the sensor heading directions to be similar. Among the links from matching
overlapping scan pairs are constructed. Some of these pairwise scan matching results have been
shown in [12]. In Fig. 8, we show (a) the unregistered scans and (b) the globally registered scans
in part (b).
Further experimental results with a variant of our algorithm are reported in [9]. Figure 9 contains
experimental results which are obtained using our global registration procedure together with a
modified version of Cox's pairwise scan matching algorithm 2 . The laser data are collected on
the RWI Pioneer platform using the SICK laser ranging device [15]. The Pioneer is a low-cost
platform with odometry error significantly higher than the much more expensive platforms used
in our other experiments. The hallway environment shown in Figure 9 is poor in features that
allow localization of the robot along the hallway. The data was collected by a robot that went
up and down the hallway several times. A large rotation error was introduced by the large turns
at the ends of the hallway.
6 Discussion
In this paper, we formulated the problem of consistent range data registration as one of optimal
pose estimation from a network of relations. The main ideas are as follows. We associate a robot
pose to a range scan to define an unambiguous object frame. By consistently maintaining the
history of robot poses, we effectively allow all object frames to be consistently registered in the
global reference frame. We use a combination of relation-based and location-based approach to
represent the world model. It can be viewed as a two-step procedure. First, spatial relations
between object frames are directly derived from odometry measurements and matching pairwise
frames. These relations, along with their uncertainties, constitute all the information in the
model. In the second step, the relations are converted to object frame locations based on an
optimization criterion. This formulation avoids the use of complex constrained optimization.
Furthermore, it does not require the assumption of known a priori covariance between object
frames.
We also derived measurement equations compatible with the formulation. It allows practical
implementation of the algorithm. We have experimentally demonstrated the effectiveness of
our estimation procedure in maintaining consistency among multiple range scans. The most
2 We are grateful to Steffen Gutmann of the AI Laboratory at the Albert-Ludwigs-Universit?t Freiburg for
providing us with these experimental results.
expensive operation, besides pairwise scan matching, is to compute the inverse of an 3n \Theta 3n
matrix. Although the number of poses n may be large for a long robot path, there are ways
to limit this size to speed up the computation. The sequential procedure enables the robot to
continuously maintain the optimal registration result.
Our approach assumes that the robot stops to collect a complete range scan at its current position.
An alternative would be to perform continuous scanning as the robot moves. Continuous scanning
would generate large amounts of data that would have to be sampled. In addition, the problem
of associating measurements with the correct robot position arises, as different parts of a scan
will have been obtained from different robot positions. Solving this problem would require an
accurate model of the robots motion. A possible solution to the problem of excessive amounts of
data is to partition the continuous scan data and transform each part to one pose on the path,
based on the odometry model. These are both worthwhile problems, which we consider outside
the scope of this paper.
Although we develop our method for mapping a 2D environment using 2D range scans, our formulation
is general and it can be applied to the 3D case as well, by generalizing pose composition
and linearization [12].
Acknowledgement
Funding for this work was provided by NSERC Canada and by the ARK project which receives its
funding from PRECARN Associates Inc., the Department of Industry, Science and Technology,
NRC Canada, the Ontario Technology Fund, Ontario Hydro, and AECL.
The authors would like to thank Steffen Gutmann, Joerg Illmann, Thomas Kaempke, Manfred
Knick, Erwin Prassler, and Christian Schlegel from FAW, Ulm for collecting range scans and
making the data available for our experiments. We thank Dr. Ingemar Cox, and the anonymous
reviewers for many constructive comments.
--R
Maintaining representations of the environment of a mobile robot.
Position referencing and consistent world modeling for mobile robots.
Blanche: An experiment in guidance and navigation of an autonomous robot vehicle.
World modeling and position estimation for a mobile robot using ultrasonic ranging.
Consistent integration and propagation of disparate sensor observa- tions
Integration, coordination and control of multisensor robot systems.
Uncertain geometry in robotics.
Map building for a mobile robot equipped with a 2D laser rangefinder.
AMOS: Comparison of scan matching approaches for self-localization in indoor environments
Stereo vision and navigation in buildings for mobile robots.
Dynamic map building for an autonomous mobile robot.
Shape Registration Using Optimization for Mobile Robot Navigation.
Robot pose estimation in unknown environments by matching 2D range scans.
Stochastic multisensory data fusion for mobile robot location and environment modelling.
SICK Laser range scanner.
On the representation and estimation of spatial uncertainty.
A geometric feature relation graph formulation for consistent sensor fusion.
--TR
--CTR
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591512 | Sensor-Based Control Architecture for a Car-Like Vehicle. | This paper presents a control architecture endowing a car-like vehicle moving in a dynamic and partially known environment with autonomous motion capabilities. Like most recent control architectures for autonomous robot systems, it combines three functional components: a set of basic real-time skills, a reactive execution mechanism and a decision module. The main novelty of the architecture proposed lies in the introduction of a fourth component akin to a meta-level of skills: the sensor-based manoeuvers, i.e., general templates that encode high-level expert human knowledge and heuristics about how a specific motion task is to be performed. The concept of sensor-based manoeuvers permit to reduce the planning effort required to address a given motion task, thus improving the overall response-time of the system, while retaining the good properties of a skill-based architecture, i.e., robustness, flexibility and reactivity. The paper focuses on the trajectory planning function (which is an important part of the decision module) and two types of sensor-based manoeuvers, trajectory following and parallel parking, that have been implemented and successfully tested on a real automatic car-like vehicle placed in different situations. | Introduction
Autonomy in general and motion autonomy in
particular has been a long standing issue in
Robotics. In the late sixties-early seventies,
Shakey (Nilsson, 1984) was one of the first robots
able to move and perform simple tasks au-
tonomously. Ever since, many authors have proposed
control architectures to endow robot systems
with various autonomous capabilities. Some
of these architectures are reviewed in x7 and compared
to the one presented in this paper. These
Institut National de Recherche en Informatique et en Automatique
approaches differ in several ways, however it is
clear that the control structure of an autonomous
robot placed in a dynamic and partially known environment
must have both deliberative and reactive
capabilities. In other words, the robot should
be able to decide which actions to carry out according
to its goal and current situation; it should
also be able to take into account events (expected
or not) in a timely manner.
The control architecture presented in this paper
aims at meeting these two requirements. It
is designed to endow a car-like vehicle moving on
the road network with motion autonomy and was
developed in the framework of the French Prax-
it'ele programme aimed at the development of a
Laugier et al.
new urban transportation system based on a fleet
of electric vehicles with autonomous motion capabilities
(Parent and Daviet, 1996). The road
network is a complex environment, it is partially
known and highly dynamic with moving obstacles
(other vehicles, pedestrians, etc.) whose future
behaviour is not known in advance. However
the road network is a structured environment with
motion rules (the highway code) and it is possible
to take advantage of these features in order to design
a control architecture that is efficient, robust
and flexible.
The control architecture is presented in this
paper as follows: in the next section, the rationale
of the architecture and its main features
are overviewed. It introduces the key concept of
sensor-based manoeuvres, i.e. general templates
that encode the knowledge of how a specific motion
task is to be performed. The model of the
car-like vehicle that is used throughout the paper
is then described (x3). One important component
of the architecture is the trajectory planner whose
purpose is to determine the trajectory leading the
vehicle to its goal. Trajectory planning for car-
Skills library
Mission Description
Sensor Data
Execution Report data
Commands
Motion Controller
Mission Monitor
Skills instantiation and execution
PMP generation and update
SBMs library
execution
Trajectory Planner
World Model
Fig. 1. The overall control architecture.
Sensor-Based Control Architecture 3
like vehicles in dynamic environments remains an
open problem and a practical solution to this intricate
problem is presented in x4. Afterwards the
concept of sensor-based manoeuvres is explored
in x5 and two types of manoeuvres are presented
in detail. These two manoeuvres have been implemented
and successfully tested on an experimental
vehicle, the results of these experiments are finally
presented in x6.
2. Overview of the Control Architecture
The control architecture is depicted in Fig. 1.
It relies upon the concept of sensor-based manoeuvres
(SBM) which is derived from the Artificial
Intelligence concept of script (Rich and
Knight, 1983). A script is a general template that
encodes procedural knowledge of how a specific
type of task is to be performed. A script is fitted
to a specific task through the instantiation of variable
parametres in the template; these parameters
can come from a variety of sources (a priori knowl-
edge, sensor data, output of other modules, etc.
Script parametres fill in the details of the script
steps and permit to deal easily with the current
task conditions.
The introduction of SBM was motivated by the
observation that the kind of motion task that a
vehicle has to perform can usually be described
as a series of simple steps (a script). A SBM is
a script, it combines control and sensing skills.
Skills are elementary functions with real-time abil-
ities: sensing skills are functions processing sensor
data whereas control skills are control programs
(open or closed loop) that generate the appropriate
commands for the vehicle. Control skills may
use data provided directly by the sensors or by the
sensing skills.
The idea of combining basic real-time skills to
build a plan in order to perform a given task can
be found in other control architectures (cf. x7);
they permit to obtain robust, flexible and reactive
behaviours. SBMs can be seen as "meta-skills",
their novelty is that they permit to encapsulate
high-level expert human knowledge and heuristics
about how to perform a specific motion task
(cf. x5). Accordingly they permit to reduce the
planning effort required to address a given motion
task, thus improving the overall response-time of
the system, while retaining the good properties of
a skill-based architecture, i.e. robustness, flexibility
and reactivity.
The control architecture features two main com-
ponents, the mission monitor and the motion con-
troller, that are described afterwards.
2.1. Mission Monitor
When given a mission description, e.g. "go park at
location l", the mission monitor (MN) generates a
parameterized motion plan (PMP) which is a set
of generic sensor-based manoeuvres (SBM) possibly
completed with nominal trajectories. The
SBMs are selected from a SBM library. A SBM
may require a nominal trajectory (it is the case
of the "Follow Trajectory" SBM). A nominal trajectory
is a continuous time-ordered sequence of
(position, velocity) of the vehicle that represents
a theoretically safe and executable trajectory, i.e.
a collision-free trajectory which satisfies the kinematic
and dynamic constraints of the vehicle.
Such trajectories are computed by the trajectory
planner by using:
ffl an a priori known or acquired model of the
vehicle environment,
ffl the current sensor data, e.g. position and velocity
of the moving obstacles, and
ffl a world prediction that gives the most likely
behaviours of the moving obstacles.
Trajectory planning is detailed in x4. The current
SBM with its nominal trajectory is passed to the
motion controller for its reactive execution.
2.2. Motion Controller
The goal of the Motion Controller (MC) is to execute
in a reactive way the current SBM of the
PMP. For that purpose, the current SBM is instantiated
according to the current execution con-
text, i.e. the variable parametres of the SBM are
set by using the a priori known or sensed information
available at the time, e.g. road curvature,
available lateral and longitudinal space, velocity
and acceleration bounds, distance to an obstacle,
etc. As mentioned above, a SBM combines control
and sensing skills that are either control pro-
4 Laugier et al.
grams or sensor data processing functions. It is
up to MC to control and coordinate the execution
of the different skills required. The sequence of
control skills that is executed for a given SBM is
determined by the events detected by the sensor
skills. When an event that cannot be handled by
the current SBM happens, MC reports a failure
to MN which updates PMP either by applying a
replanning procedure (time permitting), or by selecting
in real-time a SBM adapted to the new
situation.
3. Model of the Vehicle
A car-like vehicle is modelled as a rigid body
moving on the plane. It is supported by four
wheels making point contact with the ground,
it has two rear wheels and two directional front
wheels. The model of a car-like vehicle that is used
is depicted in Fig. 2. The configuration, i.e. the
position and orientation of the vehicle, are characterized
by the triple
and are the coordinates of the rear axle
midpoint and the orientation of the vehi-
cle, i.e. the angle between the x axis and the main
axis of the vehicle. The motion of the vehicle is
described by the following equations:!
sin OE
(1)
is the steering angle, i.e. the average
orientation of the two front wheels of the
vehicle. is the locomotion velocity of
the front axle midpoint and L is the wheelbase.
(OE; v), the steering angle and locomotion veloc-
ity, are the two control commands of the vehicle.
Since the steering angle of a car is mechanically
limited, the following constraint also holds (max-
imum curvature constraint):
Eqs. (1) correspond to a system with non-holonomic
kinematic constraints because they involve
the derivatives of the coordinates of the vehicle
and are non-integrable (Latombe, 1991). They
are valid for a vehicle moving on flat ground with
perfect rolling assumption (no slippage between
the wheels and the ground) at relatively low speed.
f
y
x
Fig. 2. Model of a car-like vehicle.
For high-speed motions, the dynamics of the vehicle
must also be considered. In the current implementation
of the architecture, only velocity and
acceleration bounds are taken into account.
4. Trajectory Planning
As mentioned earlier, trajectory planning is an important
function in the control architecture pro-
posed. Its purpose is to compute a nominal trajectory
leading the vehicle to its goal. A trajectory is
a continuous time-ordered sequence of states, i.e.
(configuration, velocity) pairs, between the current
state of the vehicle and its goal. A trajectory
must be collision-free and satisfy the kinematic
and dynamic constraints of the vehicle.
In order to plan a trajectory that avoids the
moving obstacles of the environment, the knowledge
of their future behaviour is required. In
most cases, this information is not a priori known.
An estimation of the most likely behaviour of the
moving obstacles is provided by a prediction func-
tion. The prediction function can be very simple
(assuming that the moving obstacles keep a constant
velocity) or more sophisticated (using models
of human driver behaviour for instance). The
quality of the prediction determines the quality
of the nominal trajectory. However keep in mind
that the trajectory planned is nominal: if the
world does not 'behave' according to the predic-
tion, the motion controller will deal with the prediction
error and react accordingly. On the other
Sensor-Based Control Architecture 5
hand, if the prediction is correct then the vehicle
will follow a trajectory that has been planned so
as to be optimal in time.
Trajectory planning for car-like vehicles in dynamic
environments remains an open problem and
a practical solution to this intricate problem is
presented in this section.
4.1. Outline of the Approach
The motion of a vehicle is subject to several types
of constraints and the nominal trajectory has to
respect them. These constraints are:
ffl Kinematic constraints: a wheeled car-like
vehicle is subject to kinematic constraints,
called non-holonomic, that restricts the geometric
shape of its motion. Such a vehicle can
move only in a direction which is perpendicular
to its rear wheel axle (non-steering wheels)
and its turning radius is lower-bounded.
ffl Dynamic constraints: these constraints arise
because of the dynamics of the vehicle and
the capabilities of its actuators (engine power,
braking force, ground-wheel interaction, etc.
They restrict the accelerations and velocities
of the vehicle.
constraints: collision with stationary
and moving obstacles of the environment
are forbidden.
A trajectory is a time-ordered sequence of states
q). It can be represented also by a geometric
path and a velocity profile along this path.
Because of the intrinsic complexity of trajectory
planning (cf. (Latombe, 1991) for complexity is-
sues), the trajectory planner addresses the problem
at hand in two complementary steps of lesser
complexity:
1. Path planning: a geometric path leading the
vehicle to its goal is computed. It is collision-free
with the stationary obstacles of the environment
and it respects the non-holonomic
kinematic constraints of the vehicle.
2. Velocity planning: the velocity profile of the
vehicle along its path is computed; this profile
respects the dynamic constraints of the vehicle
and yields no collisions between the vehicle
and the moving obstacles of the environment.
Path planning is illustrated in the left-hand side
of Fig. 3. It depicts an example path between two
configurations. This collision-free path is a curve
whose curvature is continuous and upper-bounded
so as to respect the kinematic constraints of a car-like
vehicle.
Velocity planning is illustrated in the right -
hand side of Fig. 3. Recall that it requires the
knowledge of the future behaviour of the moving
obstacles (this information is provided by the
prediction function). In the current implementa-
tion, a simple prediction function that assumes
constant velocity for the moving obstacles is used.
The right-hand side of Fig. 3 depicts a space-time
diagram (the horizontal axis being the position
along the path and the vertical one the time di-
mension). The curve represents the motion of the
vehicle through time whereas the thick black lines
are the traces left by moving obstacles when they
cross the path of the vehicle.
The next two sections respectively present the
path planning and the velocity planning steps.
4.2. Path Planning
As mentioned earlier, a car-like vehicle is subject
to non-holonomic kinematic constraints: it can
move only along a direction perpendicular to its
rear wheels axle (continuous tangent direction),
and its turning radius is lower-bounded (maxi-
mum curvature). In the past ten years, numerous
works, e.g. (Barraquand and Latombe, 1989; Laumond
et al., 1994; -
Svestka and Overmars, 1995),
have tackled the problem of computing feasible
paths for this type of vehicle. Almost all of them
compute paths made up of circular arcs connected
with tangential line segments. The key reason
for that is that these paths are the shortest ones
that respect the non-holonomic kinematic constraints
of such a vehicle (Dubins, 1957; Reeds and
Shepp, 1990). However their curvature profile is
not continuous. Accordingly a vehicle following
such a path has to stop at each curvature discon-
tinuity, i.e. at each transition between a segment
and an arc, in order to reorient its front wheels.
This is hardly acceptable for a vehicle driving on
the road. A solution to this problem is therefore
to plan paths with a continuous curvature profile.
In addition, a constraint on the curvature deriva-
6 Laugier et al.
x
s
y
Fig. 3. (a) Path planning and (b) velocity planning.
Fig. 4. Examples of continuous curvature paths.
tive is introduced; it is upper-bounded so as to
reflect the fact that the vehicle can only reorient
its front wheels with a finite velocity.
Addressing a similar problem (but without
the maximum curvature constraint), (Boissonnat
et al., 1994) proves that the shortest path between
two vehicle's configurations is made up of
line segments and clothoids 1 of maximum curvature
derivative. Unfortunately, (Kostov and
later proves that these
shortest paths are, in the general case, made up
of an infinity of clothoids. These results also apply
to the problem including the maximum curvature
constraint. Therefore, in order to come up
with a practical solution to the problem at hand,
a set of paths that contain at most eight parts,
each part being either a line segment, a circular
arc, or a clothoid, has been defined. It is shown
in (Scheuer and Laugier, 1998) that such paths are
sub-optimal in length. They are used to design a
local path planner, i.e. a non-complete collision-free
path planner, which in turn is embedded in
a global path planning scheme. The result is the
first path planner for a car-like vehicle that generates
collision-free paths with continuous curvature
and upper-bounded curvature and curvature
derivative. The reader is referred to (Scheuer and
Fraichard, 1997) for a complete presentation of
the continuous curvature path planner. Various
experimental results are depicted in Fig. 4.
Sensor-Based Control Architecture 7
s
search graph
moving obstacles
trajectory
Fig. 5. An example of velocity planning.
4.3. Velocity Planning
Given the nominal path generated by the path
planner, the problem is to determine the trajectory
of the vehicle along this path, i.e. its velocity
profile; this profile must respect the dynamic
constraints of the vehicle and yields no collision
between the vehicle and the moving obstacles of
the environment.
To address these two issues, i.e. moving obstacles
and dynamic constraints, the concept of
state-time space, has been introduced. It stems
from two concepts that have been used before in
order to deal respectively with moving obstacles
and dynamic constraints, namely the concepts of
configuration-time space (Erdmann and Lozano-
Perez, 1987), and state space, i.e. the space of
the configuration parameters and their deriva-
tives. Merging these two concepts leads naturally
to state-time space, i.e. the state space augmented
of the time dimension (Fraichard, 1993). In this
framework, the constraints imposed by both the
moving obstacles and the dynamic constraints are
represented by static forbidden regions of statetime
space. Besides a trajectory maps to a curve
in state-time space hence trajectory planning in
dynamic workspaces simply consists in finding a
curve in state-time space, i.e. a continuous sequence
of state-times between the current state
of the vehicle and a goal state. Such a curve
must obviously respect additional constraints due
to the fact that time is irreversible and that velocity
and acceleration constraints translate to geometric
constraints on the slope and the curvature
along the time dimension. However it is possible
to extend previous methods for path planning in
configuration space in order to solve the problem
at hand. In particular, a method derived from the
one originally presented in (Canny et al., 1988)
has been designed to solve the problem at hand.
It follows the paradigm of near-time-optimization:
the search for the solution trajectory is performed
over a restricted set of canonical trajectories hence
the near-time-optimality of the solution. These
canonical trajectories are defined as having piece-wise
constant acceleration that change its value at
given times. Besides the acceleration is selected so
as to be either minimum, null or maximum (bang
controls). Under these assumptions, it is possible
to transform the problem of finding the time-optimal
canonical trajectory to finding the shortest
path in a directed search graph embedded in
the state-time space.
An example of velocity planning is depicted in
Fig. 5. There are two windows: a trace window
showing the part of the search graph which
has been explored and a result window displaying
the final trajectory. Any such window represents
the s\Thetat plane (the position axis is horizontal
while the time axis is vertical; the frame origin
is at the upper-left corner). The thick black
8 Laugier et al.
segments represent the trails left by the moving
obstacles and the little dots are nodes of the underlying
state-time search graph. The obstacles
are assumed to keep a constant velocity. The
vehicle starts from position 0 (upper-left corner)
with a null velocity, it is to reach position 1 (right
border) with a null velocity. The reader is referred
to (Fraichard, 1993) and (Fraichard and
Scheuer, 1994) for more details about velocity
planning.
5. Sensor-Based Manoeuvres
Recall that the control architecture proposed relies
upon the concept of sensor-based manoeuvres
(SBM). At a given time instant, the vehicle
is carrying out a particular SBM that has been
instantiated to fit the current execution context
(see x2). SBMs are general templates encoding
the knowledge of how a given motion task is to
be performed. They combine real-time functions,
control and sensing skills, that are either control
programs or sensor data processing functions.
This section describes the two SBMs that have
been developed and integrated in the control architecture
proposed: trajectory following and parallel
parking. These two manoeuvres have been
implemented and successfully tested on a real automatic
vehicle, the results of these experiments
are presented in x6. The Orccad tool (Simon et
al., 1993) has been selected to implement both
SBMs and skills. "Robot procedures" (in the Or-
ccad are used to encode SBMs while
"robot-tasks" encode skills. Robot procedures
and robot tasks can both be represented as finite
automata or transition diagrams. The "trajectory
and "parallel parking" SBMs are depicted
in Fig. 6 as transition diagrams. The control
skills are represented by square boxes, e.g.
"find parking place", whereas the sensing skills
appear as predicates attached to the arcs of the
diagram, e.g. "parking place detected", or conditional
statements, e.g. "obstacle overtaken?". The
next two sections describe how the two manoeuvres
illustrated in Fig. 6 operates.
5.1. Trajectory Following
The purpose of the trajectory following SBM is
to allow the vehicle to follow a given nominal trajectory
as closely as possible, while reacting appropriately
to any unforeseen obstacle obstructing
the way of the vehicle. Whenever such an obstacle
is detected, the nominal trajectory is locally
modified in real time, in order to avoid the colli-
sion. This local modification of the trajectory is
done, in order to satisfy a set of different motion
constraints: collision avoidance, time constraints,
kinematic and dynamic constraints of the vehicle.
In a previous approach, a fuzzy controller combining
different basic behaviours (trajectory tracking,
obstacle avoidance, etc.) was used to perform trajectory
following (Garnier and Fraichard, 1996).
However this approach proved unsatisfactory: it
yields oscillating behaviours, and does not guarantee
that all the aforementioned constraints are
always satisfied.
The trajectory following SBM makes use of local
trajectories to avoid the detected obstacles. These
local trajectories allow the vehicle to move away
from the obstructed nominal trajectory, and to
catch up this nominal trajectory when the (sta-
tionary or moving) obstacle has been overtaken.
All the local trajectories verify the motion con-
straints. This SBM relies upon two control skills,
trajectory tracking and lane changing (cf. Fig. 6),
that are detailed now.
5.1.1. Trajectory Tracking The purpose of this
control skill is to issue the control commands
that will allow the vehicle to track a given nominal
trajectory. Several control methods for non-holonomic
robots have been proposed in the lit-
erature. The method described in (Kanayama et
al., 1991) ensures stable tracking of a feasible trajectory
by a car-like robot. It has been selected for
its simplicity and efficiency. The vehicle's control
commands are of the following
where represents the error between
the reference configuration q ref and the current
configuration q of the vehicle (q
' ref and v R;ref are the reference velocities,
is the rear axle midpoint velocity, k x ,
Sensor-Based Control Architecture 9
Updating
Goal reached ?
obstacle detection
Free-lane reached
no
no
yes
no
Trajectory tracking
Lane changing
Obstacle avoidance
possible
Nominal Trajectory
Trajectory tracking
Lane changing
reached ?
Nominal trajectory
yes
no
yes no
Nominal trajectory
to reach
backward & forward motions
Parallel Parking
SUCCESS
SUCCESS
Success ?
Generation-execution of appropriate
Find a parking place
building
local map
Reaching an appropriate start location
Free parking place detected
Free-space parameters
Free-lane description yes
yes
Trajectory Following
Fig. 6. The "parallel parking" and "trajectory following" SBMs.
are positive constants (the reader is referred
to (Kanayama et al., 1991) for full details about
this control scheme).
5.1.2. Lane Changing This control skill is applied
to execute a lane changing manoeuvre. The
lane changing is carried out by generating and
traffic lane
nominal trajectory
obstacle
traffic lane
Fig. 7. Generation of smooth local trajectories for avoiding
an obstacle.
tracking an appropriate local trajectory. Let T
be the nominal trajectory to track, d T be the distance
between T and the middle line of the free
lane to reach, s T be the curvilinear distance along
T between the vehicle and the obstacle (or the selected
end point for the lane change), and
be the curvilinear abscissa along T since the starting
point of the lane change (cf. Fig. 7).
A feasible smooth trajectory for executing a
lane change can be obtained using the following
quintic polynomial (cf. (Nelson, 1989)):
s
s
s
In this approach, the distance d T is supposed to
be known beforehand. Then the minimal value
required for s T can be estimated as follows:
Laugier et al.
where C max stands for the maximum allowed curvature
is the maximum allowed lateral acceleration,
is an empirical constant, e.g.
in our experiments.
At each time t from the starting time T 0 , the
reference position p ref is translated along the vector
represents the unit normal
vector to the nominal velocity vector along T ;
the reference orientation ' ref is converted into
@d
@s
, and the reference velocity
v R;ref is obtained using the following equation:
\Deltat
where dist stands for the Euclidean distance. As
shown in Fig. 6, this type of control skill can also
be used to avoid a stationary obstacle, or to overtake
another vehicle. As soon as the obstacle has
been detected by the vehicle, a value s T;min is computed
according to (6) and compared with the distance
between the vehicle and the obstacle. The
result of this computation is used to decide which
behaviour to apply: avoid the obstacle, slow down
or stop. In this approach, an obstacle avoidance
or overtaking manoeuvre consists of lane changing
manoeuvre towards a collision-free "virtual"
parallel trajectory(see Fig. 7). The lane changing
skill operates the following way:
1. Generate a smooth local trajectory - 1 which
connects T with a collision-free local trajec-
B22
parking place
traffic lane
parking lane
border of the parking lane
traffic direction
Fig. 8. Situation at the beginning of a parallel parking
manoeuvre.
tory - 2 "parallel" to T (- 2 is obtained by
translating appropriately the involved piece of
2. Track - 1 and - 2 until the obstacle has been
overtaken.
3. Generate a smooth local trajectory - 3 which
connects - 2 with T , and track - 3 .
5.2. Parallel Parking
Parallel parking comprises three main steps
(cf. Fig. 6): localizing a free parking place, reaching
an appropriate start location with respect to
the parking place, and performing the parallel
parking manoeuvre using iterative backward and
forward motions until the vehicle is parked. During
the first step, the vehicle moves slowly along
the traffic lane and uses its range sensors to build
a local map of the environment and detect obsta-
cles. The local map is used to determine whether
parking space is available to park the vehicle.
A typical situation at the beginning of a parallel
parking manoeuvre is depicted in Fig. 8. The
autonomous vehicle A1 is in the traffic lane. The
parking lane with parked vehicles B1, B2 and a
parking place between them is on the right-hand
side of A1. L1 and L2 are respectively the length
and width of A1, and D1 and D2 are the distances
available for longitudinal and lateral displacements
of A1 within the place. D3 and D4
are the longitudinal and lateral displacements of
the corner A13 of A1 relative to the corner B24
of B2.
Distances D1, D2, D3 and D4 are computed
from data obtained by the sensor systems. The
length (D1 \Gamma D3) and wide (D2 \Gamma D4) of the free
parking place are compared with the length L1
and width L2 of A1 in order to determine whether
the parking place is sufficiently large.
During parallel parking, iterative low-speed
backward and forward motions with coordinated
control of the steering angle and locomotion velocity
are performed to produce a lateral displacement
of the vehicle into the parking place.
The number of such motions depends on the distances
and the necessary parking
depth (that depends on the width L2 of the vehicle
A1). The start and end orientations of the
vehicle are the same for each iterative motion.
Sensor-Based Control Architecture 11
For the i-th iterative motion (but omitting the
index "i"), let the start coordinates of the vehicle
be x 0
and the end
coordinates be x
where T is duration of the motion. The "parallel
parking" condition means that
where admissible error in orientation
of the vehicle.
The following control commands of the steering
angle OE and locomotion velocity v provide
the parallel parking manoeuvre (Paromtchik and
Laugier, 1996b):
are the admissible
magnitudes of the steering angle and locomotion
velocity respectively, k corresponds to
a right side (+1) or left side (-1) parking place relative
to the traffic lane, k corresponds to
forward (+1) or backward (-1) motion,
4-t
. The shape of the type
of paths that corresponds to the controls (12) and
is shown in Fig. 9.
The commands (10) and (11) are open-loop
in the (x; ')-coordinates. The steering wheel
servo-system and locomotion servo-system must
execute the commands (10) and (11), in order to
provide the desired (x; y)-path and orientation '
of the vehicle. The resulting accuracy of the motion
in the (x; ')-coordinates depends on the
accuracy of these servo-systems. Possible errors
are compensated by subsequent iterative motions.
For each pair of successive motions
the coefficient k v in (11) has to satisfy the equation
that alternates between forward
and backward directions. Between successive
motions, when the velocity is null, the steering
wheels turn to the opposite side in order to
obtain a suitable steering angle OE max or \GammaOE max
to start the next iterative motion.
In this way, the form of the commands (10) and
(11) is defined by (12) and (13) respectively. In
order to evaluate (10)-(13) for the parallel parking
manoeuvre, the durations T and T , the magnitudes
OE max and v max must be known.
The value of T is lower-bounded by the kinematic
and dynamic constraints of the steering
wheel servo-system. When the control command
is applied, the lower bound of T is
s
OE
OE max are the maximal admissible
steering rate and acceleration respectively for
the steering wheel servo-system. The value of
min gives duration of the full turn of the steering
wheels from \GammaOE max to OE max or vice versa, i.e. one
can choose T
min .
The value of T is lower-bounded by the constraints
on the velocity v max and acceleration
v max and by the condition T ! T . When the
control command (11) is applied, the lower bound
of T is
ae
oe
tion, serves to provide a smooth motion of the
vehicle when the available distance D1 is small.
The computation of T and OE max aims to obtain
the maximal values such that the following "longi-
tudinal" and "lateral" conditions are still satisfied:
Fig. 9. Shape of a parallel forward/backward motion.
12 Laugier et al.
Using the maximal values of T and OE max assures
that the longitudinal and, especially, lateral displacement
of the vehicle is maximal within the
available free parking space. The computation is
carried out on the basis of the model (1) when
the commands (10) and (11) are applied. In this
computation, the value of v max must correspond
to a safety requirement for parking manoeuvres,
e.g. empirically.
At each iteration i the parallel parking algorithm
is summarized as follows:
1. Obtain available longitudinal and lateral displacements
respectively by processing
the sensor data.
2. Search for maximal values T and OE max by
evaluating the model (1) with controls (10),
so that conditions (16), (17) are still satisfied
3. Steer the vehicle by controls (10), (11) while
processing the range data for collision avoidance
4. Obtain the vehicle's location relative to environmental
objects at the parking place. If the
"parked" location is reached, stop; else, go to
step 1.
When the vehicle A1 moves backwards into the
parking place from the start location shown in
Fig. 8, the corner A12 (front right corner of the
vehicle) must not collide with the corner B24
(front left corner of the place). The start location
must ensure that the subsequent motions will
be collision-free with objects limiting the parking
place. To obtain a convenient start location, the
vehicle has to stop at a distance D3 that will ensure
a desired minimal safety distance D5 between
the vehicle and the nearest corner of the parking
place during the subsequent backward mo-
tion. The relation between the distances D1,
D2, D3, D4 and D5 is described by a function
This function can
not be expressed in closed form, but it can be estimated
for a given type of vehicle by using the
model (1) when the commands (10) and (11) are
applied. The computations are carried out off-line
and the results are stored in a look-up table
which is used on-line, to obtain an estimate of
D3 corresponding to a desired minimal safety distance
D5 for given D1, D2 and D4 (Paromtchik
and Laugier, 1996a). When the necessary parking
"depth" has been reached, clearance between the
vehicle and the parked ones is provided, i.e. the
vehicle moves forwards or backwards so as to be in
the middle of the parking place between the two
parked vehicles.
6. Experimental Results
The approach described in the paper has been implemented
and tested on our experimental automatic
vehicle (a modified Ligier electric car). This
vehicle is equipped with the following capabilities:
1. a sensor unit to measure relative distances between
the vehicle and environmental objects,
2. a servo unit to control the steering angle and
the locomotion velocity and
3. a control unit that processes data from the
sensor and servo units in order to "drive"
the vehicle by issuing appropriate servo commands
This vehicle can either be manually driven, or
it can move autonomously using the control unit
based on a Motorola VME162-CPU board and
a transputer net. A VxWorks real-time operating
system is used. The sensor unit of the
vehicle makes use of a belt of ultrasonic range
sensors (Polaroid 9000) and of a linear CCD-
camera. The servo unit consists of a steering
wheel servo-system, a locomotion servo-system for
forward and backward motions, and a braking
servo-system to slow down and stop the vehicle.
The steering wheel servo-system is equipped with
a direct current motor and an optical encoder
to measure the steering angle. The locomotion
servo-system of the vehicle is equipped with a
asynchronous motor and two optical encoders
located onto the rear wheels (for odometry
data). The vehicle has an hydraulic braking servo-
system. The Motion Controller monitors the current
steering angle, locomotion velocity, travelled
distance, coordinates of the vehicle and range data
from the environment, calculates an appropriate
local trajectory and issues the required servo com-
mands. The Motion Controller has been implemented
using the Orccad software tools (Simon
et al., 1993) running on a Sun workstation. The
Sensor-Based Control Architecture 13
compiled code is transmitted via Ethernet to the
VME162-CPU board.
The experimental car is equipped with 14 ultrasonic
range sensors (Polaroid 9000), eight of
them (a minimal configuration) are used for the
current version of the automatic parking system:
three ultrasonic sensors are at the front of the car
(looking in the forward direction), two sensors are
situated on each side of the car and one ultrasonic
sensor is at the rear of the car (looking in the
backward direction). The measurement range is
10:0m, the sampling rate is 60ms. The sensors
are activated sequentially (four sensors are
emitting/receiving signals at each instant (one for
each side of the car). This sensor system is intended
to test the control algorithms only and for
low-speed motion only. Certainly, a more complex
sensor system, e.g. a combination of vision
and ultrasonic sensors, must be use to ensure reliable
operation in a dynamic environment.
An experimental run of the "follow trajectory"
SBM with obstacle avoidance on circular road
(roundabout) is shown in Fig. 10. In this experi-
ment, the Ligier vehicle follows a nominal trajectory
along the curved traffic lane, and it finds on
its way another vehicle moving at a lower velocity
(see Fig. 10a). When the moving obstacle is de-
tected, a local trajectory for a right lane change
is generated by the system, and the Ligier performs
the lane changing manoeuvre, as illustrated
in Fig.10b. Afterwards, the Ligier moves along a
trajectory parallel to its nominal trajectory, and
a left lane change is performed as soon as the obstacle
has been overtaken (Fig. 10c). Finally the
Ligier catches up its nominal trajectory, as illustrated
in Fig. 10d.
The corresponding motion of the vehicle is depicted
in Fig. 11a. The steering and velocity controls
applied during this manoeuvre are shown in
Fig. 11b and Fig. 11c. It can be noticed in this example
that the velocity of the vehicle has increased
when moving along the local "parallel" trajectory
(Fig. 11c); this is due to the fact that the vehicle
has to satisfy the time constraints associated to
its nominal trajectory.
An experimental run of the parallel parking
SBM in a street is shown in Fig. 12. This manoeuvre
can be carried out in environments including
moving obstacles, e.g. pedestrians or some
other vehicles (cf. the video (Paromtchik and
Laugier, 1997)). In this experiment, the Ligier
was manually driven to a position near the parking
place, the driver started the autonomous parking
mode and left the vehicle. Then, the Ligier
moved forward autonomously in order to localize
the parking place, obtained a convenient start
location, and performed a parallel parking ma-
noeuvre. When, during this motion a pedestrian
crosses the street in a dangerous proximity to the
vehicle, as shown in Fig. 12a, this moving obstacle
is detected, the Ligier slows down and stops
to avoid the collision. When the way is free, the
Ligier continues its forward motion. Range data is
used to detect the parking bay. A decision to carry
out the parking maneuver is made and a convenient
start position for the initial backward movement
is obtained, as shown in Fig. 12b. Then, the
Ligier moves backwards into the bay, as shown
in Fig. 12c. During this backward motion, the
front human-driven vehicle starts to move back-
wards, reducing the length of the bay. The change
in the environment is detected and taken into ac-
count. The range data shows that the necessary
"depth" in the bay has not been reached, so further
iterative motions are carried out until it has
been reached. Then, the Ligier moves to the middle
between the rear and front vehicles, as shown
in Fig. 12d. The parallel parking maneuver is
completed.
The corresponding motion of the vehicle is depicted
in Fig. 13a where the motion of the corners
of the vehicle and the midpoint of the rear wheel
axle are plotted. The control commands (10) and
for parallel parking into a parking place situated
at the right side of the vehicle are shown
in Fig. 13b and Fig. 13c respectively. The length
of the vehicle is 2:5m, the width is
1:4m, and the wheelbase is 1:785m. The
available distances are
relative to the start location of the vehicle. The
lateral distance measured by the
sensor unit. The longitudinal distance
was estimated so as to ensure the minimal safety
distance 0:2m. In this case, five iterative
motions are performed to park the vehicle.
As seen in Fig. 13, the durations T of the iterative
motions, magnitudes of the steering angle
OE max and locomotion velocity v max correspond to
the available displacements D1 and D2 within the
14 Laugier et al.
a b c d
Fig. 10. Snapshots of trajectory following with obstacle avoidance in a roundabout: (a) following the nominal trajectory,
(b) lane changing to the right and overtaking, (c) lane changing to the left, (d) catching up with the nominal trajectory.
a
y
x [m]
motion direction
nominal trajectory
local trajectory
-0.4
angle
time
velocity
time [s]
Fig. 11. Motion and control commands in the "roundabout" scenario: (a) motion, (b) steering angle and (c) velocity
controls applied.
a b c d
Fig. 12. Snapshots of a parallel parking: (a) localizing a free parking place, (b) selecting an appropriate start location, (c)
performing a backward parking motion; (d) completing the parallel parking.
a
y
x [m]
start location
location
-0.4
angle
time
-0.4
velocity
time [s]
Fig. 13. Motion and control commands in the parallel parking scenario: (a) motion, (b) steering angle and (c) velocity
controls applied.
parking place (e.g. the values of T , OE max and v max
differ for the first and last iterative motion).
7. Related Works
As mentioned in x1, motion autonomy has been a
long standing issue in Robotics hence the important
number of works presenting control architectures
for robot systems. All these architectures
are not reviewed here, the main trends are indicated
instead.
Three main functions are to be found in
any control architecture: perception, decision
and action (hence the 'perception-decision-action'
paradigm). After a careful examination of the
existing control architectures, it appears that, to
some extent, the difference between them lies in
the decision function. Two types of approaches of
completely opposite philosophy have appeared:
Sensor-Based Control Architecture 15
deliberative approaches: in this type of ap-
proach, complex models of the environment of
the robot are built from sensory data or a priori
knowledge. These models are then used
to perform high-level reasoning, i.e. planning,
in order to determine which action to under-
take. Maintaining these models and reasoning
about them is, in most cases, a time-consuming
process that makes these methods
unable to deal with dynamic and uncertain
environments. (Moravec, 1983; Nilsson, 1984)
and (Waxman et al., 1985) are good examples
of this type of control architectures.
reactive approaches: the philosophy of this
type of approach is just the opposite: they
favor reactivity. The decision function is reduced
to a minimum. Action follows perception
closely, almost like a reflex. This
type of approach is most appropriate to dynamic
and uncertain environments since unexpected
events can be dealt with as soon as
they are detected by the sensors of the robot.
One drawback however, high-level reasoning
is very difficult to achieve (if not impossible).
(Brooks, 1990) is the canonical sensor-based
control architecture; other examples are given
in (Khatib and Chatila, 1995) or (Zapata et
al., 1990).
In an attempt to combine the advantages of both
deliberative and reactive approaches, several authors
have tried to combine high and low-level reasoning
functions within a single control architec-
ture. This idea permits to obtain hybrid control
architectures with both high-level reasoning capabilities
and reactivity.
The first hybrid architectures were obtained by
simply putting together a deliberative and a re-active
component. For instance, (Arkin, 1987)
integrates a simple motion planner to a reactive
architecture whereas (Gat et al., 1990) sends the
output of a task planner to a simple reactive execution
controller: when a problem is detected at
execution time, a reflex action is performed and
the task planner is reinvoked. The performance of
these approaches in terms of robustness, flexibility
and reactivity are far from satisfactory. Better architectures
have been proposed since, e.g. (Alami
et al., 1998; Gat, 1997) or (Simmons, 1994), they
all combine three functional components:
ffl A set of elementary real-time functions (con-
trol loops, sensor data processing functions,
etc. A task is performed through the activation
of such functions.
ffl A reactive execution mechanism that control
and coordinates the execution of the real-time
functions.
ffl A decision module that produces the task plan
and supervises its execution. It may react to
events from the execution function.
The control architecture presented in this paper
clearly falls into this class of hybrid architectures.
Skills are the real-time functions, the motion controller
is the execution mechanism while the mission
monitor is the decision module. With regard
to these architectures, the main novelty of the approach
proposed lies in the introduction of a meta-level
of real-time functions, the sensor-based ma-
noeuvres, that encapsulate high-level expert human
knowledge and heuristics about the motion
tasks to be performed, that permit to reduce the
planning effort required to address a given motion
task and thus to improve the overall response-time
of the system.
8. Conclusion
This paper has presented an integrated control
architecture endowing a car-like vehicle moving
in a dynamic and partially known environment
(the road network) with autonomous motion ca-
pabilities. Like most recent control architectures
for autonomous robot systems, it combines three
functional components: a set of basic real-time
skills, a reactive execution mechanism and a decision
module. The main novelty of the architecture
proposed lies in the introduction of a fourth
component akin to a meta-level of skills: the
sensor-based manoeuvres, i.e. general templates
that encode high-level expert human knowledge
and heuristics about how a specific motion task
is to be performed. The concept of sensor-based
manoeuvres permit to reduce the planning effort
required to address a given motion task, thus improving
the overall response-time of the system,
while retaining the good properties of a skill-based
architecture, i.e. robustness, flexibility and reactivity
Laugier et al.
After a general overview of the architecture pro-
posed, the paper has covered in more details the
trajectory planning function (which is an important
part of the decision module) and two types
of sensor-based manoeuvres: trajectory following
and parallel parking. Experimental results with
a real automatic car-like vehicle in different situations
have been reported to demonstrate the
efficiency of the approach. Future works will include
the development and testing of other types
of sensor-based manoeuvres.
Acknowledgements
This work was partially supported by the Inria-
Inrets 2 Praxit'ele programme on urban public
transport [1994-1997], and the Inco-Copernicus
project "Multi-agent robot
systems for industrial applications in the transport
domain" [1997-1999]. The authors would like
to thank E. Gauthier for his valuable contribution
to the final version of the paper.
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2. Institut National de Recherche sur les Transports et leur
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Christian Laugier received the M.
"Mo- tion planning for a non-holonomic mobile in a dynamic workspace"
Philippe Garnier received the B.
His research interests include motion control for autonomous car-like vehicles in dynamic and structured environments
Igor Paromtchik received the M.
Computer Systems and Robotics of the university of Karlsruhe
Alexis Scheuer entered the Ecole Normale Sup'erieure de Lyon
--TR | motion autonomy;control architecture;car-like vehicle |
591532 | A Robotic Excavator for Autonomous Truck Loading. | Excavators are used for the rapid removal of soil and other materials in mines, quarries, and construction sites. The automation of these machines offers promise for increasing productivity and improving safety. To date, most research in this area has focussed on selected parts of the problem. In this paper, we present a system that completely automates the truck loading task. The excavator uses two scanning laser rangefinders to recognize and localize the truck, measure the soil face, and detect obstacles. The excavators software decides where to dig in the soil, where to dump in the truck, and how to quickly move between these points while detecting and stopping for obstacles. The system was fully implemented and was demonstrated to load trucks as fast as human operators. | Introduction
The surface mining of metals, quarrying of rock, and construction
of highways require the rapid removal and handling
of massive quantities of soil, ore, and rock. Typically,
explosive or mechanical techniques are used to pulverize
the material, and digging machines such as excavators load
the material into trucks for haulage to landfills, storage ar-
eas, or processing plants. As shown in Figure 1, an excavator
sits atop a bench and loads material into trucks that
queue up to its side. The operator is responsible for designating
where the truck should park, digging material from
the face and depositing it in the truck bed, and stopping for
people and obstacles in the loading zone.
Figure
1. Excavator loading a truck with soil in a typical mass
excavation work scenario.
The opportunities for automation are immense. Typical-
ly, loading a truck requires several passes, each of which
takes 15 to 20 seconds. Reducing the time of each loading
pass by even a second translates into an enormous gain
across the entire job. The operator's performance peaks
early in the work shift and degrades as the shift wears on.
Scheduled idle times, such as lunch and other breaks, also
diminish average production across a shift. All of these factors
are areas where automation can improve productivity.
Safety is another opportunity. Excavator operators are
most likely to be injured when mounting and dismounting
the machine. Operators tend to focus on the task at hand
and may fail to notice other site personnel or equipment entering
the loading zone. Automation can improve safety by
removing the operator from the machine and by providing
complete sensor coverage to watch for potential hazards
entering the work area.
Numerous researchers have addressed aspects of automated
earthmoving (Singh, 1997). The lowest and most
common level of automation has been teleoperation. Typi-
cally, the operator is removed from the scene for reasons of
safety. Teleoperated excavators are used in applications
that pose a danger to humans, such as the uncovering of
buried ordnance (Nease and Alexander, 1993) and waste
(Burks et al., 1992; Wohlford et al., 1990), or excavation
around buried utilities. A higher level of autonomy is
achieved by systems that share control of the excavation
cycle with a human operator. Typically, these systems
(Bradley et al., 1993; Bullock and Oppenheim, 1989; Huang
and Bernold, 1994; Lever et al., 1994; Rocke, 1994; Sakai
and Cho, 1988; Salcudean et al., 1997; Sameshima and
Tozawa, 1992; Seward et al., 1992) concentrate on the process
of digging. An operator chooses the starting location
for the excavator's bucket and a control system takes over
the process of filling the bucket using force and/or joint position
feedback to accomplish the task. At the next level of
autonomy are systems that automatically select where to
dig. Such systems measure the topology of the terrain using
ranging sensors (Feng et al., 1992; Singh, 1995; Takahashi
et al., 1995) and compute dig trajectories that maximize excavated
volume. At the highest level of autonomy are systems
that sequence digging operations over a long period
(Bullock et al., 1990; Romero-Lois et al., 1989; Singh,
1998).
The prior work addresses many subproblems important
for autonomous truck loading, however in order to field a
fully automated system that performs at the level of its
manually operated equivalent, a much broader set of problems
must be solved than just digging. Sensors are needed
to sense the dig face, recognize and localize the truck, and
detect obstacles in the workspace. Perception algorithms
are needed to process the sensor data and provide information
about the work environment to the planning algo-
rithms. Planning and control algorithms are needed to
decide how to work the dig face, deposit material in the
truck, and move the bucket between the two.
We have developed a complete system for loading trucks
fully autonomously with soft materials such as soil. The
Autonomous Loading System (ALS) was implemented and
demonstrated on a 25-ton hydraulic excavator and succeeded
in loading trucks as fast as an expert human operator.
The rest of the paper describes the ALS and presents results
from experimental trials.
2. System Overview
The Autonomous Loading System uses two scanning laser
rangefinders that are mounted on either side of the boom
(see
Figure
to sense the dig face, truck, and obstacles in
the workspace. Two scanners are needed for full coverage
of the workspace and to enable concurrent sensing opera-
tions. Each sensor has a sample rate of 12 kHz, and a motorized
mirror sweeps the beam circularly in a vertical
plane. Additionally, each scanner can pan at a rate of up to
degrees per second, enabling this circle to be rotated
about the azimuth, as shown in Figure 3. The scanner positioned
over the operator's cab is called the ``left scanner'',
and it is responsible for sensing the workspace on the left
hand side of the excavator. The "right scanner", which is
located at a symmetric position on the right side of the
boom, is responsible for sensing the workspace on the right
hand side of the excavator.
Figure
2. Sensors mounted on excavator.
Figure
3. Two axis scanning sensor configuration.
The excavator uses its scanners in the following fashion
when loading a truck (Figure 4). While the excavator digs
its first bucket, the left scanner pans left from the dig face
across the truck both to detect obstacles and to recognize,
localize, and measure the dimensions of the truck. Using
this information, a desired location in the truck to dump the
material is planned, and the bucket swings toward the
truck. During this swing motion, the right scanner pans left
across the dig face to measure its new surface, and the next
location to dig is calculated. The right scanner continues to
pan toward the truck. After the soil is dumped into the
truck, the right scanner pans back across the dig face to detect
obstacles in the way of the implements. The excavator
swings back to the next dig point. During this swing, the
left scanner pans across the truck to measure the soil distribution
in the truck bed, and the next desired dump location
is calculated. This process repeats for each subsequent
loading pass until the truck is full, with the exception that
truck recognition is only necessary for the first pass for
each new truck. Typically, six passes are needed to load our
twenty-ton truck with our excavator testbed.
Left scanner
Right
scanner
Pan axis
Vehicle cab
Rotating
reflector
Distance
Scan axis
Obstacle or terrain
Scan plane
sensor
measuring
Figure
4. Top view of sensor configuration.
Information from the scanners is processed using an on-board
array of four MIPs processors. The software architecture
is shown in Figure 5. The boxes are software modules
that can run on one of the system processors. Circles
are hardware components such as sensors. Lines represent
communication channels. The sensor interfaces receive
data from the two scanners and control the panning motion
of the devices. Sensor data from the interfaces are passed to
scanline processors, where they are converted from spher-
ical, sensor coordinates to Cartesian, world coordinates using
corresponding data from the position system. These
three-dimensional range points are then made available to
whatever perception software modules require them.
One consumer of this processed sensor data is the truck
recognizer, which recognizes the truck and measures both
its dimensions and location. Two others are the dig point
planner, which plans a sequence of dig points for eroding
the dig face, and the dump point planner, which plans a sequence
of dump points for loading soil into the truck bed.
The digging motion planner controls the excavator during
digging at the specified location. The dumping motion
planner dumps the bucket of soil into the truck and returns
to the dig face. The sensor motion planner controls the panning
for both scanners to coordinate scanner and excavator
motion, following the scenario described above. The obstacle
detector processes sensor data from the scanner that is
sweeping in advance of excavator's motion and stops the
machine if an obstacle is detected in its path. The machine
controller interface communicates commands to the low
level machine joint controller, which executes the commands
and sends excavator state information back to the
planning modules.
Figure
5. ALS software architecture.
3. Hardware Subsystem
The ALS hardware subsystem consists of the servo-controlled
excavator, on-board computing system, perception
sensors, and associated electronics. In this paper we focus
on the perceptual sensors which provide the data from
which the truck is identified, the dig location is chosen, obstacles
are avoided, and ultimately the mass excavation
process is achieved.
With the target application of earthmoving, we focussed
on developing a laser based scanning system that would be
able to penetrate a reasonable amount of dust and smoke in
the air. The laser itself would need to be able to accurately
measure range from a variety of target materials (e.g., met-
als, wood, dirt, rock, snow, ice, and water), colors and tex-
tures. We also needed a system that would be robust to dust
and dirt accumulating on the protective "exit" window
(glass or plastic which protects the laser and optics from
weather and dirt, though permits the beam to pass).
Over the past decade, a variety of laser based scanners
have been produced. With the exceptions of the Dornier
(Shulz, 1997) and Schwartz (Schwartz) scanners, most
have either been research devices or limited to indoor us-
age. None that we know of addresses the problems of dust
penetration or a partially occluded (i.e., dirty) exit window.
We have developed two different time-of-flight scanning
ladar systems that are impervious to ambient dust condi-
tions. The first uses a "last-pulse" technique that observes
the waveform of the returned light and rejects early returns
that can arise from internal reflections off of a dirty exit
Left sensor
Right sensor
Dig face
Truck
scan plane
scan plane
Right
scanner
Left scanner
left sensor
interface
right sensor
interface
left scanline
processor
right scanline
processor
left
sensor
right
sensor
position
system
truck
recognizer
dig point
planner
dump point
planner
digging
motion
planner
dumping
motion
planner
sensor
motion
planner
machine
controller
interface
obstacle
detector
excavator
states commands
position data
sensor data sensor data
sensor data
sensor data
dig pt.
dump pt.
truck info.
sensor
sensor
commands
states
sensor
data
to obst. det.
window, or from a dust cloud obscuring the target (see Figure
6). In general, the next-to-last pulse returns are due to
dust in the scene and are indicative of what a normal "first
pulse" rangefinder would see. For instance, in Figure 6, a
first pulse rangefinder would detect the dirty exit window
and would be unable to "see" the target. Even if the window
were clean, the first pulse unit would still "see" the
dust cloud instead of the target. Since reflections off the
exit window are rejected with the last pulse technique, the
unit can be environmentally sealed using an inexpensive
transparent cover that does not have to be optically perfect
or clean. Another advantage is that the laser system can
also report when multiple returns occur, giving a warning
that dust is present. This is important because overall ranging
reliability and accuracy is decreased in dusty condi-
tions, so an autonomous machine might need to adopt a
slower, more conservative motion strategy.
Figure
6. Last pulse detection concept.
Figure
7. Trailing edge detection of target when target is
obscured in dust cloud.
There is, however, a limitation to last-pulse rangefind-
ing. When the target is within the dust cloud, the receiver
electronics can have difficulty separating the dust and target
returns (see Figure 7). We have built a second dust penetrating
scanner system that identifies that target by
locating the "trailing edge" of the last return signal as is
shown in Figure 7. Like the last pulse system, this device is
also robust to occlusions on the exit window making it ideal
for construction and mining environments. Though the
trailing edge detection technique forgoes some range accu-
racy, we believe it is a superior approach for environments
where the dust may frequently surround the target.
Figure
8. Last pulse vs. trailing edge detection when target is
within dust cloud.
The television monitor pictured in Figure 8 shows range
points plotted from a single scanline for both the last pulse
and trailing edge scanners. Range increases from the left to
the right of the monitor. The top monitor screen shows
scans of the rear of a dump truck. The bottom screen shows
scans of the same truck but shrouded in a heavy dust cloud.
Note that the last pulse device is unable to separate the dust
cloud from the truck and reports the front of the cloud. The
trailing edge device correctly reports range to the truck regardless
of the presence of dust.
It is important to note that both dust penetrating techniques
are physically limited by very heavy dust levels that
attenuate the return target signal below the point of detectability
4. Software Subsystem
The software subsystem consists of several software modules
that process sensor data, recognize the truck, select
digging and dumping locations, move the excavator's
joints, and guard against collision. In this section, the algorithms
employed by key software modules in the software
architecture are described.
4.1. Truck Recognition
In order to properly load a truck, an excavator operator
must verify that it is a loadable vehicle, determine its loca-
tion, and determine its dimensions. This information is essential
for calculating a loading strategy and for planning
the sequence of joint motions that implements this strategy.
Dirty exit window Dust cloud Target
Laser
source
Threshold
(using last pulse)
Return
signal
Target range
Dirty exit window Dust cloud Target
Laser
source
Return
signal
Threshold
(w/trailing edge)
Target range (w/last pulse)
In some scenarios, such as surface mining, the loaders are
serviced by a mine-owned fleet of haulage trucks. An automated
system could acquire this information by equipping
each truck with a global positioning system (GPS) sensor
and an identification transponder. However, in other scenarios
such as highway construction, the loaders are serviced
by a variety of independently-owned, on-highway
trucks of varying dimensions, so equipping each and every
truck with such sensors could be infeasible. For such sce-
narios, an automated system could acquire the necessary
information using rangefinder data.
Figure
9. Raw range data of a truck.
Figure
10. Truck model fit to segmented data.
The truck recognizer uses sensor data to automatically
recognize, localize, and dimension haulage trucks. As the
excavator digs its first bucket of soil, the left scanner pans
across the truck, which is assumed to be parked to the ex-
cavator's side. The raw sensor data are shown in Figure 9.
Each rotation of the mirror returns one vertical scanline of
data, created by intersecting a vertical plane with the truck.
Each scanline is processed into line segments which are
grouped with coplanar line segments from other scanlines
to form planar regions.
Using an interpretation tree approach (Grimson, 1990),
the simple model for a truck bed, shown in Figure 10, is
matched to the segmented data region by region. Depth-first
search is used to hypothesize model-to-scene region
matches. At each level in the tree, constraints are used to
prune the search and to check for consistency with previously
hypothesized matches. The interpretation that matches
most of the model regions and survives the verification
stage is selected as the correct one. In order for the truck
recognizer to recognize a class of truck models rather than
just a single model, the model in Figure 10 uses parameter
ranges rather than single parameter values. Ranges are used
on the sizes of the planar regions in the model, the locations
of their centroids relative to each other, and the angles between
the planes. These parameter ranges are checked for
consistency at every level in the interpretation process to
prune the search. This specification allows the truck recognizer
to identify trucks of varying sizes and truck bed
shapes.
For each complete interpretation (i.e. an attempt to match
all model regions to scene regions), the truck recognizer
performs a verification. The verification consists of finer-grained
consistency checking of truck parameters, and the
identification of the four "corner points" in the sensor data
that define the opening of the truck bed. For the selected in-
terpretation, the corner points are used to calculate the position
and orientation of the truck bed. This information is
passed to other modules in the system for producing a
dumping strategy. Figure 10 shows the model matched to
the planar regions segmented from the raw sensor data.
4.2. Coarse-to-Fine Dig Point Planning
Automated earthmoving operations such as leveling a
mound of soil are distinguished from typical planning
problems in two important ways. First, soil is diffuse and
therefore a unique description of the world requires a very
large number of variables. Second, the interaction between
the robot and the world is very complex and only approximate
models that are also computationally tractable are
available. The large state space and complex robot-world
interaction imply that only locally optimal planners (i.e. per
dig) can be created. In order to deal with the practical issues
of excavating large volumes of earth in applications, we
have developed a multi-resolution planning and execution
scheme. At the highest level is a coarse planning scheme
that uses the geometry of the site and the goal configuration
of the terrain to plan a sequence of "dig-regions." In turn,
each dig region is searched for the "best" dig that can be executed
in that region. Finally, the selected dig is executed
by a force based closed loop control scheme (Rocke, 1994).
Treatment of the problem at three levels meets different ob-
jectives. The coarse planner ensures even performance over
a large number of digs. The refined planner chooses digs
that meet geometric constraints (reachability and colli-
sions) and which locally optimize a cost function (e.g. vol-
ume, energy, time). At the lowest level is a control scheme
that is robust to errors in sensing the geometry of the terrain
Figure
11 shows the process of coarse to fine planning
for the excavator.
Figure
11. Coarse to fine planning strategy.
Figure
12. Coarse plan for an excavator.
The coarse planner takes as input processed sensor data
which it places in a terrain map (a 2-D grid of height val-
ues). The output is a sequence of dig regions, each of which
is in turn sent to a refined planner. Figure 12 shows a strategy
for removing material that was recommended by an expert
excavator operator. Each box indicates a region, and
the number within the box indicates the order in which the
region is provided to the refined planner. In this strategy,
material is removed from left to right, and from the top of
the face to the bottom. There are several reasons for choosing
this strategy. In most cases, the truck is parked on the
operator's left hand side so that the operator has an unobstructed
view of it. By digging from left to right, the implements
do not need to be raised as high to clear material
when swinging to the truck. In digging from top to bottom,
less force is required from the implements because it is not
necessary then to work against the weight of the material
up above. In addition, clearing material away from the top
minimizes the range shadows cast on the face of the terrain
given a scanning range sensor that is mounted on the cab.
The refined planner operates on an abstract representation
of an atomic action (i.e. a single dig). Rather than
searching for a bucket trajectory, the refined planner
searches through compact task parameters within the
bounds specified by the coarse planner. In order to select
the best digging action, the refined planner evaluates candidates
through the use of a forward model that simulates
the result of choosing an action (in our case the starting location
of the bucket). An evaluation function scores the trajectory
resulting from each action, and the action that meets
all constraints and optimizes the cost function is chosen.
This process is shown in Figure 13.
Figure
13. Operation of refined planner.
4.3. Template Based Dump Planning
The truck must be loaded evenly and completely. Because
of uncertainty in soil settlement, the dumping strategy may
need to be revised for each successive bucket load. The
dump point planner applies a template-based approach to
robustly find the low regions of soil distribution in the truck
bed.
Sensor data are gathered after each bucket of soil has
been dumped in the truck as the excavator is swinging back
to the dig face. Like the dig point planner, the sensor data
are placed in a 2-D terrain map. The dump point planner
also requires information about the location of the truck,
provided by the truck recognizer module, so it can filter out
any irrelevant sensor data that are outside of the truck bed.
The terrain map is then smoothed using a simple Gaussian
filter to eliminate any sensor noise. The current grid cell
resolution of the truck bed terrain map is 15 cm, with a typical
map containing on the order of 500 cells.
Occlusion of the deposited soil by the truck bed walls is
a serious problem. Rather than assuming that nothing is in
the unseen regions of the truck bed, the dump point planner
fills in any unknown grid cells with the average elevation
of the known grid cells. This results in some slight inaccuracies
in the perceived soil distribution at first, but they diminish
as more soil is placed in the truck bed.
Finally, a specific terrain shape template is convolved
Coarse
Planner
Refined
Planner
Closed Loop
Executor
Region
Goal
Candidate
for each dig:
(starting
Constraints
(kinematics,
coarse plan
Digs
bucket angle,
location)
Forward Dig
Model Evaluation
swept volume,
trajectory,
over the entire truck bed terrain map to produce a score for
each grid cell. This small 5x5 or 7x7 grid cell template
looks for a certain profile of the material in the truck bed,
such as a slope or a hole. Simple templates of constant elevations
can be used to find the lowest elevation in the truck
bed terrain map as well. The convolution operator produces
a score which represents how well the template matched
the particular region in the truck bed, and the location of the
cell with the best score is returned as the desired dump location
4.4. Script Based Motion Planning
The motion planning software coordinates the motions of
the excavator's joints for each loading pass, beginning immediately
after digging a bucket of soil and ending when
the bucket has returned to the next dig point. The main objectives
of the motion planner are to plan motions which
place the soil at the desired dump location, avoid all known
obstacles in the workspace such as the truck, and execute
each loading cycle as quickly as possible.
Figure
14. Truck loading script for an excavator.
Because of power constraints and joint coupling effects
of the excavator's hydraulic system, as well as the difficulty
in accurately modeling the dynamics of such a machine,
more traditional optimal trajectory generation schemes do
not work well. Instead, recognizing the fact that the exca-
vator's motions are highly repetitive and very similar from
loading cycle to loading cycle, and that it operates in a relatively
small portion of its total workspace, a script based
approach to motion planning was adopted (Rowe and
Stentz, 1997). A script is a set of rules which define the
general motions of the excavator's joints for a certain task,
in this case loading trucks. These rules contain a number of
variables, known as script parameters, which get instantiated
on every different loading pass.
The rules of script were designed with the input of an expert
human excavator operator and implicitly constrain
what the excavator is and is not allowed to do. For example,
if it was advised that moving two particular joints simultaneously
was a bad idea, then the rules of the script make
that motion impossible. The left hand side of the rules are
functions of the excavator's state, and the right hand side of
the rules are the commands which the planner sends to the
excavator's low level joint controllers. Thus, when the left
hand side of a particular rule evaluates to true, its corresponding
command gets sent to the excavator. The rules get
re-evaluated at a fixed rate, 10 Hz for example, during the
execution of the excavator's motion.
Figure
14 shows the script rules for the truck loading
task. The numbers in boldface are one example set of script
parameters, which will be described in more detail below.
The q's are the excavator's state, in this case the angular
positions of the joints. The commands are desired angular
joint positions. Notice that each joint has its own separate
script. Therefore, only one rule per joint may be active at a
time.
The script parameters are computed before each loading
pass starts using the information about the truck's location
and the desired dig and dump points. There are two types
of script parameters, those which appear in the left hand
side of the script rules and affect which commands are sent
by the planner, and the joint commands themselves which
appear on the right hand side of the rules.
The command script parameters in the right hand side of
the rules are primarily computed by geometric and kinematic
means. For example, consider the command of
from step 1 of the boom joint's script in Figure 14. That is
the boom angle which is required for the excavator's bucket
to safely clear the top of the truck, and is a kinematic
function of the height and location of the truck relative to
the excavator. Similarly, the stick joint commands are computed
using knowledge about the radial distance of the
truck from the excavator, and the swing joint's commands
are found from the desired dig and dump points.
The script parameters in the left hand side of the rules are
found through a combination of simple excavator dynamic
models and heuristics. These simple dynamic models capture
first order effects of the excavator's closed loop behavior
when given desired angular position commands. These
models provide information about the velocities, accelera-
tions, and command latencies for each joint, which are used
to intelligently coordinate the different joint motions, resulting
in faster loading times. As an example, consider the
case when the excavator has finished digging, and the
bucket is raising up out of the ground. The excavator does
not need to wait until the bucket has raised to its full clearance
height before swinging to the truck. Instead, it can begin
swinging at some earlier point as the bucket is still
Joint 1: Swing
digging finishes, wait
swing to truck
swing to dig
Joint 3: Stick
digging finishes, wait
move to spill point
move to dump point
move to dig
Command
Command
Joint 2: Boom
digging finishes, raise
lower to dig
Joint 4: Bucket
digging finishes, curl
move to dig
Command
Command
raising, but it must have the knowledge provided by the dynamic
models about how much time it will take to swing to
the truck and to raise the bucket so it can safely couple the
two motions to avoid a collision.
4.5. Obstacle Detection
A major requirement for automated loading is detecting
and stopping for people and other obstacles which pose a
threat for collision. Obstacle detection software has been
developed which uses sensor data to perceive objects in the
excavator's workspace, and simple dynamic models to predict
where the excavator's linkage will be for a short time
in the future as the excavator swings back and forth between
the dig face and the truck. The predicted excavator
linkage locations are compared to the sensor data, and if
there is an intersection, the excavator is immediately commanded
to stop. It is crucial that the sensors scan far
enough ahead of the excavator's motion, and the prediction
is far enough in the future, for the excavator to have enough
time and space to come to a complete stop and avoid hitting
the obstacle. This look-ahead distance is a function of the
swing joint's maximum velocity and was found through
experimentation to be between 40 to 50 in front of the ex-
cavator's swing joint.
The prediction of the excavator's location is done using
the simplified models of the excavator's closed loop dynamic
behavior. Not only is the obstacle detection algorithm
predicting what the excavator itself will do, it must
also simulate what the motion planner will do using the
predicted excavator state. It performs this prediction at the
same rate of the script rule base update, 10 Hz for instance.
The final result is a list of predicted excavator linkage
states for some amount of time. The look-ahead time was
found empirically to be between 2 - 3 seconds.
Figure
15. Depiction of the points that are calculated on the
underside of the linkage.
For each predicted linkage state, the coordinates of
points on the envelope underneath the linkage are comput-
ed. This is done using the forward kinematics of the excavator
and simple linear models of the shapes of the
linkages. This is shown in Figure 15. Each point on the underside
of the linkage for each predicted linkage state is
then compared to the 2-D elevation map of sensor data. If
any point on the underside of the linkage is lower than the
elevation of the grid cell that coincides with it, then a predicted
collision is reported and the excavator is commanded
to stop.
Figure
16. Typical dig for truck loading.
5. Results
Figure
shows the excavator after digging a bucket of
soil, and Figure 17 shows the truck after it has been loaded
with six buckets of soil. To date, we have autonomously
loaded our truck hundreds of times. The typical loading
times are 15 to 20 seconds per pass, with six passes needed
to load the truck. This rate is very close to the loading times
logged by an expert operator manually loading trucks in the
same configuration using the same excavator.
Figure
17. Truck is loaded after six passes.
6. Conclusion
We have demonstrated an autonomous loading system for
excavators which is capable of loading trucks with soft ma-
underside
linkage
envelope
terial at the speed of expert human operators. The system
uses two scanning laser rangefinders to recognize the truck,
measure the soil on the dig face and in the truck, and to detect
obstacles in the workspace. The system modifies both
its digging and dumping plans based on settlement of soil
as detected by its sensors. Expert operator knowledge is encoded
into templates called scripts which are adjusted using
simple kinematic and dynamics rules to generate very
fast machine motions. We believe ours to be the first fully
autonomous system to load trucks for mass excavation.
Acknowledgments
This paper summarizes the work of the Autonomous
Loading System team. This team consists of Stephannie
Behrens, Scott Boehmke, Howard Cannon, Lonnie Devier,
Frazier, Tim Hegadorn, Herman Herman, Alonzo
Kelly, Murali Krishna, Keith Lay, Chris Leger, Oscar Lu-
engo, Bob McCall, Ryan Miller, Richard Moore, Jorgen
Pedersen, Chris Ravotta, Les Rosenberg, Wenfan Shi, and
Hitesh Soneji in addition to the authors.
--R
Artificial intelligence in the control and operation of construction plant-the autonomous robot excavator
A Laboratory Study of Force-Cognitive Excavation
Force and Geometry Constraints in Robot Excavation.
Remote excavation using the telerobotic small emplacement excava- tor
Research on Control Method of Planning Level for Excavation Robot.
Object Recognition by Computer: The Role of Geometric Constraints
Control Model for Robotic Backhoe Excavation and Obstacle Handling.
Robotics for Challenging Environments.
Intelligent Excavator Control for a Lunar Mining System.
A Strategic Planner for Robot Excavation.
Operation System for Hydraulic Excavator for Deep Trench Works.
Impedance control of a teleoperated mini excavator
of the 8th IEEE International Conference on Advanced Robotics (ICAR)
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Schwartz Electro-optics Inc
Imaging Ladar Camera for Washing Robots.
Synthesis of Tactical Plans for Robotic Excava- tion
State of the Art in Automation of Earthmoving
Autonomous shoveling of rocks by using image vision system on LHD.
New capability for remotely controlled excavation.
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--CTR
Joaqun Gutirrez , Dimitrios Apostolopoulos , Jos Luis Gordillo, Numerical comparison of steering geometries for robotic vehicles by modeling positioning error, Autonomous Robots, v.23 n.2, p.147-159, August 2007 | laser rangefinder;manipulator;dig planning;autonomous excavation;software architecture;robotic excavator;integrated robotic system |
591811 | Grounded Symbolic Communication between Heterogeneous Cooperating Robots. | In this paper, we describe the implementation of a heterogeneous cooperative multi-robot system that was designed with a goal of engineering a grounded symbolic representation in a bottom-up fashion. The system comprises two autonomous mobile robots that perform cooperative cleaning. Experiments demonstrate successful purposive navigation, map building and the symbolic communication of locations in a behavior-based system. We also examine the perceived shortcomings of the system in detail and attempt to understand them in terms of contemporary knowledge of human representation and symbolic communication. From this understanding, we propose the Adaptive Symbol Grounding Hypothesis as a conception for how symbolic systems can be envisioned. | Introduction
The Behavior-based approach to robotics has proven that
it is possible to build systems that can achieve tasks
robustly, react in real-time and operate reliably. The
sophistication of applications implemented ranges from
simple reactivity to tasks involving topological map
building and navigation. Conversely, the classical AI
approach to robotics has attempted to construct symbolic
representational systems based on token manipulation.
There has been some success in this endeavor also.
While more powerful, these systems are generally slow,
brittle, unreliable and do not scale well - as their
'symbols' are ungrounded.
In this paper, we present an approach for
engineering grounded symbolic communication between
heterogeneous cooperating robots. It involves designing
behavior that develops shared groundings between them.
We demonstrate a situated, embodied, behavior-based
multi-robot system that implements a cooperative
cleaning task using two autonomous mobile robots.
They develop shared groundings that allow them to
ground a symbolic relationship between positions
consistently. We show that this enables symbolic
communication of locations between them.
The subsequent part of the paper critically examines
the system and its limitations. The new understanding of
the system we come to shows that our approach will not
scale to complex symbolic systems. We argue that it is
impossible for complex symbolic representational
systems to be responsible for appropriate behavior in
situated agents. We propose the Adaptive Symbol
Grounding Hypothesis as a conception of how systems
that communication symbolically can be envisioned.
Before presenting the system we have developed,
the first section briefly discusses cooperation and
communication generally and looks at some instances of
biological cooperation in particular. From this, we
determine the necessary attributes of symbolic systems.
2. Cooperation and Communication
Cooperation and communication are closely tied.
Communication is an inherent part of the agent
interactions underlying cooperative behavior, whether
implicit or explicit. If we are to implement a concrete
cooperative task that requires symbolic level
communication, we must first identify the relationship
between communication and cooperative behavior. This
section introduces a framework for classifying
communication and uses it to examine some examples of
cooperative behavior in biological systems. From this
examination, we draw conclusions about the mechanisms
necessary to support symbolic communication. These
mechanisms are utilized in our implementation of the
cooperative cleaning system, as described in the
subsequent section.
It is important to realize that 'cooperation' is a word
- a label for a human concept. In this case, the concept
refers to a category of human and possibly animal
behavior. It does not follow that this behavior is
necessarily beneficial to the agents involved. Since
evolution selects behavioral traits that promote the genes
that encourage them, it will be beneficial to the genes but
not necessarily the organism or species. Human
cooperative behaviour, for example, is a conglomerate of
various behavioural tendencies selected for different
reasons (and influenced cultural knowledge). Because
the design of cooperative robot systems is in a different
context altogether, we need to understand which aspects
are peculiar to biological systems.
2.1 Some Characteristics of Communication
Many authors have proposed classifications for the types
of communication found in biological and artificial
systems (e.g. see Arkin and Hobbs, 1992b; Balch and
Arkin, 1994; Cao et al., 1995; Dudek et al., 1993; Kube
and Zhang,# 1997a). Like any
classification, these divide the continuous space of
communication characteristics into discrete classes in a
specific way - and hence are only useful within the
context for which they are created. We find it necessary
to introduce another classification here.
A communicative act is an interaction whereby a
signal is generated by an emitter and 'interpreted' by a
receiver. We view communication in terms of the
following four characteristics.
. Interaction distance - This is the distance
between the agents during the communicative
interaction. It can range from direct physical
contact, to visual range, hearing range, or long
range.
. Interaction simultaneity - The period between
the signal emission and reception. It can be
immediate in the case of direct contact, or
possibly a long time in the case of scent markers,
for example.
. Signaling explicitness - This is an indication of
the explicitness of the emitter's signaling
behavior. The signaling may be a side effect of
an existing behavior (implicit), or an existing
behavior may have been modified slightly to
enhance the signal through evolution or
learning. The signaling may also be the result
of sophisticated behavior that was specifically
evolved, learnt, or in the case of a robot,
designed, for it.
. Sophistication of interpretation - This can be
applied to either the emitter or the receiver. It is
an indication of the complexity of the
interpretation process that gives meaning to the
signal. For example, a chemical signal may
invoke a relatively simple chain of chemical
events in a receiving bacterium. It is possible
that a signal has a very different meaning to the
emitter and receiver - the signal may have no
meaning at all to its emitter. Conversely, the
process of interpretation of human language is
the most complex example known.
2.2 Representation
It is not possible to measure the sophistication of the
interpretive process by observing the signal alone.
Access to the mechanics of the process within an agent
would be necessary. Unfortunately, our current
understanding of the brain mechanisms underlying
communication in most animals is poor, at best.
Therefore, our only approach is to examine the structure
of the communicated signals. Luckily, there appears to
be some correlation between the structural complexity of
communicated signals and the sophistication of their
interpretive processes. Insect mating calls are simple in
structure and we posit a simple interpretive process. At
the other end of the spectrum, human language has a
complex structure and we consider its interpretation
amongst the most sophisticated processes known. Bird
song and human music are possible exceptions, as they
are often complex in structure, yet have a relatively
simple interpretation. This is due to other evolutionary
selection pressures, since song also provides fitness
information about its emitter to prospective mates and, in
the case of birds, serves to distinguish between members
of different species.
Science, through the discipline of linguistics, has
learned much about the structure of the signals generated
by humans that we call language (see Robins, 1997). We
utilize a small part of that here by describing a
conception of the observed structure of language. Using
Deacon's terms we define three types of reference, or
levels of representation: - iconic, indexical and symbolic
(Deacon, 1997).
Iconic representation is by physical similarity to
what it represents. The medium may be physically
external to the agent - for example, as an orange disc
painted on a cave wall may represent the sun.
Alternatively, it may be part of the agent, such as some
repeatable configuration of sensory neurons, or "internal
analog transforms of the projections of distal objects on
our sensory surfaces" (Shepard and Cooper, 1982).
Indexical reference represents a correlation or
association between icons. All animals are capable of
iconic and indexical representation to varying degrees.
For example, an animal may learn to correlate the icon
for smoke with that for fire. Hence, smoke will come to
be an index for fire. Even insects probably have limited
indexical capabilities. Empirical demonstrations are a
mechanism for creating indexical references in others.
Pointing, for example, creates an association between the
icon for the physical item being indicated and the object
of a sentence. The second part of this paper describes
how we have used empirical demonstration to enable the
communication of locations between robots.
Indexical
Iconic
Symbolic
Figure
of representation
The third level of representation is symbolic. A
symbol is a relationship between icons, indices and other
symbols. It is the representation of a higher-level pattern
underlying sets of relationships. It is hypothesized that
language can be represented as a symbolic
hierarchy (Newell and Simon, 1972). We will use the
term sub-symbolic to refer to representations that need
only iconic and indexical references for their
interpretation. If the interpretation of a symbol requires
following references that all eventually lead to icons, the
symbol is said to be grounded. That is, the symbols at
the top of Figure 1 ultimately refer to relationships
between the icons at the bottom. A symbol's grounding
is the set of icons, indices and other symbols necessary to
it.
The problems associated with trying to synthesize
intelligence from ungrounded symbol systems - the
classical AI approach - have been well documented in
the literature. One such problem is termed the frame
problem (see Ford and Hayes, 1991; Pylyshym, 1987).
The importance of situated and embodied agents has
been actively espoused by members of the behavior-based
robotics community, in recognition of these problems, for
many years (see Brooks,# 1992a;
Pfeifer, 1995; Steels, 1996 for a selection). This
hypothesis is called the physical grounding hypothesis
(Brooks, 1990). Consequently, we adopted the behavior-based
approach for our implementation.
2.3 Cooperative biological systems
In this subsection we describe five selected biological
cooperative systems and classify the communication each
employs using the scheme introduced above. From
these, in the following subsection, we identify the
necessary mechanisms for symbolic communication,
which were transferred to the implementation of the
cooperative multi-robot cleaning system.
2.3.1 Bacteria
Cooperation between simple organisms on earth is
almost as old as life on earth itself. Over a billion years
ago bacteria existed similar to contemporary bacteria
recently observed to exhibit primitive cooperation.
Biologists have long understood that bacteria live in
colonies. Only recently has it become evident that most
bacteria communicate using a number of sophisticated
chemical signals and engage in altruistic behavior
(Kaiser and Losick, 1993). For example, Mycobacteria
assemble into multi-cellular structures known as fruiting
bodies. These structures are assembled in a number of
stages each mediated by different chemical signal
systems. In these cases the bacteria emit and react to
chemicals in a genetically determined way that evolved
explicitly for cooperation. Hence, we classify the
signaling as explicit. The interaction distance is
moderate compared to the size of a bacterium, and the
simultaneity is determined by the speed of chemical
propagation. The mechanism for interpretation is
necessarily simple in bacteria.
We can consider this communication without
meaning preservation - the meaning of the signal is
different for emitter and receiver. The emitter generates
a signal without interpreting it at all (hence, it has no
meaning to the emitter). The receiver interprets it
iconically 1 . This type of communication has been
implemented and studied in multi-robot systems. For
example, Balch and Arkin have implemented a collective
multi-robot system, both in simulation and with real
robots, to investigate to what extent communication can
1 The stereotypical way a particular chemical receptor on the bacteria's
surface triggers a chain of chemical events within, is an icon for the
presence of the external chemical signal.
increase their capabilities (Balch and Arkin, 1994). The
tasks they implemented were based on eusocial insect
tasks, such as forage, consume, and graze. One scheme
employed was the explicit signaling of the emitter's state
to the receiver. They showed that this improves
performance, as we might expect. Specifically it
provides the greatest benefit when the receiver cannot
easily sense the emitter's state implicitly. This finding
was also observed by Parker in the implementation of a
puck moving task where each robot broadcast its state
periodically (Parker, 1995); and by Kube and Zhang with
their collective box pushing system (Kube and
Zhang, 1994). The second part of this paper will
demonstrate that the result also holds for our system.
2.3.2 Ants
Of the social insect societies, the most thoroughly studied
are those of ants, termites, bees and wasps
(Wilson, 1971; Wilson, 1975; Crespi and Choe, 1997).
Ants display a large array of cooperative behaviors. For
example, as described in detail by Pasteels et al. (Pasteels
et al., 1987), upon discovering a new food source, a
worker ant leaves a pheromone trail during its return to
the nest. Recruited ants will follow this trail to the food
source with some variation while laying their own
pheromones down. Any chance variations that result in
a shorter trail to the food will be reinforced at a slightly
faster rate, as the traversal time back and forth is less.
Hence, it has been shown that a near optimal shortest
path is quickly established as an emergent consequence
of simple trail following with random variation.
In this case, the interaction distance is local - the
receiver senses the pheromone at the location it was
emitted. As the signal persists in the environment for
long periods, there may be significant delay between
emission and reception. The signaling mechanism is
likely to be explicit and the interpretation, while more
complex than for bacteria, is still relatively simple. The
ants also communicate by signaling directly from
antennae to antennae.
Since both emitter and receiver can interpret the
signal in the same way, we consider it communication
with meaning preservation. The crucial element being
that both agents share the same grounding for the signal.
In this case, the grounding is probably genetically
determined - through identical sensors and neural
processes. This mechanism can also be applied to multi-robot
systems. For example, if two robots shared
identical sensors, they could simply signal their sensor
values. This constitutes an iconic representation, and it
is grounded directly in the environment for both robots
identically. Nothing special needs to be done to ensure a
shared grounding for the signal.
2.3.3 Wolves
A social mammal of the Canine family, wolves are
carnivores that form packs with strict social hierarchies
and mating systems (Stains, 1984). Wolves are
territorial. Territory marking occurs through repeated
urination on objects on the periphery of and within the
territories. This is a communication scheme reminiscent
of our ants and their chemical trails. Wolves also
communicate with pheromones excreted via glands near
the dorsal surface of the tail.
Wolves hunt in packs. During a pack hunt,
individuals cooperate by closely observing the actions of
each other and, in particular, the dominant male who
directs the hunt to some extent. Each wolf knows all the
pack members and can identify them individually, both
visually and by smell. Communication can be directed
at particular individuals and consists of a combination of
specific postures and vocalizations. The interaction
distance in this case is the visual or auditory range
respectively, and the emission and reception is effectively
simultaneous. The signals may be implicit, in the case of
observing locomotory behavior, for example; or more
explicit in the case of posturing, vocalizing and scent
marking. It seems likely that the signals in each of these
cases are interpreted similarly by the emitter and
receiver.
Again, this is an instance of communication with
meaning preservation. A significant difference is that
the shared grounding enabling the uniform interpretation
of some signals (e.g. vocalizations and postures) is not
wholly genetically determined. Instead, a specific
mechanism exists such that the grounding is partially
learnt during development - in a social environment
sufficiently similar to both that a shared meaning is
ensured.
2.3.4 Non-human primates
Primates display sophisticated cooperative behavior. The
majority of interactions involve passive observation of
collaborators via visual and auditory cues, which are
interpreted as actions and intentions. As Bond writes in
reference to Vervet monkeys, "They are acutely and
sensitively aware of the status and identity of other
monkeys, as well as their temperaments and current
dispositional states" (Bond, 1996). Higher primates are
able to represent the internal goals, plans, dispositions
and intentions of others and to construct collaborative
plans jointly through acting socially (Cheney and
Seyfarth, 1990). In this case, the interaction is
simultaneous and occurs within visual or auditory range.
The signaling is implicit but the sophistication of
interpretation for the receiver is considerable. Some
explicit posing and gesturing is also utilized, which is
used to establish and control ongoing cooperative
interactions.
As with the Wolves, we observe communication
with meaning preservation through a shared grounding
that is developed through a developmental process. In
this case, the groundings are more sophisticated, as is the
developmental process required to attain them.
2.3.5 Humans
In addition to the heritage of our primate ancestors,
humans make extensive use of communication, both
written and spoken, that is explicitly evolved or learnt.
There is almost certainly some a priori physiological
support for language learning in the developing human
brain (Bruner, 1982). Humans cooperate in many and
varied ways. We display a basic level of altruism toward
all humans and sometimes animals. We enter into
cooperative relationships - symbolic contracts - with
mates, kin, friends, organizations, and societies whereby
we exchange resources for mutual benefit. In many
cases, we provide resources with no reward except the
promise that the other party, by honoring the contract,
will provide resources when we need them, if possible.
We are able to keep track of all the transactions and the
reliability with which others honor contracts (see
Deacon, 1997 for a discussion).
Humans also use many types of signaling for
communication. Like our primate cousins, we make
extensive use of implicit communication, such as
posturing (body language). We also use explicit
gesturing - pointing, for example. Facial expressions
are a form of explicit signaling that has evolved from
existing expressions to enhance the signaling reliability
and repertoire. Posturing, gesturing and speaking all
involve simultaneous interaction. However, with the
advent of symbolic communication we learned to utilize
longer-term interactions. A physically realized icon,
such as a picture, a ring or body decoration, is more
permanent. The ultimate extension of this is written
language. The coming of telephones, radios and the
Internet have obviously extended the interaction
distances considerably.
While symbolic communication requires
considerable sophistication of interpretation, humans
also use signals that can be interpreted more simply. For
example, laughter has the same meaning to all humans,
but not to other animals. We can make the necessary
connection with the emotional state since we can hear
and observe others and ourselves laughing - we share the
same innate involuntary laugh behavior.
The developmental process that provides the shared
groundings for human symbolic communication -
cultural language learning - can be seen as an extension
of the processes present in our non-human primate
ancestors (Hendriks-Jansen, 1996). The major
differences being in the complexity due to the sheer
number of groundings we need to learn and the intrinsic
power of symbolic representation over exclusively
indexical and iconic representation. Symbolic
representations derive their power because they provide a
degree of independence from the symbolic, indexical and
iconic references that generated the relationship
represented. New symbols can be learnt using language
metaphor (Lakoff and Johnson, 1980; Johnson, 1991).
2.4 Symbolic communication and its prerequisites
Evolution does not have the luxury of being able to make
simultaneous independent changes to the design of an
organism and also ensure their mutual consistency (in
terms of the viability of the organism). For this reason,
once a particular mechanism has been evolved, it is built
upon rather than significantly re-designed to effect a new
mechanism. It is only when selection pressures change
enough to render things a liability that they may be
discarded. This is why layering is observed in natural
systems (e.g. Mallot, 1995).
In the examples above, we can perceive a layering
of communication mechanisms that are built up as we
look at each in turn - from bacteria to humans. Each
leveraging the mechanism developed in the previous
layer. The ant's use of chemical pheromone trails to
implement longer duration interactions is supported by
direct-contact chemical communication, pioneered by
their distant bacterial ancestors. Wolves also employ this
type of communication, which provides an environment
that supports the developmental process for learning
other shared groundings. The sophistication of such
developmental processes is greater in non-humans
primates and significantly so in humans. However, even
for humans, these processes still leverage the simpler
processes that provide the scaffolding of shared iconic
and indexical groundings (see Thelen and Smith, 1994;
Hendriks-Jansen, 1996).
We believe such layering is integral to the general
robustness of biological systems. If a more sophisticated
mechanism fails to perform, the lesser ones will still
operate. We emulate the layering in the implementation
of our system for this reason.
From our examination, the following seem to be
necessary for symbolic communication between two
agents.
. Some iconic representations in common (e.g. by
possessing some physically identical sensory-motor
apparatus).
. Either a shared grounding for some indexical
representations, a common process that develops
shared indexical groundings, or a combination
of both (e.g. a mechanism for learning the
correlation between icons - such as correlating
'smoke' with `fire').
. A common process that develops shared
groundings (e.g. mother and infant
'innate' behavior that scaffolds language
development - turn-taking, intentional
interpretation, mimicking etc.)
Additionally, unless the symbol repertoire is to be fixed
with specific processes for acquiring each symbol, it
seems necessary to have:
. A mechanism for learning new symbols by
communicating known ones (e.g. interpretation
and learning through metaphor).
The implementation of this last necessity in a robot
system is currently beyond the state-of-the-art. However,
the first three are implemented in the cooperative
cleaning system, as described in the following section.
3. The System
Our research involved the development of an architecture
for behavior-based agents that supports cooperation
(Jung, 1998; Jung and Zelinsky, 1999) 2 . To validate the
architecture we implemented a cooperative cleaning task
using the two Yamabico mobile robots pictured in Figure
(Yuta et al., 1991). The task is to clean our laboratory
floor space. Our laboratory is a cluttered environment,
so the system must be capable of dealing with movable
obstacles, people and other hazards.
3.1 The Robots
As we are interested in heterogeneous cooperation, we
built each robot with a different set of sensors and
actuators, and devised the cleaning task such that it
cannot be accomplished by either robot alone. One of
the robots, 'Joh', has a vacuum cleaner that can be
turned on and off via software. Joh's task is to vacuum
piles of 'litter' from the laboratory floor. As our aim was
not to design a high performance cleaning system per se,
chopped Styrofoam serves as 'litter'. Joh cannot vacuum
close to walls or furniture, as the vacuum is mounted
between the drive wheels. It has the capability to 'see'
piles of litter using a CCD camera and a video
transmitter that sends video to a Fujitsu MEP tracking
vision system. The vision system uses template
correlation, and can match about 100 templates at frame
rate. The vision system can communicate with the robot,
via a UNIX host, over a radio modem. Visual obstacle-avoidance
behavior has been demonstrated at speeds of
up to 600mm/sec (Cheng and Zelinsky, 1996).
Figure
- The two Yamabicos 'Flo' and `Joh'
The other robot, 'Flo', has a brush tool that is
dragged over the floor to sweep distributed litter into
larger piles for Joh to pick-up. It navigates around the
perimeter of the laboratory where Joh cannot vacuum
and deposits the litter in open floor space. Sensing is
primarily using four specifically developed passive tactile
'whiskers' (Jung and Zelinsky, 1996a). The whiskers
provide values proportional to their angle of deflection.
Both robots are also fitted with ultrasonic range sensors
and wheel encoders.
3.2 A layered solution
We implemented the cleaning task by layering solutions
involving more complex behavior over simpler solutions.
This provides a robust final solution, reduces the
complexity of implementation and allows us to compare
the system performance at intermediate stages of
development.
The first layer involves all the basic behavior
required to clean the floor, but does not include any
capacity to purposefully navigate, explicitly
communicate or cooperate. Flo sweeps up litter and
periodically deposits it into piles where it is accessible by
Joh. Joh uses the vision to detect the piles and vacuum
them up. Therefore, the signaling - depositing litter
piles - is implicit in this case, as it is normal cleaning
behavior. The interaction is not simultaneous, as Joh
doesn't necessarily see the piles as soon as they are
deposited. The interaction distance ranges over the size
of the laboratory. Flo doesn't interpret the piles of litter
as a signal at all - and in fact has no way of sensing
them. Joh has a simple interpretation - the visual iconic
representation of the pile acts as a releaser to vacuum
over it.
no awarness of each other
implicit visual communication
of likely litter position
Layer 3
explicit communication
of litter relative positions
Layer 4
communication
of litter locations
Figure
solution
Figure
visually tracking Flo (no vacuum attached)
The second layer gives Joh an awareness of Flo.
We added the capability for Joh to visually detect and
track the motion of Flo. This is another communication
mechanism that provides state information about Flo to
Joh. In this case, the signaling is again implicit, the
interaction distance is visual range and the interaction is
simultaneous. Joh uses the visual iconic representation
of Flo to ground an indexical reference for the likely
location of the pile of litter deposited. Figure 4 shows
Joh visually observing Flo via a distinctive pattern.
Details of the implementation the visual behavior we
employed can be found in (Jung et al., 1998a). A top
view of typical trajectories of the robot is shown in
Figure
5.
The third layer introduces explicit communication.
Specifically, upon depositing a pile of litter, Flo signals
via radio the position (distance and orientation) of the
pile relative to its body and the relative positions of the
last few piles deposited. Flo and Joh both have identical
wheel encoders, so we are ensured of a shared grounding
for the interpretation of the communicated relative
distance and orientation to piles. Although odometry has
a cumulative error, this can be ignored over such short
distances. The catch is that the positions are relative to
Flo. Hence, Joh must transform them to egocentric
positions based on the observed location of Flo. If Flo is
not currently in view, the information is ignored. A
typical set of trajectories is shown in Figure 6.
Joh
Flo
Figure
trajectories when Joh can observe Flo
depositing litter (Layer 2)
Joh
Flo
Litter
Figure
trajectories when explicit communication is
utilized (Layer
The fourth and final layer involves communication
of litter locations by Flo to Joh even when Flo cannot be
seen. This is accomplished by using a symbolic
interpretation for a specific geometric relationship of
positions to each other. What is communicated to
convey a location is analogous to 'litter position is
<specific-geometric-relation-between> <position-A>
<position-B> '. The positions
are indexical references that are themselves grounded
through a shared process, a location-labeling behavior,
described below. The distance and direction are in fact
raw encoder data, hence an iconic reference, relying on
the shared wheel encoders. There is no signal
communicated for the symbolic relation itself (like a
word), since there is only one symbol in the system, it is
unambiguous. Obviously, if more symbols were known,
or a mechanism for leaning new symbols available,
labels for the symbols would need to be generated and
signaled (and perhaps syntax established).
First, we describe the action selection scheme
employed, as it is the basis for the navigation and map
building mechanism, which in turn is the basis for the
location-labeling behavior.
3.3 Action Selection
We needed to design an action selection mechanism that
is distributed, grounded in the environment, and employs
a uniform action selection mechanism over all behavior
components. Because the design was undertaken in the
context of cooperative cleaning, we also required the
mechanism to be capable of cooperative behavior and
communication, in addition to navigation. Each of these
requires some ability to plan. This implies that the
selection of which action to perform next must be made
in the context of which actions may follow - that is,
within the context of an ongoing plan. In order to be
reactive, flexible and opportunistic, however, a plan
cannot be a rigid sequence of pre-defined actions to be
carried out. Instead, a plan must include alternatives,
have flexible sub-plans and each action must be
contingent on a number of factors. Each action in a
planned sequence must be contingent on internal and
external circumstances including the anticipated effects
of the successful completion of previous actions. Other
important properties are that the agent should not stop
behaving while planning occurs and should learn from
experience.
There were no action selection mechanisms in the
literature capable of fulfilling all our requirements. As
our research is more concerned with cooperation than
action selection per se, we adopted Maes' spreading
activation algorithm and modified it to suit our needs.
Her theory "models action selection as an emergent
property of an activation/inhibition dynamics among the
actions the agent can select and between the actions and
the environment" (Maes, 1990a).
3.3.1 Components and Interconnections
The behavior of a system is expressed as a network that
consists of two types of nodes - Competence Modules
and Feature Detectors. Competence modules (CMs) are
the smallest units of behavior selectable, and feature
detectors information about the external or
internal environment. A CM implements a component
behavior that links sensors with actuators in some
arbitrarily complex way. Only one CM can be executing
at any given time - a winner-take-all scheme. A CM is
not limited to information supplied by FDs - the FDs are
only separate entities in the architecture to make explicit
the information involved in the action selection
calculation.
FD
FD
CM
CM
CM
FD
Key:
(sucessor, predecessor or conflictor)
+ve Correlation
-ve Correlation
Activation Link
Precondition
Figure
Network components and interconnections
The graphical notation is shown above where
rectangles represent CMs and rounded rectangles
represent FDs. Although there can be much exchange of
information between CMs and FDs the interconnections
shown in this notation only represent the logical
organization of the network for the purpose of action
selection.
Each FD provides a single Condition with a
confidence [0.1] that is continuously updated from the
environment (sensors or internal states). Each CM has
an associated Activation and the CM selected for
execution has the highest activation from all Ready CMs
whose activations are over the current global threshold.
A CM is Ready if all of its preconditions are satisfied.
The activations are continuously updated by a spreading
activation algorithm.
The system behavior is designed by creating CMs
and FDs and connecting them with precondition links.
These are shown in the diagram above as solid lines
from a FD to a CM ending with a white square. It is
possible to have negative preconditions, which must be
false before the CM can be Ready. There also exist
correlation links, dotted lines in the figure, from a CM to
a FD. The correlations can take the values [-1.1] and
are updated at run-time according to a learning
algorithm. A positive correlation implies the execution
of the CM causes, somehow, a change in the
environment that makes the FD condition true. A
negative correlation implies the condition becomes false.
The designer usually initializes some correlation links to
bootstrap learning.
Together these two types of links, the precondition
links and the correlation links, completely determine
how activation spreads thought the network. The other
activation links that are shown in Figure 7 are
determined by these two and exist to better describe and
understand the network and the activation spreading
patterns. The activation links dictate how activation
spreads and are determined as follows.
. There exists a successor link from CM p to CM s
for every FD condition in s's preconditions list
that is positively correlated with the activity of p.
. There exists a predecessor link in the opposite
direction of every successor link.
. There exists a conflictor link from CM x to CM y
for every FD condition in y's preconditions list
that is negatively correlated with the activity of x.
The successor, predecessor and conflictor links resulting
from the preconditions and correlations are shown in
Figure
7.
In summary, a CM s has a predecessor CM p, if p's
execution is likely to make one of s's preconditions true.
A CM x has a conflictor CM y, if y's execution is likely
to make one of x's preconditions false.
3.3.2 The Spreading of Activation
A rigorous description of the spreading activation
algorithm is beyond the scope of this paper. The
algorithm has been detailed in previous publications
(Jung, 1998; Jung and Zelinsky, 1999). The activation
rules can be more concisely described in terms of the
activation links. The main spreading activation rules
can be simply stated:
. Unready CMs increase the activation of
predecessors and decrease the activation of
conflictors, and
. Ready CMs increase the activation of successors.
In addition, these special rules change the activation of
the network from outside in response to goals and the
current situation:
. Goals increase the activation of CMs that can
satisfy them and decrease the activation of those
that conflict with them, and
. FDs increase the activation of CMs for which they
satisfy a precondition.
To get a feel for how it works, we describe part of a
network that implements the cleaning task for Flo, as
shown in Figure 8. With some of the components
shown, a crude perimeter-following behavior is possible.
The rectangles are basic behaviors (CMs), the ovals
feature detectors (FDs), and only the correlation and
precondition links are shown (the small circles indicate
negation of a precondition). The goal is Cleaning.
This occurs when Flo roughly follows the perimeter of
the room by using Follow to follow walls and
ReverseTurn to reverse and turn away from the
perimeter when an obstacle obstructs the path.
Periodically the litter that has accumulated in the
sweeper is deposited away from the perimeter by
DumpLitter.
The spreading activation algorithm 'injects'
activation into the network CMs via goals and via FDs
that meet a precondition. Therefore, the Cleaning goal
causes an increase in the activation of Follow,
DumpLitter and ReverseTurn. Suppose Flo is in a
situation where its left whiskers are against a wall
(ObstacleOnLeft is true) and there are no obstacles in
front (ObstacleAhead and FrontHit both false). In
this case, the activation of Follow will be increased by
all the FDs in its precondition set (including Timer
which is false before being triggered). Being the only
CM ready, it is scheduled for execution until the
situation changes. Once the Timer FD becomes true,
Follow is no longer ready, but DumpLitter becomes
ready and is executed. Follow and DumpLitter also
decrease each other's activation as they conflict - each is
correlated with the opposite state of Timer.
Although, the selection of CMs in this example
depends mainly on the FD states, when the selection of
CMs depends more on the activation spread from other
CMs, the networks can exhibit 'planning' - as Maes has
shown. This is the basis for action planning in our
networks, and gives rise to path planning as will be
described below.
Figure
Partial network for Flo (produced by our GUI).
From the rules we can imagine activation spreading
backward through a network, from the goals, through
CMs with unsatisfied preconditions via the precondition
links until a ready CM is encountered. Activation will
tend to accumulate at the ready CM, as it is feeding
activation forward while its successor is feeding it
backward. Eventually it may be selected for execution,
after which its activation is reset to zero. If its execution
was successful, the precondition of its successor will
have been satisfied and the successor may be executed (if
it has no further unsatisfied preconditions). We can
imagine multiple routes through the network, activation
building up faster via shorter paths. These paths of
higher activation represent 'plans' within the network.
The goals act like a 'homing signal' filtering out through
the network and arriving at the current 'situation'.
One important difference between our and Maes'
networks is that in ours the flow of activation is weighted
according to the correlations - which are updated
continuously at run-time according to previous
experience. The mechanism for adjusting the correlation
between a given CM-FD pair is simple. Each time the
CM becomes active, the value of the FD's condition is
recorded. When the CM is subsequently deactivated, the
current value of the condition is compared with the
recorded value. It is classified as one of: Became True,
Became False, Remained True or Remained False. A
count of these cases is maintained (B t , B f , R t , R f ). The
correlation is then:
corr
Where the total samples N B B R R
To keep the network plastic, the counts are decayed so
recent samples have a greater effect than historic ones.
3.4 Navigation and map building
3.4.1 Spatial and Topological path planning
There are two main approaches to navigational path
planning. One method utilizes a geometric
representation of the robot environment, perhaps
implemented using a tree structure. Usually a classical
path planner is used to find shortest routes through the
environment. The distance transform method falls into
this category (Zelinsky et al., 1993). These geometric
modeling approaches do not fit with the behavior-based
philosophy of only using categorizations of the robot-
environment system that are natural for its description,
rather than anthropocentric ones. Hence, numerous
behavior-based systems use a topological representation
of the environment in terms only of the robot's behavior
and sensing (e.g. see # 1992). While these
approaches are more robust than the geometric modeling
approach, they suffer from non-optimal performance for
shortest path planning. This is because the robot has no
concept of space directly, and often has to discover the
adjacency of locations.
Consider the example below, where the robot in (a)
has a geometric map and its planner can directly
calculate the path of least Cartesian distance, directly
from A to D. However, the robot in (b) has a topological
map with nodes representing the points A, B, C and D,
connected by a follow-wall behavior. Since it has never
previously traversed directly from A to D, the least path
through its map is A-B-C-D.
Figure
9 - (a) Geometric vs (b) Topological Path Planning
Consequently, our aim was to combine the benefits
of geometric and topological map representations in a
behavior-based system using our architecture.
3.4.2 A self-organizing map
In keeping with the behavior-based philosophy, we found
no need to explicitly specify a representation for a map
or a specific mechanism for path planning. Instead, by
introducing the key notion of location feature detectors
(location FDs), the correlation learning and action
selection naturally gave rise to map building and path
planning - for 'free'.
A location feature detector is a component of our
architecture specialized to respond when the robot is in a
particular location (the detector's characteristic location).
We employ many detectors and the locations to which
they respond are non-uniformly distributed over the
laboratory floor space. Each location FD contains a
vector v, whose components are elements of the robot
state vector:
non-location FD values
The variable g contains global Cartesian coordinates and
orientation estimated from wheel encoders and a model
of the locomotion controller. The sensors include
ultrasonic range readings and in Flo's case, tactile
whisker values. The fds component contains the
condition values of all FDs in the system, except for the
location FDs themselves. For example, in Joh's case this
includes visual landmark FDs.
The condition confidence value of each location FD
is updated by comparing it to the current state of the
robot's sensors and other non-location FDs. A weighted
Euclidean norm N w is used - with the (x,y) coordinate
weights dominating.
Hence, the vector of the location FD whose condition is
true with highest confidence is considered to represent
the 'current location' of the robot. The detectors are
iconic representations of locations (see Figure 10).
The location FD vectors v are initialized such that
the (x,y) components are distributed as a regular grid
over the laboratory floor space, and the other components
are randomly distributed over the vector space. During
operation of the system, the location FD vectors are
updated using Kohonen's self-organizing map (SOM)
algorithm (Kohonen, 1990). This causes the spatial
distribution of the location FD vectors to approximate
the frequency distribution of the robot's state vector over
time.
Figure
shows how the detectors have organized
themselves to represent one of our laboratories. One
useful property of a SOM is that it preserves topology -
nodes that are adjacent in the representation are
neighboring locations in the vector space.
Since the location FD vectors v are continuously
matched with the robot state vector x, in which the (x,y)
coordinates are estimated via odometry, there is a major
drawback. The odometry error in (x,y) is cumulative.
We remedy this by updating the robot state vector
coordinates. Specifically, the system has feature
detectors for various landmark types that are
automatically correlated with the location FDs by the
correlation learning described above. If it should happen
that a landmark FD becomes true with high confidence
that is strongly correlated with a location FD
neighboring the location FD for the 'current location',
then the state vector (x,y) component is updated. The
coordinates are simply moved closer to the coordinates of
the location FD to which the landmark is correlated.
Assuming the landmarks don't move over moderate
periods, this serves to keep the location FD (x,y)
components registered with the physical floor space.
Now
Flo
Location feature detectors
(iconic refererences to position)
Indexical reference to
current location
Figure
location detector SOM and current
location index
The system also maintains an indexical reference
that represents the robot's current location. Recall that
an indexical reference is a correlation between icons.
The robots each have a sense of time - in terms of the
ordering relation between sensed events (which is shared
to the extent that the ordering of external events is
perceived to be the same by both robots). Hence, the
current location index is an association between the most
active location detector and the current time.
It is clear this mechanism fulfills our requirement
for spatial mapping. The topological mapping derives
again from the correlation learning in the architecture.
Specifically, the system learns by experience that a
particular behavior can take the robot from one state to
another - for example by changing the current location
index in a consistent way. Over time, behavior such as
becomes correlated with the start and end
locations of a wall segment. The spreading activation
will cause the behavior to be activated when the system
needs to 'plan' a sub-path from the start to the end.
Similarly, simple motion behavior becomes correlated
with moving the robot from one location to one of its
neighbors.
3.4.3 Navigation
Once we have feature detectors that respond to specific
locations, it is straightforward to add spatial and
topological navigation. Each time a behavior (a CM) is
activated, the identity of the current location FD before
and after its execution is recorded. A new instance of the
CM is created, and initialized with the 'source' location
FD as a precondition and the 'destination' as a positive
correlate. Hence, the system remembers which behavior
can take it from one specific location to another. If the
CM does not consistently do this, its correlation with the
destination location FD will soon fall. If it falls to zero,
the CM is removed from the network. Changes in the
environment also cause correlations to change, thus
allowing the system to adapt.
With this mechanism, the system learns topological
adjacency of locations in terms of behavior. For
example, if the activation of the Follow CM
consistently takes the robot from the location FD
corresponding to the start of a wall, to the end of the
wall, then the links shown below will be created.
FL
FL
Figure
Behavioral adjacency of locations via Follow
The spreading activation algorithm for action selection is
able to plan a sequence of CM activations to achieve
navigation between any arbitrary locations.
Spatial navigation is achieved by initializing the
network so that a simple Forward behavior links each
location FD with its eight neighbors in both directions.
Hence, initially the system 'thinks' it can move in a
straight line between any locations that are neighbors in
the SOM. If presence of an obstacle blocks the straight-line
path from one location to its neighbor, then this will
be learnt through a loss of correlation between the
corresponding Forward CM and the 'destination' FD.
The mechanisms described here for map building and
navigation are presented in detail in (Jung, 1998; Jung
and Zelinsky, 1999a).
3.5 A shared grounding for locations
For layer 4 of the implementation, we wanted to add the
capability for Flo to communicate the locations of litter
piles in a more general way. In such a way that it would
be useful to Joh if Flo were not in view or even in
another room. In the system as described thus far, Flo
and Joh do not share any representations except the
iconic representations of their shared sensors (odometry
and ultrasonic). The location feature detectors may be
correlated with visual landmarks in Joh's map, and
whisker landmarks in Flo's (among other information).
Hence, before we can communicate Flo's
representation for location we need a procedure to
establish a shared grounding with Joh. For this purpose,
we have implemented a location labeling procedure.
Location labeling is essentially behavior whereby Flo
teaches Joh a location by empirical demonstration. It
proceeds as follows.
If Joh is tracking Flo in its visual field at a
particular time and there are no previously labeled
locations near by, then Joh signals Flo indicating that
Flo's current location should be labeled. Although an
arbitrary signal could be generated and communicated to
serve as a common labeling icon for the location, in this
specific case no signal is necessary. Because there are
only two robots, the time ordering of the labeling
procedures is identical to each. Hence, a time ordered
sequence number maintained by each serves as the
labeling icon with a shared grounding. The first location
is labeled '1 st Label', the next `2 nd Label', etc. If Joh
receives a confirmation signal from Flo, it associates the
label icon with Flo's current location. Joh calculates
Flo's location based on its own location and a calculation
of Flo's range from visual tracking information. Flo also
labels its own location index in the same way. This
procedure creates an indexical representation of specific
locations that are associations between a location
detector icon and the label icon (the shared sequence
number). Although the locations themselves are not
represented using the same icons by both Flo and Joh,
they represent the same physical location. Figure 12
shows the situation after the labeling procedure has
occurred four times (the symbol is explained below).
3.6 A symbol for a relationship between locations
The next step is to endow both Joh and Flo with the
ability to represent an arbitrary location in relationship to
already known locations. Recall that a symbol is defined
as a relationship between other symbolic, indexical and
iconic references.
Ideally, symbols should be learnt, as in biological
systems. The relationship a symbol represents is a
generalization from a set of observed 'exemplars' -
specific relationships between other symbols, indices and
icons. How this can be accomplished is still an open
research area. For this reason, and because we only need
a single symbol that will not be referenced by higher-level
symbols, we chose to simply provide the necessary
relationship. We can consider the symbol a 'first-level
as it is not dependent on any other symbols, but
grounded directly to iconic and indexical representations.
As symbol systems go, ours is as impoverished as it can
be.
The relationship represented by the symbol is
between two known location indices and a distance and
orientation in terms of wheel encoder data. The two
known locations define a line segment that provides an
origin for position and orientation. The wheel encoder
data then provides a distance and orientation relative to
this - which together defines a unique location (see
Figure
13). For example, a pile could be specified as
being approximately 5m away from the 2 nd labeled
location at an angle of relative to the direction of the
st labeled location from the 2 nd . The top of Figure 12
shows the symbol in the context of the overall system.
Indexical
references to
shared labeled
locations
Represented
position
(wheel encoder data)
iconic distance and orientation
Figure
- Schematic of the <specific-geometric-relation-
between> symbol used to communication locations
3.7 Symbolic communication
Finally, we are in a position to see how a location can be
symbolically communicated from Flo to Joh. With a
particular pile location in mind, Flo first calculates the
representation for it using the symbolic relationship
above. It selects the two closest locations, previously
labeled, as the indexical references and computes the
corresponding iconic wheel encoder data that will yield
the desired pile location. This information is then
signaled to Joh by signaling the labels for each of the
known locations in turn, followed by the raw encoder
data. This signal is grounded in both robots, as the
1st label
2nd label
3rd label
4th label
Indexical references to
shared labeled locations
Now
Indexical
Iconic
Symbolic
Encoder
data
Symbol (represents relationship for
describing a location index
in relation to two known
location indices and iconic
encoder data)
Flo
current
location
Figure
references that represent sensory data, Indexical references that associate pairs of icons (a label with a
location) and a symbol (see text). The fine lines between location feature detectors show their adjacency in the SOM; the pairs of
arrow headed lines from indexical references define which two icons they associate; and the two sets of arrow headed lines from
the symbol designate two 'exemplars' (see text).
labels were grounded through the location labeling
procedure, and the wheel encoders are a shared sense.
Hence, the meaning is preserved. Joh can recover the
location by re-grounding the labels and reversing the
computation.
3.8 Results
The typical trajectories in Figure 14 show that Joh is able
to successfully vacuum the litter in the pile to the left.
This occurs after the location of the pile has been
communicated symbolically by Flo. The pile was
initially obscured by the cardboard box, but Joh was able
to correctly compute its location and plan a path around
the box using its map. This can be contrasted with the
layer 3 solution shown in Figure 6, where no symbolic
communication or map was utilized. If the box were
blocking the straight-line path to the litter pile in that
case, Joh would not have been able to navigate to within
visual range to locate it.
As the system was not designed as a floor cleaning
system per-se, rigorous experiments to record its
cleaning performance were not conducted. However, we
did run experiments that seem to show that the addition
of symbolic communication does improve cleaning
performance. We expect this intuitively, as the
governing factor in vacuuming performance is the path
length between litter piles. The ability to navigate
purposively from one known litter pile location to the
next, instead of having to rely on an obstacle free path,
or chance discovery of the pile locations, shortens the
average path length.
Joh
Flo
Litter
Figure
14 - Typical trajectories during cooperation
We also ran experiments utilizing each layer in turn
(including the lower ones on which it builds). We
recorded the percentage of the floor cleaned every two
minutes from 3-15 minutes. It was difficult to run all of
the experiments consistently for more than 15 minutes
due to problems with hardware reliability. The results
are plotted in Figure 15. Initially, about 30% of the
'litter' was distributed around the perimeter and the
remainder scattered approximately uniformly over the
rest of the floor. The percentage cleaned was estimated
by dividing the floor into a grid and counting how many
tiles had been cleaned.1030507090
Time (mins)
Cleaned
Layer 1 Layers 1&2 Layers 1-3 Layers 1-4
Figure
Performance of layered cleaning solutions
Clearly, the addition of each layer improves the
cleaning performance. In particular, layer 4, utilizing
initially falls behind as some
time is used to perform location labeling rather than
cleaning. This starts to pay off later after a number of
locations have been labeled.
This experiment also shows the robustness gained
by layering the solution. The implementation of layer 1
is robust due to its simplicity. If any of the mechanisms
employed in the subsequent layers were to fail, we have
demonstrated that the system will continue to perform
the cleaning task, although not as quickly.
4. A Critical Examination
4.1 The limitations of our system
The are two obvious limitations to the approach we have
described for developing grounded symbolic
communication between robots. The first is that the
common process by which a shared symbol grounding is
developed is the design process. That is, the shared
grounding was established by identical design and
implementation of the mechanism for its interpretation.
This is an impractical way to develop sophisticated
systems, as the mechanism for the interpretation
of each symbol must be designed in turn.
Is this just a practicality problem, or it is impossible
in principle? When we designed the system, we believed
that it was possible, if impractical, to build general
symbol systems in this way - by explicitly designing the
process of interpretation for each symbol. We
hypothesized that all that was missing was a mechanism
to learn the symbolic representations - to effectively
automate the process. However, we argue below that is it
in fact impossible in principle (for all but the simplest
systems - like the one presented).
The second obvious limitation is a related one. The
approach doesn't include a mechanism for learning new
symbols, even if it had an existing symbol repertoire
designed in.
4.2 Symbols revisited
Our definition of grounded from section 2.2 contained a
hidden assumption. We defined a symbol to be grounded
if its interpretation required following references that all
eventually lead to icons. Recall that, symbols and the
structure of their relationships to each other and to
indices and icons, is a linguistic one. It is the
empirically observed structure of the signals that humans
generate and interpret. This grammatical structure of
spoken and written language is a relatively persistent one
(ignoring the fact that languages change slowly over
time). The hidden assumption, which we now believe to
be incorrect, was that this somehow implies that a
similarly persistent analogous structure must be present
within the mind of the humans that generate signals
conforming to the structure. That is, just because there is
a relatively persistent symbolic system present in human
cultural artifacts - such as books, paintings, buildings,
music, etc. - this does not imply that any symbol system
persists within the human mind. It was with this invalid
assumption that we proceeded to construct just such a
system within the robots, by representing and
relating icons, indices and symbols.
We believe there is ample evidence that no
persistent symbolic structure within the human mind that
mirrors the structure of human language exists - but this
remains to be seen. Dennett has argued strongly against
the idea of a Cartesian theater - a place in the mind
where all the distributed information is integrated for a
central decision-maker (Dennett, 1993). It seems that
distributed information about the external world
(possibly contradictory) need not be integrated unless a
particular discrimination is necessary for performance
(for example to speak or behave). Even then, only the
information necessary for the discrimination need be
integrated.
Even if humans don't use the equivalent of a
persistent cognitive grammar to reason about the world,
why can't robots use one?
4.3 Symbolic representation is not situated
A symbol represents a discrete category in the continuous
space of sensory-motor experience. Hence it defines a
boundary such that points in the space lie either within
the category or outside of it - there are no gray areas.
Therefore, a symbol system is a way of characterizing
sensory-motor experience in terms of membership of the
categories it defines. Symbols derive their power by
conferring a degree of independence from the context
dependent, dynamic and situated experiences from which
they are learnt. This allows symbolic communication to
preserve its meaning when the interaction is extended in
time (e.g. the period between these words being written
and you reading them).
Suppose we build a robot for a particular task that
necessitates symbolic communication, and endow it with
a symbolic representation system according to the
approach we have outlined, whereby static symbol
groundings are designed in. The robot is situated in the
sense that the task for which is it designed provides a
context for its interaction with the environment (from the
theory of situated action - Mills, 1940; Suchman, 1987).
The robot is an embodied agent and has a grounded
symbol system. It satisfies the criteria of the physical
grounding hypothesis (Brooks, 1990).
We argue that this approach to building a robot will
not necessarily work, except in the simplest cases. The
task in which the robot is situated dictates the
discriminations it must make in order to behave
appropriately - it must behave in terms of its affordances
(Gibson, 1986). Since the discriminations it can make
are determined by the categories defined by its symbol
system, which is necessarily static, it will only work if
the task very specific - ensuring the appropriate
discriminations don't change. This is precisely the
situation in which our system operates - in the situated
context defined by a statically specified cleaning task.
A robot capable of operating flexibly in a dynamic
situated context must continually adapt the
discriminations it makes. If using a symbolic
representation system, this implies the categories defined
by the symbols, and hence the meaning of the symbols
themselves, must change 3 . However, a dynamic symbol
system looses its power for communication - one of the
main reasons for endowing the robot with a symbol
system in the first place.
Consequently, a robot that utilizes a static symbolic
representation system (like the one we presented) cannot
be situated if its task is to behave flexibly in a dynamic
context. Hence, our approach of designing in the robot's
groundings does not scale from systems designed
to achieve simple specific tasks, to more general flexible
behavior.
We also see a more pragmatic way in which larger
systems built via our approach can become
unsituated. In order to manage complexity in the design
process, we often structure a system by categorizing and
apply linguistic labels to design components (i.e. we
need to name elements of our designs). Although this
activity is logically independent from the way the system
3 It may be possible in principle for an agent to use a static symbol system
that covers all possible categorizations and hence accommodates any
possible discrimination needed for appropriate behavior in any situated
context. However, we dismiss this as impossible in practice due to
computation intractability.
functions, the anthropocentric groundings we use in our
interpretation of the linguistic labels inevitably effect the
design.
For example, by naming a behavior component
WallFollowing, we may accidentally allow hidden
assumptions from our understanding of 'walls' to come
into play, despite being aware of this pitfall. If the robot
possesses anything that could be called a concept for a
'wall', it is surely impoverished compared to our human
understanding of 'walls'. We contend that avoiding this
pitfall becomes harder, to the point of practical
impossibility, as the symbol systems become more
complex and the discrepancy between our labels and the
robot's representations grow.
4.4 Adaptive Symbol Grounding Hypothesis
There is increasing evidence that humans do not reason
about the world and behave using symbolic
representations (Hendriks-Jansen, 1996 provides a
thorough argument). Instead, like other biological
systems, we represent 4 the world in terms of changing
affordances - dictated by our situatedness. We make
only the discriminations necessary to behave
appropriately. The symbols we use to communicate seem
to be generated during language production and
interpretation by a dynamic process that grounds them in
our adaptive internal representations while preserving
their static, public, statistically persistent meaning.
Hence, the symbols we generate are influenced by our
situated representations during production and they have
the power to influence them during interpretation. The
representations themselves are only transient.
We refer to this conception as the Adaptive Symbol
Grounding Hypothesis.
By this conception, we envisage the process of
learning new concepts as follows. A process within the
emitter wishing to communicate a new concept
dynamically generates a transient symbolic
representation that best approximates it by matching the
internal representation with learnt static linguistic
relationships. This structure is reflected in the
signal. The interpretation process within the receiver
causes a similar transient symbolic structure to emerge.
Again, an approximate match is made between the
symbolic structure and the internal representation -
which influences the representations. In this case, the
influence causes a new concept to be discovered. The
structure provides the scaffolding necessary to
get the receiver thinking in the right way to discover the
new concept.
4 We do not mean to imply that biological agents represent the world to
themselves. Of course any observations of the internal states of an agent
can be said to represent something - if we as scientific observers interpret
it, it represents something to us.
So the essential points of the Adaptive Symbol
Grounding Hypothesis can be summarized as follows.
. The persistent relationships between icons,
indices and symbols that comprise the
hierarchical structure of language (e.g. grammar)
are only observed in the communicated signals.
. Agents engaging in symbolic communication do
not need to maintain an explicit representation
analogous to the symbolic structure of the
language.
. Symbol grounding is transient and adaptive.
Explicit symbolic representations and their
situated groundings only persist during the
generation and interpretation of the signals of
communication. The specific
groundings with which icons for particular
symbols are associated depend upon a history of
use. The mapping adapts both to the immediate
context and to track long-term common usage
within a community of language users.
4.5 Implication for cooperative robotics
In the future, we will require increasingly complex tasks
to be carried out by multi-robot teams. Hence, the
behavioral sophistication of the individual robots will be
greater. If we wish to engineer multi-robot systems that
can cooperate in complex ways, they will eventually
require symbolic communication.
The Adaptive Symbol Grounding Gypothesis
implies that all symbols are learnt. Hence, we advocate
the ubiquitous use of learning in engineering all robotic
systems. Without it, we don't believe symbolic
communication of significance is possible.
Multi-robot systems are usually classified as either
homogeneous or heterogeneous. This is usually based
upon physical attributes, such as sensors and actuators;
but can be equally applied to the computational and
behavioral ability of the robots. A robot system is
classified as heterogeneous if one or more agents are
different from the others. Balch proposes a metric to
measure the diversity in multi-robot systems he calls
social entropy - which also recognizes physically
identical robots that differ only in their behavioral
repertoire (Balch, 1997).
If robots are engineered with an emphasis on
learning and are consequently more a product of their
experience, as we suggest above, then even physically
homogeneous teams will have significant social entropy.
The teams will necessarily be heterogeneous in terms of
their representation of the world and hence behavior.
Therefore, we don't envisage homogeneous multi-robot
systems playing a large role in the cooperative robotics
domain in the long term.
5.
Summary
In the first part of the paper, we defined what we mean
by grounded and provided a framework for talking about
symbols in terms of indexical and iconic references. We
also introduced the classification scheme for
communication involving the characteristics interaction
distance, interaction simultaneity, signaling explicitness
and sophistication of interpretation. We discussed
cooperation and communication in bacteria, ants,
wolves, primates and humans in these terms to deduce
some prerequisites for symbolic communication.
If we are not interested in preserving the meaning of
a signal between emitter and receiver, then the
implementation is straightforward. If we wish to
preserve meaning, then we have to ensure a shared
grounding between the agents. In the case of iconic
representations, as they are essentially grounded directly
in sensory information, this can only be ensured if the
sensors are identical between the agents. In the case of
indexical and symbolic representations, a specific
mechanism for establishing a shared grounding is
needed. For indexical representations, an empirical
demonstration can serve to ground them to appropriate
icons. The location labeling procedure we implemented
on our robots takes this form.
We described the implementation of the cooperative
cleaning system, including the spreading activation
action-selection mechanism and purposive navigation in
order to provide an understanding for the communication
mechanism. The symbolic communication relies on:
. the shared grounding of icons through common
sensors,
. the shared grounding for locations, developed
through a specific process - the location labeling
behavior, and
. the shared grounding for the symbol representing
a specific relationship between locations -
provided by design.
In the final part of the paper, we critically examined
the system and its limitations. Specifically, one obvious
limitation is that the system only contains a single
symbol, and it was provided at design time - with no
mechanism for learning further symbols. By looking
again at the notion of a symbol we were able to
understand that this approach cannot scale to larger
systems.
We argued that situated, embodied agents cannot
use symbolic representations of the world to interactively
behave in it. The Adaptive Symbol Grounding
Hypothesis was introduced as an alternative conception
for how symbol system might be used in situated agents.
Finally, we concluded that symbol grounding must be
learnt. Consequently, we advocate the ubiquitous use of
learning in heterogeneous multi-robot systems, because
without it symbolic communication is not possible. We
believe this would be a severe limitation to the
sophistication of cooperation in the future.
--R
Dimensions of Communication and Social Organization in Multi-Agent Robotic Systems
Simulation of Adaptive Behavior 92
Communication in Reactive Multiagent Robotic Systems
Social Entropy: a New Metric for Learning Multi-robot Teams
An Architectural Model of the Primate Brain
of Computer Science
Elephants Don't Play Chess
Intelligence Without Reason
"The Analysis of Action"
Cooperative Mobile Robotics: Antecedents and Directions
How Monkeys see the world
The Evolution of Social Behaviour in Insects and Arachnids
The Symbolic Species: The co-evolution of language and the human brain
Consciousness Explained
A taxonomy for swarm robots
Reasoning Agents in a Dynamic World: The Frame Problem
The Ecological Approach to Visual Perception
Catching Ourselves in the Act
Knowing through the body
Range and Pose Estimation for Visual Servoing on a Mobile Robotic Target
An architecture for distributed cooperative planning in a behaviour-based multi-robot system
Integrating Spatial and Topological Navigation in a Behavior-Based Multi-Robot Application
How and Why Bacteria Talk to Each Other
The self-organising map
Collective Robotics: From Social Insects to Robots
Metaphors we Live By
Situated Agents Can Have Goals.
Layered Computation in Neural Networks
Situated actions and vocabularies of motive
Human problem solving
The Effect of Action Recognition and Robot Awareness in Cooperative Robotic Teams
"From individual to collective behavior in social insects"
The Robot's Dilemma.
A Short History of Linguistics
Mental images and their transformations
"Orders and Families of Recent Mammals of the World"
The origins of intelligence
Plans and Situated Actions: The Problem of Human-Machine Communication
A Dynamic Systems Approach to the Development of Cognition and Action
The Insect Societies: Their Origin and Evolution
Sociobiology: The New Synthesis
Implementation of a small size experimental self-contained autonomous robot - sensors
LAAS/CNRS.
A Qualitative Approach to Achieving Robust Performance by a Mobile Agent
--TR
--CTR
David Hurt , Paul Tarau, An empirical evaluation of communication effectiveness in autonomous reactive multiagent systems, Proceedings of the 2005 ACM symposium on Applied computing, March 13-17, 2005, Santa Fe, New Mexico
Ariel Felner , Yaron Shoshani , Yaniv Altshuler , Alfred M. Bruckstein, Multi-agent Physical A* with Large Pheromones, Autonomous Agents and Multi-Agent Systems, v.12 n.1, p.3-34, January 2006
Luca Iocchi , Daniele Nardi , Maurizio Piaggio , Antonio Sgorbissa, Distributed Coordination in Heterogeneous Multi-Robot Systems, Autonomous Robots, v.15 n.2, p.155-168, September
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591902 | Theory of Mind for a Humanoid Robot. | If we are to build human-like robots that can interact naturally with people, our robots must know not only about the properties of objects but also the properties of animate agents in the world. One of the fundamental social skills for humans is the attribution of beliefs, goals, and desires to other people. This set of skills has often been called a theory of mind. This paper presents the theories of Leslie (1994) and Baron-Cohen (1995) on the development of theory of mind in human children and discusses the potential application of both of these theories to building robots with similar capabilities. Initial implementation details and basic skills (such as finding faces and eyes and distinguishing animate from inanimate stimuli) are introduced. I further speculate on the usefulness of a robotic implementation in evaluating and comparing these two models. | Introduction
Human social dynamics rely upon the ability to correctly attribute beliefs, goals, and percepts to other people.
This set of metarepresentational abilities, which have been collectively called a "theory of mind" or the ability to
"mentalize", allows us to understand the actions and expressions of others within an intentional or goal-directed
framework (what Dennett [15] has called the intentional stance). The recognition that other individuals have knowl-
edge, perceptions, and intentions that differ from our own is a critical step in a child's development and is believed
to be instrumental in self-recognition, in providing a perceptual grounding during language learning, and possibly
in the development of imaginative and creative play [9]. These abilities are also central to what defines human
interactions. Normal social interactions depend upon the recognition of other points of view, the understanding of
other mental states, and the recognition of complex non-verbal signals of attention and emotional state.
Research from many different disciplines have focused on theory of mind. Students of philosophy have been
interested in the understanding of other minds and the representation of knowledge in others. Most recently, Dennett
[15] has focused on how organisms naturally adopt an "intentional stance" and interpret the behaviors of others
as if they possess goals, intents, and beliefs. Ethologists have also focused on the issues of theory of mind. Studies
of the social skills present in primates and other animals have revolved around the extent to which other species
are able to interpret the behavior of conspecifics and influence that behavior through deception (e.g. Premack [33],
Povinelli and Preuss [32], and Cheney and Seyfarth [12]). Research on the development of social skills in children
have focused on characterizing the developmental progression of social abilities (e.g. Fodor [17], Wimmer and
Perner [37], and Frith and Frith [18]) and on how these skills result in conceptual changes and the representational
capacities of infants (e.g. Carey [10], and Gelman [19]). Furthermore, research on pervasive developmental disorders
such as autism have focused on the selective impairment of these social skills (e.g. Perner and Lang [31],
Karmiloff-Smith et. al. [24], and Mundy and Sigman [29]).
Researchers studying the development of social skills in normal children, the presence of social skills in primates
and other vertebrates, and certain pervasive developmental disorders have all focused on attempting to decompose
the idea of a central "theory of mind" into sets of precursor skills and developmental modules. In this
paper, I will review two of the most popular and influential models which attempt to link together multi-disciplinary
research into a coherent developmental explanation, one from Baron-Cohen [2] and one from Leslie [27]. Section
4 will discuss the implications of these models to the construction of humanoid robots that engage in natural human
social dynamics and highlight some of the issues involved in implementing the structures that these models
propose. Finally, Section 5 will describe some of the precursor components that have already been implemented
by the author on a humanoid robot at the MIT Artificial Intelligence lab.
Leslie's Model of Theory of Mind
Leslie's [26] theory treats the representation of causal events as a central organizing principle to theories of object
mechanics and theories of other minds much in the same way that the notion of number may be central to object
Brian Scassellati
(a)
(b)
(c)
(d)
Fig. 1. Film sequences used by Leslie [25] to study perception of causality in infants based on similar tests in adults performed
by Michotte [28]. The following six events were studied: (a) direct launching - the light blue brick moves off immediately after
impact with the dark red brick; (b) delayed reaction - spatially identical to (a), but a 0.5 second delay is introduced between
the time of impact and the movement of the light blue brick; (c) launching without collision - identical temporal structure but
without physical contact; (d) collision with no launching - identical result but without causation; (e) no contact, no launching
another plausible alternative. Both adults and infants older than six months interpret events (a) and (e) as different from the
class of events that violate simple mechanical laws (b-d). Infants that have been habituated to a non-causal event will selectively
dishabituate to a causal event but not to other non-causal events. Adapted from Leslie [25].
representation. According to Leslie, the world is naturally decomposed into three classes of events based upon
their causal structure; one class for mechanical agency, one for actional agency, and one for attitudinal agency.
Leslie argues that evolution has produced independent domain-specific modules to deal with each of these classes
of event. The Theory of Body module (ToBY) deals with events that are best described by mechanical agency, that
is, they can be explained by the rules of mechanics. The second module is system 1 of the Theory of Mind module
which explains events in terms of the intent and goals of agents, that is, their actions. The third module
is system 2 of the Theory of Mind module (ToMM-2) which explains events in terms of the attitudes and beliefs
of agents.
The Theory of Body mechanism (ToBY) embodies the infant's understanding of physical objects. ToBY is
a domain-specific module that deals with the understanding of physical causality in a mechanical sense. ToBY's
goal is to describe the world in terms of the mechanics of physical objects and the events they enter into. ToBY in
humans is believed to operate on two types of visual input: a three-dimensional object-centered representation from
high level cognitive and visual systems and a simpler motion-based system. This motion-based system accounts
for the causal explanations that adults give (and the causal expectations of children) to the "billiard ball" type
launching displays pioneered by Michotte [28] (see figure 1). Leslie proposed that this sensitivity to the spatio-temporal
properties of events is innate, but more recent work from Cohen and Amsel [13] may show that it develops
extremely rapidly in the first few months and is fully developed by 6-7 months.
ToBY is followed developmentally by the emergence of a Theory of Mind Mechanism (ToMM) which develops
in two phases, which Leslie calls system-1 and system-2 but which I will refer to as ToMM-1 and ToMM-2
after Baron-Cohen [2]. Just as ToBY deals with the physical laws that govern objects, ToMM deals with the
psychological laws that govern agents. ToMM-1 is concerned with actional agency; it deals with agents and the
goal-directed actions that they produce. The primitive representations of actions such as approach, avoidance, and
escape are constructed by ToMM-1. This system of detecting goals and actions begins to emerge at around 6 months
of age, and is most often characterized by attention to eye gaze. Leslie leaves open the issue of whether ToMM-1
is innate or acquired. ToMM-2 is concerned with attitudinal agency; it deals with the representations of beliefs
and how mental states can drive behavior relative to a goal. This system develops gradually, with the first signs
of development beginning between months of age and completing sometime near 48 months. ToMM-
Intentionality
Detector (ID)
Eye Direction
Detector (EDD)
Shared Attention
Mechanism (SAM)
Theory of Mind
Mechanism (ToMM)
Stimuli with
self-propulsion
and direction
Eye-like
stimuli
Dyadic
representations
(desire, goal)
Dyadic
representations
(sees)
Triadic
representations
Full range of mental
state concepts,
expressed in
M-representations
Knowledge of the
mental, stored and
used as a theory
Fig. 2. Block diagram of Baron-Cohen's model of the development of theory of mind. See text for description. Adapted from
[2].
2 employs the M-representation, a meta-representation which allows truth properties of a statement to be based
on mental states rather than observable stimuli. ToMM-2 is a required system for understanding that others hold
beliefs that differ from our own knowledge or from the observable world, for understanding different perceptual
perspectives, and for understanding pretense and pretending.
3 Baron-Cohen's Model of Theory of Mind
Baron-Cohen's model assumes two forms of perceptual information are available as input. The first percept describes
all stimuli in the visual, auditory, and tactile perceptual spheres that have self-propelled motion. The second
percept describes all visual stimuli that have eye-like shapes. Baron-Cohen proposes that the set of precursors to a
theory of mind, which he calls the "mindreading system," can be decomposed into four distinct modules.
The first module interprets self-propelled motion of stimuli in terms of the primitive volitional mental states of
goal and desire. This module, called the intentionality detector (ID) produces dyadic representations that describe
the basic movements of approach and avoidance. For example, ID can produce representations such as "he wants
the food" or "she wants to go over there". This module only operates on stimuli that have self-propelled motion,
and thus pass a criteria for distinguishing stimuli that are potentially animate (agents) from those that are not
(objects). Baron-Cohen speculates that ID is a part of the innate endowment that infants are born with.
The second module processes visual stimuli that are eye-like to determine the direction of gaze. This module,
called the eye direction detector (EDD), has three basic functions. First, it detects the presence of eye-like stimuli
in the visual field. Human infants have a preference to look at human faces, and spend more time gazing at the
eyes than at other parts of the face. Second, EDD computes whether the eyes are looking at it or at something else.
Baron-Cohen proposes that having someone else make eye contact is a natural psychological releaser that produces
pleasure in human infants (but may produce more negative arousal in other animals). Third, EDD interprets gaze
direction as a perceptual state, that is, EDD codes dyadic representational states of the form "agent sees me" and
"agent looking-at not-me".
The third module, the shared attention mechanism (SAM), takes the dyadic representations from ID and EDD
and produces triadic representations of the form "John sees (I see the girl)". Embedded within this representation
is a specification that the external agent and the self are both attending to the same perceptual object or event. This
shared attentional state results from an embedding of one dyadic representation within another. SAM additionally
can make the output of ID available to EDD, allowing the interpretation of eye direction as a goal state. By
allowing the agent to interpret the gaze of others as intentions, SAM provides a mechanism for creating nested
representations of the form "John sees (I want the toy)".
4 Brian Scassellati
The last module, the theory of mind mechanism (ToMM), provides a way of representing epistemic mental
states in other agents and a mechanism for tying together our knowledge of mental states into a coherent whole as
a usable theory. ToMM first allows the construction of representations of the form "John believes (it is raining)".
ToMM allows the suspension of the normal truth relations of propositions (referrential opacity), which provides a
means for representing knowledge states that are neither necessarily true nor match the knowledge of the organism,
such as "John thinks (Elvis is alive)". Baron-Cohen proposes that the triadic representations of SAM are converted
through experience into the M-representations of ToMM.
Baron-Cohen's modules match a developmental progression that is observed in infants. For normal children,
ID and the basic functions of EDD are available to infants in the first 9 months of life. SAM develops between 9
and months, and ToMM develops from months to 48 months. However, the most attractive aspects of this
model are the ways in which it has been applied both to the abnormal development of social skills in autism and to
the social capabilities of non-human primates and other vertebrates.
Autism is a pervasive developmental disorder of unknown etiology that is diagnosed by a checklist of behavioral
criteria. Baron-Cohen has proposed that the range of deficiencies in autism can be characterized by his model.
In all cases, EDD and ID are present. In some cases of autism, SAM and ToMM are impaired, while in others only
ToMM is impaired. This can be contrasted with other developmental disorders (such as Down's syndrome) or
specific linguistic disorders in which evidence of all four modules can be seen.
Furthermore, Baron-Cohen attempts to provide an evolutionary description of these modules by identifying
partial abilities in other primates and vertebrates. This phylogenetic description ranges from the abilities of hog-
nosed snakes to detect direct eye contact to the sensitivities of chimpanzees to intentional acts. Roughly speaking,
the abilities of EDD seem to be the most basic and can be found in part in snakes, avians, and most other vertebrates
as a sensitivity to predators (or prey) looking at the animal. ID seems to be present in many primates, but the
capabilities of SAM seem to be present only partially in the great apes. The evidence on ToMM is less clear, but it
appears that no other primates readily infer mental states of belief and knowledge.
4 Implications of these Models to Humanoid Robots
A robotic system that possessed a theory of mind would allow for social interactions between the robot and humans
that have previously not been possible. The robot would be capable of learning from an observer using normal
social signals in the same way that human infants learn; no specialized training of the observer would be necessary.
The robot would also be capable of expressing its internal state (emotions, desires, goals, etc.) through social
interactions without relying upon an artificial vocabulary. Further, a robot that can recognize the goals and desires
of others will allow for systems that can more accurately react to the emotional, attentional, and cognitive states
of the observer, can learn to anticipate the reactions of the observer, and can modify its own behavior accordingly.
The construction of these systems may also provide a new tool for investigating the predictive power and validity
of the models from natural systems that serve as the basis. An implemented model can be tested in ways that
are not possible to test on humans, using alternate developmental conditions, alternate experiences, and alternate
educational and intervention approaches.
The difficulty, of course, is that even the initial components of these models require the coordination of a
large number of perceptual, sensory-motor, attentional, and cognitive processes. In this section, I will outline the
advantages and disadvantages of Leslie's model and Baron-Cohen's model with respect to implementation. In the
following section, I will describe some of the components that have already been constructed and some which are
currently designed but still being implemented.
The most interesting part of these models is that they attempt to describe the perceptual and motor skills that
serve as precursors to the more complex theory of mind capabilities. These decompositions serve as an inspiration
and a guideline for how to build robotic systems that can engage in complex social interactions; they provide a
much-needed division of a rather ambiguous ability into a set of observable, testable predictions about behavior.
While it cannot be claimed with certainty that following the outlines that these models provide will produce a
robot that has the same abilities, the evolutionary and developmental evidence of sub-skills does give us hope
that these abilities are critical elements of the larger goal. Additionally, the grounding of high-level perceptual
abilities to observable sensory and motor capabilities provides an evaluation mechanism for measuring the amount
of progress that is being made.
From a robotics standpoint, the most salient differences between the two models are in the ways in which
they divide perceptual tasks. Leslie cleanly divides the perceptual world into animate and inanimate spheres, and
allows for further processing to occur specifically to each type of stimulus. Baron-Cohen does not divide the
perceptual world quite so cleanly, but does provide more detail on limiting the specific perceptual inputs that each
Fig. 3. Cog, an upper-torso humanoid robot with twenty-one degrees of freedom and sensory systems that include visual,
auditory, tactile, vestibular, and kinesthetic systems.
module requires. In practice, both models require remarkably similar perceptual systems (which is not surprising,
since the behavioral data is not under debate). However, each perspective is useful in its own way in building a
robotic implementation. At one level, the robot must distinguish between object stimuli that are to be interpreted
according to physical laws and agent stimuli that are to be interpreted according to psychological laws. However,
the specifications that Baron-Cohen provides will be necessary for building visual routines that have limited scope.
The implementation of the higher-level scope of each of these models also has implications to robotics. Leslie's
model has a very elegant decomposition into three distinct areas of influence, but the interactions between these
levels are not well specified. Connections between modules in Baron-Cohen's model are better specified, but
they are still less than ideal for a robotics implementation. Issues on how stimuli are to be divided between the
competencies of different modules must be resolved for both models. On the positive side, the representations that
are constructed by components in both models are well specified.
5 Implementing a Robotic Theory of Mind
Taking both Baron-Cohen's model and Leslie's model, we can begin to specify the specific perceptual and cognitive
abilities that our robots must employ. Our initial systems concentrate on two abilities: distinguishing between
animate and inanimate motion and identifying gaze direction. To maintain engineering constraints, we must focus
on systems that can be performed with limited computational resources, at interactive rates in real time, and
on noisy and incomplete data. To maintain biological plausibility, we focus on building systems that match the
available data on infant perceptual abilities.
Our research group has constructed an upper-torso humanoid robot with a pair of six degree-of-freedom arms,
a three degree-of-freedom torso, and a seven degree of freedom head and neck. The robot, named Cog, has a visual
system consisting of four color CCD cameras (two cameras per eye, one with a wide field of view and one with
a narrow field of view at higher acuity), an auditory system consisting of two microphones, a vestibular system
consisting of a three axis inertial package, and an assortment of kinesthetic sensing from encoders, potentiometers,
and strain gauges. (For additional information on the robotic system, see [7]. For additional information on the
reasons for building Cog, see [1, 6].)
In addition to the behaviors that are presented in this section, there are also a variety of behavioral and cognitive
skills that are not integral parts of the theory of mind models, but are nonetheless necessary to implement the desired
functionality. We have implemented a variety of perceptual feature detectors (such as color saliency detectors,
motion detectors, skin color filters, and rough disparity detectors) that match the perceptual abilities of young in-
fants. We have constructed a model of human visual search and attention that was proposed by Wolfe [38]. We have
also implemented motor control schemes for visual motor behaviors (including saccades, smooth-pursuit tracking,
6 Brian Scassellati
and a vestibular-occular reflex), orientation movements of the head and neck, and primitive reaching movements
for a six degree-of-freedom arm. We will briefly describe the relevant aspects of each of these components so that
their place within the larger integrated system can be made clear.
5.1 Pre-attentive visual routines
Human infants show a preference for stimuli that exhibit certain low-level feature properties. For example, a four-
month-old infant is more likely to look at a moving object than a static one, or a face-like object than one that has
similar, but jumbled, features [16]. To mimic the preferences of human infants, Cog's perceptual system combines
three basic feature detectors: color saliency analysis, motion detection, and skin color detection. These low-level
features are then filtered through an attentional mechanism before more complex post-attentive processing (such
as face detection) occurs. All of these systems operate at speeds that are amenable to social interaction (30Hz).
Color content is computed using an opponent-process model that identifies saturated areas of red, green, blue,
and yellow [4]. Our models of color saliency are drawn from the complementary work on visual search and attention
from Itti, Koch, and Niebur [22]. The incoming video stream contains three 8-bit color channels (r, g, and
b) which are transformed into four color-opponency channels (r 0 , Each input color channel is first
normalized by the luminance l (a weighted average of the three input color channels):
r
l
l
l
(1)
These normalized color channels are then used to produce four opponent-color channels:
nb n kr n g n k (5)
The four opponent-color channels are thresholded and smoothed to produce the output color saliency feature map.
This smoothing serves both to eliminate pixel-level noise and to provide a neighborhood of influence to the output
map, as proposed by Wolfe [38].
In parallel with the color saliency computations, The motion detection module uses temporal differencing and
region growing to obtain bounding boxes of moving objects [5]. The incoming image is converted to grayscale and
placed into a ring of frame buffers. A raw motion map is computed by passing the absolute difference between
consecutive images through a threshold function
This raw motion map is then smoothed to minimize point noise sources.
The third pre-attentive feature detector identifies regions that have color values that are within the range of
skin tones [3]. Incoming images are first filtered by a mask that identifies candidate areas as those that satisfy the
following criteria on the red, green, and blue pixel components:
The final weighting of each region is determined by a learned classification function that was trained on hand-
classified image regions. The output is again median filtered with a small support area to minimize noise.
5.2 Visual attention
Low-level perceptual inputs are combined with high-level influences from motivations and habituation effects by
the attention system (see Figure 4). This system is based upon models of adult human visual search and attention
[38], and has been reported previously [4]. The attention process constructs a linear combination of the input
feature detectors and a time-decayed Gaussian field which represents habituation effects. High areas of activation
in this composite generate a saccade to that location and compensatory neck movement. The weights of the feature
detectors can be influenced by the motivational and emotional state of the robot to preferentially bias certain
stimuli. For example, if the robot is searching for a playmate, the weight of the skin detector can be increased to
cause the robot to show a preference for attending to faces.
Frame Grabber
Eye Motor Control
inhibit reset
Motivations, Drives
and Emotions
Color Detector
Motion Detector
Habituation
Skin Detector
Attention Process
Fig. 4. Low-level feature detectors for skin finding, motion detection, and color saliency analysis are combined with top-down
motivational influences and habituation effects by the attentional system to direct eye and neck movements. In these images,
the robot has identified three salient objects: a face, a hand, and a colorful toy block.
5.3 Finding eyes and faces
The first shared attention behaviors that infants engage in involve maintaining eye contact. To enable our robot
to recognize and maintain eye contact, we have implemented a perceptual system capable of finding faces and
eyes [35]. Our face detection techniques are designed to identify locations that are likely to contain a face, not
to verify with certainty that a face is present in the image. Potential face locations are identified by the attention
system as locations that have skin color and/or movement. These locations are then screened using a template-based
algorithm called "ratio templates" developed by Sinha [36].
The ratio template algorithm was designed to detect frontal views of faces under varying lighting conditions,
and is an extension of classical template approaches [36]. Ratio templates also offer multiple levels of biological
plausibility; templates can be either hand-coded or learned adaptively from qualitative image invariants [36]. A
ratio template is composed of regions and relations, as shown in Figure 5. For each target location in the grayscale
peripheral image, a template comparison is performed using a special set of comparison rules. The set of regions
is convolved with an image patch around a pixel location to give the average grayscale value for that region.
Relations are comparisons between region values, for example, between the "left forehead" region and the "left
temple" region. The relation is satisfied if the ratio of the first region to the second region exceeds a constant value
(in our case, 1.1). The number of satisfied relations serves as the match score for a particular location; the more
relations that are satisfied the more likely that a face is located there. In Figure 5, each arrow indicates a relation,
with the head of the arrow denoting the second region (the denominator of the ratio). The ratio template algorithm
has been shown to be reasonably invariant to changes in illumination and slight rotational changes [35].
Locations that pass the screening process are classified as faces and cause the robot to saccade to that target
using a learned visual-motor behavior. The location of the face in peripheral image coordinates is then mapped into
foveal image coordinates using a second learned mapping. The location of the face within the peripheral image
can then be used to extract the sub-image containing the eye for further processing (see Figure 6). This technique
has been successful at locating and extracting sub-images that contain eyes under a variety of conditions and from
many different individuals. These functions match the first function of Baron-Cohen's EDD and begin to approach
8 Brian Scassellati
Fig. 5. A ratio template for face detection. The template is composed of 16 regions (the gray boxes) and 23 relations (shown by
arrows). Darker arrows are statistically more important in making the classification and are computed first to allow real-time
rates.
Fig. 6. A selection of faces and eyes identified by the robot.Faces are located in the wide-angle peripheral image. The robot
then saccades to the target to obtain a high-resolution image of the eye from the narrow field-of-view camera.
the second and third functions as well. We are currently extending the functionality to include interpolation of gaze
direction using the decomposition proposed by Butterworth [8] (see section 6 below).
5.4 Discriminating animate from inanimate
We are currently implementing a system that distinguishes between animate and inanimate visual stimuli based
on the presence of self-generated motion. Similar to the findings of Leslie [25] and Cohen and Amsel [13] on the
classification performed by infants, our system operates at two developmental stages. Both stages form trajectories
from stimuli in consecutive image frames and attempt to maximize the path coherency. The differences between
the two developmental states lies in the type of features used in tracking. At the first stage, only spatio-temporal
features (resulting from object size and motion) are used as cues for tracking. In the second stage, more complex
object features such as color, texture, and shape are employed. With a system for distinguishing animate from
inanimate stimuli, we can begin to provide the distinctions implicit in Leslie's differences between ToBY and
ToMM and the assumptions that Baron-Cohen requires for ID.
Computational techniques for multi-target tracking have been used extensively in signal processing and detection
domains. Our approach is based on the multiple hypothesis tracking algorithm proposed by Reid [34] and
implemented by Cox and Hingorani [14]. The output of the motion detection module produces regions of motion
Humanoids2000 9
and their respective centroids. These centroid locations form a stream of target locations fP 1
t g with k
targets present in each frame t. The objective is to produce a labeled trajectory which consists of a set of points,
one from each frame, which identify a single object in the world as it moves through the field of view:
However, because the number of targets in each frame is never constant and because the existence of a target
from one frame to the next is uncertain, we must introduce a mechanism to compensate for objects that enter and
leave the field of view and to compensate for irregularities in the earlier processing modules. To address these
problems, we introduce phantom points that have undefined locations within the image plane but which can be
used to complete trajectories for objects that enter, exit, or are occluded within the visual field. As each new
point is introduced, a set of hypotheses linking that point to prior trajectories are generated. These hypotheses
include representations for false alarms, non-detection events, extensions of prior trajectories, and beginnings of
new trajectories. The set of all hypotheses are pruned at each time step based on statistical models of the system
noise levels and based on the similarity between detected targets. This similarity measurement is based either
purely on distances between points in the visual field (a condition that represents the first developmental stage
described above) or on similarities of object features such as color content, size, visual moments, or rough spatial
distribution (a condition that reflects a sensitivity to object properties characteristic of the second developmental
stage). At any point, the system maintains a small set of overlapping hypotheses so that future data may be used
to disambiguate the scene. Of course, the system can also produce the set of non-overlapping hypotheses that are
statstically most likely.
We are currently developing metrics for evaluating these trajectories in order to classify the stimulus as either
animate or inanimate using the descriptions of Michotte's [28] observations of adults and Leslie's [25] observations
of infants. The general form of these observations indicate that self-generated movement is attributed to stimuli
whose velocity profiles change in a non-constant manner, that is, animate objects can change their directions and
speed while inanimate objects tend to follow a single acceleration unless acted upon by another object.
6 Ongoing Work
The systems that have been implemented so far have only begun to address the issues raised by Leslie's and Baron-
Cohen's models of theory of mind. In this section, three current research directions are discussed: the implementation
of gaze following; the extensions of gaze following to deictic gestures; and the extension of animate-inanimate
distinctions to more complex spatio-temporal relations such as support and self-recognition.
6.1 Implementing gaze following
Once a system is capable of detecting eye contact, three additional subskills are required for gaze following:
extracting the angle of gaze, extrapolating the angle of gaze to a distal object, and motor routines for alternating
between the distal object and the caregiver. Extracting angle of gaze is a generalization of detecting someone gazing
at you, but requires additional competencies. By a geometric analysis of this task, we would need to determine not
only the angle of gaze, but also the degree of vergence of the observer's eyes to find the distal object. However, the
ontogeny of gaze following in human children demonstrates a simpler strategy.
Butterworth [8] has shown that at approximately 6 months, infants will begin to follow a caregiver's gaze to the
correct side of the body, that is, the child can distinguish between the caregiver looking to the left and the caregiver
looking to the right (see Figure 7). Over the next three months, their accuracy increases so that they can roughly
determine the angle of gaze. At 9 months, the child will track from the caregiver's eyes along the angle of gaze
until a salient object is encountered. Even if the actual object of attention is further along the angle of gaze, the
child is somehow "stuck" on the first object encountered along that path. Butterworth labels this the "ecological"
mechanism of joint visual attention, since it is the nature of the environment itself that completes the action. It is
not until 12 months that the child will reliably attend to the distal object regardless of its order in the scan path.
This "geometric" stage indicates that the infant can successfully determine not only the angle of gaze but also the
vergence. However, even at this stage, infants will only exhibit gaze following if the distal object is within their
field of view. They will not turn to look behind them, even if the angle of gaze from the caregiver would warrant
such an action. Around months, the infant begins to enter a "representational" stage in which it will follow gaze
angles outside its own field of view, that is, it somehow represents the angle of gaze and the presence of objects
outside its own view.
months: Representational stage
6 months: Sensitivity to field
9 months: Ecological stage
months: Geometric stage
Fig. 7. Proposed developmental progression of gaze following adapted from Butterworth (1991). At 6 months, infants show
sensitivity only to the side that the caregiver is gazing. At 9 months, infants show a particular strategy of scanning along the
line of gaze for salient objects. By one year, the child can recognize the vergence of the caregiver's eyes to localize the distal
target, but will not orient if that object is outside the field of view until months of age.
Implementing this progression for a robotic system provides a simple means of bootstrapping behaviors. The
capabilities used in detecting and maintaining eye contact can be extended to provide a rough angle of gaze. By
tracking along this angle of gaze, and watching for objects that have salient color, intensity, or motion, we can
mimic the ecological strategy. From an ecological mechanism, we can refine the algorithms for determining gaze
and add mechanisms for determining vergence. Once the robot and the caregiver are attending to the same object,
the robot can observe both the vergence of its own eyes (to achieve a sense of distance to the caregiver and to the
target) and the pupil locations (and thus the vergence) of the caregiver's eyes. A rough geometric strategy can then
be implemented, and later refined through feedback from the caregiver. A representational strategy will require the
ability to maintain information on salient objects that are outside of the field of view including information on their
appearance, location, size, and salient properties.
6.2 Extensions of gaze following to deictic gestures
Although Baron-Cohen's model focuses on the social aspects of gaze (primarily since they are the first to develop
in children), there are other gestural cues that serve as shared attention mechanisms. After gaze following, the next
most obvious is the development of imperative and declarative pointing.
Imperative pointing is a gesture used to obtain an object that is out of reach by pointing at that object. This
behavior is first seen in human children at about nine months of age, and occurs in many monkeys [11]. However,
there is nothing particular to the infant's behavior that is different from a simple reach - the infant is initially
as likely to perform imperative pointing when the caregiver is attending to the infant as when the caregiver is
looking in the other direction or when the caregiver is not present. The caregiver's interpretation of infant's gesture
provides the shared meaning. Over time, the infant learns when the gesture is appropriate. One can imagine the
child learning this behavior through simple reinforcement. The reaching motion of the infant is interpreted by the
adult as a request for a specific object, which the adult then acquires and provides to the child. The acquisition
of the desired object serves as positive reinforcement for the contextual setting that preceded the reward (the
reaching action in the presence of the attentive caregiver). Generation of this behavior is then a simple extension
of a primitive reaching behavior.
Declarative pointing is characterized by an extended arm and index finger designed to draw attention to a
distal object. Unlike imperative pointing, it is not necessarily a request for an object; children often use declarative
pointing to draw attention to objects that are clearly outside their reach, such as the sun or an airplane passing
overhead. Declarative pointing also only occurs under specific social conditions; children do not point unless there
is someone to observe their action. I propose that imitation is a critical factor in the ontogeny of declarative pointing.
This is an appealing speculation from both an ontological and a phylogenetic standpoint. From an ontological
perspective, declarative pointing begins to emerge at approximately 12 months in human infants, which is also the
same time that other complex imitative behaviors such as pretend play begin to emerge. From the phylogenetic
Humanoids2000 11
perspective, declarative pointing has not been identified in any non-human primate [33]. This also corresponds to
the phylogeny of imitation; no non-human primate has ever been documented to display imitative behavior under
general conditions [21]. I propose that the child first learns to recognize the declarative pointing gestures of the
adult and then imitates those gestures in order to produce declarative pointing. The recognition of pointing gestures
builds upon the competencies of gaze following and imperative pointing; the infrastructure for extrapolation from
a body cue is already present from gaze following, it need only be applied to a new domain. The generation of
declarative pointing gestures requires the same motor capabilities as imperative pointing, but it must be utilized in
specific social circumstances. By imitating the successful pointing gestures of other individuals, the child can learn
to make use of similar gestures.
6.3 Extensions of animate-inanimate distinctions
The simple spatio-temporal criteria for distinguishing animate from inanimate has many obvious flaws. We are
currently attempting to outline potential extensions for this model. One necessary extension is the consideration of
tracking over longer time scales (on the order of tens of minutes) to allow processing of a more continuous object
identity. This will also allow for processing to remove a subset of repetitively moving objects that are currently
incorrectly classified as animate (such as would be caused by a tree moving in the wind).
A second set of extensions would be to learn more complex forms of causal structures for physical objects,
such as the understanding of gravity and support relationships. This developmental advance may be strongly tied
to the object concept and the physical laws of spatial occupancy [20].
Finally, more complex object properties such as shape features and color should be used to add a level of
robustness to the multi-target tracking. Kalman filters have been used to track complex features that gradually
change over time [14].
7 Conclusion
While theory of mind studies have been more in the realm of philosophy than the realm of robotics, the requirements
of humanoid robotics for building systems that can interact socially with people will require a focus on the
issues that theory of mind research has addressed. Both Baron-Cohen and Leslie have provided models of how
more complex social skills can be developmentally constructed from simpler sensory-motor skill sets. While neither
model is exactly suited for a robotic implementation, they do show promise for providing the basis of such
an implementation. I have presented one initial attempt at building a framework of these precursors to a theory of
mind, but certainly much more work is required. However, the possibility of a robotic implementation also raises
the questions of the use of such an implementation as a tool for evaluating the predictive power and validity of
those models. Having an implementation of a developmental model on a robot would allow detailed and controlled
manipulations of the model while maintaining the same testing environment and methodology used on human
subjects. Internal model parameters could be varied systematically as the effects of different environmental conditions
on each step in development are evaluated. Because the robot brings the model into the same environment
as a human subject, similar evaluation criteria can be used (whether subjective measurements from observers or
quantitative measurements such as reaction time or accuracy). Further, a robotic model can also be subjected to
controversial testing that is potentially hazardous, costly, or unethical to conduct on humans. While this possibility
does raise a host of new questions and issues, it is a possibility worthy of further consideration.
--R
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Cynthia Breazeal , Daphna Buchsbaum , Jesse Gray , David Gatenby , Bruce Blumberg, Learning From and About Others: Towards Using Imitation to Bootstrap the Social Understanding of Others by Robots, Artificial Life, v.11 n.1, p.31-62, January 2005 | social interaction;humanoid robots;visual perception |
591962 | Searching for a Global Search Algorithm. | We report on a case study to assess the use of an advanced knowledge-based software design technique with programmers who have not participated in the techniques development. We use the KIDS approach to algorithm design to construct two global search algorithms that route baggage through a transportation net. Construction of the second algorithm involves extending the KIDS knowledge base. Experience with the case study leads us to integrate the approach with the spiral and prototyping models of software engineering, and to discuss ways to deal with incomplete design knowledge. | Introduction
Advanced techniques to support software construction
will only be widely accepted by practitioners if
they can be successfully used by software engineers
who were not involved in their development and did
not get on-site training by their inventors. Experience
has to be gained how knowledge-based methods can
be integrated into the practical software engineering
process.
We report on experience made with the application
of the approach to algorithm design underlying the
Kestrel Interactive Development System (KIDS) [14]
to the construction of control software for a simplified
baggage transportation system at an airport. In this
paper, we use the term "KIDS approach" to denote the
concepts that have been implemented in the system
The KIDS approach has been applied to a number
of case studies at Kestrel Institute. In par-
ticular, it has been used in the design of a transportation
scheduling algorithm with impressive performance
[15, 16]. We wished to find out if we were
able to use this method based on the available publi-
We did not use the implemented system KIDS in the case
study.
cations and produce satisfactory results with reason-able
effort. A second goal of this work has been to
study how a knowledge-based approach can be integrated
into the overall software engineering process.
As a case study we chose a non-trivial abstraction of
a practically relevant problem to make our experience
transferable to realistic applications.
In the following, we elaborate on two issues: a process
model we found useful to support application of
the KIDS approach, and the merits and shortcomings
we encountered when we explored alternative solutions
to the transportation scheduling problem.
We have integrated the spiral and prototyping models
of software engineering [2] with the KIDS approach.
We developed the first formal specification and a prototype
implementation in parallel. The prototype
served to validate the specification and to improve understanding
of the problem domain.
In the KIDS approach, global search algorithms are
constructed by specializing global search theories that
abstractly describe the shape of the search tree set up
by the algorithm. For the case study, we wished to
explore two alternative search ideas. While we found
a theory suitable for the first one in the literature,
the second one could not be realized with the documented
design knowledge. This lead us to develop a
new global search theory that needs a slightly modified
specialization procedure.
In Section 2, we introduce the baggage transportation
problem. Section 3 provides a brief review of
the global search theory and the KIDS approach. We
present its integration into a process model in Section
4. The design of two transportation schedulers is described
in Section 5. Optimizations are sketched in
Section 6, where we also discuss the resulting algo-
rithms. We summarize our experience with the approach
in Section 7.
Baggage Transportation Scheduling
We wish to develop a controller for the transportation
of baggage at an airport. Pieces of baggage
are transported from check-in counters to gates, from
gates to other gates, or from gates to baggage delivery
points. The controller must schedule the baggage
through the network in such a way that each piece
arrives at its destination in due time.
To simplify the problem, we do not consider on-line
scheduling of a continuous flow of baggage fed
into the system at the check-in counters, but schedule
all baggage checked-in at a particular point in time.
2.1 Domain Model
We model the transportation net as a directed
graph as shown in Figure 1. Check-in counters and
baggage delivery counters, gates and switches are represented
by nodes. We classify these in three kinds:
input nodes, transportation nodes and output nodes.
Check-in counters correspond to input nodes, switches
to transportation nodes and baggage delivery counters
to output nodes. Since gates serve to load and unload
airplanes, we represent them by an input and an output
node.
The edges of the graph model conveyor belts. The
capacity of a belt is the maximum amount of baggage
("total weight") it can carry at a time. Its velocity is
the time it takes to carry a piece from the start to the
node.
gate G2
gate G1 gate G3 gate G4
delivery D1 delivery D2
checkin C1 checkin C2 ceckin C3 checkin C4 checkin C5
Figure
1: Transportation network
A piece of baggage is described by its weight, source
and destination nodes, and its due time. Source and
destination are input and output nodes, respectively.
Weight and due time are positive natural numbers.
Due times are specified relative to the beginning of
the transportation process.
2.2 Problem Specification
Our task is to assign, to each piece of baggage, a
route through the network leading from its source to
its destination node in due time. To keep things sim-
ple, we require an acyclic network without depots at
Function transport plan( g : graph
where acyclic(g)-
returns
dom(q):feasible path(g; b; snd(q(b)))g))
Figure
2: Problem Specification
the transportation nodes. Thus, the only way to re-solve
scheduling conflicts that arise if capacities of conveyor
belts are exceeded is to delay baggage at source
nodes. A route therefore is a pair of a delay and a path
through the network. A plan maps pieces of baggage
to routes.
Attempting to find a plan for a particular set of
baggage makes sense only if there exists a feasible path
for each piece. This is a path p through the network
g leading from the source to the destination nodes of
a piece of baggage b.
feasible path(g; b; p) ()
is path(g; p) -
(1)
Note that we do not require a punctual schedule
to exist. We wish to find a plan in every case, punctual
and without delaying baggage at input nodes if
possible. Thus, we define a cost function based on
the criteria if all baggage is delivered in time and if
baggage is delayed at input nodes.
cost all baggage
in time no delays
1 yes no
no yes
3 no no
For example, imagine we have a suitcase b 1 at gate
and another one b 2
at C 4
in
Figure
1. Both have
weight 1. They are checked in for the same flight,
starting at gate G 2 . Let the transportation time of
each belt be one time unit and its capacity also be
one unit. As we have to avoid exceeding the capacity
of the belt leading to G 2 , a solution is to delay b 1 by
one time unit. This gives us the transportation plan:
Now we can set up the problem specification as
shown in Figure 2. For each acyclic graph g and set
of baggage bs with feasible paths through the graph,
we are interested in plans q such that all baggage is
scheduled dom(q)), the total amount of baggage
on a belt at any time does not exceed its capacity
(capacity bounded), and all assigned paths are feasi-
ble. From this set, we select a plan p with minimal
cost.
3 Design of Global Search algorithms
The basic idea of algorithm design in the KIDS approach
[12, 14] is to represent design knowledge in
design theories. Such a theory is a logical characterization
of problems that can be solved by an algorithmic
paradigm like "divide and conquer" or "global
search". Algorithm design consists of showing that a
given problem is an instance of some design theory. In
the following, we briefly summarize how global search
algorithms are developed in the KIDS approach. For
a full account, we refer the reader to [13, 14].
3.1 Design Theory
A problem specification is a quadruple
hD; R; I; Oi where D is the input domain and R is
the output range of the function f to synthesize.
The predicate I : D ! Bool describes the admissible
inputs and O describes the input/output behavior
of f . Hence, f : D ! R is a solution to P
design theory extends
a problem specification by additional functions.
It states properties of these functions sufficient to formulate
a schematic algorithm that correctly solves the
problem.
The global search theory of Figure 3 describes the
search for an optimal solution with respect to a cost
function c. The basic idea is to split search spaces containing
candidate solutions into "smaller" ones until
solutions are directly extractable.
The
R is the type of search space descriptors, -
I
defines legal descriptors. For an input x, -
r 0 and Split
describe the search tree for solutions z with O(x; z):
its root is -
r 0 (x), the initial search space; a descendent
relation on nodes is given by Split:
if - s is a (direct) subspace of - r for an input x. The
solutions obtainable by looking at a single node - r of
the search tree are described by Extract(z; - r).
By axiom GS3, Satisfies(z; -
r) describes the solutions
z contained in a search space - r that can be found with
finite effort. There must exist a finite path in the
search tree from - r to a search space -
s from which z
can be extracted. Split is defined by
Since we wish to find a globally optimal solution,
GS2 requires that all feasible solutions are contained in
Sorts D;R; -
Operations
R
Satisfies
R \Theta -
Extract
Axioms
GSC: - is total ordering on C
Function
where I(x)
returns
Function
returns
Figure
3: Global search theory and algorithm schema
the initial search space. Axioms GS0 and GS1 ensure
that all considered search spaces are legal.
The program shown on the bottom of Figure 3 provides
a schematic algorithm consistent with the global
search theory. The function F computes an optimal
solution z for a given input x by initiating a global
search in the initial search space -
(x). The actual
search algorithm is implemented in f gs. It minimizes
over the solutions that are directly extractable from
the input search space -
r and the ones found by recursive
search in spaces - s obtained by splitting - r.
Necessary filters provide the basis to optimize the
code gained by instantiating the algorithm schema.
A necessary filter \Phi is used to prune branches of the
search tree that cannot contain solutions. It satisfies
the implication
r) (2)
3.2 Algorithm Design
How can we find a global search algorithm for a
given problem specification? We have to find a search
space description -
R and operations -
and Extract such that the global search axioms are ful-
filled. In the KIDS approach this is done by referring
to knowledge about search strategies on concrete data
structures that is formalized in a library of general
global search theories 2 . Examples are theories to enumerate
all sequences over a finite set and to enumerate
all mappings between finite sets. A global search
theory for a given problem is constructed by specializing
a theory from the library. A problem theory
specializes to a problem theory
Constructively verifying the existential quantifier in
(3) gives us a systematic way to find a global search
theory for the problem at hand. In this way, the structure
of the search tree expressed in A is adapted to B
and gives us an algorithm for B that is an instance
of the algorithm schema in Figure 3. In general this
algorithm is very inefficient, but has a high potential
for optimization which can be exploited by deriving
necessary filters, by program transformation, and by
data structure refinement.
4 A Process Model
Our presentation of the application domain theory
and problem specification in Section 2 only describes
the final result of the specification effort. To develop
the domain theory is one of the major tasks if not the
most complex and time consuming one in the KIDS
approach. Much of its complexity stems from two requirements
we demand of the domain theory: it must
not only make precise the informal - usually incomplete
and sometimes inconsistent - ideas about the
nature and context of the problem, but it must also
be formulated so as to aid and not impede the subsequent
design process.
As a consequence, it is very unlikely that a satisfying
domain theory can be built from scratch. This observation
led us to integrate the KIDS approach with
the prototyping and spiral models of software engineering
[2]. One cycle of development, as sketched in
Figure
4, has three phases. The first is concerned with
establishing or enhancing the domain theory, the second
produces code, and in the third phase code and
theory are tested and validated.
We found it useful to build the first draft of the domain
theory in parallel with a prototype. In this early
phase, shaded gray in Figure 4, the domain theory is
not rich enough to apply algorithm design knowledge
from design theories. Building a prototype enables us
to get a deeper understanding of the problem and the
We used Appendix A of [13]
code
prototype
validation / test domain theory
algorithm design
optimization
specification
validation
theory
validation
domain
application
test
Figure
4: Process model
essential properties of the application area. It helps
us to build a complete domain theory and to avoid
dead-end developments.
The way in which the domain model is expressed,
the data structures used, and the properties stated,
can have much influence on the ease with which algorithm
design can be carried out. Thus, what seems
to be one cycle of design in Figure 4, may in practice
require several rounds of refining the domain theory
until the formalized notions smoothly fit with the design
theory we wish to use.
One example from the baggage transportation case
study is the way we modeled delays in routes (cf. Section
2.2). In an early version of the domain theory,
we described them by repetitions of the input nodes
in paths, each occurrence of the input node denoting
a delay of one time unit. This forced us to introduce
predicates to characterize legal routes, and we could
not use an acyclic graph model. When we decided to
reformulate the theory and make delays explicit the
theory became much more elegant and further design
was much easier.
The process of theory refinement perpetuates as we
derive filters and optimize code. The validation and
test phases also serve us to validate the code with respect
to properties that are not captured by the design
knowledge put to our disposal in the KIDS approach.
5 Two Ways to Find Transportation
Plans
Looking at the sort of transportation plans,
nat \Theta seq(vertex))
suggests two strategies to search for solutions to our
scheduling problem.
1. Domain Extension. Start with the initially empty
map and successively extend it by assigning possibly
delayed feasible paths to baggage.
2. Image Modification. Start with the map that assigns
their source nodes and no delay to all bag-
successively modify the assigned routes by
extending paths or increasing delays.
Both strategies enumerate all feasible transportation
plans. In the KIDS approach, search strategies
are provided in a library of general global search the-
ories. Algorithm design proceeds by specializing one
of these to the problem at hand. The first condition
in (3), RB ' RA , suggests to match the output domains
of the problem specification with the ones of the
library theories to find candidates to specialize.
When we began algorithm design for the transportation
problem, our initial idea was to use the
image modification strategy, but there is no general
global search theory documented in the KIDS library
[13] that models image modification. Instead, we
found a theory that describes domain extension. This
motivated us to explore both approaches.
5.1 Domain Extension
The global search theory gs finite mappings enumerates
all maps from a finite set U to a finite set
.
F 7! gs finite mappings
R 7! map(ff; fi)
I 7! -hU; V i:jU
O 7! -hU; V
R 7! set(ff) \Theta set(ff) \Theta map(ff; fi)
I 7! -hU; V
Satisfies 7!
Extract 7!
A search space is described by a partition of U into
two sets S and T , and by a map M from S to V . M
can be completed to a map from U to V by assigning
elements of V to all elements of T . Split performs one
of these assignments: it picks arbitrary elements a and
b of T and V , respectively, and extends M by a 7! b.
We find instantiations for the type variables in
gs finite mappings by unifying its output domain
with the one of transport plan.
ff 7! baggage
route
We can specialize gs finite mappings to
transport plan if we can constructively verify
(3) for these theories, i.e. we must find expressions in
and bs for U and V to prove
dom(M):feasible path(g; b; snd(M(b)))
The predicate Map is defined by
Comparing the right hand side of this definition with
and the definition (1) of feasible paths
suggests to use the sets
U 7! bs
to specialize gs finite mappings.
The use of the upper bound md(g; bs) on delays
makes the set of routes finite, and thus ensures termination
of the algorithm. We discuss termination
further in Section 7.
Since there are feasible paths for all baggage in
bs (cf. the precondition of transport plan in Figure
2), we can assign to md(g; bs) the sum of the times
needed to traverse a feasible path for each piece of
baggage. Applying the substitution for ff, fi, U , and
V to gs finite mappings gives us a global search theory
for transport plan.
R 7! set(baggage) \Theta set(baggage) \Theta plan
I 7!-hg;
Satisfies 7!
Extract 7!
The resulting search strategy assigns complete
routes to one piece of baggage after another. Without
further optimization, Split assigns arbitrary routes to
pieces of baggage and only when a complete plan can
be extracted is tested whether the assigned routes are
feasible. An obvious way to prevent infeasible assignments
in the first place is to develop a necessary filter.
Instantiation of (2) and the fact that capacity bounded
is monotonic in domain extensions of M gives us
5.2 Image Modification
There is no global search theory documented in
[11, 13, 14] that supports searching for maps by image
modification. So we developed a new theory for this
purpose.
Abstracting from the concrete scheduling problem,
the image modification strategy can be sketched as
follows: The images of a given map (the initial sched-
ule) are increased along the various degrees of freedom
that are given by the range type of the map. A suitable
successor relation on the elements of the range
type can be used to describe the "direction" in which
to increase the images of the map. This idea is formalized
in gs parallel mappings 3 .
F 7! gs parallel mappings
D 7! map(ff; fi) \Theta set(fi \Theta fi)
R 7! map(ff; fi)
I 7!-hM;Si:
O 7!-hM;Si; N:
R 7! map(ff; fi) \Theta set(fi \Theta fi)
I
Satisfies 7! -N; hM;Si:(8x 2 dom(M):M(x) S N(x))
Extract 7! -N;
The inputs are a map M and a successor relation
S. The search enumerates all maps with the same
domain as M and images that are extended "along"
S, i.e. they are in the reflexive and transitive closure
S of S. The domain of M must be finite, and S must
be non-dense to ensure that GS3 holds. The relation S
is anti-reflexive to ensure progress in the search. Note
that the search spaces (of sort -
R) do not only contain
the map built so far but also the relation S. This
is necessary to describe (in Satisfies) if a solution is
contained in a search space.
5.2.1 Data Type Driven Specialization
Having captured our search idea in
gs parallel mappings, we want to specialize this
theory to the transportation planning problem. It
turns out that the corresponding instance of (3)
does not help much in systematically finding M and
S, because we cannot refer to the structure of the
3 x S y is a notation for hx; yi 2S.
range type fi in gs parallel mappings. Still, it is not
desirable to develop a global search theory for the
specific structure of the range type of the transport
schedules because the newly developed theory should
be sufficiently abstract so as to be applicable to a
broad range of problems. To solve this dilemma, we
propose to specialize gs parallel mappings in two
steps. The first step determines a suitable successor
relation S while the second step finds a substitution
for M .
To determine S, we first find substitutions for the
type variables ff and fi. For the transportation problem
we get
ff 7! baggage
Now we can analyze the range type nat \Theta
seq(vertex) to find a canonical successor relation on its
elements based the basic types it is composed of. We
know the usual successor function on natural numbers,
and a canonical way to extend sequences is to append
an element. In analogy to lexicographical orderings on
pairs, we construct a successor relation by extending
either element of a pair. Thus, we define S by
hn;
We substitute (5) for S in (3). After simplification we
get
capacity
As in Section 5.1, we can now easily determine a substitution
for M and get the map that assigns to each
b in bs the non-delayed path only consisting of the
source node of b.
Like the algorithm of Section 5.1, the one working
with image modification has a high potential for
optimization and its development proceeds by filter
construction and optimization.
6 Optimization and Results
The algorithms resulting from the instantiation of
the scheme in Figure 3 are so inefficient that optimization
is absolutely necessary. The efficiency of the
resulting code heavily depends on the adequate choice
of filters and program transformations.
Important optimizations on the resulting algorithm
are the introduction of a priority queue on search
spaces and simplifications on often used predicates,
such as capacity bounded. We have also eliminated
common subexpressions and introduced an analysis of
the transportation net with respect to the input baggage
to eliminate vertices that do not lie on feasible
paths.
The ordering on search spaces used in the priority
queue encodes a heuristic to determine which node of
the search tree to consider next. We have used one
based on cost, total delay, and length of the paths in
a plan.
Test runs of the algorithms show that, despite the
optimizations, performance of both is still poor. They
also reveal that the algorithm based on image modification
is about a factor of two faster than the one
based on domain extension.
The size of theories and programs are summarized
in
Table
1.
Document Lines
Library of Specifications 490
Domain Theory 240
Algorithm Theory 350
Implementation 920
Table
1: Size of theories and programs
All specifications are written in the specification
language SPECTRUM [3]. We use an existing library
of basic spectrum specifications such as natural
numbers, sets, sequences, directed graphs and
others. Based on these modules, the domain theory
comprises about 240 lines. The algorithms extend the
domain theory by 110 lines. Both schedulers are implemented
in the functional language OPAL [10]. The
translation of specifications into executable code and
its optimization produces about 920 lines of code for
the OPAL program. The code is well-structured and
highly reusable. Both implementations share most of
the code which facilitates exploring alternatives.
The case study required an effort of approximately
9 person months. We spent about one third of that
time to learn the KIDS approach. Approximately 75%
of the remaining time were devoted to building the
domain theory.
There exists a number of approaches to algorithm
design and program synthesis, e.g. [1, 4, 8, 9]. We have
chosen the KIDS approach for our case study because
it provides design steps that reflect significant design
decisions of programmers and are described precisely
by logical theories. In this way, they are good reference
points for software engineers who wish to learn and
use them. However, we did not use the KIDS system
because we wanted to have full control over the design
process and adapt it to our needs if necessary.
Transportation scheduling. Our case study relates
to the research on design of transportation schedulers
at Kestrel [15, 16]. They study schedulers that
assign trips to resources like planes, ships, and trucks
to meet movement requirements. In this setting, trips
fully occupy resources for an interval of time, i.e. the
load of a resource cannot be extended during a trip.
Furthermore, a trip changes the availability of a re-
source: the destination of one trip becomes the source
of the next one. In baggage transportation, how-
ever, load of resources can continually change as baggage
flows through the net, but source and destination
points of a resource remain fixed in time.
Another difference lies in the focus of our work. For
several years, a highly specialized theory on transportation
scheduling has been developed at Kestrel
with the aim to produce extremely efficient schedulers.
Recently, this has even led to a refinement of the abstract
global search theory [16]. The purpose of our
case study, in contrast, has been to study in how far
the KIDS approach as documented in the literature
can support programmers who have no particular experience
with the approach, to design algorithms for
a non-trivial problem.
Process model. The steps in designing a global
search algorithm: specializing a theory, deriving fil-
ters, and applying optimizing program transforma-
tions, provide a clear separation of concerns during
design. Specialization determines the basic structure
of the search, necessary filters exploit properties of
the application domain, and only the final program
transformations and data type refinements eliminate
redundancies in the code and "fuse" filters with the
basic search structure to gain efficiency.
Each of these tasks corresponds to one cycle in the
process model that we introduced in Section 4. Thus
the model helps programmers to focus activities on a
particular task and to avoid introducing certain design
ideas at the "wrong" time into the development. In
early attempts to design the algorithm of Section 5.2,
we tried to introduce optimizations too early - trying
to generate delayed routes only if necessary - which
totally messed up our design.
The first phase of the development, before a sufficiently
complete application domain theory is avail-
able, is the most complex part of the process. We
found prototyping useful to understand the problem
domain, but techniques to guide theory development
remain to be established.
With a domain theory at hand, the KIDS approach
is well suited to construct prototypes in little time.
This supports exploring alternative designs.
Termination. In the global search theory we have
used, the issue of termination of the constructed algorithms
is not addressed. This lead us to the somewhat
unnatural introduction of the upper bound md(g; bs)
on delays (cf. Section 5).
Termination of global search algorithms can be
spoiled in two ways. There may be branches of the
search tree with infinite length, or there may be nodes
with infinitely many children. In [14], a well-founded
ordering is introduced into the abstract global search
theory to prevent infinite chains of Split-operations.
Kreitz [7] has formalized global search in the Nuprl
type theory [4]. He prevents infinite branchings of
the search tree by using only finite sets in his for-
malization. He introduces wf-filters to prune infinite
branches and proposes to provide a collection of wf-
filters for each theory.
There are methodological reasons not to require
termination of all global search theories. Both theories
used in Section 5 enumerate an infinite number
of maps. Still, we would appreciate a systematic way
that relieves programmers of dealing with termination
on-the-fly.
Dealing with incomplete design knowledge. As
we have seen in Section 5.2, it is not unlikely that the
knowledge expressed in design theories fails to support
a particular design idea. Although we are not aware
of systematic support for constructing new theories in
KIDS, it is still worthwhile to stick to the approach
and develop a new design theory that describes the
desired search strategy in an abstract way. In [5], we
decided to construct the problem specific algorithm
theory of Section 5.2 in one step and to manually verify
it against the abstract global search theory. This
decision was mainly due to lack of experience and increased
the complexity of the task considerably. More-
over, it led to a less efficient algorithm.
It seems to be unlikely to find "practically com-
plete" knowledge bases for software construction sys-
tems. Such systems should be designed to ease routine
extension of their knowledge bases. In [6], a generic
system architecture based on the notion of strategies
is proposed. Strategy modules have a clearly defined
interface to the system kernel, so new ones can be
integrated into the system in a routine way. The system
Specware [17] under development at Kestrel also
seems to allow for a modularized and easily extendible
knowledge base.
Constructing a new global search theory is a non-trivial
task and deserves support if the approach shall
be applied routinely. A starting point may be the
observation that search strategies often seem to derive
from the structure of the output domain R.
Acknowledgement
. We would like to thank David
Basin, Maritta Heisel and Burkhart Wolff for fruitful
discussions. Klaus Didrich and Maritta provided
comments on a draft of this paper.
--R
Artificial Intelligence
A spiral model of software development and enhancement.
The requirement and design specification language Spectrum.
Implementing Mathematics with the Nuprl Proof Development System.
Eine Fallstudie zur Entwicklung korrek- ter Software: Steuerung einer Gep-ackf-orderanlage
Tool support for formal software development: A generic ar- chitecture
Deriving Programs that Develop Programs.
Automating Software Design.
A deductive approach to program synthesis.
The programming language OPAL.
Kestrel Interactive Development System
Algorithm theories and design tactics.
Structure and design of global search al- gorithms
KIDS: A semiautomatic program development system.
Transformational approach to transportation scheduling.
Synthesis of high-performance transportation schedulers
Specware: formal support for composing software.
--TR | program synthesis;KIDS;scheduling;formal methods |
592010 | Extending Design Environments to Software Architecture Design. | Designing a complex software system is a cognitively challenging task; thus, designers need cognitive support to create good designs. Domain-oriented design environments are cooperative problem-solving systems that support designers in complex design tasks. In this paper we present the architecture and facilities of Argo, a domain-oriented design environment for software architecture. Argos own architecture is motivated by the desire to achieve reuse and extensibility of the design environment. It separates domain-neutral code from domain-oriented code, which is distributed among active design materials as opposed to being centralized in the design environment. Argos facilities are motivated by the observed cognitive needs of designers. These facilities extend previous work in design environments by enhancing support for reflection-in-action, and adding new support for opportunistic design and comprehension and problem solving. | Figure
1.Design environment facilities of Janus, adapted from Fischer, 1994 .
nizational guidelines, and the opinions of fellow project stakeholders and domain
experts.
Design environments suchasFramer Lemke and Fischer, 1990 , Janus Fischer,
1994 , Hydra Fischer et al., 1993 , and VDDE Voice Dialog Design Environment
Sumner, Bonnardel, and Kallak, 1997 support re ection-in-action. Figure 1 shows
facilities of this family of design environments. The domain-oriented construction
kit facility allows users to visualize and manipulate a design. The construction analyzer
facility critiques the design to give design feedback that is linked to hypertext
argumentation. The goal speci cation facility helps to keep critics relevanttothe
designer's objectives. Re ection-in-action is also supported by simulation facilities
that allow what-if analysis as a further design evaluation. A catalog of example
designs can be accessed via the catalog explorer facility.
Designers will gain the most from design feedback that is both timely and relevant
to their current design task. Design environments can address timeliness by linking
critics to a model of the design process. For instance, Framer uses a checklist
to model the process of designing a user interface window. At a given time the
designer works on one checklist item and only critics relevant to that item are
active. Design environments can address relevance by linking critics to speci cations
of design goals. For instance, Janus and Hydra allow the designer to specify goals
for kitchen oorplans, and thus activate only those critics relevant to stated design
goals. Furthermore, Hydra uses critiquing perspectives i.e., explicit critiquing
modes to activate critics relevanttoany given set of design issues and deactivate
irrelevant critics.
2.2. Architectural Styles
Work on software architecture Perry and Wolf, 1992 has focused on representing
systems as composed of software components and connectors Garlan and Shaw,
1993 . Architectural styles constrain and inform architectural design by de ning
the types of components and connectors available and the ways in which they may
be combined Abowd, Allen, and Garlan, 1993 . Styles can be expressed as a set of
style rules. A simple architectural style is pipe-and- lter which de nes components
to be batch processes with standard input and output streams, and connectors to
be data pipes. One pipe-and- lter style rule is that the architecture should contain
no cycles.
Argo supports the C2 architectural style Taylor et al., 1996 . C2 is a component-
and message-based style designed to model applications that have a graphical user
interface. The style emphasizes reuse of UI components such as dialogs, structured
graphics models, and constraint managers Medvidovic, Oreizy, and Taylor, 1997 .
The C2 style can be informally summarized as a layered network of concurrent
components that communicate via message broadcast buses. Components may only
send messages requesting operations upward, and noti cations of state changes
downward. Buses broadcast messages sent from one component to all components
in the next higher or lower layer. Each component has a top and bottom interface.
The top interface of a component speci es the noti cations that it handles, and
the requests it emits upward. The bottom interface of a component speci es the
noti cations that it emits downward, and the requests it handles.
2.3. Software Architecture Design Tools
Design tools in the domain of software architecture have tended to emphasize analysis
of well-formedness and code generation. The Aesop system allows for a style-
speci c design tool to be generated from a speci cation of the style Garlan, Allen,
and Ockerbloom, 1994 . The DaTE system allows for construction of a running
system from an architectural description and a set of reusable software components
Batory and O'Malley, 1992 . Although not a software architecture tool,
AMPHION is similar in that it allows users to enter a graphical speci cation from
which the system can generate a running program Lowry et al., 1994 . Eachof
these systems provides support for design representation, manipulation, transforma-
tion, and analysis, but none of them explicitly supports architects' cognitive needs.
Argo can generate code to combine software components into a running system;
however, the main contribution of Argo to the software architecture communityis
its emphasis on cognitive needs.
KBSA ADM Benner, 1996 is a software design environment that embodies the
results of many research projects stemming from a seminal vision of knowledge-based
software development support Green et al., 1983 . KBSA ADM has many
features in common with Argo, including critics, a to do" list, multiple coordinated
models of the system under design, and process modeling. KBSA ADM is intended
to package previous research results into a full-featured software developmentenvi-
ronment. In contrast, Argo is intended to explore possible features that explicitly
support identi ed cognitive needs. Support for cognitive needs in both KBSA ADM
and Argo is inspired by previous work in design environments, however we believe
that Argo has a more integrated, reusable, and scalable infrastructure that yields
better cognitive support.
6 ROBBINS, HILBERT, AND REDMILES
Critics with Design Knowledge
Feedback
Control
Situated
Analysis
Internal Design Architect
Representation Perspectives
Design Interactions
Figure
2.Design environment facilities of Argo.
3. Overview of Argo
To Do
Decision
Model
Process
Model
Figure
2 provides an overview of selected facilities of the Argo software architecture
design environment. The architect uses multiple, coordinated design perspectives
Figure
3 to view and manipulate Argo's internal representation of the architecture
which is stored as an annotated, connected graph. Critics monitor the partial
architecture as it is manipulated, placing their feedback in the architect's to do"
list
Figure
4 . Argo's process model Figure 5 serves the architect as a resource
in carrying out an architecture design process, while the decision model lists issues
that the architect is currently considering. Criticism control mechanisms use that
decision model to ensure the relevance and timeliness of feedback from critics.
For comparison, Figure 1 shows facilities of the Janus family of design environ-
ments. Like Janus, Argo provides a diverse set of facilities to support re ection-in-
action including construction and critiquing mechanisms. Argo, however, extends
these facilities byintegrating them with a exible process model and to do" list to
explicitly support opportunistic design, and multiple, coordinated design perspectives
to aid in comprehension and problem solving. Each of these cognitive theories
and the facilities that support them are discussed in Section 5.
The subsections below describe each of Argo's facilities. The last subsection
provides a usage scenario that describes howanarchitect mightinteract with Argo.
3.1. Critics
Critics support decision making by continuously and pessimistically analyzing partial
architectures and delivering design feedback. Each critic performs its analysis
independently of others, checking one predicate, and delivering one piece of design
feedback. Critics provide domain knowledge of a varietyoftypes. Correctness
critics detect syntactic and semantic aws. Completeness critics remind the architect
of incomplete design tasks. Consistency critics point out contradictions within
Figure
3. Three architecture design perspectives: the Component Component perspective top
shows conceptual component communication, the Classes perspective middle shows modular
structure, and the Resource Component perspective bottom shows machine and operating system
resource allocation. The small window in the lower left shows the running KLAX game,
represented by this architecture.
8 ROBBINS, HILBERT, AND REDMILES
the design. Optimization critics suggest better values for design parameters. Alternative
critics present the architect with alternatives to a given design decision.
Evolvability critics consider issues, such as modularization, that a ect the e ort
needed to change the design over time. Presentation critics look for awkward use of
notation that reduces readability. Tool critics inform the architect of other available
design tools at the times when those tools are useful. Experiential critics provide reminders
of past experiences with similar designs or design elements. Organizational
critics express the interests of other stakeholders in the development organization.
These types serve to describe and aggregate critics so that they may be understood
and controlled as groups. Some critics maybeofmultiple types, and new types
may be de ned, as appropriate, for a given application domain. Altogether, we
have authored over fty critics, including examples of eachtype. Some examples
of architecture critics are given in Table 1.
We expect critics to be authored by project stakeholders for various reasons. An
initial set of critics is developed by a domain engineer when constructing a domain-oriented
design environment. Practicing architects may de ne critics to capture
their experience in building systems and distribute those critics to other architects
in their organization, or keep them for their own use in the future. A similar
authoring activitywas observed by Gantt and Nardi who found that groups of
CAD tool users often had members they called gardeners" that assumed the role of
codifying solutions to local problems Gantt and Nardi, 1992 . Practicing architects
may also re ne existing critics by adding special cases to their predicates or by
modifying their feedback. For example, one way for an architect to resolve criticism
is to suggest a modi cation to the critic that raised the issue. Researchers may also
de ne critics to support an architectural style. Existing literature on architectural
styles and system design is a rich source of advice that can be made active via critics.
Many organizations already have design guidelines that currently require designers
to manually check their design. Software componentvendors may de ne critics
to add value to the components that they sell and to reduce support costs. For
example, a critic supplied with an ASCII spell checking component might suggest
upgrading to a Unicode version if the architect declares that internationalization is
a goal.
Interactions among stakeholders in the design community can guide the evolution
of critics. If the architect does not understand a particular critic's feedbackor
believes it to be incorrect, he or she may send structured email through Argo to
the author of that critic. This opens a dialog between knowledge providers i.e.,
domain experts and consumers i.e., practicing architects so that the critics may
be revised to be more relevant and timely. In this way critics can be thoughtofas
pro-active answers" in an organizational memory Terveen, Selfridge, and Long,
1993; Ackerman and McDonald, 1996 . Possible improvements to Argo's support
for organizational memory include associating multiple experts with each critic,
prioritizing experts based on organizational distance, and tracking email dialogs so
that requests for changes are not forgotten.
Table
1. Selected Argo Architectural Critics
Name of Critic
Critic Type Decision Category
Missing Memory Rqmts
Completeness Machine Resources
Component Choice
Alternative Component Selection
Too Many Components
Evolvability Topology
Hard Combination to Test
Organizational Component Selection
Generator Limitation
Tool Component Selection
Not Enough Reusable Components
Consistency Reuse
Avoid Overlapping Nodes
Presentation Readability
Questionable
Experiential Portability
Explanation of Problem
The memory required to run this component
has not been speci ed.
There are other components that could t" in
place of what you have: list of components.
There are too many components at the same level
of decomposition to be easily understood.
If you need to use these components together,
please make arrangements with the testing manager.
The default code generator cannot make
full use of this component.
The fraction of components marked as being
reusable is belowyour stated goals.
Overlapping nodes does not haveany meaning in
this notation and obscures node labels.
Your colleague, name of person, had di culty
using this component under name of OS.
3.2. Criticism Control Mechanisms
Formalizing the analyses and rules of thumb used by practicing software architects
could produce hundreds of critics. To provide the architect with a usable amountof
information, a subset of applicable critics must be selected for execution at any given
time. Critics must be controlled so as to make e cient use of machine resources,
but our primary focus is on e ectiveinteraction with the architect.
Criticism control mechanisms are predicates used to limit execution of critics to
when they are relevant and timely to design decisions being considered by the ar-
chitect. For example, critics related to readability should not be active when the
architect is trying to concentrate on machine resource utilization. Computing relevance
and timeliness separately from critic predicates allows critics to focus entirely
on identifying problematic conditions in the product i.e., the partial architecture
while leaving cognitive design process issues to the criticism control mechanisms.
This separation of concerns also makes it possible to add value to existing critics
by de ning new control mechanisms.
3.3. The To Do" List
Design feedback from large numbers of critics must be managed so as not to overwhelm
or distract the architect. The to do" list user interface Figure 4 presents
Figure
4.The architect's to do" list.
design feedback to the architect in a non-disruptiveway. When a to do" item is
added to the list, the architect may act on it immediately,ormay continue manipulating
the design uninterrupted. To do" items come from several sources: critics
post items presenting their analyses, the process model posts items to remind the
architect to nish tasks that are in progress, and the architect may post items as
reminders to return to deferred design tasks. Architects may address items in any
order. Tabs on the to do" list lter items into categories.
Each to do" item is tied into the design context in whichitwas generated. That
context includes the state of the design, background knowledge about the domain,
and experts to contact within the design community. When the architect selects
an item from the upper pane of the To Do List" window, the lower pane displays
details about the open design issue and possible resolutions. Double-clicking on an
item highlights the associated or architectural elements in all visible
design perspectives. Once an item is selected, the architect may manipulate the
critic that produced that item, send email to its author, or followhyperlinks to
background information.
3.4. Design Perspectives
A design perspective de nes a projection or subgraph of the design materials
and relationships that represent a software architecture. Perspectives are chosen to
present only architectural elements relevant to a limited set of related design issues.
Figure
3 shows three perspectives on a system modeled in Argo. The systemshown is a simple video game called KLAX in which falling, colored tiles must be
arranged in rows and columns. In the Component Component perspective, nodes
represent software components and connectors, while arcs represent communication
pathways. Small circles on the components represent the communication ports
of each component. The Resource Component perspective hierarchically groups
modules into operating system threads and processes. The Classes perspective
maps conceptual components to classes in the hierarchy of programming language
classes that implement them.
3.5. Process Model
Argo uses a process model to support architects in carrying out design processes.
Design processes are di cult to state prescriptively because they are exploratory,
tend to be driven by exceptions, and often change when new requirements, con-
straints, or opportunities are uncovered Cugola et al., 1995 . Rather than address
traditional process modeling concerns e.g., scheduling and enactment , our approach
focuses on cognitive issues of the design process by annotating each task
with the types of decisions that the architect must consider during that task. We
use a simpli ed version of the IDEF0 process notation IFIP, 1993 that models
dependencies between tasks without prescribing a temporal ordering.
To support cognitive needs, Argo must maintain a model of some aspects of the
architect's state of mind. Speci cally, Argo's decision model lists decision types that
the architect is currently considering. This information is used to control critics
so that they are relevant and timely to the tasks at hand. The primary source
of information used to determine the state of the decision model is decision type
annotations on tasks in the process model. The architect may edit the decision
model directly. Design manipulations performed by the architect can also indicate
which decisions are currently being considered.
Figure
5 shows an example coarse-grained architecture design process model. Two
of the tasks are to choose machine resources Choose Rsrcs and to choose reusable
components Choose Comps . The second task is annotated with the decision type
Reuse. When the architect indicates that he or she is working on choosing reusable
components, these annotations cause Argo to enable critics that support reuse
decisions. The design process model shown in Figure 5 is a fairly simple one, partly
because the C2 style does not impose any explicit process constraints, and partly
because this example does not consider issues of organizational policy. In practice,
the process would be more complex.
3.6. Usage Scenario
In this section we describe howanarchitect mightinteract with Argo while working
through several steps in transforming the basic KLAX game shown in Figure 3
Figure
5.A model of the design process.
into a multi-player spelling game. The basic KLAX game uses sixteen separate
components, including components that generate colored tiles, display those tiles,
and determine when the player has aligned matching tiles. The spelling game
variation will use the same basic architecture with new components to generate
and display letters and to determine when the player has aligned letters to spell a
word.
While working on the architecture of the basic KLAX game, the architect places
the TileArtist component in the architecture. Shortly thereafter, an alternative
critic posts a to do" item indicating that another component from the company's
library, LetterArtist, de nes the same interface and should be considered as an
alternative. The architect knows that LetterArtist is not appropriate for basic
KLAX and takes no action, but the suggestion inspires the idea of building a
spelling variation, so he or she leaves the item on the to do" list. Later, when
basic KLAX is completed, the architect reviews any remaining to do" items and
is reminded to investigate the spelling variation. He or she replaces TileArtist
with LetterArtist and de nes new components for NextLetter and Spelling
to replace NextTile and Matching, respectively. While the architect is replacing
these components the architecture is temporarily in an inconsistent state. Critics
that check for consistency between componentinterfaces may post to do" items
describing these interface mismatches, but those items are automatically removed
when the new components are connected and their interfaces are fully de ned.
Adding two new components to the architecture may cause a consistency critic to
re if the current percentage of reused components falls below stated reuse goals.
Satis ed with the choice of components and their communication relationships,
the architect uses Argo's process model to decide what to do next. The process
model contains a task for choosing reusable components and a task for allocating
machine resources which depends on its output. At this point the architect decides
to work on machine resource allocation and marks that task as in progress. Doing so
enables critics that support design decisions related to machine resource allocation,
and three new to do" items are posted indicating that the three new components
Figure
6.Architect's workspace after modifying KLAX.
have not been allocated to any host or operating system process. The architect
then turns to the Resource Component design perspective and nds that the nodes
representing TileArtist, NextTile, and Matching have been removed and new
nodes for LetterArtist, NextLetter, and Spelling have been added but not
connected to any process or host. The architect connects the new components
as the old ones were connected. At this point it occurs to the architect that a
server-side Spelling component might be too slow in a future multi-player product,
so he or she connects Spelling to the game client process instead. By viewing
the Component Component perspective and Resource Component perspective the
architect is able to understand interactions between two aspects of the architecture.
Figure
6 shows what the architect would see at this point: two design perspectives
are open and several new potential problems have been reported by critics. The
selected to do" item arose because the Spelling component requires more memory
than is available.
In this usage scenario the architect has engaged in a constructive dialog with
design critics: critics prompted the architect with new possibilities and pointed
out inconsistencies. The architect used Argo's process model to help decide which
design task to address next, and used two design perspectives to visualize and
manipulate aspects of the architecture relevanttotwo distinct design issues. These
14 ROBBINS, HILBERT, AND REDMILES
later two aspects of the scenario highlight new facilities of Argo that are not found
in previous work on DODEs.
4. Implementation
This section discusses the implementation of Argo. We base our discussion on two
prototypes: an initial prototype implemented in Smalltalk and the currentversion
implemented in Java. First we discuss the core elements of our
criticism control mechanisms, perspectives, and processes. We then describe Argo's
own architecture and the representation of architectures being designed with Argo.
4.1. Critics
In Argo, a critic is implemented as a combination of 1 an analysis predicate,
type and decision category attributes for determining relevance, and 3 a to
do" list item to be given as design feedback. The stored to do" list item contains
a headline, a description of the issue at hand, contact information for the critic's
author, and a hyperlink to more information. We encode critics as programming
language predicates. Determining which languages are best suited for expressing
critics is a topic for future research. Each critic is associated with one type of
design material and is applied to all instances of that type. Critics may access the
attributes of the design materials they are applied to, and traverse relationships
to other design materials. Critic predicates are written pessimistically: unspeci ed
design attributes are assumed to havevalues that cause the critic to re. Table 2
presents one critic in detail.
Argo provides a critic run-time system that executes critics in a background
thread of control. Critics may be run periodically or be triggered by speci c architecture
manipulations. During execution a critic applies its analysis predicate to
evaluate the design and posts a copy of its to do" item, if appropriate. Another
thread of control periodically examines each item on the architect's to do" list and
removes items that are no longer applicable.
4.2. Criticism Control Mechanisms
Criticism control mechanisms are implemented as predicates that determine if each
critic should be enabled. Argo uses several criticism control mechanisms, any one
of which can disable a critic. In each of the following examples, criticism control
mechanisms decide which critics should be enabled by comparing information
provided by the architect to attributes on the critics. Architects may hush" individual
critics, rendering them temporarily disabled, if their feedback is felt to be
inappropriate or too intrusive. This allows architects to defer the issues raised bya
particular critic without risk of leaving the critic disabled inde nitely. Argo's user
Table
2. Details of the Invalid Connection critic
Attribute Value
Name Invalid Connection
Design Material Component
Types f Correctness g
Decision Categories f Component Selection, Message Flows g
Hushed False
Smalltalk Predicate :comp invalidServices
inputs , comp outputs
select: :s s isSatisfied not .
invalidServices isEmpty not.
Feedback This component needs the following messages be sentor
received, but they are not present: a list of messages
Author jrobbins ics.uci.edu
MoreInfo http: www.ics.uci.edu pub arch argo v05 docs .
interface allows groups of critics to be enabled or disabled bytype. This allows the
architect to control groups of critics easily. Another control mechanism checks the
critic's decision types against those listed in the decision model. This keeps critics
relevant to the tasks at hand.
Criticism control mechanisms normally enhance relevance and timeliness. How-
ever, relevance and timeliness can be reduced if criticism control mechanisms use
incorrect information. For example, if the architect mistakenly indicates that a
given issue is not of interest, then the architect will see no feedback related to that
issue and might mistakenly assume that the architecture has no problems. This
situation can be avoided byhushing critics instead of disabling them and by using
awell designed process that reminds the architect to review all the issues. Argo
advises the architect to check the decision model when the to do" list becomes
overly full or if too many to do" items are being suppressed. The number of suppressed
to do" items is computed by occasionally running disabled critics without
presenting their feedback.
4.3. Design Perspectives
In Argo, perspectives are objects that de ne a subgraph of the design materials in
the current design. Twotypes of perspectives are de ned in Argo: predicate and ad-
hoc. Predicate perspectives contain a predicate that selects a subgraph of the design.
Ad-hoc perspectives contain an explicit list of design materials and relationships.
This latter mechanism allows for manual construction of perspectives via a diagram
editor. When a new design material instance is added to the design, predicate
perspectives automatically include it if appropriate, whereas ad-hoc perspectives
will only contain the new material if it is explicitly added to that perspective.
4.4. Process Model
Argo's process modeling plug-in provides a simpli ed process modeling notation
based on IDEF0 Figure 5 . The design process is modeled as a task network,
where each task in the design process works on input produced by upstream tasks
and produces output for use bydownstream tasks. No control model is mandated:
tasks can be performed in any order provided needed inputs are available ; tasks
can be repeated; and anynumber of tasks can be in progress at a given moment.
Each task is marked with a status: future, in progress,or nished. Each task is
also marked with a list of decision types. Status information is shown graphically
via color in the process diagram. These attributes are used to update the decision
model. When the architect indicates that a task is considered nished, the design
environment can use this cue to generate additional criticism, perhaps marking the
task as still in progress if there are high priority to do" items pending.
The process of de ning and evolving the process referred to as the meta-process
is itself a complex, evolutionary task for which architects may need support. The
process model in Argo is rst-class: it is represented as a connected graph of active
design materials and the architect may de ne and modify the process model via the
same facilities used to work on architectures. Multiple perspectives may be de ned
to view the process. Critics may operate on the process model to check that it is
well-formed and guide its construction and modi cation, e.g., the output of this task
should be used by another task. The same techniques used to control architecture
critics can be used on process critics, including modeling the meta-process so that
process critics will be relevant and timely. While the abilitytochange the process
gives exibility to individual architects, process critics can communicate or enforce
external process constraints.
4.5. Design Environment Architecture
Figures
1 and 2 indicate what facilities are available to architects, but they give
little indication of how the design environment is implemented. Janus and similar
systems have tended to have one major software component for each facility. Those
components form a knowledge-rich design environment with tight user interface,
data, and control integration Thomas and Nejmeh, 1992 . Our interest in software
architecture motivated us to seek a more exible and extensible architecture, while
retaining a fairly high level of integration.
Figure
7 presents Argo's architecture as a virtual machine. The lowest layer
provides domain-neutral infrastructure and user interface components including
support for representing connected graphs, multiple perspectives, the critic run-time
system, to do" list, and logging facilities. Domain-speci c plug-ins are built
Shared SoftArch
Document
User's Active
Design Document
Shared Process
Document
SoftArch Plug-in: adds support
for code generation, simulation, .
Process Plug-in: adds
control over decision model
Domain-Neutral Kernel: connected graphs, perspectives, rationale log,
critic run-time, "to do" list, decision model, reusable interface elements
Active design documents
store architectures or
reusable design templates
with palettes of active
design materials, critics,
code generation templates,
simulation parameters, etc.
Figure
7.Argo's architecture presented as a virtual machine.
on top of that infrastructure if needed. These plug-ins typically provide pervasive
functionality that cannot be built into any particular design material. For example,
code generation support is useful for all design materials in the software architecture
domain. Most domain-oriented artifacts are stored in active documents" in the
top layer. These documents are active in that they contain design materials e.g.,
software components that carry their own domain knowledge and behavior in the
form of critics, simulation routines, and code generation templates. Documents
may contain palettes of design materials, designs, reusable design templates, process
fragments, or other supporting artifacts.
One advantage of this architecture is that artifacts from various supporting domains
may be used. Here, a domain is a coherent body of concepts and relationships
found in a given problem area, and a supporting domain is a domain for a problem
area of secondary concern to the designer, but is useful in completing the design
task. For example, a software architect's primary design domain is the construction
of systems from software components, whereas recording and managing design
rational is a domain of concern that is important to architects but secondary to con-
struction. In Argo, plug-ins for software architecture, process modeling, and design
rationale may all be available simultaneously, providing software architects with
rst-class supporting artifacts for process and rationale. Each supporting artifact
may be manipulated, visualized, and critiqued.
In designing this architecture we shift away from a monolithic, knowledge-rich design
environment that manipulates passive design materials to a modular, domain-neutral
infrastructure that allows the architect to interact with knowledge-rich,
active design materials. The same trend toward distributing knowledge and behavior
to the objects of interest can be observed in the general rise of object-oriented
and component-based approaches to software design. Active design materials can
be thought of as rst-class objects with local attributes and methods. The analysis
predicates of critics can be thought of as methods. Critics that cannot easily be
associated with any one design material may be associated with one or more design
perspectives. For simplicity, Figure 2 presents critics as looking down on the design
from above; a more literal presentation would show critics associated with each
node, looking around at their neighbors.
The advantages of this shift include increased extensibility, scalability, and separation
of concerns in the design environment, and stronger encapsulation of design
materials. Encapsulation is enhanced because attributes needed for analysis can be
made local, or private, to the design materials, thus supporting local name spaces
and data typing conventions. This increases extensibility because each design material
may be packaged with its own analyses, and thus de ne its own semantics,
which need not be anticipated by the design environment builder. Scalability in the
number of critics is increased because there is no central repository of critics critics
simply travel with design materials. Concerns are separated because the design environment
only provides infrastructure to support analyses packaged as critics and
need not perform any analysis itself. All of these advantages support the evolution
of architectures, design environments, and software architecture communities over
time.
ective support for diverse design decisions depends on the architect's ability
to obtain and manage large numbers of critics. In the Javaversion of Argo, design
materials and critics may be dynamically loaded over the Internet. For example, in
a software component marketplace, an architect mightdownload several component
design materials, try them in the current architecture, consider the resulting design
feedback, and make an informed component selection.
5. Cognitive Theories and Extensions to the DODE Approach
Our extensions to previous design environment facilities are motivated by theories
of designers' cognitive needs. Speci cally,we extend previous design environment
facilities by enhancing support for re ection-in-action and adding new support for
cognitive needs identi ed in the theories of opportunistic design and comprehension
and problem solving. These theories identify the cognitive needs of designers and
serve to de ne requirements on design environments. In the subsections below
we describe how Argo addresses these requirements. Table 3 summarizes Argo's
support for cognitive needs.
5.1. Re ection-In-Action
5.1.1. Theory
As discussed in Section 2.1, Schoen's theory of re ection-in-action indicates that
designers iteratively construct, re ect on, and revise eachintermediate, partial
design. Guindon, Krasner, and Curtis note the same e ect as part of a study of
software developers Guindon, Krasner, and Curtis, 1987 . Calling it serendipitous
design," they noted that as the developers worked hands-on with the design, their
mental model of the problem situation improved, hence improving their design.
Software architectures are evolutionary artifacts in that they are constructed incrementally
as the result of manyinterrelated design decisions made over extended
periods of time. We visualize design as a process in which a path is traced through
a space of branching design alternatives. A particular software architecture can be
Table
3. Argo features and the cognitive theories that they support.
Critics
ProLPMcroeuwslsteibnmpatlseror,dfieoervlsdebtrolacapkuptihnogrship
CRProeoemnscteinsstduaeoptriueossrnstoapcneordcintipeicvsesesslsfimistic
Kept relevent and timely
Produce feedback with links
ToAPCdrulolsocltiweosmtsiczhraiobtilces
DesPirgoncepsesrespdietcintigves
Reflection-in-action
Diversity of knowledge X
Evaluation during design XX
Providing missing knowledge X X
Opportunistic design
Timliness X
Reminding
Process flexibility X XX
Process guidance X X
Process visibility X X
Comp. & problem solving
Dividing complexity X
Multiple perspectives that
match multiple mental models
thought of as a product of one of the possible paths through this space. Choices
at any point can critically a ect alternatives available later, and every decision has
the potential of requiring earlier decisions to be reconsidered.
Traditional approaches to software architecture analysis require architects to make
numerous design decisions before feedback is provided. Such analyses evaluate
the products of relatively complete paths through design space, without providing
much guidance at individual decision points. As a result, substantial e ort may
be wasted building on poor decisions before feedbackisavailable to indicate the
existence of problems, and fewer design alternatives can be explored. Furthermore,
when analysis is performed only after extended design episodes, it may be di cult
to identify where exactly in the decision path the architect initially went wrong.
Diverse analyses are required to support architects in addressing diverse design
issues, such as performance, security, fault-tolerance, and extensibility. Researchto
date has produced a diverse set of architectural analysis techniques. They include
static techniques, such as determining deadlock based on communication protocols
between components Allen and Garlan, 1994 and checking consistency between
architectural re nements Moriconi, Qian, and Riemenschneider, 1995 , as well as
dynamic techniques, such as architecture simulation Luckham and Vera, 1995 .
The need for diversity in analysis is further driven by the diversity in project
stakeholders and the potentially con icting opinions of experts in the software architecture
eld itself Garlan, 1995 . Curtis, Krasner, and Iscoe note con icting
requirements and thus design evaluation criteria as a major problem for software
development in general Curtis, Krasner, and Iscoe, 1988 . Con ict will naturally
arise in architecture design, and analysis techniques should be capable of accommodating
it. Accommodating con ict in analysis yields more complete support,
whereas forbidding con ict essentially prevents architects from being presented with
multiple sides of a design issue. Consider architectural styles, which provide design
guidance by suggesting constraints on component and connector topology: a given
architecture may satisfy the rules of several diverse styles simultaneously.Feedback
items related to each of those styles can be useful, even if they contain con icting
advice.
5.1.2. Support in Argo
Argo supports re ection-in-action with critics and the to do" list. Critics deliver
knowledge needed to evaluate design decisions. The to do" list serves as a knowledge
in-box" by presenting knowledge from various sources. We will soon add
visual indicators to draw the architect's attention to design materials with pending
criticism Silverman and Mezher, 1992; Terveen, Stolze, Hill, 1995 . The to
do" list and informative assumption described below together support decision
making by allowing the architect to browse potential design problems, guideline
violations, and expert opinions.
Existing software analysis techniques are extremely powerful for detecting well-
de ned properties of completed systems, such as memory utilization and perfor-
mance. These approaches adhere to what we call the authoritative assumption:
they support architectural evaluation by proving the presence or absence of well-
de ned properties. This allows them to give de nitive feedback to the architect,
but may limit their application to late in the design process, after the architect has
committed substantial e ort building on unanalyzed decisions.
Such approaches also tend to use an interaction model that places a substantial
cognitive burden on architects. For example, architects are usually required to know
of the availability of analysis tools, recognize their relevance to particular design
decisions, explicitly invoke them, and relate their output back to the architecture.
This model of interaction draws the architect's attention away from immediate
design goals and toward the steps required to get analytical feedback. Explicit
invocation of external tools scales well in terms of machine resources, but not in
terms of human cognitive ability.We believe the cognitive burden of interacting
with external tools may be enough to prevent their e ective use.
Argo follows the DODE tradition in using what we call the informative assump-
tion: architects are ultimately responsible for making design decisions, and analysis
is used to support architects by informing them of potential problems and pending
decisions. Critics are pessimistic: they need not go so far as to prove the presence
of problems; in fact, formal proofs are often not possible, or even meaningful, on
partial architectures.
Heuristic analyses can identify problems involving design details that may not be
explicitly represented in the architecture, either because the model is too abstract,
or because the architecture is only partially speci ed. Critics can pessimistically
predict problems before they are evident in the partial design, and positively detect
problems very quickly after they are evident in the partial design, typically
within seconds of the design manipulation that introduces the problem. Criticism
control mechanisms help trade early detection for relevance to current goals and
concerns. In cases where all relevant design details are speci ed, critics can produce
authoritative feedback.
Unfortunately, for most design issues, there are inherit trade-o s that prevent
achieving both informative and authoritative feedback. There will always be a gap
between the making of a decision and the analysis of that decision. That gap allows
the passing of time, expenditure of e ort, and loss of cognitive context. When one
decision is analyzed in isolation, the gap may be small, but the feedbackisat
best informative because that decision interacts with others that have not yet been
made. When analysis is deferred until groups of interrelated decision have all been
made, the gap is necessarily larger, but the feedbackmay be more authoritative
because more interactions are known.
However, there is a compromise for the informative vs authoritative tradeo :
existing analysis tools can be modi ed to make pessimistic assumptions in cases
where partial architectures lack information needed for authoritative analysis; and
existing critics can be controlled so as to achieve a certain degree of con dence
before providing feedback. Alternatively, external batch analysis tools can be paired
with tool critics that remind the architect when those tools would be useful. For
example, a tool critic could watch for modi cations that a ect the result of the batch
analysis and check that the architecture is in a state that can be analyzed i.e., it
has no syntax errors that would prevent that particular analysis , then re-run the
batch tool, and parse its output into to do" items with links back to the design
context. In this case the critic's knowledge is of tools available in the development
environment and when they are applicable, whereas the tools themselves provide
architectural or domain knowledge.
Reusing existing analysis tools is one way to produce new critics, but we expect
most critics to be written and modi ed by domain engineers, domain experts, ven-
dors, or practicing architects. Argo's approach helps to ease critic authoring in
that critics are pessimistic, critic authors need not coordinate their activities with
22 ROBBINS, HILBERT, AND REDMILES
other authors to avoid giving con icting advice, and critics need not consider relevance
and timeliness. Argo's infrastructure eases critic authoring by providing a
framework for implementing critics, a user interface for managing critics and their
feedback, and templates for critics and their More Info" web pages.
5.2. Opportunistic Design
5.2.1. Theory
It is customary to think of solutions to design problems in terms of a hierarchical
plan. Hierarchical decomposition is a common strategy to cope with complex design
situations. However, in practice, designers have been observed to perform tasks in
an opportunistic order Hayes-Roth and Hayes-Roth, 1979; Guindon, Krasner, and
Curtis, 1987; Visser, 1990 . The cognitive theory of opportunistic design explains
that although designers plan and describe their work in an ordered, hierarchical
fashion, in actuality, they choose successive tasks based on the criteria of cognitive
cost. Simply stated, designers do not followeven their own plans in order, but
choose steps that are mentally least expensive among alternatives.
The cognitive cost of a task depends on the background knowledge of designers,
accessibility of pertinent information, and complexity of the task. Designers' background
knowledge includes their design strategies or schemas Soloway et al., 1988 .
If they are lacking knowledge about how to structure a solution or proceed with
a particular task, they are likely to delay this task. Accessibility of information
may also cause a deviation in planned order. If designers must search for information
needed to complete a task, that task might be deferred. Complexity of a task
roughly corresponds to the number of smaller tasks that comprise it.
Priority or importance of a step is the primary factor that supersedes the least cost
criteria. Priority or importance may be set by external forces, e.g., an organizational
goal or a contract. Designers may also set their own priorities. In some observations,
designers placed a high priorityonoverlooked steps or errors Visser, 1990 .
Thus, the theory of opportunistic design outlines a natural" design process in
which designers choose their next steps to minimize cognitive cost. However, there
are inherent dangers in this natural" design process. Mental context switches
occur when designers change from one task to another. When starting a new step
or revisiting a former one, designers must recall schemas and information needed
for the task that were not kept in mind during the immediately preceding task.
Inconsistencies can evolve undetected. Some requirements maybeoverlooked or
forgotten as the designer focuses on more engaging ones. E ciency is lost because
of many context switches. Guindon, Krasner, and Curtis observed the following
di culties.
The main breakdowns observed are: 1 lack of specialized design schemas;
2 lack of a meta-schema about the design process leading to poor allocation
of resources to the various design activities; 3 poor prioritization of issues
leading to poor selection of alternative solutions; 4 di culty in considering
all the stated or inferred constraints in de ning a solution; 5 di cultyin
performing mental simulations with many steps or test cases; 6 di culty
in keeping track and returning to subproblems whose solution has been post-
poned; and 7 di culty in expanding or merging solutions from individual
subproblems to form a complete solution. Guindon, Krasner, and Curtis,One implication is that designers would bene t from the use of process modeling.
Common process models support stakeholders in carrying out prescribed activities,
e.g., resolving a bug report. Software process research has focused on developing
process notations and enactment tools that help ensure repeatable execution of
prescribed processes. However, in their focus on repeatable processes, process tools
have tended to be restrictive in their enforcement of process steps.
Design environments can allow the bene ts of both an opportunistic and a prescribed
design process. They should allow, and where possible augment, human
designers' abilities to choose the next design task to be performed. They can help
designers by providing information so they do not make a context switch. Process
support should exhibit the following characteristics to accommodate the real design
process as described by the theory of opportunistic design and address the problems
ed by Guindon, Krasner, and Curtis.
Visibility helps designers orient themselves in the process, thus supporting the
designer in following a prescribed process while indicating opportunities for choice.
The design process model should be able to represent what has been done so far
and what is possible to do next. Visibility enables designers to take a series of
excursions into the design space and re-orient themselves afterwards to continue
the design process.
Flexibility allows designers to deviate from a prescribed sequence and to choose
which goal or problem is most e ective for them to work on. Designers must be able
to add new goals or otherwise alter the design process as their understanding of the
design situation improves. The process model should serve primarily as a resource
to designers' cognitive design processes and only secondarily as a constrainton
them. Allowing exibility increases the need for guidance and reminding.
Guidance suggests which of the many possible tasks the designer should perform
next. Opportunistic design indicates that cognitive costs are lower when tasks are
ordered so as to minimize mental context switching. Guidance sensitive to priorities
e.g., schedule constraints must also be considered. Guidance can include simple
suggestions and criticisms. It may also include elaborate help, such as explanations
of potential design strategies or arguments about design alternatives.
Reminding helps designers revisit incomplete tasks or overlooked alternatives.
Reminding is most needed when design alternatives are many and when design
processes are complex or driven by exceptions.
Timeliness applies to the delivery of information to designers. If information
and design strategies can be provided to designers in a timely fashion, some plan
deviations and context switches maybeavoided. Achieving timeliness depends on
anticipating designers' needs. Even an approximate representation of designers'
planned steps can aid in achieving timeliness.
5.2.2. Support in Argo
Motivated by the theory of opportunistic design, wehave attempted to move from
prede ned processes that force a certain ordering of design decisions to exible
process models with the properties outlined above. We extend previous work in
design environments byintroducing an explicit model of the design process with
progress information and a more exible to do" list user interface for presenting
design feedback.
Argo's process model supports visibilityby displaying the process and the archi-
tect's progress in it. Visibility is further supported by the availabilityofmultiple
perspectives on the process. For example, an architect maychoose a perspective
that shows only parts of the process that lead to a certain goal. Furthermore, the
to do" list presents a list of issues that the architect may consider next.
Several authors have noted that traditional, sequential work- ow systems do not
adequately support exibility and proposed the use of constraint-based process
models Dourish et al., 1996; Glance, Pagani, and Pereschi, 1996 . In Argo, exibility
is allowed by the simple fact that Argo does not use the process model to
constrain the architect's actions: the architect may address any to do" item or
perform any design manipulation at any time. Furthermore, exibilityissupported
by the architect's ability to modify the process model to better represent their
mental model of the design process. Process critics, process perspectives, and a
meta-process all support the architect in devising a good design in the process
domain.
In the currentversion of Argo, guidance is provided only implicitly by the layout
of the process model and the prioritization of the to do" items. However, the
theory of opportunistic design suggests that guidance should be based, in part, on
the mental context required to perform each task. Pending to do" items could be
prioritized based on a rough estimate of the cognitive cost of addressing them.
The to do" list and process model together support reminding by showing the
issues that are yet to be addressed. The to do" list reminds the architect of issues
that can be addressed immediately while the process model shows tasks that must
be addressed eventually. Critics and to do" items remind the architect of issues
that need to be reconsidered as problems arise. Beyond the knowledge contained in
the critics and the process model, the architect can also create to do" items that
contain arbitrary text and links as personal reminders.
The continuous application of critics enables Argo to provide timely feedback.
Criticism control mechanisms help make continuous critiquing practical and reduce
distractions i.e., unneeded context switches due to irrelevant feedback. In
addition to improving design decisions, timely feedback helps the architect make
timely process decisions, e.g., is this design excursion complete?" and does a past
decision need reconsideration?"
5.3. Comprehension and Problem Solving
5.3.1. Theory
The theory of comprehension and problem solving observes that designers must
bridge a gap between their mental model of the problem or situation and the formal
model of a solution or system Kintsch and Greeno, 1985; Fischer, 1987 . The
situation model consists of designers' background knowledge and problem-solving
strategies related to the current problem or design situation. The system model
consists of designers' knowledge of an appropriate formal description. Problem
solving or design proceeds through successive re nements of the mapping between
elements in the design situation and elements in the formal description. Successive
re nements are equated with increased comprehension, hence the name of the
theory.
In the domain of software, designers must map a problem design situation onto
a formal speci cation or programming language Pennington, 1987; Soloway and
Ehrlich, 1984 . In this domain, the situation model consists of knowledge of the
application domain and programming plans or design strategies for mapping appropriate
elements of the domain into a formal description. The system model
consists of knowledge of the speci cation or programming language's syntax and
semantics. Programming plans or design strategies enable designers to successively
decompose the design situation, identify essential elements and relationships, and
compose these elements and relationships into elements of a solution. At successive
steps, designers can acquire new information about the situation model or about
the system model.
Pennington observed that programmers bene ted from multiple representations
of their problem and iterative solutions Pennington, 1987 . Namely multiple representations
such as program module decomposition, state, and data ow, enabled
programmers to better identify elements and relationships in the problem and solution
and, thus, more readily to create a mapping between their situation models
and working system models. Kintsch and Greeno's research indicated that familiar
aspects of a situation model improved designers' abilities to formulate solutions
Kintsch and Greeno, 1985 . These two results were applied and extended in Red-
miles' research on programmers' behavior, where again multiple representations
supported programmers' comprehension and problem solving when working from
examples Redmiles, 1993 .
Dividing the complexity of the design into multiple perspectives allows each perspective
to be simpler than the overall design. Moreover, separating concerns into
perspectives allows information relevant to certain related issues to be presented
together in an appropriate notation Robbins et al., 1996 . Design perspectives
mayoverlap: individual design materials may appear in multiple perspectives. Co-ordination
among design perspectives ensures that materials and relationships presented
in multiple perspectives may be consistently viewed and manipulated in any
of those perspectives. Overlapping, coordinated perspectives aid understanding of
26 ROBBINS, HILBERT, AND REDMILES
new perspectives because new design materials are shown in relationship to familiar
ones Redmiles, 1993 .
Good designs usually have organizing structures that allow designers to locate
design details. However, in complex designs the expectation of a single unifying
structure is a naive one. In fact, complex software system development is driven bya
multitude of forces: human stakeholders in the process and product, functional and
non-functional requirements, and low-level implementation constraints. Alternative
decompositions of the same complex design can support the organizing structures
that arise from these forces and the di erent mental models of stakeholders with
di ering backgrounds and interests. Using diverse organizing structures supports
communication among stakeholders with diverse backgrounds and mental models
whichiskey to developing complex systems that are robust and useful.
It is our contention that no xed set of perspectives is appropriate for every possible
design; instead perspective views should emphasize what is currently important
in the project. When new issues arise in the design, it may be appropriate to use
a new perspective on the design to address them. While we emphasize the evolutionary
character of design perspectives, an initial set of useful, domain-oriented
perspectives can often be identi ed ahead of time Fischer et al., 1994 .
5.3.2. Support in Argo
Multiple, overlapping design perspectives in Argo allow for improved comprehension
and problem solving through the decomposition of complexity, the leveraging
of the familiar to comprehend the unfamiliar, and the use of notations appropriate
to multiple stakeholders' interests. Supporting the mental models of a particular
domain must be done by domain engineers, practicing architects, and other stake-holders
who apply Argo to a speci c domain. Architects and other stakeholders may
de ne their own perspectives in the course of design. Presentation and evolvability
critics advise architects in de ning and using perspectives.
Soni, Nord, and Hofmeister identify four architectural views: 1 conceptual software
architecture describes major design elements and their relationships; 2 modular
architecture describes the decomposition of the system into programming language
modules; 3 execution architecture describes the dynamic structure of the
system; and 4 code architecture describes the way that source code and other artifacts
are organized in the developmentenvironment Soni, Nord, and Hofmeister,
1995 . Their experience indicates that separating out the concerns of each view
leads to an overall architecture that is more understandable and reusable.
The 4+1 View Model Kruchten, 1995 consists of four main views: 1 the logical
view describes key abstractions classes and their relationships, e.g., is a and
instantiates; 2 the process view describes software components, how they are
grouped into operating system processes, and how those processes communicate;
3 the development view describes source code modules and their dependencies;
4 the physical view describes how the software will be distributed among processors
during execution. These four views are supplemented with scenarios and use
cases that describe essential requirements and help relate elements of the various
views to each other. The views provide a well-de ned model of the system, but
more importantly they identify and separate major concerns in software develop-
ment. The Uni ed Modeling Language UML also uses multiple perspectives to
visualize various aspects of a design Fowler and Scott, 1997 . In demonstrating
Argo, wechose perspectives similar to those described in these approaches; how-
ever, we believe that the choice of perspectives depends on the type of software
being built and the tasks and concerns of design stakeholders.
Argo supports multiple, coordinated perspectives with customization. In addition
to the perspectives described in this paper, Argo allows for the construction
of new perspectives and their integration with existing perspectives. Architects
who are given a xed set of formal notations often revert to informal drawings
when those notations are not applicable Soni, Nord, and Hofmeister, 1995 . One
goal of Argo is to allow for the evolution of new notations as new needs are rec-
ognized. In addition to the structured graphics representing the architecture and
process, we allow architects to annotate perspectives with arbitrary, unstructured
graphics as demonstrated in Figure 3 . Customizable presentation graphics are
needed because the unifying structures of the system under construction must be
communicated convincingly to other architects and system implementors. Tobe
convincing, the style of presentation must t the professional norms of the development
organization: it should look like a presentation, not an architect's scratch
pad. Furthermore, ad-hoc annotations that are found to be useful can be incrementally
formalized and incorporated into the notations of future designs Shipman and
McCall, 1994 . We expect that Argo's low barrier to customization will encourage
evolution from unstructured notations to structured ones as recurring formalization
needs are identi ed.
6. Evaluation
The preceding section has provided theoretical evaluation of our extensions to the
DODE approach. Also, the implementation of Argo described in Section 4 provides
a proof-of-concept that many of the desired features for Argo can be realized. This
section outlines our plans to further evaluate Argo as a working tool. Argo's architecture
and infrastructure can be evaluated in terms of howwell they support
construction of design environments in new domains. Argo's support for cognitive
needs can be evaluated by measuring qualities of design processes and products.
6.1. Application to New Domains
The process of applying Argo to a new domain consists of de ning new design
materials with critics, a design process, and design perspectives. Belowwe describe
our experience in carrying out these tasks for three domains.
28 ROBBINS, HILBERT, AND REDMILES
In the domain of C2-style software architectures, there are two basic design ma-
terials: software components and connectors. The basic relationship between these
materials describes how they communicate. This basic model was extended to include
design materials for operating system threads, operating system processes,
and source code modules. We rapidly authored approximately twenty critics that
check for completeness and consistency of the representation and adherence to
published C2 style guidelines Taylor et al., 1996 . The number of needed critics
is small because the C2 style addresses only system topology and simple communication
patterns. The C2 design process started with tasks to create each of the
design material types, and was re ned by splitting activities based on possible design
material attributes, e.g., reused components vs. new components. We started
with two perspectives discussed in previous work on C2, conceptual and implemen-
tation, and later included a perspective to visualize relationships between software
components and the program modules that implement them.
Wehave also adapted the Argo infrastructure to implement a design environment
for object-oriented design that supports a subset of the Object Modeling Technique
Rumbaugh et al., 1991 . Since this domain is well de ned and described in a single
book, it was straightforward to identify the design materials, relationships,
graphical notations, and perspectives. Our OMT subset includes the object model,
behavioral model, and information model, but excludes the more advanced features
of each. A core set of ten critics that address correctness and completeness
of the design was also straightforward to implement, e.g., an abstract class with
no subclasses indicates an incomplete design. Additional critics were inspired by
a book on OO design heuristics Riel, 1996 , e.g., a base class should not make
direct references to its subclasses because that means that adding new subclasses
requires modi cations to the base class. Some of these heuristics were more di cult
to specify as critics because they rely on information not present in the represen-
tation, e.g., semantically related data and behavior should be kept together. In
this example, an authoritative answer cannot be given because the OMT design
representation does not contain enough semantic information; however, critics may
apply pessimistic heuristics to identify when this issue might be a problem. The
provided process and perspectives collected various process fragments described in
these two books.
Wehave begun to apply Argo to software requirements speci cations using the
CoRE methodology Faulk et al., 1992 in the avionics application domain. CoRE
is based on the SCR requirements methodology Henninger, 1980 with extensions
that deal with the modular decomposition of the requirements document. As with
OMT, existing documents describe the design materials, standard notations, and
analyses. Existing tools cover essentially all analyses that can be performed on the
requirement speci cation without considering the application domain, e.g., identifying
non-deterministic transitions in a mode-transition table. In implementing
a CoRE design environmentwe will demonstrate added value over existing tools
byintegrating analysis more tightly with the cognitive process of devising a speci
cation, and by providing heuristics to support modularization of requirements
documents in the avionics domain, e.g., autopilot control modes are largely inde-
pendentofcockpit display modes and should be speci ed in separate requirements
modules, however there should be certain constraints between the two. To date
wehave implemented twenty critics that check correctness and completeness of
CoRE requirements speci cations and integrated them into an independently developed
requirements editing tool. Doing so has given us additional con dence in
our critiquing infrastructure.
Argo's architecture and infrastructure have provided satisfactory support for the
initial implementations of domain-oriented design environments in three domains.
We plan to evaluate howwell our infrastructure extends in three dimensions:
larger domains with more critics and more complex designs and processes,
2 addition of new domain-oriented plug-ins e.g., design rationale , and 3 use of
the infrastructure by people outside of our research group e.g., an avionics software
development group .
6.2. Evaluating Cognitive Support
Toevaluate Argo's support for the cognitive needs of designers, user testing will
focus on how Argo a ects the productivity of the designer and the qualityofthe
resulting product. Our support for re ection-in-action should increase productiv-
ityby decreasing time spent reworking design decisions, lead to better designs in
cases where critics provide knowledge that the designer lacks, shorten the lifespan
of errors, reduce the number of missing design attributes, and strengthen the de-
signer's con dence in the nal design because more issues will have been raised and
addressed. We expect our support for opportunistic design will allow designers to
rely less on mental or paper notes, and to make better process choices. Comprehension
of a sample design should increase when the designer's mental models match
one or more design perspectives. Some experimental data can be automatically col-
lected, e.g., the lifespan of errors, while others will rely on human observation and
interviews. Experimental subjects will use Argo with all features enabled, while
control subjects will use Argo with some features disabled. We plan to evaluate the
resulting designs with the help of blind judges, as described by Murray Murray,
1991 . Tests that measure our hypotheses will indicate the degree to which identied
cognitive needs are supported by Argo's features, and thereby suggest weights
for the associations in Table 3.
A related task is devising a methodology for on-going evaluation of the qualityof
the knowledge provided by critics, the guidance contained in process models, and
the mental models suggested by perspectives. This methodology should support
on-going maintenance of the design environment and periodic reorganization and
reseeding" of domain knowledge Fischer et al., 1994 . Structured email between
designers and knowledge providers is one source of data for this evaluation. We are
also investigating event monitoring techniques that capture data to help evaluate
howwell provided knowledge impacts actual usage. Examples of quantities that
could be automatically collected include the number of critics that re, how many
to do" items the designer views, and how many errors are xed as a result of
viewing feedback from critics.
A recentevaluation of VDDE Voice Dialog Design Environment raised several
questions about the character of the impact of design critics Sumner, Bonnardel,
and Kallak, 1997 . The study found that designers preempted critical feedbackby
anticipating criticisms and avoiding errors that the critics could identify. Designers
assessed the relevance of each criticism before taking action, and in cases where
experienced designers disagreed with criticism they usually added design rationale
describing their decision. Sumner, Bonnardel, and Kallak suggest that evaluation of
critiquing systems should explicitly consider designers of di ering skill levels. They
further suggest that future critiquing systems use alternativeinterface metaphors
that users will perceive as cooperative rather than adversarial. In our own experiments
we plan to group subjects by experience and watch closely for anticipation
of criticism.
7. Conclusions and Future Work
Designing a complex system is a cognitively challenging task; thus, designers need
cognitive support to create good designs. In this paper wehave presented the
architecture and facilities of Argo, our software architecture design environment.
Argo's architecture is motivated by the desire for reuse and extensibility. Argo's
facilities are motivated by the observed cognitive needs of designers. The architecture
separates domain-neutral code from domain-oriented code and active design
materials. The facilities extend previous work in design environments by enhancing
support for re ection-in-action, and adding new support for opportunistic design,
and comprehension and problem solving.
In future work, we will continue exploring the relationship between cognitive theories
and tool support. Further identi cation of the cognitive needs of designers
will lead to new design environment facilities to support those needs. Also, we will
seek ways to better support the needs that wehave identi ed in this paper, e.g., a
process model that approximates the cognitive cost of switching design tasks. Fur-
thermore, we will investigate ways of better supporting and using design rationale.
For example, the architect's interactions with the to do" list is a potentially rich
source of data for design rationale: items are placed on the list to identify open
issues, and removed from the list as those issues are resolved. Design rationale is
an important part of design context and to do" items should reference relevant
past items when possible.
Our current prototype of Argo is robust enough for experimental usage. It is our
goal to develop and distribute a reusable design environment infrastructure that
others may apply to new application domains. Successful use of our infrastructure
by others will serve to inform and evaluate our approach. A Javaversion of Argo
with documentation, source code, and examples is available from the authors.
Acknowledgments
The authors would like to thank Gerhard Fischer CU Boulder , David Morley
Rockwell International , and Peyman Oreizy, Nenad Medvidovic, the other members
of the Chiron research team at UCI, and the anonymous reviewers.
sponsored by the Defense Advanced Research Projects Agency, and Air
Force Research Laboratory, Air Force Materiel Command, USAF, under agree-
mentnumber F30602-97-2-0021 and F30602-94-C-0218, and by the National Science
Foundation under Contract Number CCR-9624846. Additional support is
provided byRockwell International. The U.S. Government is authorized to reproduce
and distribute reprints for Governmental purposes notwithstanding any copyright
annotation thereon. The views and conclusions contained herein are those of
the authors and should not be interpreted as necessarily representing the
policies or endorsements, either expressed or implied, of the Defense Advanced Re-search
Projects Agency, Air Force Research Laboratory or the U.S. Government.
Approved for Public Release Distribution Unlimited.
Notes
1. KLAX is a trademark of Atari Games.
--R
Answer Garden 2: Merging Organizational Memory with Collaborative Help.
Using Style to Understand Descriptions of Software Architecture.
Beyond De nition Use: Architectural Interconnection.
The Design and Implementation of Hierarchical Software Systems with Reusable Components.
Addressing Complexity
How to Deal with Deviations During Process Model Enactment.
A Field Study of the Software Design Process for Large Systems.
Free ow: Mediating Between Representations and Action in Work ow Systems.
A Conceptual Framework for the Augmentation of Man's Intellect.
Cognitive View of Reuse and Redesign.
Supporting Software Designers with Integrated Domain-Oriented Design Environments
Construction Kits and Design Environments: Steps Toward Human Problem-Domain Communication
Embedding Computer-Based Critics in the Contexts of Design
UML Distilled: Applying the Standard Object Modeling Language.
Gardeners and gurus: patterns of cooperation among CAD users.
Exploiting Style in Architectural Design Envi- ronments
An Introduction to SoftwareArchitecture: Advances in Software Engineering and Knowledge Engineering
Generalized Process Structure Grammars GPSG for Flexible RepresentationsofWork.
Report on a Knowledge-Based Software Assistant
Requirements and Design of DesignVision
Breakdown and Processes During Early Activities of Software Design by Professionals.
A Cognitive Model of Planning.
Specifying Software Requirements for Complex Systems: New Techniques and Their Application.
Understanding and Solving Word Arithmetic Problems.
A Cooperative Problem Solving System for User Interface Design.
An Event-Based Architecture De nition Language
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KI: A Tool for Knowledge Integration.
Program transformation systems.
Stimulus Structures and Mental Representations in Expert Comprehension of Computer Programs.
Foundations for the study of software architecture.
Reducing the Variability of Programmers' Performance Through Explained Examples.
Cooperative Software.
Visual Language Features Supporting Human-Human and Human-Computer Communication
The Re ective Practitioner: How Professionals Think in Action.
Designing as Re ective Conversation with the Materials of a Design Situation.
Supporting Knowledge-Base Evolution with IncrementalFor- malization
Expert critics in engineering design: lessons learned and research needs.
Empirical Studies of Programming Knowledge.
Designing Documentation to Compensate for Delocalized Plans.
Software Architecture in Industrial Applications.
The Cognitive Ergonomics of Knowledge-Based Design Support Systems
" to Living Design Memory."
" to Magic World"
De nitions of Tool Integration for Environments.
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592026 | Efficient Specification-Based Component Retrieval. | In this paper we present a mechanism for making specification-based component retrieval more efficient by limiting the amount of theorem proving required at query time. This is done by using a classification scheme to reduce the number of specification matching proofs that are required to process a query. Components are classified by assigning features that correspond to necessary conditions implied by the component specifications. We show how this method of feature assignment can be used to approximate reusability relationships between queries and library components. The set of possible classification features are formally defined, permitting automation of the classification process. The classification process itself is made efficient by using a specialized theorem proving tactic to prove feature implication. The retrieval mechanism was implemented and evaluated experimentally using a library of list manipulation components. The results indicate a better response time than existing formal approaches. The approach provides higher levels of consistency and automation than informal methods, with comparable retrieval performance. | Introduction
The concept of component reuse is fundamental to all engineering disciplines. Components provide levels of
abstraction used to effectively construct increasingly complex systems. Software Engineering is no exception,
where a main focus has been providing languages and methodologies to help software designers create useful
and reusable abstractions. In fact, software reuse was recently described by Mili et al. as the "only realistic
approach" to meet the needs of the software industry [23].
The potential for software reuse is not limited to source code, but includes algorithms, architectures,
domain models, design decisions, program transformations, documentation - virtually every possible aspect
of a software system. In addition, the benefits of software reuse extend beyond the design phase to the
analysis and maintenance phases of development. While this article focuses on functional components, the
methodology presented is compatible with any software reuse artifacts where a refinement ordering exists
between specifications. This includes source code, high-level components [1], software architectures [9, 26],
and program transformations [4]. Note also that component retrieval is not confined to library reuse; it
can also be considered in the context of integrating components into generic software architectures, either
statically [1] or dynamically [27].
While the potential benefits of software reuse are far reaching, in practice software reuse has not flourished.
There are both managerial and technical reasons for this. One major technical barrier has been providing
tools to automate the reuse process. To understand the reason for the technical difficulty in automating
reuse, it is helpful to decompose the library reuse process into three activities:
Retrieval - Specifying a query and locating potential reuse candidates within a software library.
Evaluation - Determining the relationship between a retrieved component and the specification of the
desired component.
Adaptation - Making changes to a component to meet reuse requirements.
Ideally, a reuse tool should provide automated assistance for all three reuse activities. These activities
are interdependent: high reuse potential (from retrieval) should indicate low reuse effort (during evaluation)
with respect to known adaptation methods. However, automation is difficult because each activity requires
different information about a component. Retrieval benefits from an abstract classification of component
function that supports efficient comparison. For automated evaluation to be useful, it must provide a designer
with both a precise relationship and a high level of assurance. Therefore, evaluation depends on a precise
and detailed description of component behavior. Adaptation requires knowledge about the structure of a
component and the functions of its sub-components. Therefore, the choice of a component representation
scheme determines what software reuse activities can be automated effectively.
Source code reuse has been the main focus of most software reuse efforts. However, automation of the reuse
processes requires understanding aspects of a component that are not described in source code. Source code
provides a description of how a component performs its function. For purposes of retrieval, we are interested
in precisely what this function is. The difference between what and how represents a semantic gap that makes
it difficult to understand the function of the code and recognize a potentially reusable component. Source
code is not a sufficient representation to support automation of the reuse processes. Therefore, a method is
required that represents the additional information necessary to automate reuse.
1.1. Specification-Based Reuse
Formal specification languages provide the expressiveness and precision necessary to capture what the function
of a component is. Specification matching [35, 45] applies theorem proving to evaluate relationships
between specifications. Given a formal definition of reusability, specification matching can be used to evaluate
the reusability of a component with respect to a requirements specification. In addition, automated
reasoning can be used to determine what changes are necessary to reuse a component and guide component
adaptation [29, 39].
To illustrate the potential of using formal methods to assist software reuse, we present a sample scenario
of a designer interacting with a specification-based retrieval system. First the designer must develop a
specification for the desired component that will be used as a query to the library. The specification defines
the domain, range, precondition and postcondition of the desired component. The specification can be
abstract and need not be complete. For example, if a designer is looking for list decomposition methods in
a library of list manipulation components, the following query might be used:
PRE true
POST FORALL (x:E)
The result of submitting a query to the component retrieval system is a list of components, a matching,
and any necessary port substitutions. Matching conditions are formal relationships between the component
and query specifications representing conditions under which a component may be reused. The matching
conditions may be associated with a set of rules or automated mechanisms for adapting a component. In
the case of the example, the retrieval system gives the following results:
Component Match Substitutions
removeFirst Weak Post rest 7! output
removeLast Weak Post rest 7! output
split Weak Post left 7! output
split Weak Post right 7! output
The Weak Post match, defined formally as I C -OC ) OP , indicates that a component provides a solution
for part (but not all) of a problem's domain. The designer would like to find a component that covers the
entire domain. However, in this case it is not obvious what should happen in the case of an empty input
list. The designer can strengthen the precondition of the query to eliminate this case:
In response to the new query, the system provides the following results:
Component Match Substitutions
removeFirst Weak Plug-in rest 7! output
removeLast Weak Plug-in rest 7! output
split Weak-post left 7! output
split Weak-post right 7! output
The Weak Plug-in Match is one of several matches that gives a formal guarantee that a component
provides a valid solution for the legal inputs of a problem. Therefore both the removeFirst and removeLast
components can be safely reused to solve the second query.
1.2. Technical Overview
Due to the overhead of automated theorem proving, specification matching is too computationally expensive
to test a large number of components in an acceptable amount of time [7, 21, 43]. To attain practical response
times, specification matching must be limited to evaluating a small number of components retrieved by a
separate mechanism. An alternative to limiting specification matching is to limit the expressibility of the
specification language, making retrieval more efficient [31]. However, this lowers the level of assurance during
evaluation because it takes into account fewer aspects of a component's behavior. Approaches that do not
distinguish retrieval from evaluation will either have inefficient retrieval or weak evaluation.
To effectively separate retrieval from evaluation, it is necessary to maintain consistency between the evaluation
criteria and the retrieval goal. Specifically, the classification scheme used for retrieval should identify
components that will match the query specification with respect to a reusability relationship. Because evaluation
is based on component semantics, classification schemes based upon syntactical measures are not
guaranteed to be consistent with the evaluation criteria.
This paper presents a method for making specification-based retrieval efficient by using the semantic
information provided by the specifications. Figure 1 depicts the flow of design information within the
retrieval system. In the diagram, boxes represent data structures, ovals represent computations and arrows
represent data flow into computations and references between data structures.
In the system, formal specifications are used to model the problem requirements and the function of
the library components. The specifications are based on abstract data types defined in an algebraic domain
theory [14, 42]. In Section 2 we present formal component specifications. Section 3 explains how specification
matching is used to determine component reusability.
Component retrieval is made efficient by layering a classification scheme above the specifications. The
classification scheme consists of a collection of formal definitions representing possible component features
in the domain. These definitions control the classification process in place of a human domain expert.
Formalizing the scheme permits automated classification of the specifications. The semantic classification
scheme is presented in Section 4.
The output of the classification phase is a set of features that are used as a query to the component library.
The library retrieval mechanism returns components that have feature sets similar to the query. Feature-based
retrieval is discussed in Section 5. The similar components are passed on to a more detailed evaluation
that uses specification matching to determine each component's precise relationship to the requirements
specification.
The retrieval system was implemented using the ML programming language [25] and the HOL [12] theorem
proving system. In Section 6 we evaluate the response time and retrieval performance experimentally and
compare it with other retrieval methods. Section 7 compares the experimental results with other published
results. Section 8 discusses the performance of semantic classification in terms of the effects on the retrieval
system. We then discuss related work and conclude.
2. Component Specifications
Software systems can be represented at the architectural level as a collection of interconnected components
[38]. The components of a system are its constituent subsystems that encapsulate part of the system's
Classification
Scheme
Specification
Matching
Classification
Retrieval
Theory
Domain Requirements
Specification
Component
Library
Component
Retrieval
System Similar
Components
Feature
Matching
Components
Domain Modeler Designer
Figure
1. Overview of Specification-Based Retrieval with Semantic Classification
RecordList
includes Record, List(Rec)
introduces
defined
asserts 8 l:List[Rec], r:Rec, k:Key
9 r (contains(l,r) - k=r.key);
r.key ? first(l).key
assumes TotalOrder(Key)
introduces
asserts
Rec partitioned by .key
prepend
first
rest
last
contains
asserts
List[E] partitioned by empty, first, rest
append(-,e) == prepend(e,-);
Figure
2. Larch Shared Language specification of a list of records
functionality. Each sub-system is, in turn, composed of a collection of sub-systems resulting in a hierarchical
system structure. Decomposition continues in this manner until reaching components that are not
implemented by component composition but by actual source code.
Formal specifications provide benefits over informal specifications and source code descriptions. First,
because the specifications are formal, they provide a precise, unambiguous description of the component's
function. Second, the description is declarative as opposed to operational in nature, meaning that it describes
what a component does without reference to how it does it. This is important issue for reuse because
dependency upon unspecified or unintended functionality can cause problems in the context of evolving
requirements and implementations.
A formal specification can be broken into two levels of abstraction: domain theories and interface specifications
[14]. The domain theory defines the vocabulary used in the specification by providing models of
the data types and operations used in the domain of interest. Interface specifications define the behavior of
system components in terms of the domain theory.
2.1. Domain Theories
We use algebraic specifications to build domain theories. An algebraic specification defines a set of abstract
data types and operations. For example, Figure 2 contains a Larch Shared Language [14] specification for a
list of records. The RecordList specification (called traits in Larch) includes both the Record specification
and a parameterized List specification. In the List specification, operators are defined for appending,
COMPONENT search IS
IMPORT RecordList;
END search;
Figure
3. Example Interface Specification
prepending, concatenating, and computing list membership and list length. The generated by clause defines
the operators that can be used to recursively construct all values of the type. The partitioned by clause
identifies a set of operators that can be evaluated to deduce the equivalence of two values of the type. Once
a domain theory has been developed for a certain problem domain its data types and operations can be
referenced by interface specifications.
2.2. Interface Specifications
An interface specification defines the behavior of a component in terms of a domain, range, precondition and
postcondition. For example, Figure 3 shows an interface specification for a search component. The domain
and range define the input and output types of the component, respectively. The precondition specifies the
set of inputs that the component's operation is defined over, called the legal inputs. The postcondition defines
the relationship that must hold between an input and a valid output. If the precondition is true when the
component begins executing, the component is guaranteed to terminate in a state where the postcondition
is true [13]. There are no restrictions or guarantees on the behavior of the component when the precondition
does not hold.
Formally, a component specification, P , can be translated into the following predicate logic axiom:
where DP and RP are the domain and range of the component and I P (x) and OP (x; z) are the precondition
and postcondition, respectively.
3. Reusability
Using formal specifications to evaluate reusability requires a formal definition of the relationship that must
exist for a component to be reused to solve a new problem. We stress this point because existing literature
on specification-based component retrieval is not consistent in the choice of evaluation conditions. We first
consider the case where a component completely satisfies a problem or query specification. Then we discuss
other relationships that may indicate components that can be adapted and reused.
A component completely solves a problem if it results in one of the problem's valid outputs for each of the
problem's legal inputs. Formally, component specification C satisfies problem specification P if the following
condition holds:
The first conjunct states that the component will accept all legal inputs to the problem. The second conjunct
states that all valid outputs of the component for a legal problem input are valid outputs of the problem.
The behavior of a component outside of the legal problem inputs is of no concern in determining its ability to
solve the problem. We assume ignoring potential subtype substitutions. Subtypes
can be supported implicitly using predicates, such as Integer(x) ) Real(x), or explicitly for more efficient
reasoning [36].
Plug-in
Weak Plug-in
\Gamma\Psi
Plug-in Post
@
@R
Satisfies
\Gamma\Psi
Weak Post
I C - OC ) OP
@
@R
\Gamma\Psi
Stronger
Weaker
Figure
4. A lattice of specification matches used for evaluating reusability
It is not always the case that a component completely satisfying the problem will exist in the library.
Therefore, it is desirable for a query to match components that can be adapted or combined to solve the
problem. Zaremski and Wing [43, 45] have identified a collection of specification matches that can be useful
in comparing specifications. Figure 4 shows a subset of these matches that we think are of interest in
determining reusability, with the addition of Satisfies match. 1 Following Zaremski and Wing, matches are
arranged in a lattice, where an arrow between two matches indicates that the match at the base of the arrow
is stronger than (logically implies) the match at the head of the arrow. The formal notation is abbreviated
by dropping the quantifiers and variable arguments for the predicates.
The three matches on the left-most path all require that the precondition of the problem be stronger than
the precondition of the component. They differ in the set of inputs that require a valid output from the
postcondition relation of the component: Plug-in match checks the whole domain, Weak Plug-in restricts the
check to the legal component inputs, and Satisfies further restricts this to the legal inputs of the problem.
Because of the logical relationship between these matches, any components matching under Plug-in or Weak
Plug-in will match under Satisfies. However, Plug-in and Weak Plug-in will cause the disregard of useful
components if used as a retrieval condition.
The Plug-in Post and Weak Post matches differ from the others by not requiring all legal problem inputs to
be legal component inputs. If a component matches a problem in one of these ways, there could be problem
inputs that cause unspecified behavior in the component. However, we do know that for any legal problem
input that is also a legal component input, the component provides a valid problem output. Therefore,
components that match in these ways can be used as a partial solution to the problem. A collection of
such components can be composed to provide a complete solution to the problem. These matches are also
useful for finding components with non-trivial preconditions without having to specify a precondition in the
query [43].
Because the specification matches are formally defined, they can be checked using an automated theorem
prover. However, due to the complexity of automated theorem proving, specification matching is too computationally
expensive to test a large number of components [7, 21, 43]. Practical specification-based retrieval
requires an efficient way to identify components that will match the query specification with respect to a
reusability relationship. The next section presents a method for doing this by classifying components using
the semantics of their specifications.
4. Semantic Classification
The efficiency required for specification-based retrieval can be achieved using a feature-based classification
scheme. Feature-based retrieval is efficient because it relies only on the syntactic matching of attribute-value
pairs, or features. The similarity of two components is measured by the number of features they have in
common.
When applying feature-based classification by hand, library components are assigned a set of features by
a domain expert. To retrieve a set of potentially useful components, the designer translates (classifies) the
problem requirements into a set of features and the corresponding class of components is retrieved from the
library. Queries can be generalized by relaxing how the feature sets are compared.
A potential problem with feature-based classification is maintaining the consistency of the classification
process [24, 33]. Effective retrieval requires the domain expert and the developers to have a common understanding
of the intended semantics of the features. It would be desirable to systematize and automate
classification to increase the confidence that classification is consistent among the domain experts and the
developers. Automatic indexing based on semantics 2 is not practical from a code reuse standpoint because it
would require a massive re-engineering effort. However, formal component specifications provide an explicit
semantic representation that can be used as a foundation for component classification.
Automation of feature-based classification requires answering two questions [17]: how do we automatically
generate features from a specification and what are the possible features that a component can have? We
answer the first question by describing a framework for assigning features to components in a way that
assists the search for reusable components. We then discuss how we define the set of possible features used
to classify components.
4.1. Feature Set Generation
The classification scheme used for retrieval should identify components that will match the query specification
with respect to the reuse matches identified in Section 3. Inspection of these specification matches reveals
a general pattern: in each case part of one specification logically implies part of another specification, such
as I P . The search for these situations can be guided using necessary
conditions.
A necessary condition for a predicate P is another predicate \Phi logically weaker than P , i.e. P ) \Phi. For
two predicates, P and Q, such that P ) Q, every necessary condition of Q will be a necessary condition of
Therefore, we can commonly describe P and Q by the fact that they both logically imply \Phi.
Necessary conditions can be used in this way to assist the search for reusable components. More specifically,
they can identify components that cannot match a specification and therefore should not undergo detailed
analysis. Both the component and query specifications are classified based on a given set of necessary
conditions. A classification scheme associates a feature with each necessary condition. A feature is assigned
to a component if its associated necessary condition is logically implied by the component's specification:
The general result of this semantic classification method is that if a component has a feature set similar
to that of the query then there is the potential for a reuse match to hold between the component and query.
Conversely, components that do not have features in common are less likely to be reusable. Therefore,
syntactic comparison of feature sets can be used to efficiently approximate the semantic relationships between
components.
In general, semantic classification cannot be guaranteed to eliminate only non-matching components. The
behavior of the system depends upon the set of necessary conditions used in the classification scheme and
which reuse matches are being approximated. In effect, approximate reasoning based on feature sets is
unsound and incomplete. However, this should not be considered as a critical flaw until it is clear what
its effects are on the practical performance of the system [5]. The impact of unsound and incomplete
classification is evaluated by experimentally measuring its effect on retrieval performance (Section 6).
4.2. Feature Definitions
The classification process is controlled by a collection of feature definitions that determine the set of necessary
condition/feature pairs used. The feature definitions capture the knowledge a domain expert would use to
classify components by hand. By formally defining the classification features and the feature assignment
process, classification can be fully automated.
Filter(T
Figure
5. Sample feature definitions for a data-flow abstraction
To characterize aspects of interface specifications, the features must represent abstract relationships between
component inputs and outputs. The feature definitions link the feature names and values to logical
predicates which specify the associated concept. Feature definitions have the following format:
The feature name provides a syntactic label for a concept. The feature value (Type1; Type2) represents
data type parameters that are instantiated based on the domain and range of the specification. If two
components perform a similar function over different data types, they will have a feature with the same
name, but different values. The variables x and y are actually metavariables that range over the set of input
or output variables for a component. The isIO() predicates, either isInput() or isOutput(), determine
which of these sets a metavariable ranges over and associates the type of the variable with the feature
value parameters. The Condition() predicate is the necessary condition associated with the feature. A
feature definition is instantiated by substituting in all combinations of input and output variables from the
component being classified. Instantiation is restricted by type checking any operators which are used when
specifying Condition().
The set of features used to classify a reuse library is domain dependent; the necessary conditions in the
feature definitions are specified in terms of the formal domain theory. This allows specific abstractions to
be made about operators that provide similar functionality over different types in the domain. For example,
the containment operator (contains()) is used to describe containment in many different situations. By
referencing this operator in a feature definition, we can draw similarities between components whose function
is specified in terms of containment. Figure 5 shows a subset of the features that are currently defined that
provide a data-flow style abstraction for the list processing domain. For example, the Select feature represents
the case where an output is an element of an input variable. Because the definition is parameterized on the
datatypes, Select is a possible feature whenever the containment operator exists between an input and output
type. This set of feature definitions is used in the following example.
4.3. Classification Example
Figure
6 shows the classification of the search component from Figure 3 based on the first two feature
definitions from Figure 5. First, the domain and range of the interface specification are substituted into the
feature definitions to create the set of feature/necessary conditions used for classification. For example, the
Domain
Theory
Instantiate
Feature/Goal
Pairs
Definitions
Feature
Proof Tactic
Specialized
Feature
Assignment
Requirements
Specification
Figure
6. Example: feature-based classification of a search component
Select and NonMember feature definitions:
are instantiated with the input variables and types from the domain (input :
(Rec) of the specification, giving:
The second and fourth instantiated definitions are eliminated due to type checking constraints, because the
Bool is not defined in the domain theory.
Next, the specialized proof tactic is used to check that the necessary conditions are implied by the specifi-
cation. For example, the following proof obligation is generated for the first instantiated definition:
In this example, the proof for the term contains(input; item) succeeds. Therefore, its associated feature,
Select(List[Rec]; Rec), is assigned to the component.
5. Retrieval
Given the feature set representation for a problem, we wish to retrieve components that match the problem
in terms of the formal specification matches. Because the features are assigned to the components based on
necessary conditions, the less features that a component has in common with a query, the less likely it is
that one of the reusability matches holds between the two. Therefore, we are interested in components that
have features in common with the query.
Because identical specifications will have identical feature sets, the initial query is for components having an
identical feature set to the problem. If there are no such components, the query is generalized by loosening the
feature comparison constraints to include a larger class of components. In the prototype, query generalization
is automated and continues until the number of retrieved components reaches a user-supplied threshold.
The first step in generalization is to weaken the requirements for feature value equivalence. Because
the feature values represent types that the component operates over, this identifies components performing
similar operations on different types. The second step is to reduce the number of the problem features
that the component must contain. This allows retrieval of components that may be partial solution to the
problem. Finally, the two types of generalization are combined to find components having any features in
common with the query. The following sample retrieval session provides an example of each type of query
generalization.
Figure
7 shows two problem specifications and a set of component specifications, all with their associated
feature sets. The specifications have all been classified using the feature definitions in Figure 5 and the
resulting feature sets are listed below each specification. Components are displayed beneath the problem
specifications for which they were retrieved. To exhibit the relationship between the feature sets and the
specification matches, the strongest match that exists between the problem and a component is listed under
the component specification.
In the first example, we wish to find the record with a specified key within a list. The component is
expected to perform correctly only if there is such a record in the list. Two components are retrieved
that have identical feature sets to find. The first, search, matches under Plug-in (and therefore Satisfies)
meaning it can be used to solve the problem. The second, binarySearch, only matches under Plug-in Post
because it requires the input to be sorted. It is possible that the input list is known to be sorted, but the
designer did not include that information in the query. Therefore, binarySearch is of possible use in solving
the problem. A third component, treeSearch can be retrieved by weakening the constraint that feature
values must be equal. The treeSearch component has both the Select and Build features, however with
different type values. This component may be useful after substituting List[Rec] for Tree and Rec for
Bucket.
The second example problem is specified more abstractly than the first. The designer is looking for
components that take a list and return a smaller list composed of elements from the first list. The manner of
selecting the elements for the new list is unspecified. This query is useful for finding the existing options for
decomposing lists. The first two components that are returned, removeFirst and removeLast, both match
the query under Weak Plug-in. This is because the precondition of the components, NOT empty(input), is
required to be true to ensure that the output list is smaller than the input list. Technically, for these two
components and this query, Weak Plug-in and Satisfies are equivalent matches because the preconditions are
logically equivalent. The third component, split, only matches under Weak Post because the component
precondition is stronger than query precondition. The split component provides a valid solution to the
problem, except in the case where length(input) = 1.
6. Empirical Evaluation
Component retrieval based on brute force specification matching attempts to match a query with every
component in the library. Therefore, it involves many individual proof attempts, most of which will fail [37].
The goal of semantic classification is to eliminate components that will lead to unsuccessful proof attempts
during evaluation, saving time and effort. The number of (matching and/or non-matching) components
eliminated depends upon the performance of the retrieval system. There were several experiments performed
to evaluate the retrieval system performance.
6.1. Implementation
The semantic classification system was implemented using the the ML programming language [25] and
HOL [12] theorem proving system. Several precautions were taken to reduce the overhead of automated reasoning
during classification [28]. First, the feature sets for the library components are calculated beforehand
and stored in an index. Second, a special purpose proof tactic was constructed in HOL to solve theorems
in the form of feature implication proofs. The proof tactic is parameterized on the set of domain axioms it
applies, making it domain independent. Finally to speed up inference, inductive proofs were eliminated by
burying the induction into proofs of lemmas and adding the lemmas to the domain theory. These precau-
2Problem Specifications:
COMPONENT find IS
IMPORT RecordList;
END find;
Features: Select(List[Rec],Rec),Build(Key,Rec)
Component Specifications:
COMPONENT subSet IS
POST FORALL (x:Rec)
IMPORT RecordList;
END subSet;
Features: Filter(List[Rec],Rec)
COMPONENT search IS
IMPORT RecordList;
END search;
Features: Select(List[Rec],Rec),Build(Key,Rec)
COMPONENT binarySearch IS
IMPORT RecordList;
END find;
Features: Select(List[Rec],Rec),Build(Key,Rec)
COMPONENT treeSearch IS
IMPORT BucketTree;
END find;
Features:
COMPONENT removeFirst IS
first : Rec;
IMPORT RecordList;
END removeFirst;
Features: Select(List[Rec],Rec),Filter(List[Rec],Rec)
COMPONENT removeLast IS
last
IMPORT RecordList;
END removeLast;
Features: Select(List[Rec],Rec),Filter(List[Rec],Rec)
COMPONENT split IS
POST NOT isEmpty(left) AND NOT isEmpty(right)
IMPORT RecordList;
END split;
Features: Split(List[Rec],Rec),Filter(List[Rec],Rec)
Figure
7. Example Problem Specification and Component Specifications with Feature Sets
tions result in an incomplete proof procedure. One goal of the experiments was evaluate the impact of this
incompleteness on retrieval performance.
6.2. The Library
The component retrieval evaluation was done using a library of specification for list manipulation components.
This library has been used in experiments with other specification-based component retrieval systems [37],
providing an opportunity for direct comparison of results. The library was designed to test whether the
specification-based retrieval can handle variation in the way that components are specified and the way that
queries are posed to the system. For example, there are 3 different specifications for a head component that
takes a list and returns a list containing only the first element of the original list.
6.3. Evaluation Method
The two traditional measures of component retrieval performance are recall and precision [23]. Recall is the
ratio of the number of relevant items retrieved to the total number of relevant items in the library. High
recall indicates that relatively few relevant components were overlooked. Precision is the ratio of the number
relevant items retrieved to the total number of items retrieved. High precision means that relatively few
irrelevant components were retrieved. In general, there is a tradeoff between precision and retrieval. The goal
is to find a practical balance between the two. The relevance condition is fundamental to the evaluation of
a retrieval system. As discussed below, experiments were conducted with two different relevance conditions.
It was also informative to observe the number of components retrieved by the system. This number
can help estimate the load that would be placed on the designer to interpret the results of a query in
an interactive system, or similarly, the search space that would be faced by an adaptation system when
considering component compositions [28].
The response time of the system was also measured to determine the practicality of the method. For each
measured quantity, the minimum, maximum and median was calculated for each of the scenarios in the
experiment.
6.4. Design of the Experiments
The experiments were designed to compare the way that several factors affected the performance of the
retrieval system. The first and foremost was the ability of the automated classification system to derive
classification features. Second, we were interested in the performance of retrieval in the context of both exact
retrieval and approximate retrieval. Finally, the nature of the library raised questions about determining an
appropriate query set for experimentation. Therefore, two different query sets were used.
6.4.1. Feature Generation The retrieval system as a whole can be separated into the classification scheme
(as defined by the feature definitions) and the classification mechanism (the specialized proof tactic and
domain theorems). In a sense, the mechanism attempts to implement the scheme. Both aspects of the
retrieval system can affect precision and recall. The classification scheme affects precision by the size and
consistency of component clusters. It affects recall because a scheme may not always contain a feature that
can be inferred from a relevant components.
The incompleteness of the classification mechanism (the specialized proof tactic is incomplete) could cause
it to generate fewer features and subsequently retrieve fewer components than intended. If the missing
components are relevant it will lower both recall and precision. The classification mechanism is sound (it is
implemented in terms of sound constructs in HOL) so it will only derive implied features.
In each experiment three scenarios were tested:
1. Signature Matching [44]: retrieval based on component signatures
2. Expected Features: retrieval based on features that would be assigned by a complete proof procedure.
3. Derived Features: retrieval based on features assigned by the implemented system.
Signature matching retrieves components with identical signatures. For this library, all of the components
have identical signatures (list ! list). Therefore, the performance of signature matching provides a profile
of the composition of the library in terms of relevant components. The expected features determine the
performance that the classification scheme, if implemented perfectly, would allow. Expected features were
determined by inspection with the aid of the lattice of specifications for the library. The implemented
retrieval mechanism (i.e., the domain theory axioms together with the feature derivation tactics) was used
to generate the derived features. The results were evaluated to see how close it comes to implementing the
classification scheme.
6.4.2. Relevance Conditions The choice of a relevance condition is fundamental in determining the significance
of precision and recall measures [22]. In our experiment, we evaluated the performance of the
system with respect to two relevance conditions. First, we use Satisfies match to be consistent with standard
specification-based retrieval experiments [21, 37]. Second, we consider a relevant component to be one
that matches the query specification with respect to any of the reuse matches identified in Section 3. This
relevance condition is important in the context of the adaptation, where a relevant component is one that
can (potentially) be adapted by the system [28, 29].
6.4.3. Query Set Following the experiments done by Schumann and Fischer [37], the library components
themselves are used as the set of queries to test the performance of the system. Using the components as
the query set makes two assumptions: (1) the component specifications represent a good sample of queries
that may be asked and (2) these queries all have the same probability of being posed to the system. To get
results that would predict the performance of the tool in a realistic setting, it would be necessary to have
a distribution of queries that represents how the tool would be used. There is no study in the component
retrieval literature that would provide this information.
The library used in the experiment has several groups of functionally equivalent specifications that would, in
practice, all point to a one component. This raises a question about what should define a unique query in the
experiment: a specification or a component with potentially many specifications. Therefore, the experiment
was run once with duplicate specifications in the query set, and once without. The difference between the
results for the two cases was negligible, indicating that the system does not favor the components with
multiple specifications [28]. We present only the results from the experiment including equivalent queries
here.
6.5. Experimental Results
The experiment involved a library of list manipulation components all having the signature list 7! list. The
library contained 63 specifications for 45 functionally unique components. The experiment was divided into
two parts. For the first part, the classification scheme of Figure 5 was used. Due to the limited signatures,
only the following features applied:
Filter(T
For the second part of the experiment, the classification scheme was extended. The extension of the scheme
was guided by placing the library components into a lattice based on the Satisfies matching condition.
The domain theory theorems that were used as parameters to the classification mechanism are shown in
Figure
8. These theorems were selected by observing the failed proofs of expected features and determining
Normalization Rules:
Normalization Implications:
Rewrite Rules:
Expansion Rules:
Figure
8. Domain Theory for List Library Experiment
Table
1. Retrieval for Satisfies Match Using the Initial Classification Scheme
Scenario Retrieved Precision Recall
Signature Match 63.00 (63 - 63) 0.11 (0.02 - 0.64) 1.00 (1.00 - 1.00)
Expected Features
Exact Match 22.71 (3 - 31) 0.20 (0.03 - 0.74) 0.84 (0.05 - 1.00)
Relaxed Match
Derived Features
Exact Match 22.27 (3 - 31)
Relaxed Match 43.57 (12 - 51) 0.12 (0.02 - 0.65) 0.86 (0.10 - 1.00)
the theorems necessary to make the proofs succeed. The theorems mainly deal with reasoning about containment
(which is used to define the feature definitions) in terms of the list type constructors CONS, APPEND
and [].
6.5.1. Satisfies Match The results of the initial part of the experiment for Satisfies match are shown in
Table
1. The table entries denote the average value with the minimum and maximum in parenthesis. The
retrieval mechanism comes very close to implementing the classification scheme: the expected feature sets
were derived for 61 of the 63 specifications. The failed classification was due to the use of a three way
conditional in the specification that was not supported directly in the domain theory. The domain theory
could be extended to support these conditionals, however an effort was made to not over-specialize the
domain theory toward supporting classification application. Therefore, this extension was not made.
The distribution of expected feature sets for the library is show in Table 2. This shows that the classification
scheme does a questionable job of clustering components in the library; nearly half of the components are
only assigned the Filter feature. In fact, the 3 features are not independent but represent a generalization
hierarchy: Route ) Permute ) Filter. While it is useful to have features that specialize other features, it
would also be useful to have other orthogonal features to provide better coverage of the library.
Table
2. Distribution of feature sets for initial classification
scheme.
Feature Set No. Specifications No. Components
fFilterg
some_total
last_total1,2
tail1,3
swap id_segment
id_front
run_max_eq
segment_ne_total
id
segment_front
id_single
segment
swap_outer
swap_outer_total
swap_total
perm_r1,2
run_max_bracket
run_eq1,2,3
run_bracket1,2
lead_total1,2
lead
segment_rear
segment_ne
perm_lr1,2
id_nil
elim_dup_lr
rot_r1,2
rot_r_total1,2,3
rot_l_total1,2,3
rotate_total
rotate
rot_l1,2
tail_total1,2
elim_dup_unique_l
elim_dup_r
last head1,3
some
id_rear
elim_dup_unique_r
perm_l1,2
elim_dup_l
no_dup
elim_dup_unique_lr
FILTER
Figure
9. Lattice of Specifications for the List Component Library
The partial-order lattice induced by the Satisfies Match on the library is a useful aid for discovering
potential features. This lattice is shown in Figure 9 with the areas covered by the Filter, Permute and Route
features shown with dashed lines. As hoped, the features tend to group components that are related in the
lattice. The lattice can be used to identify groups of components that are closely related, and then their
specifications may be inspected to identify a logical feature that they share. 3
For example, the some component is the root of a tree containing last, head, some total, last total
and head total. The some component has the following specification:
POST EXISTS(i:Rec) mem1 mem2 .
This specification is very similar to the definition of the Select feature, only Select looks for an element as
an output, not a singleton list. Therefore, if the definition of Select is modified to identify a singleton list,
it should be an expected feature of all of these components.
Other useful feature can be identified using the lattice. For example, associating a feature with the id nil
component:
and its descendants provides coverage for the id segment tree an and also divides the components with the
Filter feature roughly in half in a fairly orthogonal manner. As a complementary feature to IdNil, a feature
NoNil could be defined stating that a component does not accept an empty input, with the intention that a
component could not imply both of these features.
These three features were formalized as follows:
They were added to the feature definitions and the experiment was rerun. On inspection of the derived
feature sets, it was immediately obvious that there were two problems: (1) the Some feature was never
derived and (2) many components were assigned both IdNil and NoNil, which was not the intention of the
scheme.
The source of the first problem was that the proof tactic failed to handle existential goals. This problem
was solved by extending the tactic to attempt to solve a goal by substituting in free variables from the goal
for the existential variable. Additionally, the expected results were wrong: as defined, the Some feature
will not hold for some total and its descendants. The components named * total are total functions that
map an empty input list to an empty output list. In this case, the output does not contain an element and
therefore, these specifications will not imply the Some feature.
The second problem with the extended scheme was that IdNil and NoNil were not complementary as
thought. In fact, NoNil implies IdNil because, in the case where the input is not empty, assuming it is empty
(which is be the first step in proving IdNil) allows the proof of IdNil to succeed trivially. This results in the
coverage of the IdNil feature to be nearly identical to the Filter feature, making it a useless extension. Taken
together, these experiences indicate the need for tools to support the construction of classification schemes,
as discussed in the future work section.
Table
3. Distribution of Feature Sets for Extended Classification Scheme.
Feature Set No. Specifications No. Components
Table
4. Retrieval for Satisfies Match Using the Extended Scheme
Scenario Retrieved Precision Recall
Signature Match 63.00 (63 - 63) 0.11 (0.02 - 0.64) 1.00 (1.00 - 1.00)
Expected Features
Exact Match 13.64 (1 - 23)
Relaxed Match
Derived Features
Exact Match 14.05 (3 - 23) 0.29 (0.04 - 1.00) 0.69 (0.05 - 1.00)
Relaxed Match 43.57 (12 - 51) 0.12 (0.02 - 0.65) 0.86 (0.10 - 1.00)
Table
5. Approximate Retrieval Using the Initial Classification Scheme
Scenario Retrieved Precision Recall
Signature Match 63.00 (63 - 63) 0.14 (0.02 - 0.81) 1.00 (1.00 - 1.00)
Expected Features
Exact Match 22.71 (3 - 31) 0.27 (0.03 - 1.00) 0.84 (0.06 - 1.00)
Relaxed Match
Derived Features
Exact Match 22.27 (3 - 31) 0.28 (0.03 - 1.00) 0.81 (0.06 - 1.00)
Relaxed Match 43.57 (12 - 51) 0.16 (0.02 - 0.96) 0.85 (0.07 - 1.00)
The experiment was run again, this time without the IdNil feature and with the expected results more
throughly evaluated. The distribution of feature sets is shown in Table 3. The extended scheme does a better
job of breaking the components into clusters. Nearly 1/3 of the components that were initially assigned only
the Filter feature are now assigned additional features.
The results of evaluating the system with the extended classification scheme are shown in Table 4. Compared
to the results of the previous classification scheme, there was a noticeable drop in the average number
of components retrieved along with an increase in precision. This was accompanied by a slight decrease in
recall. Once again, the classification mechanism came very close to implementing the classification scheme.
6.5.2. Approximate Retrieval The experiments were repeated while considering a relevant component to
be one that matches the query specification with respect to any of the reuse matches identified in Section 3.
Because this relevance condition is logically weaker than the Satisfies match (it is the disjunction of Satisfies
and Weak Post) the relevant components for a query will be a superset of those relevant to Satisfies.
Table
6. Approximate Retrieval Using the Extended Scheme
Scenario Retrieved Precision Recall
Signature Match 63.00 (63 - 63) 0.14 (0.02 - 0.83) 1.00 (1.00 - 1.00)
Expected Features
Exact Match 14.97 (1 - 25) 0.32 (0.04 - 1.00) 0.57 (0.04 - 1.00)
Relaxed Match
Derived Features
Exact Match 14.05 (3 - 23) 0.31 (0.04 - 1.00) 0.54 (0.06 - 1.00)
Relaxed Match 43.57 (12 - 51) 0.16 (0.02 - 0.96) 0.84 (0.07 - 1.00)
The results of the experiment using the initial classification scheme are shown in Table 5. The precision of
this experiment was higher, because more of the retrieved components are relevant. Recall remained the same
indicating that the same percentage of new relevant components is retrieved. The implementation of the
classification scheme continues to come close to the performance of a complete and consistent implementation.
The results of the experiment using the extended classification scheme are shown in Table 6. For the exact
match there was an increase in precision over the initial scheme comparable with the increase seen in the
Satisfies match experiment. However, there was a larger drop in recall. This is caused by the exact match
not retrieving many of the new relevant components. This means that the exact feature match using the
extended scheme does not approximate the relevance condition very well in terms of recall. The relaxed
match results remain consistent with the other experiments. The results indicate that relaxed feature match
is more appropriate for approximate retrieval than exact feature match.
6.5.3. Response Time The response time of the classification system during the experiments ranged from
0.15 to 0.66 seconds with an average of 0.35 seconds on a 200MHz PentiumPro processor running Linux.
Database access accounted for only 0.025 seconds, on average, therefore the bulk of the time was spent
classifying queries. This is well within the acceptable response time for an interactive system.
7. Comparison of Results
7.1. Specification-Based Retrieval
Most specification-based component retrieval systems are in the "proof of concept" stage and therefore have
not been evaluated over a sizable component library. A notable exception is the NORA/HAMMR system of
Fischer and Schumann [37]. The NORA/HAMMR retrieval system is set up as a chain of filters. The initial
filter is signature matching and they become more restrictive as they progress. The final filter in the chain
is full scale specification matching. The NORA/HAMMR system was evaluated using the same library as
our experiments using all of the specifications (with duplicates) as the query set.
As an intermediate filter, they use the MACE model checker in several configurations to select a subset
of components from the library to undergo specification matching. Using a 20 second time limit for model
checking computation, they observed average recall rates between 74.7% and 81.3% with precision between
18.5% and 16.5%. This is comparable to the results achieved with the initial classification scheme in our
experiments, however, our response time was 0.66 seconds in the worst case.
They have experimented with several automated theorem provers to do specification matching as the final
stage of retrieval. For example, using the SETHEO prover and a time limit of 20 seconds, the results were
a recall rate of 61.2% and precision of 100%. The high precision is due to the fact that SETHEO's proof
procedure is sound. The loss in recall is due to a lack of completeness that comes from a technique for
approximating induction, restricting the set of axioms available for inference and the 20 second time limit.
Lowering the time limit to 1 second causes the recall to drop below 50% [6]. Using semantic classification as
a filter prior to specification matching could reduce the load on specification matching and allow this time
limit to be raised, increasing recall.
7.2. Information Retrieval Methods
The majority of component retrieval tools used in practice are based on information retrieval methods.
Frakes and Pole conducted an extensive study of representation methods for reusable components [8]. They
did comparison of the retrieval performance of attribute-value, enumerated, faceted and keyword based
representations. The relevance determination was made by two domain experts. There is no mention of
the (manual) method used to classify the components. The recall and precision measurements were in the
30-40% and 50-100% ranges, respectively for all of the methods. Statistical analysis showed no significant
difference among the methods. Our retrieval results are consistent with these numbers.
The benefit that semantic classification provides over these methods is consistency and automation of the
classification process. While the uses have to be familiar with the specification language, they do not have to
be familiar with the organization of the component library. In addition, specification-based retrieval provides
a precise relationship that exists between a retrieved component and a query, increasing the utility of the
retrieval results. All of this is achieved with similar performance results.
Girardi and Ibrahim [10, 11] evaluate a method based on syntactic and semantic analysis of natural language
descriptions (ROSA). They use normalized versions of the precision and recall formulas because the system
returns ranked results. They used a library of 418 general purpose Unix commands and 20 queries derived
from user descriptions of frequently used commands. They report recall values in the 99-100% range with
precision values in the 90-92% range. However they do not state their relevance condition, making the results
rather non-informative. Their selection of the query set is also important because it does not evaluate the
method in the less frequent cases.
8. Discussion
8.1. System Wide Effects
The retrieval performance of semantic classification must be considered in the context of its role in the retrieval
system. The results of retrieval are passed on to an evaluation phase (based on specification matching)
that has perfect precision and recall, assuming a sound and complete proof procedure is used. The point is
that relevant components cannot be added, but only removed by specification matching. This means semantic
classification acts as a filter that sets an upper bound on the recall of the combined retrieval/evaluation
system. Limiting recall will shrink the design space that can searched, potentially lowering the quality of
the designs created by the system.
In contrast, the precision of the retrieval phase has no effect on the precision of the combined system. The
precision effects the number of proofs that must be attempted during specification matching, and therefore
has an effect on the system response time. Therefore, the recall/precision tradeoff in the feature-based
retrieval phase translates into a design quality/response time tradeoff in the context of the entire retrieval
system.
8.2. Limitations
In the experiments, the major reason for feature generation failure was due to specification that was broken
into cases based on conditions that were not supported by rules in the domain theory. In these specifications
the postcondition has the form:
To prove that a feature is implied by a specification of this form requires a proof by cases approach. To
facilitate this, there must be an axiom in the domain theory of the form:
If there are no rules in the domain theory to support the specific case decomposition used in the specification,
the feature proofs cannot succeed. This can be fixed by either guiding the user during specification to use
a set of conditions that are supported by the domain theory, or by allowing conditional features that are
assigned if a feature holds under any (rather than all) of the conditions specified.
Feature derivation runs into a similar problem in the case of partial specifications, where component
behavior is not defined for all legal inputs. A feature is only assigned to a component if it can be derived in
terms of the behavior of the component for every legal input value. Partial specifications are too logically
weak to allow a feature to be derived in the case of the undefined behavior. This can be fixed by (1)
disallowing partial specifications, (2) strengthening the postcondition for the purposes of feature generation,
or (3) allowing conditional features as described above.
There was also a problem where two components we specified in such a way that their specifications cause
non-termination of rewriting. For example, a head component with input l and output m can be specified
as:
If this statement is used as a left-to-right rewrite rule, the rewriting system will not terminate. One possible
solution is to build a timeout option into the rewriting system, similar to that used by Schuman and
Fischer [37].
Finally, it should be noted that, in general, specification-based component retrieval is susceptible to loss
of recall due to the semantic gap between a component and a specification [22]. A component is associated
with a specification if the component correctly implements a specification. However, there is a gap that
a query may fall into: it is possible that a component may satisfy a query that its specification does not
satisfy. The effects of this situation cannot be evaluated in the experiments because we are working only
with specifications. However, it should be noted as a potential limitation of the method.
8.3. Building Classification Schemes
Discovering features is not an easy process. Formalizing an abstract concept that is shared among several
components is difficult and picking a collection of useful ones is even harder. Several times during this
investigation, the intuitive concept of a feature was disproved by the system. Recognizing the utility of
hierarchical and complementary features is a good start toward building better schemes. However, tool
support would be necessary to scale the method to larger libraries and more sophisticated classification
schemes.
There are several ways that the formalized classification framework provides a foundation for automated
tools. The formal definitions of the necessary conditions can support analysis of the scheme. For example,
a set of features can be proven to be mutually exclusive while their disjunction is a tautology. Therefore,
every component would be assigned exactly one of these features. These features would provide coverage
similar to the "facets" in faceted classification [34].
It is also possible to provide support for extending classification schemes within the framework. For
example, if two distinguishable components are classified identically, it may be possible to identify the parts
of the specifications that are distinguishable and automatically derive a new feature that represents this
difference.
8.4. Signatures vs. Semantics
In most specification-based component retrieval systems, the first filter used to reduce the library is signature
matching [7, 43, 44]. Signature matching uses the types from a component's interface to determine its
compatibility with a query. The guiding assumption is that if the types do not match there is no reason to
further examine a component's behavior. We believe there are two cases where semantic classification has
potential for greater recall than signature-based approaches.
The retrieval performance of signature matching degrades when considering relevant components that could
be combined or adapted to satisfy a query. Standard signature matching will eliminate relevant components
with partial matches to an interface. Allowing partial matches would allow too many component to match,
greatly decreasing the retrieval precision of signature matching. Because feature definitions are not required
to constrain all inputs and outputs in a component, semantic classification does not have this problem. In
fact, components are retrieved that provide slices of the appropriate behavior, independent of the type used.
The type information (the feature values) is used to identify type substitutions before evaluation.
Using signatures has been suggested to discover combinations of components in a library that match a
query signature [15, 23]. We take a semantic approach to the problem of combining components [28, 29].
classification assists this approach by retrieving components that provide pieces of the appropriate
behavior. Therefore, it can locate partial solutions even when the remainder of the solution is not in the
library.
9. Related Work
The use of formal specifications to assist software component retrieval has been widely proposed projects [3,
37, 16, 21]. 4 Zaremski and Wing [45] provided a foundation for studying the more general activity of specification
matching, the verification of logical relationships between specifications. There are many approaches
to making specification-based component retrieval more efficient. However, only a few of these methods
make use of the semantics provided by formal specifications.
The NORA/HAMMR deductive retrieval tool built by Fischer and Schumann uses a series of filters to
identify reusable components [37]. This tool is novel in its use of model checking as one of the search
filters. Before running the theorem prover to check the match condition, the conditions are checked by
searching for a model in a small part of the specification theory. Since it is necessary for such a model
to exist for the conditions to hold, only those components that pass the model checking stage need to be
checked in the entire theory. They also use various techniques to improve the performance of the theorem
prover during specification matching, such as reducing the set of axioms used and parallel proof attempts.
Evaluation of prototype implementations on libraries of list manipulation functions showed very encouraging
results. However, the recall (retrieved components/useful components) of the prototype was limited by
signature matching. The notion of using filters to restrict the search space is consistent with our use of
necessary conditions to eliminate non-matching component. Because a semantic-based classification scheme
can provide better recall than signature matching, it could be used as a preliminary filter and potentially
increase recall.
The Inquire retrieval mechanism [32] within the Inscape [31] environment supports retrieval based on
component specifications. Preconditions and postconditions for components are formulated in terms of a
given set of formally defined logical predicates. An inference mechanism is used during the retrieval process
to retrieve components that provide the various predicates. The formal predicate definitions are useful as
unambiguous descriptions of the predicate vocabulary. The prototype implementation is reported to work
very well, but with a restricted specification language. The restricted language reduces the inference power
necessary for component retrieval. The methodology presented here uses a reversed approach from Inscape.
Interface specifications are defined in full first-order logic. The formally defined features (predicates) are then
assigned to the specifications if they are logically implied by the specification. In this case, the predicates are
not the complete specification of the component, but represent various aspects of the component's function.
Because feature predicates do not have to be useful as specification predicates, they can be more abstract,
allowing more flexibility in the types of similarity that can be represented. In addition, the more expressive
specification language allows precise evaluation of reusability.
Deductive program synthesis [20, 40] also makes use of formal methods to automate software reuse. For
example, the Amphion system [18, 41] successfully uses deductive synthesis to construct software from a
subroutine library for solar system geometry. In these systems, components (language primitives or sub-
routines) are represented as mathematical functions and their behavior specified via axioms. A program
is synthesized by proving that, for any valid input, there exists an output that satisfies the specification.
The occurrence of a primitive function in the proof constructed during deductive synthesis corresponds to a
call to the associated subroutine. Therefore, a component is effectively "retrieved" when its corresponding
axioms are used during the proof process. This means that the domain theory axioms and the tactics used in
decomposing proofs will determine which components are used. Ongoing research is exploring the integration
of architectural decomposition tactics with our current component retrieval system to support automated
component adaptation and integration [28, 29, 30].
10. Conclusion
Software reuse and formal specification are two methodologies that show high potential impact on software
productivity and reliability. Used together, they permit increased automation and assurance in the reuse
process. We presented how component reusability and similarity can be described formally as logical relationships
between the problem specification and a component specification. However, it is too computationally
expensive to formally verify these relationships in the quantities required for practical component retrieval.
Therefore, specification-based retrieval would benefit from a method to approximate these relationships and
identify a subset of the library to undergo verification.
In this paper, we described a method for classifying components based on their formal specifications.
Features are assigned to components based on specific necessary conditions that are implied by the component
specifications. The logical form of the specification matches determining reusability ensures that components
with similar feature sets are more likely to match. The collection of necessary conditions that controls the
classification scheme is defined formally to allow automated classification. The theorem proving required
during classification is applied in such a way that complexity of component classification is much less than
that of applying multiple specification matches. Once classified, components are retrieved via syntactic
comparison of feature sets.
The results of empirical evaluation on a library of list components show that the method can provide
retrieval performance comparable to existing methods. The benefits are a faster response time than other
formal approaches. The method improves upon informal methods by providing higher levels of consistency
and automation.
Our future work will focus on integrating specification-based component retrieval with support for automated
component adaptation and integration [28]. A long-term goal is to develop support for run-time
component integration in high-assurance component-based systems [19]. We are also investigating tools to
support development and maintenance of formal classification schemes.
Acknowledgments
We would like to thank Bernd Fischer, Gary Leavens, Amy Moormann Zaremski, Ali Mili, Santos Lazzeri
and Dale Martin for helpful comments on the development and evaluation of this research. We also thank the
anonymous reviewers of the current and earlier versions of this work for their suggestions for the presentation
and evaluation of the work. Support for this work was provided in part by the Advanced Research Projects
Agency and monitored by Wright Labs under the RAASP Technology Program contract number F33615-
93-C-1316 and the CEENSS Technology Program contract number F33615-93-C-4304.
Notes
1. We do not use Zaremski and Wing's method for identifying reuse matches based on syntactic
patterns. We select matches based on formalization of intuitive notions of reusability, and their
utility in component retrieval.
2. As opposed to free text indexing of source code and/or comments, which would not satisfy the
high level of assurance required in this application.
3. It should be noted that distances in the lattice are meaningless; the lattice was arranged by
hand to minimize the crossing of links.
4. For a general overview of component retrieval methods for assisting software reuse, see the
survey by Mili et al [23]
--R
Validating component compositions in software system generators.
Perlis, editors. Software Reusability - Concepts and Models
Program development as a formal activity.
Two theses of knowledge representation: languages restrictions
NORA/HAMMR: Making deduction-based software component retrieval practical
An empirical study of representation methods for reusable software components.
Formal Foundations for the Specification of Software Architecture.
Automatic indexing of software artifacts.
Using english to retrieve software.
HOL: A proof generating system for higher-order logic
The Science of Programming.
Languages and Tools for Formal Specification.
Generalized behavior-based retrieval
Using formal methods to construct a software library.
A formal approach to domain-oriented software design environments
Fundamentals of deductive program synthesis.
A refinement based system.
A survey of software reuse libraries.
Reusing software: Issues and research directions.
Another nail to the coffin of facated controlled-vocabulary component classification and retrieval
The Definition of Standard ML.
Correct architecture refinement.
Automated Component Retrieval and Adaptation Using Formal Specifications.
Toward automated component adaptation.
Declarative specification of software architectures.
The Inscape environment.
Rub'en Prieto
Rub'en Prieto
Specifications as search keys for software libraries.
Automated deduction and formal methods.
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David Hemer, Semi-Automated Component-Based Development of Formally Verified Software, Electronic Notes in Theoretical Computer Science (ENTCS), 187, p.173-188, July, 2007
Robert G. Bartholet , David C. Brogan , Paul F. Reynolds, Jr., The computational complexity of component selection in simulation reuse, Proceedings of the 37th conference on Winter simulation, December 04-07, 2005, Orlando, Florida
Sofien Khemakhem , Khalil Drira , Mohamed Jmaiel, SEC: a search engine for component based software development, Proceedings of the 2006 ACM symposium on Applied computing, April 23-27, 2006, Dijon, France
Brandon Morel , Perry Alexander, SPARTACAS Automating Component Reuse and Adaptation, IEEE Transactions on Software Engineering, v.30 n.9, p.587-600, September 2004 | component retrieval;formal specification;software reuse |
592038 | Efficient Implementations of Software Architectures via Partial Evaluation. | The notion of flexibility (that is, the ability to adapt to changing requirements or execution contexts) is recognized as a key concern in structuring software, and many architectures have been designed to that effect. However, the corresponding implementations often come with performance and code size overheads. The source of inefficiency can be identified to be in the loose integration of components, because flexibility is often present not only at the design level but also in the implementation. To solve this flexibility vs. efficiency dilemma, we advocate the use of partial evaluation, which is an automated technique to produce efficient, specialized instances of generic programs. As supporting case studies, we consider several flexible mechanisms commonly found in software architectures: selective broadcast, pattern matching, interpreters, software layers, and generic libraries. Using Tempo, our specializer for C, we show how partial evaluation can safely optimize implementations of those mechanisms. Because this optimization is automatic, it preserves the original genericity and extensibility of the implementation. | Introduction
What is partial evaluation?
Partial evaluation is a technique to partially execute a program, when only some
of its input data are available. Consider a program p requiring two inputs, x 1
and x 2 . When specific values d 1 and d 2 are given for the two inputs, we can
run the program, producing a result. When only one input value d 1 is given,
we cannot run p, but can partially evaluate it, producing a version p d1 of p
specialized for the case where x Partial evaluation is an instance of
program specialization, and the specialized version p d1 of p is called a residual
program.
For an example, consider the following C function power(n, x), which computes
x raised to the n'th power.
int n,x;
f int p;
while (n ?
if (n % 2 ==
else
Given values 7, we can compute power(5,7), obtaining the result
exploits that x even integers n).
Suppose we need to compute power(n, x) for a great many
different values of x. Then we can partially evaluate the function for
obtaining the following residual function:
int x;
f int p;
We can compute power 5(7) to obtain the result 7 In fact, for
any input x, computing power 5(x) will produce the same result as computing
Department of Computer Science, University of Copenhagen, Universitetsparken
2 Department of Mathematics and Physics, Royal Veterinary and Agricultural University,
Thorvaldsensvej 40, DK-1871 Frederiksberg C, Denmark
power(5,x). Since the value of variable n is available for partial evaluation, we
say that n is static; conversely, the variable x is dynamic because its value is
unavailable.
This example shows the strengths of partial evaluation: in the residual program
power 5, all tests and all arithmetic operations involving n have been
eliminated. The flow of control (that is, the conditions in the while and if
statements) in the original program was completely determined by the static
variable n.
Now suppose we needed to compute power(n,7) for many different values
of n. This is the opposite problem of the above: now n is dynamic (unknown)
and x is static (known). There is little we can do in this case, since the flow of
control is determined by the dynamic variable n. One could imagine creating a
table of precomputed values of 7 n for some values of n, but how are we to know
which values are relevant?
In many cases some of the control flow is determined by static variables, and
in these cases substantial speed-ups can be achieved by partial evaluation.
1.1 Notation
We can consider a program in two ways: as a function transforming inputs to
outputs, and also as a data object, being input to or output from other programs.
We need to distinguish the function computed by a program from the program
text itself.
Writing p for the program text, we write for the function computed by
or when we want to make explicit the language L in which p is written.
Consequently, denotes the result of running program p with input d on
an L-machine.
Now we can assert that power 5 is a correct residual program (in C) for
power and given input 5:
1.2 Interpreters and compilers
An interpreter Sint for language S, written in language L, satisfies for any S-
program s and input data d:
That is, running s with input d on an S-machine gives the same result as using the
interpreter Sint to run s on an L-machine. This includes possible nontermination
of both sides.
A compiler STcomp for source language S, generating code in target language
T , and written in language L, satisfies
That is, p can be compiled to a target program p 0 such that running p 0 on a
T-machine with input d gives the same result as running p with input d on
an S-machine. Though the equation doesn't specify this, we normally assume
compilation to always produce a target program.
Partial evaluators
2.1 What is a partial evaluator?
A partial evaluator is a program which performs partial evaluation. That is, it
can produce a residual program by specializing a given program with respect to
part of its input.
Let p be an L-program requiring two inputs x 1 and x 2 as above. A residual
program for p with respect to x is a program p d1 such that for all values
d 2 of the remaining input,
A partial evaluator is a program peval which, given a program p and a part d 1
of its input, produces a residual program p d1 . In other words, a partial evaluator
peval must satisfy:
peval
This is the so-called partial evaluation equation, which reads as follows: If partial
evaluation of p with respect to d 1 produces a residual program p d1 , then running
d1 with input d 2 gives the same result as running program p with input [d 1
As for compilers, the equation does not guarantee termination of the left-hand
side of the implication. In contrast to compilers we will, however, not always
assume partial evaluation to succeed. While it is desirable for partial evaluation
to always terminate, this is not guaranteed by a large number of existing partial
evaluators. See sections 2.2 and 5.5 for more about the termination issue.
Above we have not specified the language L in which the partial evaluator is
written, the language S of the source programs it accepts, or the language T of
the residual programs it produces. These languages may be all different, but for
notational simplicity we assume they are the same, . Note that
opens the possibility of applying the partial evaluator to itself (see below).
For an instance of the partial evaluation equation, consider
5, then from must follow that power(5,7)
2.2 What is achieved by partial evaluation?
The definition of a partial evaluator does not stipulate that the specialized program
must be any better than the original program. Indeed, it is easy to write a
program peval which satisfies the partial evaluation equation in a trivial way, by
prepending a new 'specialized' function power 5 to the original program. The
new function simply calls the original one with the given argument:
int n,x;
f int p;
while (n ?
if (n % 2 ==
else
int x;
f return(power(5, x)); g
While this program is a correct residual program, it is no faster than the original
program, and quite possibly slower. Even so, the construction above can be used
to prove existence of partial evaluators, a proof similar to Kleene's (1952) proof
of the s-m-n theorem [63], a theorem that essentially stipulates the existence of
partial evaluators in recursive function theory.
But, as the example in the introduction demonstrated, it is sometimes possible
to obtain residual programs that are arguably faster than the original pro-
gram. The amount of improvement depends both on the partial evaluator and
the program being specialized. Some programs do not lend themselves to spe-
cialization, as no computation can be done before all input is known. Sometimes
choosing a different algorithm may help, but in other cases the problem itself is
ill-suited for specialization. An example is specializing the power function to a
known value of x, as discussed in the introduction.
Looking at the definition of power, one would think that specialization with
respect to a value of x would give a good result: the assignments,
do not involve n, and as such can be executed
during specialization. The loop is, however, controlled by n. Since the termination
condition is not known, we cannot fully eliminate the loop. But x and p
will have different values in different iterations of the loop, so we cannot replace
them by constants. Hence, we find that we cannot perform the computations on
x and p anyway. We could force unfolding of the loop to keep the values of x
and p known, but since there is no bound on the number of different values x
and p can obtain, no finite amount of unfolding can eliminate x and p from the
program.
This conflict between termination of specialization and quality of residual
program is common. The partial evaluator must try to find a balance that
ensures termination often enough to be interesting (preferably always) while
yielding sufficient speed-up to be worthwhile. Due to the undecidability of the
halting problem, no perfect strategy exists, so a suitable compromise must be
found. See Section 5.5 for more on this subject.
3 Another approach to program specialization
A generating extension of a two-input program p is a program p gen which, given
a value d 1 for the first input of p, produces a residual program p d1 for p with
respect to d 1 . In other words,
The generating extension takes a given value d 1 of the first input parameter x 1
and constructs a version of p specialized for x
As an example, we show below a generating extension of the power program
from the introduction:
int n;
f
printf("int x;"n");
printf("f int p;"n");
while (n ?
if (n % 2 ==
else f printf("
printf(" return(p);"n");
Note that power-gen closely resembles power: those parts of power that depend
only on the static input n are copied directly into power-gen, and the parts that
also depend on x are made into strings, which are printed as part of the residual
program. Running power-gen with input yields the following residual
program:
int x;
f int p;
This is almost the same as the one shown in the introduction. The difference is
because we have now made an a priori distinction between static variables (n)
and dynamic variables (x, p). Since p is dynamic, all assignments to it are made
part of the residual program, even was executed at specialization
time in the example shown in the introduction.
Later we shall see that a generating extension can be constructed by applying
a sufficiently powerful partial evaluator to itself. One can even construct a
generator of generating extensions that way.
4 Partial evaluation, interpreters, and compila-
tion
4.1 Compilation using a partial evaluator
In Section 1.2 we defined an interpreter as a program taking two inputs: a
program to be interpreted and input to that program:
We often expect to run the same program repeatedly on different inputs. Hence,
it is natural to partially evaluate the interpreter with respect to a fixed, known
program and unknown input to that program. Using the partial evaluation
equation we get
[s; d] for all d
Using the definition of the interpreter we get
The residual program is thus equivalent to the source program. The difference
is the language in which the residual program is written. If the input and output
languages of the partial evaluator are identical, then the residual program is
written in the same language L as the interpreter Sint. Hence we have compiled
s from S, the language that the interpreter interprets, to L, the language in
which it is written.
4.2 Compiler generation using a self-applicable partial eval-
uator
We have seen that we can compile programs by partially evaluating an in-
terpreter. Typically we will want to compile many different programs, which
amounts to partially evaluating the same interpreter repeatedly with respect to
different programs. This situation calls for optimization by yet another application
of partial evaluation. Hence we use a partial evaluator to specialize a partial
evaluator peval with respect to a program Sint, but without the argument s of
Sint. Using the partial evaluation equation we get:
peval ]][peval;
peval Sint
Using the results from above, we get
peval Sint
for which we have
d for all d
We recall the definition of a compiler from Section 1.2:
We see that peval Sint
fulfills the requirements for being a compiler from S to T .
In the case where the input and output languages of the partial evaluator are
identical, the language in which the compiler is written and the target language
of the compiler are both the same as the language L in which the interpreter
is written. Note that we have no guarantee that partial evaluation terminates,
neither when producing the compiler nor when using it. Experience has shown
that while this may be a problem, it is often the case that if compilation terminates
for a few general programs then it terminates for all.
Note that the compiler peval Sint is a generating extension of the interpreter
Sint , according to the definition shown in section 3. This generalizes to any
program, not just interpreters: partially evaluating a partial evaluator peval
with respect to a program p yields a generating extension p
program.
4.3 Compiler generator generation
Having seen that it is interesting to partially evaluate a partial evaluator, we
may want to do this repeatedly: to partially evaluate a partial evaluator with
respect to a range of different programs (e.g., interpreters). Again, we may
exploit partial evaluation:
peval peval
Since which is a generating extension of p, we can see
that peval peval is a generator of generating extensions. The program peval peval
is itself a generating extension of the partial evaluator: peval peval peval .
In the case where p is an interpreter, the generating extension p gen is a compiler.
Hence, peval gen is a compiler generator, capable of producing a compiler from
an interpreter.
4.4
Summary
: The Futamura projections
Instances of the partial evaluation equation applied to interpreters, directly
or through self-application of a partial evaluator, are collectively called the
Futamura projections. The three Futamura projections are:
The first Futamura projection: compilation
peval
The second Futamura projection: compiler generation
peval ]][peval;
The third Futamura projection: compiler generator generation
The first and second equations were devised by Futamura in 1971 [40], and the
latter independently by Beckman et al. [12] and Turchin et al. [94] around 1975.
5 Techniques for partial evaluation
5.1 Polyvariant specialization
Polyvariant specialization is a technique for partial evaluation which works for
a range of languages. A program is thought of as a collection of program points ,
connected by control-flow edges . In a flow-chart language, program points and
control-flow edges are, respectively, labelled basic blocks and jumps ; in a functional
language, they are defined functions and function calls ; in a logic language,
they are predicates and predicate applications (atoms).
Polyvariant specialization constructs a residual program by creating zero
or more specialized variants of each program point, and connecting them by
residual control-flow edges.
5.1.1 The exponentiation example revisited
To illustrate polyvariant specialization, consider the power function from Section
1 in flowchart form, with explicitly labelled basic blocks:
lab1: if (n !=
if (n % 2 !=
goto lab1
goto lab1
lab3: return(p)
A program on this form is specialized to given values of the static variables by
specializing the basic blocks. For each basic block, and for each set of static
variable values with which it may be executed, one creates a specialized basic
block in the residual program. This is polyvariant specialization [24, 57].
For instance, the basic block labelled lab1 may be executed with static
Hence one creates a specialized basic block, whose label lab1 fn=5;p=1g
consists of the original label and bindings for the static variables. The body of
the specialized basic block consists of the specialized residual commands from
the original basic block. Naturally, the specialized version of a jump goto lab
is itself a jump goto lab f:::g
to a specialized (decorated) version of lab.
To see how this works, let us specialize the above program with the known
value and an unknown value for x. First we get to lab1 with
Using this information to specialize that basic block, we perform the conditional
ifs statically, find that (n != 0) is false and (n % 2 != 0) is true, and so must
jump to lab2, still with We create a residual goto command, and
a new specialized label lab2 fn=5;p=1g
goto lab2 fn=5;p=1g
The corresponding specialized basic block is the block at lab2 specialized with
respect to 1. The assignment specializes to the residual
since x is dynamic. This means that p is no longer static.
The assignment can be executed because n is static. The new static
environment has 4. Hence the goto lab1 specializes to goto lab1 fn=4g
and we get:
goto lab1 fn=4g
Note that we had to generate a constant expression 1 to represent the static
value 1 of p in the residual program. We say that the static value of p has been
lifted to appear in the residual program.
Next we must specialize the basic block at lab1 with respect to 4, and so
on. This process continues until specialized basic blocks have been created for
all specialized labels occurring in the residual program. In total, the following
residual program is obtained:
goto lab2 fn=5;p=1g
goto lab1 fn=4g
goto lab1 fn=2g
goto lab1 fn=1g
goto lab2 fn=1g
goto lab1 fn=0g
goto lab3 fn=0g
return(p)
This can be simplified by replacing jumps (gotos) with the code they jump to;
this is called transition compression or unfolding . The result is almost as in
Section 1:
return(p)
The technique of polyvariant specialization turns out to work for other languages
too; this is demonstrated in later sections for functional languages and logic
languages.
The specialization process builds a graph whose nodes are specialized program
points (labels), and whose edges are residual control-flow edges (jumps).
This may be done by maintaining a set pending of the specialized program
points still to be created, and a mapping out from specialized program points
to specialized program code fragments (basic blocks). One repeatedly chooses
and removes a program point pp from pending, constructs the corresponding
specialized program code fragment code pp , and extends the mapping out with
[pp 7! code pp ]. Moreover, one extends the set pending by any new specialized
labels reachable from code pp . More precisely, pending is extended with the set
is the set of program points
pp 0 to which there is a jump goto pp 0 from code pp .
To begin with, pending contains just the program's entry point together
with the initial values of its static variables, and out is empty. The procedure
terminates if and when pending becomes empty, in which case out contains the
residual program. This process may fail to terminate, as discussed in Section 5.5
below.
5.2 Online versus offline partial evaluation
There are two types of partial evaluators. An online partial evaluator is a kind
of generalized interpreter, which needs no a priori division of variables into
static and dynamic. During partial evaluation, the environment maps static
variables to concrete values, and dynamic variables to symbolic expressions.
When processing an expression e, the partial evaluator makes an online decision
whether to evaluate it (giving a concrete value), or to residualize it (giving a
residual expression), based on the current bindings of the variables appearing
in e.
An offline partial evaluator, by constrast, works in two phases. The first
phase is a binding-time analysis , which classifies the program's variables into
(definitely) static and (possibly) dynamic, and similarly classifies all operations.
The second phase is the specialization proper. This phase simply uses the
static/dynamic classification of variables and operations when processing an ex-
pression; all evaluate/residualize decisions have been made offline. It never uses
the actual value of a variable or expression, unless the binding-time analysis
guarantees that it is static and hence indeed is a concrete value.
Since offline partial evaluators rely on a program analysis, they are usually
more conservative than online partial evaluators, missing some opportunities for
specialization. On the other hand, offline specializers have a simpler structure,
and may exploit the global knowledge about the program gained by the binding-time
analysis. Experience shows that it is harder to construct self-applicable
online specializers than offline ones.
Hybrids of online and offline specializers have been constructed. For instance,
one may use a three-valued binding-time analysis, which classifies variables and
expressions as 'definitely static', `definitely dynamic', or 'undecided' [92]. The
specialization phase will just obey the static and dynamic annotations, but use
the actual (specialization time) value of variables to decide whether to evaluate
or residualize.
A generator of generating extensions is similar to an offline partial evaluator,
since a generating extension embodies an a priori distinction between early
(static) inputs and late (dynamic) inputs. A generator of generating extension
usually includes a binding-time analysis.
Online partial evaluation has been studied for Scheme by Ruf and Weise
[82, 100] and by many researchers in the logic programming community.
5.3 Binding-time analysis
The classification of variables into static and dynamic is called a division. The
division must be congruent : if the value of some dynamic expression e may be
assigned to a variable y, then y must be made dynamic. The expression e is
static if it contains no dynamic variables.
Considering again the flow-chart version of the power function in Section 5.1,
we see that if n is static and x is dynamic from the outset, then p must be
classified as dynamic because p x is assigned to p, whereas n remains static: n
is never assigned a dynamic value.
A simple binding time analysis may be performed by means of an abstract
interpretation in which each variable and expression takes one of the abstract
values S (for static) or D (for dynamic). One builds an initial division in which
all variables are S, except for the dynamic input parameters. Now all assignments
in the program are abstractly executed, possibly reclassifying variables as
dynamic to satisfy the congruence requirement, until no more variables need to
be reclassified as dynamic.
Alternatively, binding-time analysis may be done by type inference with sub-
types, where S is considered a subtype of D, meaning that S may be coerced
to D (corresponding to the lifting of a static value). This kind of binding-time
analysis may be implemented efficiently by constraint solving [53].
When composite data structures (tuples, records, lists) are considered, a data
structure may be partially static. For instance, the value of a variable may be a
whose first component is static, and whose second component is dynamic.
This may be described by the binding-time S \Theta D. Similarly, a list of such
pairs may be described by the binding-time (S \Theta D) list. The type inference
approach to binding-time analysis is especially useful for handling partially static
data structures in strongly typed languages, such as Standard ML, Pascal, or C.
Latently typed languages, such as Scheme, are handled essentially by considering
dynamic expressions to be untyped.
When a division has been computed by the binding-time analysis, one must
decide for each operation in the program whether it must be evaluated or re-
sidualized (producing residual code) during partial evaluation. An arithmetic
operation must be residualized unless all its operands are static. An if statement
must be residualized unless the condition is static. We shall assume that
an assignment will be residualized unless the assigned variable is static. We also
assume that all gotos are residualized (any excess gotos may be removed by
subsequent transition compression).
To visualize the classification of operations, we annotate the dynamic operations
by underlining. For the power function, the annotation would be:
lab1: if (n !=
if (n % 2 !=
goto lab1
goto lab1
lab3: return(p)
Doing polyvariant specialization of this program with blindly following
the annotations, we obtain (after transition compression):
return(p)
which is just the result obtained by the generating extension in Section 3. This
is because the generator of generating extensions presuppose the a priori distinction
between the static (n) and dynamic (x; p) variables.
5.4 Residual programs containing loops
The residual program generated above contains no loops; all conditionals were
statically decidable and all transitions could be compressed. However, the machinery
in Section 5.1 suffices for creating residual programs containing loops.
Consider the following contrived example:
while (n ?
else
Written as a flow-chart, the program is
lab1: if (n !=
goto lab3
lab2:
goto lab3
lab3:
goto lab1
lab4: return(sum)
Let us specialize it with respect to static dynamic sum and n. Specializing
the basic block at lab1 with respect to must create a residual
version of the first conditional, because n is dynamic, whereas the second conditional
can be reduced, because k is static and non-zero, giving a residual jump
to lab2 fk=3g
if (n !=
goto lab2 fk=3g
Next we specialize the code at label lab2 with respect to
goto lab3 fk=3g
Continuing in this manner, we obtain this residual program:
if (n !=
goto lab2 fk=3g
goto lab3 fk=3g
goto lab1 fk=3g
return(sum)
After transition compression, we get:
if (n !=
goto lab1 fk=3g
return(sum)
The decorated labels lab1 fk=3g
and lab4 fk=3g
may be replaced by simple ones,
such as lab1r and lab4r. Then we see that partial evaluation has eliminated the
tests on k inside the loop; effectively, they were found to be loop-invariant. The
loop is recreated in the residual program simply because the jump at lab3 fk=3g
goes back to the specialized program point lab1 fk=3g
at the beginning of the
program.
5.5 Termination of partial evaluation
Transition compression should be applied with care in the program just shown.
An attempt to (repeatedly) unfold all remaining occurrences of goto lab1 fk=3g
would never terminate. Infinite looping due to transition compression is avoided
fairly easily; either by unfolding a jump to a (residual) label only if there is
exactly one way to reach that label [21], or by ascertaining that unfolding must
stop due to some descending chain condition [87].
Termination problems caused by infinite specialization are harder to deal
with. For illustration, consider again the power program in Section 5.1, but now
with static straightforward application of
polyvariant specialization will attempt to produce an infinite residual program:
if (n !=
if (n % 2 !=
goto lab1 fx=49;p=1g
goto lab1 fx=7;p=7g
if (n !=
if (n % 2 !=
goto lab1 fx=2401;p=1g
goto lab1 fx=49;p=49g
This program is incomplete, and it cannot be completed using a finite number
of program points, if we insist on keeping x and p static.
In an online partial evaluatior, one may recognize that the configuration
is 'similar' to the previously encountered
and that the two program points should therefore be merged into a single more
general one, e.g. by making x dynamic (which eventually forces p to be dynamic
also). This process is called generalization.
In an offline partial evaluator, one may recognize after binding-time analysis
that the tests are dynamic (not decided by the static variables), and that static
data are constructed under dynamic control. This is a sign of danger, indicating
that x and p should be made dynamic too, making specialization completely
trivial (but safe).
Holst developed a finiteness analysis and used it to ensure termination of
polyvariant specialization [54].
5.6 Generalized partial evaluation
One more lesson may be learnt from the (partially constructed) residual program
just shown. The basic block labelled lab3 fx=49;p=1g
is superfluous. Reaching it
would require the tests (n != 0) and (n % 2 != 0) to fail and the test ((n/2
!= to succeed, which is impossible for integral n; the former two imply that
2.
The superfluous basic block is created because the static environment (as
outlined above) takes into account only the values of static variables (x and p),
not the outcome of previously encountered dynamic tests (on n). Polyvariant
specialization may be enhanced to do so, giving generalized partial evaluation.
Then a theorem prover is required to decide static conditionals and to decide
whether two static environments are equivalent [41]. In certain data domains
and applications, less powerful methods may suffice [46].
6 Partial evaluation for other languages
6.1 Functional languages
6.1.1 First-order languages
Partial evaluation of a first-order functional language may be done by polyvariant
specialization as described in Section 5.1 above. The notions of label , basic block ,
and global variable must be replaced by the notions of function name, function
definition, and function parameter. Henceforth a specialized program point is a
specialized function name, and a residual program is a collection of specialized
function definitions.
For illustration, consider a functional version of the power program from
Section 1, here using Standard ML syntax:
else if n mod
else x * power(n-1, x)
Specializing the function power with respect to static dynamic x, we
obtain
fun power fn=5g
and power fn=4g
and power fn=2g
and power fn=1g
and power fn=0g
Note that a specialized function name power fn=5g
consists of an original function
name power together with a binding for the static parameters, here just n.
The residual program may be simplified by unfolding trivial function calls (and
reducing the subexpression (x * x) * 1 arising from this unfolding):
fun power fn=5g
and power fn=2g
This residual program is equivalent to that generated for the C version of power
in Section 1.
Binding-time analysis may proceed as for a flow-chart language. For each
application (f e) where e may be dynamic, reclassify the formal parameter of f
to dynamic. Since the language is first-order, f must be a known function.
As for flow-chart languages, a partial evaluator may either be offline or online.
An offline partial evaluator will perform a binding-time analysis of the program,
to classify all parameters as either static or dynamic, before embarking on the
specialization phase proper. A complete description of a simple offline self-
applicable partial evaluator for a first-order functional language may be found
in [58, Chapter 5 and Appendix A].
6.1.2 Higher-order functional languages
Polyvariant specialization can be applied to higher-order functional languages
(in which functions may be passed around as values) as well. The main new
challenges are: how to represent static functional values during partial evalu-
ation, how to lift functional values from static to dynamic, how to specialize
with respect to functional values, and how to do binding-time analysis.
A functional value may be represented by a closure (g; vs) consisting of a
function name g together with the values vs of the static free variables in g's
body.
Lifting of a (partially) static functional value to a dynamic value is complicated
and is usually avoided in offline partial evaluators, by requiring that every
(partially) static functional value must be applied to an argument. Any functional
value occurring in a dynamic context will be reclassified as dynamic by
the binding-time analysis.
Specializing a function f with respect to a fully static functional closure
(g; vs) is simple; just specialize with respect to the function name g and the
values vs of the (static) free variables.
Specializing f with respect to a partially static (g; vs) is more involved, since
the body of g may have dynamic free variables. These variables may be free also
in the residual expression resulting from applying (g; vs). Hence the dynamic
must be lifted out of g's body at specialization time, and must be
passed as extra parameters to the residual function f (g;vs) .
A higher-order functional program may contain applications
evaluates to some function. A closure analysis can provide an approximation to
the set of functions that e 1 may evaluate to; using this information, binding-time
analysis may proceed as for a first-order language [58, Chapter 15]. Alternatively,
the binding-time analysis may be based on type inference [53]; this is preferable
if one wants to permit partially static data structures also.
Self-applicable partial evaluators exist for realistic higher-order functional
languages such as Scheme [20, 21, 28, 29, 31] and Standard ML [70] as well as
for the call-by-value lambda calculus with some extensions [51], and for the pure
lambda calculus [74, 75]. For further information, see e.g. [58, Chapter 10]; for
full details, see the above-mentioned papers.
6.2 Logic programming languages (Prolog)
A distinguishing feature of Prolog and other logic programming languages is
the ability to run with incomplete input. While this seems similar to partial
evaluation, there are a number of differences:
ffl The result of running a Prolog program with incomplete input is a (pos-
sibly) infinite list of instantiations of both input and output variables.
Though this can be considered a list of facts, and hence a restricted form
of program, we generally want a partial evaluator to be able to produce
non-trivial residual programs, possibly containing loops.
ffl Prolog has some non-logical features that means that running a program
with incomplete input is not a generalization of running with complete
input. As an example, calling the predicate defined by
with partially instantiated input p(A,a) returns the result
running with complete input p(a,a) would fail.
Most of the research in partial evaluation (or partial deduction, as it is often
called) of logic languages has tended to avoid the second issue by working with
pure logic languages [43, 68]. Some systems, however, deal with non-logical
features of Prolog [76, 84].
Partial evaluation of logic languages is typically done using the same basic
techniques as for functional languages: call unfolding and polyvariant specializ-
ation, program points being predicates. A major source of speed-up in partially
evaluating logic programs is the ability to detect failing computations at specialization
time, and cut these away in the residual program. This way, not only
static computations but also dynamic computations in failing branches can be
eliminated by partial evaluation. This makes the potential speed-up by partial
evaluation greater in logic languages than in functional or imperative languages.
Online specialization has been the preferred technique in the logic language
community, usually combined with powerful techniques for avoiding non-termination
[23, 71]. In logic languages, online specialization presents more opportunities
for specialization than offline specialization, because unification will often
instantiate otherwise dynamic variables. When self-application has been a major
goal, offline specialization has been used also [52, 76].
An example of Prolog specialization is shown below. It specializes a program
for regular expression matching
accepts(R,[]) :- nullable(R).
accepts(R,[C-S]) :- first(R,C), next(R,C,R1), accepts(R1,S).
The program takes a regular expression and a string as arguments. If the string
is empty, the regular expression is tested for nullability (acceptance of empty
string). If the string starts with a character C, it is tested whether this is among
the first set of the regular expression. If this is the case, a new regular expression
for matching the rest of the string is produced by next. The predicates
nullable, first and next are not shown, but note that the set of Cs for which
first succeeds is determined by R. Hence, partial evaluation of first with respect
to a known R and unknown C will yield a number of instantiations of C.
Specializing the program above with respect to R being the regular expression
(ajb) aba yields the following residual program:
accepts-3([]).
Since nullable depends only on static values, it is completely eliminated, only
visible as failed or true cases for the empty string. The call to first has
instantiated C with a or b. This instantiation has made it possible to fully
evaluate next, which has yielded a total of four different regular expressions,
each giving rise to a specialized version of accepts. For accepts-0, the regular
expression is (ajb) aba, for accepts-1 it is (ajb) abajba, for accepts-2 it is
(ajb) abaja, and for accepts-3 it is (ajb) abajbajffl.
6.3 A full imperative language (C)
We have seen that polyvariant specialization suffices for partial evaluation of
flow-chart languages, and hence for simple imperative languages. A realistic
imperative language, such as C, includes composite data structures (records and
indexed arrays), pointers and dynamic data structures, functions which may
have side effects on global variables, etc.
An offline partial evaluator for C needs a sophisticated binding-time analysis
to deal with pointers and composite data structures. For instance, a pointer
variable p may be dynamic, or the pointer may be static but point to a dynamic
object, or both the pointer and the pointed-to object may be static.
The binding-time analysis may require programs to be 'well-behaved'. Assume
that a is an array, and that the program contains an assignment of the form
e, where e is dynamic. Then in principle any variable in the program
may become dynamic as a result of this assignment, in case the address a[n] is
outside the allocated array a. This would be too conservative, making partial
evaluation trivial. Instead, one should require programs to be well-behaved, so
that any such address is indeed inside a.
For a taste of the difficulties caused by the combination of non-local side
effects and (recursive) functions, consider a function which has a side effect on
a static global variable, but where the side effect is controlled by some dynamic
expression dyn:
int global;
int
stmts
else
After the call to foo, the value of global may be either 1 or -1, but we cannot
know which one at partial evaluation time, because dyn is dynamic. The simplest
solution is to reclassify global as dynamic, but this wastes static information
which might be useful when partially evaluating stmts. Another solution is to
unfold the call to foo, giving a residual program of this form:
int global;
int
f if dyn f stmts fglobal=1g ; g
else f stmts fglobal=\Gamma1g ; g
Here stmts fglobal=1g
is a specialized version of stmts. However, when function
foo is recursive, such unfolding is impossible. A third solution is to introduce a
(dynamic) continuation variable cont, which will be assigned a different value
in each branch of the residual version foo' of foo. In function main, after the
call to foo', there will be a switch on cont:
int global;
int
switch (cont) f
case 1: stmts fglobal=1g ; break;
case 2: stmts fglobal=\Gamma1g ; break;
However, this will not work when foo is recursive, and the recursion is under
dynamic control, since the number of paths through foo will not be statically
bounded in that case. Hence for recursive procedures, the only feasible option
may be to reclassify global as dynamic.
These and many other problems were studied by Lars Ole Andersen, who
constructed two systems for specialization of C programs. The first one is a
self-applicable partial evaluator for a C subset, including procedures as well as
pointers and arrays [6, 8]. The second one is a generator of generating extensions
for all of ANSI C [9]; the latter system can be licensed from the University of
Copenhagen.
The techniques for C should carry over to e.g. Ada, Modula, or Pascal with
little modification, but to our knowledge this has not been done.
7 Partial evaluation in perspective
7.1 Program specialization without a partial evaluator
So far we have focused mainly on specialization using a partial evaluator. But
the ideas and methods presented here can be, and indeed have been, used without
using a partial evaluator.
Specialization by hand
It is quite common for programmers to hand-tune code for particular cases.
Often this amounts to doing partial evaluation by hand. As an example, here is
a quote from a paper [80] about the programming of a video-game:
How Nevryon manages to keep up its speed Basically there are
two ways to write a routine:
It can be one complex multi-purpose routine that does everything,
but not quickly. For example, a sprite routine that can handle any
size and flip the sprites horizontally and vertically in the same piece
of code.
Or you can have many simple routines each doing one thing. Using
the sprite routine example, a routine to plot the sprite one way,
another to plot it flipped vertically and so on.
The second method means more code is required but the speed advantage
is dramatic. Nevryon was written in this way and had about
separate sprite routines, each of which plotted sprites in slightly
different ways.
specialization is used. But it is doubtful that a general purpose partial
evaluator was used to do the specialization. Instead the specialization has been
performed by hand, possibly without ever explicitly writing down the general
purpose routine that forms the basis for the specialized routines.
Using hand-written generating extensions
We saw in Section 3 how a generating extension for the power function was
easily produced from the original code, using knowledge about which variables
contained values known at specialization time. While it is not always quite so
simple as in this example, it is often not particularly difficult to write generating
extensions of small to medium sized procedures or programs.
In situations where no partial evaluator is available, this is often a viable way
to obtain specialized programs. Using a generating extension instead of writing
the specialized versions by hand is useful when either a large number of variants
must be generated, or when it is not known in advance what values the program
will be specialized with respect to.
A common use of hand-written generating extensions is for run-time code
generation, where a piece of specialized code is generated and executed, all at
run-time. As in the sprite example above, one often generates specialized code
for each plot operation when large bitmaps are involved. The typical situation is
that a general purpose routine is used for plotting small bitmaps, but special code
is generated for large bitmaps. The specialized routines can exploit knowledge
about the alignment of the source bitmap and the destination area with respect
to word boundaries, as well as clipping of the source bitmap. Other aspects such
as scaling, differences in colour depth etc. have also been targets for run-time
specialization of bitmap-plotting code.
Hand-written generating extensions have been used for optimizing parsers by
specializing with respect to particular tables [78], and for converting interpreters
into compilers [77].
Handwritten generating extension generators
In recent years, it has become popular to write a generating extension generator
instead of a partial evaluator [9, 16, 55], but the approach itself is quite old [12].
A generating extension generator can be used instead of a traditional partial
evaluator as follows. To specialize a program p with respect to data d, first
produce a generating extension p gen , then apply p gen to d to produce a specialized
program p d .
Conversely, a self-applicable partial evaluator can produce a generating extension
generator (cf. the third Futamura projection), so the two approaches
seem equally powerful. So why write a generating extension generator instead
of a self-applicable partial evaluator? Some reasons are:
ffl The generating extension generator can be written in another (higher level)
language than the language it handles, whereas a self-applicable partial
evaluator must be able to handle its own text.
ffl For this reason, among others, it may be easier to write a generating
extension generator than a self-applicable partial evaluator.
ffl A partial evaluator must contain an interpreter, which may be problematic
for typed languages, as explained below. Neither the generating extension
generator, nor the generating extensions, need to contain an interpreter.
When writing an interpreter for a strongly typed language, one must use a single
type in the interpreter to represent an unbounded number of types used in the
programs that are interpreted. The same is true for a partial evaluator: a single
universal type must be used for the static input to the program that will be
specialized. Hence, the static input must be coded. This means that the partial
evaluation equation must be modified to take this coding into account:
peval
where overlining means that a value is coded, e.g. d 1 is the coding of the value
of d 1 .
When self-applying the partial evaluator, the static input is a program. The
program is normally represented in a special data type that represents program
text. This data type must now be coded in the universal type:
peval ]][peval;
This double encoding is space- and time-consuming, and has been reported to
make self-application intractable, unless special attention is paid to make the
encoding compact [67]. A generating extension produced by self-application
must also use the universal type to represent static input, even though this will
always be of the same type.
This observation leads to the idea of making generating extensions accept
uncoded static input. To achieve this, the generating extension generator simply
copies the type declarations of the original program into the generating extension.
The generating extension generator takes a single input: a program, and need
not deal with arbitrarily typed data. A generating extension handles values from
a single program, the types of which are known when the generating extension
is constructed. Hence, neither the generator of generating extensions, nor the
generating extensions themselves, need to handle arbitrarily typed values. The
equation for specialization using a generating extension generator is shown below.
Note the absence of coding.
We will usually expect generator generation to terminate, but, as for normal
partial evaluation, allow the construction of the residual program (performed by
gen ) to loop.
7.2 When is partial evaluation worthwhile?
In Section 2.2 we saw that we cannot always expect speed-up from partial eval-
uation. Sometimes no significant computations depend on the known input only,
so virtually all the work is postponed until the residual program is executed.
Even if computations appear to depend on the known input only, evaluating
these during specialization may require infinite unfolding (as seen in Section 2.2)
or just so much unfolding that the residual programs become intractably large.
On the other hand, the example in Section 1 manages to perform a significant
part of the computation at specialization time. Even so, partial evaluation will
only pay off if the residual program is executed often enough to amortize the
cost of specialization.
So, we must have two conditions before we can expect any benefit from
partial evaluation:
There are computations that depend only on static data.
These are executed repeatedly, either by repeated execution of the program
as a whole, or by repetition (looping or recursion) within a single execution
of the program.
The static (known) data can be obtained in several ways: it may be constants
appearing in the program text or it can be part of the input.
It is quite common that library functions are called with some constant para-
meters, such as format strings, so in some cases partial evaluation may speed
up programs even when no input is given. In such cases the partial evaluator
works as a kind of optimizer, often achieving speed-up when most optimizing
compilers would not. On the other hand, most partial evaluators may loop or
create an excessive amount of code while trying to optimize programs, and hence
are ill-suited as default optimizers.
Specialization with respect to partial input is the most common situation.
Here there are often more opportunities for speed-up than just exploiting constant
parameters. In some cases (e.g., when specializing interpreters) most of the
computation can be done during partial evaluation, sometimes yielding speed-ups
by an order of magnitude or more, similar to the speed difference between
interpreted and compiled programs. When you have a choice between running
a program interpreted or compiled, you will choose the former if the program
is only executed a few times and contains no significant repetition, whereas you
will want to compile it if it is run many times or involves much repetition. The
same principle carries over to specialization.
Partial evaluation often gets most of its benefit from replication: loops are
unrolled and the index variables exploited in constant folding, or functions are
specialized with respect to several different static parameters. In some cases this
replication can result in enormous residual programs, which may be undesirable
even if much computation is saved. In the example in Section 1 the amount
of unrolling and hence the size of the residual program is proportional to the
logarithm of n, the static input. This expansion is small enough that it doesn't
become a problem. If the expansion were linear in n, it would be acceptable for
small values of n. Specialization of interpreters typically yield residual programs
that are proportional to the size of the source program, which is reasonable. On
the other hand, quadratic or exponential expansion is hardly ever acceptable.
It may be hard to predict the amount of replication caused by a partial
evaluator. In fact, seemingly innocent changes to a program can dramatically
change the expansion done by partial evaluation, or even make the difference
between termination or nontermination of the specialization process. Similarly,
small changes can make a large difference in the amount of computation that is
performed during specialization and hence the speed-up obtained. This is similar
to the way parallelizing compilers are sensitive to the way programs are writ-
ten. Hence, specialization of off-the-shelf programs often require some (usually
minor) modification to get optimal benefit from partial evaluation. Ideally, the
programmer should write his program with partial evaluation in mind, avoiding
the structures that can cause problems, just like programs for parallel machines
are best written with the limitations of the compiler in mind.
7.3 Partial evaluation, optimizing compilers, and modern
machines
Many compilers perform transformations such as constant folding and inlining
(of small functions) to improve target programs. These transformations are
similar to some of those performed by a partial evaluator. However, in contrast
to a partial evaluator, a compiler rarely produces more than one specialized
version of a given piece of code (except possibly by inlining). This kind of
specialization is essential in partial evaluators, and must be handled correctly
also in the presence of loops and recursive procedures.
With the complex memory hierarchies of modern computer hardware it is
hard to know when a program modification actually achieves a speed-up. Exploiting
the memory hierarchy well (data registers and instruction pipeline, two
or more levels of cache, main memory, and virtual memory) is crucial for the
performance of modern machines. Hence it may be detrimental to unroll a loop
so that it does not fit in the cache, but beneficial to inline a procedure if this
replaces indirect jumps by linear code sequences. How much unrolling, inlining,
or replication to perform is machine dependent, and you often see optimizations
that improve performance on one machine but degrade it on others.
With the increasing degree of micro-parallelism in modern microprocessors,
one may even get no benefit from eliminating the static computations, as they
may not be part of the critical path and hence may be executed in parallel
with the dynamic computations. On the other hand, the elimination of variables
by specialization reduces register pressure, and unrolling of loops and inlining
of functions increase basic block size, giving more opportunities for low-level
optimization.
This means that it is hard to predict the amount of speed-up obtained by
partial evaluation. Examples exist where a residual program is twice as fast as
the original program on one machine is, and slower than the original on another
machine. The speed-up is also affected by the optimizations performed when
compiling the residual programs.
8 Applications of partial evaluation
We saw in Section 4 that partial evaluation can be used to compile programs
and to generate compilers. This has been one of the main practical uses of
partial evaluation. Not for making compilers for C or similar languages, but for
rapidly obtaining implementations of acceptable performance for experimental
or special-purpose languages. Since the output of the partial evaluator typically
is in a high-level language, a traditional compiler is used as a back-end for the
compiler generated by partial evaluation [1, 14, 25, 27, 30, 33, 61]. In some
cases, the compilation is from a language to itself. In this case the purpose is to
make certain computation strategies explicit (e.g., continuation passing style) or
to add extra information (e.g., for debugging) to the program [20, 42, 83, 93].
Many types of programs, e.g. scanners and parsers, use a table or other
data structure to control the program. It is often possible to achieve speed-up
by partially evaluating the table-driven program with respect to a particular
table [7, 78]. However, this may produce very large residual programs, as tables
(unless sparse) often represent the information more compactly than does code.
These are examples of converting structural knowledge representation to procedural
knowledge representation. The choice between these two types of representation
has usually been determined by the idea that structural information
is compact and easy to modify but slow to use while procedural information
is fast to use but hard to modify and less compact. Automatically converting
structural knowledge to procedural knowledge can overcome the disadvantage
of difficult modifiability of procedural knowledge, but retains the disadvantage
of large space usage.
Section 7.1 mentioned a few applications of specialization to computer graph-
ics. This has been one of the areas that have seen most applications of partial
evaluation. An early example is [49], where an extended form of partial evaluation
is used to specialize a renderer used in a flight simulator.
In a flight simulator the same landscape is viewed repeatedly from different
angles. Though the occlusion of surfaces depend on the angle of view, it is
often the case that the knowledge that a particular surface occludes (or not)
another can decide the occlusion question of other pairs of surfaces. Hence, the
partial evaluator simulates the sorting of surfaces and when it cannot decide
which of two surfaces must be plotted first, it leaves that test in the residual
program. Furthermore, it uses the inequalities of the occlusion test as positive
and negative constraints in the branches of the conditional it generates. These
constraints are then used to decide later occlusion tests (by attempting to solve
the constraints by the Simplex method). Each time a test cannot be decided
more information is added to the constraint set (which effectively constrains the
view-angle), allowing more later tests to be decided. Goad reports that for a
typical landscape with 1135 surfaces (forming a triangulation of the landscape)
the typical depth of paths in the residual decision tree was 27, compared to the
more than 10000 comparisons needed for a full sort [49]. This rather extreme
speed-up is due to the nature of landscapes: many surfaces are almost parallel,
and hence can occlude each other only in very narrow viewing angles.
Another graphics application has been ray-tracing. In ray-tracing, a scene is
rendered by tracing rays (lines) from each pixel on the screen into an imaginary
world behind the scene, testing which objects these rays hit. The process is
repeated for all rays using the same fixed scene. Since there may be millions of
pixels (and hence rays) in a typical ray-tracing application, specialization with
respect to a fixed scene but unknown ray can give speed-up even for rendering
single pictures. Speed-ups of more than 6 have been reported for a simple ray-tracer
[73]. For a more realistic ray-tracer, speed-ups in the range 1.5 to 3 have
been reported [10]. The speed-up is gained from several sources: the ray/object
intersection routine is specialized for each object and the (highly parametrized)
shading (colouring) function is specialized for each object. Furthermore, the
representation of the scene is converted to procedural form.
Figure
1 is an example of a ray-traced picture made by the ray-tracer from
[73]. The picture shows a 3D diagram of the process of partial evaluation: A
program P and one of its inputs x are fed to the partial evaluator PE yielding
a residual program P x .
Partial evaluation has also been applied to numerical computation, in particular
simulation programs. In such programs, part of the model will be constant
during the simulation while other parts will change. By specializing with respect
to the fixed parts of the model, some speed-up can be obtained. An example
is the N-body problem, simulating the interaction of moving objects through
gravitational forces. In this simulation, the masses of the objects are constant,
whereas their position and velocity change. Specializing with respect to the
mass of the objects can speed up the simulation. Berlin reports speed-ups of
more than 30 for this problem [15]. However, the residual program is written
in C whereas the original one was in Scheme, which may account for part of
the speed-up. In another experiment, specialization of some standard numerical
algorithms gave speed-ups ranging from none at all to about 5 [47].
When neural networks are trained, they are usually run several thousand
times on a number of test cases. During this training, various parameters will
be fixed, e.g. the topology of the net, the learning rate and the momentum.
By specializing the trainer to these parameters, speed-ups of 25 to 50% are
reported [56].
This list of applications is not exhaustive, but should give an impression of
the range of possibilities.
9 Further reading
Here we first sketch the history from 1952 to 1984, then give a number of pointers
to the literature on partial evaluation and some related topics. The book by
Jones, Gomard, and Sestoft [58] includes more material on the subjects mentioned
above, and a large bibliography; the updated source text for that bibliography
is available for anonymous ftp from ftp.diku.dk as file
pub/diku/dists/jones-book/partial-eval.bib.Z.
9.1 History
Kleene's s-m-n theorem (1952) asserts the feasibility of partial evaluation [63],
and his constructive proof provides the design for a partial evaluator. This design
did not, and was not intended to, provide any improvement of the specialized
program. Such improvement, by symbolic reductions or similar, has been the
goal in all subsequent work in partial evaluation.
Lombardi is probably the first one to use the term 'partial evaluation' [69].
Futamura is the first researcher to consider a partial evaluator as a program as
well as a transformer, and thus to consider the application of the partial evaluator
to itself [40]. Futamura's paper gives the equations for compilation and compiler
generation by partial evaluation, but not for compiler generator generation. The
Figure
1: Partial evaluation in action
three equations were called the Futamura projections by Andrei Ershov [38].
Futamura's early ideas were not implemented.
Around 1975, Beckman, Haraldsson, Oskarsson, and Sandewall developed
a partial evaluator called Redfun for a substantial subset of Lisp [12], and described
the possibility of compiler generator generation by double self-application.
Turchin and his group also worked with partial evaluation in the early 1970s,
in the context of the functional language Refal, and gave a description of self-application
and double self-application [94]. The history of that work is briefly
summarized in English in [95].
Andrei Ershov worked with imperative languages, and used the term mixed
computation to mean roughly the same as partial evaluation [34, 35].
In 1984, Jones, Sestoft, and S-ndergaard constructed a self-applicable partial
evaluator for a simple first-order functional language [59, 60, 86]; until then
neither single nor double self-application had been carried out in practice.
At the same time the interest in partial evaluation in logic programming and
other areas was increasing. This was the background for the 1987 Workshop on
Partial Evaluation and Mixed Computation [19, 39]. Subsequent proceedings on
partial evaluation may be found in [2, 3, 4, 5, 32, 88].
9.2 Partial evaluators
Imperative languages: Early papers on partial evaluation for imperative languages
include [34, 36, 37]. Bulyonkov and Ershov reported a self-applicable
partial evaluator for a flow chart language [25]; so did Gomard and Jones [50].
Gl-uck et al. created a (non-self-applicable) specializer for numeral algorithms in
Fortran [11, 47]. Andersen [6, 8, 9] developed two systems for specialization of
C programs; see Section 6.3.
Lisp and Scheme: The first major partial evaluator for Lisp was Redfun,
reported by Beckman et al. [12]. Weise et al. constructed a fully automatic online
partial evaluator for a subset of Scheme [100]. Jones et al. constructed a self-
applicable partial evaluator for a first-order functional language [59, 60]; Romanenko
improved it in various respects [81]. Consel constructed the self-applicable
partial evaluator Schism for a Scheme subset, handling partially static structures
and polyvariant binding times [28, 29, 31]. Bondorf and Danvy constructed the
self-applicable partial evaluator Similix for a subset of Scheme [20, 21].
Standard ML: Danvy, Heintze, and Malmkjaer developed the partial evaluator
Pell-Mell [70]. Birkedal and Welinder created a generator of generating
extensions [17, 18].
Refal and supercompilation: Turchin created the Refal language and developed
the program transformation techniques of driving and supercompilation,
which generalize partial evaluation [95, 96, 97]. A number of recent surveys on
driving and supercompilation exist [48, 89, 90, 91].
Prolog partial evaluation was pioneered by Komorowski [64, 65]; subsequent
work on Prolog includes [13, 44, 45, 66, 93, 98, 99]. Sahlin constructed a practical
but non-self-applicable partial evaluator for full Prolog [84, 85]. Bondorf and
Mogensen [76] constructed a self-applicable partial evaluator for a Prolog subset,
Gurr one for the logic language G-odel [52]. J-rgensen and Leuschel created a
generator of generating extensions for Prolog [62].
9.3 Related topics
McCarthy used program transformation rules in calculational proofs for recursive
functional programs [72]. Boyer and Moore automated some proofs of this
kind [22].
Burstall and Darlington viewed 'manual' program transformation as the application
of a few types of meaning-preserving program rewritings: definition,
instantiation, unfolding, folding, abstraction, and laws [26].
Partial evaluation specializes a program forwards by using knowledge about
available input. Conversely, program slicing specializes a program backwards,
using knowledge about the demand for output [79].
--R
A compiler based on partial evalu- ation
New Haven
Partial evaluation of C and automatic compiler generation (extended abstract).
Program Analysis and Specialization for the C Programming Language.
Partial evaluation applied to ray tracing.
Partial evaluation of numerical programs in Fortran.
A partial evaluator
A partial evaluation procedure for logic pro- grams
Compiling scientific code using partial evaluation.
Partial evaluation applied to numerical computation.
Partial evaluation of Standard ML.
Partial Evaluation and Mixed Computation.
Automatic autoprojection of higher order recursive equations.
Proving theorems about Lisp functions.
A general criterion for avoiding infinite unfolding.
Polyvariant mixed computation for analyzer programs.
How do ad-hoc compiler constructs appear in universal mixed computation processes? In
A transformation system for developing recursive programs.
Compiling or-parallelism into and-parallelism
New insights into partial evaluation: The Schism experiment.
Binding time analysis for higher order untyped functional lan- guages
The Schism Manual
Dagstuhl Seminar on Partial Evaluation
On compiling embedded languages in Lisp.
On the partial computation principle.
Mixed computation in the class of recursive program schemata.
On the essence of compilation.
Mixed computation: Potential applications and problems for study.
On Futamura projections.
Special Issue: Selected Papers from the Workshop on Partial Evaluation and Mixed Computation
Partial evaluation of computation process - an approach to a compiler-compiler
Generalized partial computation.
Transforming logic programs by specialising interpreters.
Tutorial on specialisation of logic programs.
Some low-level source transformations for logic programs
Specialisation of Prolog and FCP programs using abstract interpretation.
Generating optimizing specializers.
Application of metasystem transition to function inversion and transformation.
Automatic construction of special purpose programs.
Compiler generation by partial evaluation.
A partial evaluator for the untyped lambda- calculus
Efficient type inference for higher-order binding-time analysis
Finiteness analysis.
Handwriting cogen to avoid problems with static typing.
Speeding up the back-propagation algorithm by partial evaluation
Automatic program specialization: A re-examination from basic principles
Partial Evaluation and Automatic Program Generation.
An experiment in partial evaluation: The generation of a compiler generator.
Generating a compiler for a lazy language by partial eval- uation
Efficiently generating efficient generating extensions in Prolog.
Introduction to Metamathematics.
A Specification of an Abstract Prolog Machine and Its Application to Partial Evaluation.
Partial evaluation as a means for inferencing data structures in an applicative language: A theory and implementation in the case of Prolog.
A Prolog partial evaluation system.
A strongly-typed self-applicable partial evaluator
Partial evaluation in logic program- ming
Lisp as the language for an incremental computer.
ML partial evaluation using set-based analysis
Ensuring global termination of partial deduction while allowing flexible polyvariance.
A basis for a mathematical theory of computation.
The application of partial evaluation to ray-tracing
Converting interpreters into compilers.
Comparative efficiency of general and residual parsers.
Program specialization via program slicing.
The realm of Nevryon.
A compiler generator produced by a self-applicable specializer can have a surprisingly natural and understandable structure
On the specialization of online program specializers.
Meta interpreters for real.
The Mixtus approach to automatic partial evaluation of full Prolog.
An Automatic Partial Evaluator for Full Prolog.
The structure of a self-applicable partial evaluator
Automatic call unfolding in a partial evaluator.
Special Issue on Partial Evaluation and
Turchin's supercompiler revisited.
An algorithm of generalization in positive supercompilation.
Towards unifying partial eval- uation
How to have your cake and eat it
Partial evaluation of Prolog programs and its application to meta programming.
Basic Refal and Its Implementation on Computers.
A supercompiler system based on the language Refal.
The concept of a supercompiler.
Program transformation with metasystem transitions.
A Prolog meta-interpreter for partial evaluation and its application to source to source transformation and query-optimisation
A partial evaluation system for Prolog: Some practical considerations.
Automatic online partial evaluation.
--TR
--CTR
Mohan Rajagopalan , Saumya K. Debray , Matti A. Hiltunen , Richard D. Schlichting, Profile-directed optimization of event-based programs, ACM SIGPLAN Notices, v.37 n.5, May 2002
Arvind S. Krishna , Aniruddha Gokhale , Douglas C. Schmidt , Venkatesh Prasad Ranganath , John Hatcliff, Towards highly optimized real-time middleware for software product-line architectures, ACM SIGBED Review, v.3 n.1, p.13-16, January 2006
Arvind S. Krishna , Aniruddha S. Gokhale , Douglas C. Schmidt, Context-specific middleware specialization techniques for optimizing software product-line architectures, ACM SIGOPS Operating Systems Review, v.40 n.4, October 2006
Yasushi Shinjo , Calton Pu, Achieving Efficiency and Portability in Systems Software: A Case Study on POSIX-Compliant Multithreaded Programs, IEEE Transactions on Software Engineering, v.31 n.9, p.785-800, September 2005
Anne-Franoise Le Meur , Julia L. Lawall , Charles Consel, Towards bridging the gap between programming languages and partial evaluation, ACM SIGPLAN Notices, v.37 n.3, p.9-18, March 2002
Jacques Carette, Gaussian elimination: a case study in efficient genericity with MetaOCaml, Science of Computer Programming, v.62 n.1, p.3-24, September 2006
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592047 | The Model-Composition Problem in User-Interface Generation. | Automated user-interface generation environments have been criticized for their failure to deliver rich and powerful interactive applications. To specify more powerful systems, designers require multiple specialized modeling notations. The model-composition problem is concerned with automatically synthesizing powerful, correct, and efficient user interfaces from multiple models specified in different notations. Solutions to the model-composition problem must balance the advantages of separating code generation into specialized code generators each able to take advantage of deep, model-specific knowledge against the correctness and efficiency obstacles that result from such separation. We present a correct and efficient solution that maximizes the advantage of separation by using run-time composition mechanisms. | Introduction
Building user interfaces (UIs) is time consuming and costly. In systems with graphical
UIs (GUIs), nearly 50% of source code lines and development time can be
attributed to the UI [14]. GUIs are usually built from a fixed set of modules
composed in regular ways. Hence, GUI construction is a natural target for automa-
tion. Automated tools have been successful in supporting the presentation aspect
of GUI functionality, but they provide only limited support for specifying behavior
and the interaction of the UI with underlying application functionality. The
model-based approach to interactive system development addresses this deficiency
by decomposing UI design into the construction of separate models, each of which
is declaratively specified [5]. Once specified, automated tools integrate the models
and generate an efficient system from them. The model-composition problem is the
need to efficiently implement and automatically integrate interactive software specified
in separate, declarative models. This paper introduces the model-composition
problem and presents a solution.
A model is a declarative specification of some single coherent aspect of a user
interface, such as its appearance or how it interfaces to and interacts with the
underlying application functionality. By focusing attention on a single aspect of
a user interface, a model can be expressed in a highly-specialized notation. This
property makes systems developed using the model-based approach easier to build
and maintain than systems produced using other approaches [23].
e
s
R
e
e UI
Synchronization
Toolkit
Module
Model
Model
Model
Dialogue
Application Presentation
Presentation
Dialogue
Application
Module Module
Module
Figure
1. Model-based code generation
The Mastermind project [5, 15] is concerned with the automatic generation of
user interfaces from three kinds of models: Presentation models represent the appearance
of user interfaces in terms of their widgets and how the widgets behave;
Application models represent which parts (functions and data) of applications are
accessible from the user interface; and Dialog models represent end-user interac-
tions, how they are ordered, and how they affect the presentation and the applica-
tion. A dialog model acts as the glue between presentation and application models
by expressing constraints on the sequencing of behavior in those models. Model-specific
compilers generate modules of code from each model, and these resulting
modules are composed into a complete user interface (Figure 1). A distinguishing
characteristic of Mastermind is that the model-specific code generators work
independently of one another.
Composing code generated from multiple models is difficult. A model, by de-
sign, represents a single aspect of a system and is neutral with respect to others [3].
Inevitably, however, functionality described in one model overlaps with or is dependent
upon functionality described in another. A button, for example, is specified in
a presentation model, but the behavior of the button influences behavior in other
models, such as when pressing the button causes other widgets to be enabled or dis-
abled. Such effects are described in a dialog model. The effect of pressing a button
can also cause an application method to be invoked. Such effects are described in
an application model. When code generated from multiple models must cooperate,
these redundancies and dependencies can be difficult to resolve. Resolving them
automatically means that behavior in different models must be correctly unified,
and the mechanism for this unification must be implemented efficiently.
The model-composition problem is concerned with automatically synthesizing
powerful, correct, and efficient user-interfaces from separate presentation, dialog,
and application models. We present a two-fold solution. First, we formalize the
three models as concurrent agents, which synchronize on common events (Sec-
tion 3). Second, we present a runtime architecture that supports the composition
of modules generated from independent model compilers (Section 4). We present
MODEL-COMPOSITION PROBLEM 3
the results of this approach on two examples and give evidence to show that it
scales up (Section 5).
2. Background
Model-based approaches to user-interface generation use models that are specified
in diverse and often incompatible notations. This characteristic complicates model
composition because the composition mechanisms in one model may not exist in another
(Section 2.1). Prior research on the architecture of user-interfaces suggests
using communicating agents to structure user-interface code (Section 2.2). Formal
models of communicating agents provide a technique called conjunction, which
is useful for composing partial specifications of a system (Section 2.5). The contribution
of this paper is an extension of conjunction as a specification-composition
operator into a runtime-composition mechanism.
2.1. Model-based generation
The model-based approach to interactive system development expresses system
analysis, design, and implementation in terms of an integrated collection of mod-
els. Unlike conventional software engineering, in which designers compose software
documentation whose meaning and relevance can diverge from that of the delivered
code, in the model-based approach, designers build models of critical system
attributes and then analyze, refine, and synthesize these models into running sys-
tems. Model-based UI generation works on the premise that development and
support environments may be built around declarative models of a system. Developers
using this approach build interfaces by specifying models that describe the
desired interface, rather than writing a program that exhibits the behavior [21].
One characteristic of model-based approaches is that, by restricting the focus
of a model to a single aspect of a system, modeling notations can be specialized
and highly declarative. The Mastermind Presentation Model [6], for example,
combines concepts and terminology from graphic design with mechanisms for describing
complex presentations using functional constraints. The Mastermind
Dialog Model [19] uses state and event constructs to describe the user-computer
conversation; the composition features include state hierarchy, concurrency, and
communication. The Mastermind Application Model combines concepts and terminology
from object-oriented design techniques [18] with mechanisms for composing
complex behavior based on method invocation.
Figure
2 compares the Mastermind models in terms of their domains of dis-
course, communication mechanisms, runtime components, and how they are com-
posed. Composition mechanisms in one model may not exist in another model.
single one of these intra-model mechanisms is sufficient for composing all three
Mastermind models. The model-composition problem can be restated as the need
to unify behavior in multiple models without violating the rules of intra-model composition
and while generating efficient code. The model-composition problem is a
declarative instance of the problem of constructing a software system where the ma-
Module Process
Implementation
Action
Implementation
Intra-module
Composition
Application Abstract
Method
Invocation
Subclassing
Aggregation
Presentation Amulet
Objects
Constraints,
Commands
Instantiation,
Aggregation
Dialog State
Machines
Synchronous
Message passing
Orthogonal
Composition
Figure
2. Multi-paradigm action implementations
jor components are expressed with programming languages from different families
or paradigms. Zave has called this the multi-paradigm programming problem [24].
2.2. Multi-agent user-interface architectures
The Mastermind approach to model composition builds on prior work in multi-agent
user-interface architectures, which provide design heuristics for structuring
interactive systems. These architectures describe interactive systems as collections
of communicating agents, which are independent computational units with identity
and behavior. Two general frameworks-Model-View-Controller (MVC) [11] and
agent roles and provide
guidance on how agents should be connected.
MVC prescribes how SmallTalk simulations can be composed by instantiating instances
of three types of agents: models (not to be confused with the Masterimind
models) describing application state, views providing presentations of models, and
controllers allowing users to affect simulation behavior. A view registers interest in
one or more attributes of a model. When an attribute changes, all registered views
are notified so that they can recompute their display if necessary.
The PAC framework more closely matches Mastermind than does MVC. In
PAC, a presentation agent maintains the state of the display and accepts input
from the user, an abstraction agent maintains a representation of the underlying
application state, and a controller agent ensures that presentation and abstraction
remain synchronized. The Mastermind Presentation, Application, and Dialog
models are descriptions of the roles played by PAC's presentation, abstraction, and
controller agents.
Since Mastermind models describe PAC agents, we chose to make Mastermind
models compose in the same manner that PAC agents compose. Specifically, the
presentation and application models define actions, which are ordered by temporal
constraints in the dialog model. To make these ideas more formal, we built upon
prior work on formal definitions of agent composition.
MODEL-COMPOSITION PROBLEM 5
2.3. Formal models of agents
The PAC framework provides heuristic definitions of user-interface agent roles and
connections. PAC agents are concurrent, and they compose by communicating control
and data messages among themselves. To generate code from the models of
these agents, we need to formalize the building blocks of agents and agent compo-
sition. We chose the terminology and definitions that have been adopted by the
various process algebras, specifically Lotos [4]. Process algebras formalize concurrency
and communication, and they have proved particularly useful for describing
UI software as a collection of agents [1, 2]. Other notations, such as StateCharts [8]
and Petri nets [16], have also been explored for modeling UI agents, as these alternative
notations also provide definitions of concurrency and communication. We
chose Lotos because composition in Lotos resembles conjunction [25], which is a
useful paradigm for composing partial specifications (Section 2.5).
We model the behavior of an agent using a Lotos abstraction called a process,
which is a computational entity whose internal structure can only be discovered by
observing how it interacts with its environment. Processes perform internal (unob-
servable) computations and interact with other, concurrently executing, processes.
The interaction between processes is synchronous: If one process tries to communicate
with a process that is not ready to communicate, the former process blocks
until the latter is ready. Thus, the act of communicating synchronizes concurrent
processes.
A process represents the state of an agent as a procedure for performing future
actions. An action is an atomic computational step taken by an individual pro-
cess. Actions of a process can be observed through the events in which the actions
participate. An event is an observable unit of multi-process communication. Multiple
processes participate in an event by simultaneously performing actions over
the same gate. A gate is a primitive synchronization device used to observe the
occurrence of an action in a process. Each action is associated with a single gate.
The gates of a composite agent are the union of the gates of its constituents. If two
or more constituents name the same gate, then any actions over that gate proceed
simultaneously. That is, the processes associated with the constituent agents synchronize
actions that share the same gate name. Thus, gates also represent a class
of possible inter-process synchronization events. During such an event, an action
can offer one or more data values that can be observed by actions in other processes
that are participating in the same event.
A complete agent is modeled by a process that represents the initial state of the
agent. A multi-agent system is modeled by a collection of concurrent, communicating
processes. When composing a system of multiple agents, the designer must
decide how to coordinate actions in the various processes that model the agents.
are coordinated by synchronizing actions labeled with identically named
gates.
6 STIREWALT AND RUGABER
2.4. Lotos
Lotos is a rich language for specifying the partial ordering of actions within a
process and the structure of multi-process interactions. Complex processes may
be expressed by either combining sub-processes using an ordering operator (e.g.,
process P is the sequential composition of sub-processes P 1 and P 2 ) or by conjoining
sub-processes so that they run independently but synchronize actions with gates.
An event allows values to flow between participating actions. Lotos also describes
the semantics of value passing with respect to synchronization.
Actions in Lotos have the following structure:
action ::= gate (inputjoutput)
input
output
gate ::= identifier
Each action names a gate and zero or more inputs and outputs. An input names a
variable in which to record a value that is offered by an action in another process. An
output is an expression for computing a value to offer to actions in other processes.
Actions concisely represent the occurrence of many possible events. Like actions,
events are associated with a particular gate. Unlike actions, events have no concept
of input or output; rather they represent unique values that flow between actions.
Events have the following structure:
event ::= gate (value)
value
Note that the values are always constants because events are unique assignments
of values during a synchronization.
In Lotos, the gates over which two conjoined processes are required to synchronize
must be specified between the vertical lines that symbolize the conjunction
operator (k). For example, given the following Lotos process definitions:
process
process
process R [
Process R behaves like P on gate g 3 and Q on gate g 4 , but R must behave like P
and Q in synchrony on gates g 1 and g 2 .
For processes with many gates, the Lotos notation quickly becomes unreadable.
In this paper, we abbreviate the conjunction operator using notational conventions
similar to those used in CSP [9]. In our abbreviated notation, we write the conjunction
of P and Q as P k Q with the understanding that P and Q must synchronize on
gates that are common to the agents whose states P and Q respectively represent.
Suppose the behavior of an agent can be described by a Lotos process B. If the
agent can perform an action by synchronizing on event e (denoted B(e)), then its
MODEL-COMPOSITION PROBLEM 7
behavior from that point on is defined by another process B(e). The systems
under study are deterministic, which means that B(e) is always unique. Moreover,
when a system is defined by conjoining sub-processes, the compositional structure is
preserved throughout the lifetime of the system. That is, if
then
where:
occurs over a gate of agent i
Any event that can be observed of a process P can also be observed of any conjunction
of P with other processes. This fact will be important when we define the
observer function (Section 3.4).
2.5. Conjunction as composition
Alexander uses conjunction to compose separately defined application and presentation
agents [2]. Abowd uses agent-based separation to illuminate usability
properties of interactive systems [1]. Both of these approaches rely on the use of
conjunction to compose agents that are defined separately but interact. In fact, conjunction
is a general operator for composing partial specifications of a system [25].
The idea is that each partial specification imposes constraints upon variables (or,
in the case of agents, events) that are mentioned in other partial specifications.
When these specifications are conjoined, the common variables must satisfy all
constraints.
We define the behavior of a system generated from Mastermind models to be
any behavior that is consistent with the conjunction of constraints imposed by the
dialog, presentation, and application models. We then extend conjunction from a
specification tool into a mechanism for composing runtime modules.
2.6.
Summary
Three issues must be addressed to solve the model-composition problem: The solution
must generate user-interfaces with rich dynamic behavior, the correctness of
module composition must be demonstrated, and the generated modules must co-operate
efficiently. In Mastermind, the rich expressive power is achieved through
special-purpose modeling notations [15, 5]. The remainder of this paper addresses
the generation of correct implementations with maximal efficiency while preserving
the expressive power of Mastermind models.
3. Model-composition theory
Recall from Figure 1 that each class of model has a code generator that synthesizes
runtime modules for models in that class. The modules are generated without
detailed knowledge of the other models. At run time, however, modules must
cooperate as prescribed by the conjunction of the models that generated them. This
section describes the relationship between model composition and the mechanism
by which the associated modules cooperate at runtime.
3.1. Notation
The subject of this paper is the automatic generation and composition of runtime
modules from design-time models. A module is a unit of code generated from a
single model. We use a third class of construct-the Lotos process-to define the
correctness of model and module composition. In formal arguments, we need to
refer to all three types of constructs; thus we distinguish the constructs by using
different fonts. We also need special functions that map models and modules into
comparable domains.
We represent the classes of Mastermind models using German letters. The symbols
D, and A represent respectively the classes of Mastermind presentation,
dialog, and application models. We use the italic font to represent Lotos processes
and the semantic models of these processes. The set P rocess represents the set of
Lotos processes. Specific processes are written in capital italic letters (e.g., P ,
D, and A, respectively). The set T raceSets defines the set of event traces over
the alphabet of gates and the space of values that can be offered and observed by
Lotos actions. The function raceSets maps a Lotos process
to the set of all event traces that can be observed of that process.
We represent runtime entities using the Sans serif font. The set Component represents
the class of all runtime components. A component is a block of code that
provides gates for observing the actions of the component. By defining components
as runtime code that provides gates for observing behavior, we can define the function
raceSets that maps a component to the set of event
traces that can be observed through the gates that the component provides.
There are two categories of component in the Mastermind architecture: the generated
modules and the synchronous composition of these modules. Instances of the
generated modules are written Pres, Dialog, and Appl, respectively. We also think
of the modules in synchronous composition as a component, which is attained by
connecting the generated modules using some synchronization infra-structure (de-
fined in Section 4). This composite component is written Synch[Pres; Dialog; Appl].
The name Synch suggests that the component is the synchronization of the three
generated modules; the brackets suggest that the generated modules fit into the
larger system and that Synch by itself is not a component.
3.2. Inter-model composition
Model-based code generators construct runtime modules from design-time mod-
els. The code generation strategy is model-specific, reflecting the specialization of
models to a particular aspect of a system. At run time, however, modules must co-
operate, and the cooperative behavior must not violate any correctness constraints
imposed by the models. There is an inherent distinction between behavior that
MODEL-COMPOSITION PROBLEM 9
is limited to the confines of a given model and behavior that affects or is affected
by other models. Inter-model composition is concerned with managing this latter
inter-model behavior.
Some behavior is highly model specific and neither influences nor is affected by
behavior specified in other models. As Figure 2 illustrates, in a Mastermind
presentation model, graphical objects are implemented using primitives from the
Amulet toolkit [13], and attribute relations are implemented as declarative formulas
that, at runtime, eagerly propagate attribute changes to dependent attributes. As
long as changes in these attributes do not trigger behavior in dialog or application
models, these aspects can be ignored when considering model composition.
In an application model, object specifications are compiled into abstract classes
under the assumption that the designer will later extend these into subclasses and
provide implementations for the abstract methods. As long as the details of these
extensions do not trigger behavior in dialog or presentation models, this application
behavior may also be ignored when defining model composition.
Within a module, entities compose according to a model-specific policy. In a
presentation model, for example, objects compose by part-whole aggregation, and
attributes compose by formula evaluation over dependent attributes. In an application
model, objects compose using a combination of subclassing, aggregation, and
polymorphism. When considering how models compose, some details of intra-model
composition can be abstracted away, but not all of them. Models impose temporal
sequencing constraints on the occurrence of inter-model actions, and models
contribute to the values computed by the entire system. These constraints and
contributions must be captured in some form and used to reason about model
composition.
We map this inter-model behavior into a semantic domain that is common across
all of the models. This domain is described by the Lotos notation, which specifies
temporal constraints on actions and data values. We assume that Lotos processes
can be derived from the text of a model specification (Section 3.4). Designers
may, for example, need to designate actions of interest to other models. Lotos
processes do not capture all of the behavior of models in composition, but they do
express the essential inter-model constraining behavior.
3.3. Example
We now present an example of inter-model behavior expressed as a Lotos process.
The dialog model being considered is for a Print/Save widget similar to those
found in the user interfaces of drawing tools, web browsers, and word processors
(See
Figure
3). These widgets allow the user to format a document for printing
either to a physical printer or to a file on disk; we call the former task printing and
the latter task saving. Options specific to printing, such as print orientation (e.g.,
portrait vs. landscape), and to saving, such as the name of the file into which to
save, are typically enabled and disabled depending upon the user's choice of task.
These ordering dependencies are reflected in the dialog model for this widget shown
by the Lotos process in Figure 4.
Figure
3. Screen shot of the Print/Save dialog box
The process P rintSave can synchronize on any of the gates that follow it in square
brackets. In this example, the gates print, save, go, cancel, layout, and kbd (line
1 in the figure) define points for synchronizing with the presentation; whereas the
gates lpr and write define points for synchronizing with the underlying application.
The process parameters lpdhost and f ilename (line 2) store the name of the default
printer and the user-selected filename, respectively. The parameter doc represents
the document to be printed or saved, and the parameter port represents the print
orientation (portrait if true, landscape if false).
The widget in Figure 3 is specified by a separate presentation model (not shown).
This model defines a pair of radio buttons labeled File and Printer and two buttons
labeled OK, and CANCEL. When these buttons are pressed, they offer the events
save, print, go, and cancel respectively. The presentation model also contains a
pair of radio buttons that specify paper orientation. These buttons display graphics
of a page in either portrait or landscape mode and, when selected, offer the event
port with a value of true if the choice is for portrait orientation and false for
landscape orientation. Finally, there is a text entry box in which the user can type
in a file name. As the user edits this name, the text box responds by offering the
contents of the string typed so far as part of the kbd event. Note that the actual
being pressed are not returned, as editing functionality is best handled in a
text widget and is not considered inter-model behavior. A separate application
model (not shown) defines procedures for issuing a print request and saving a file
to disk. These procedures are responsive to the events lpr and write respectively.
Actions that synchronize on these events offer a number of values including printer
name (lpdhost) and filename (f ilename).
MODEL-COMPOSITION PROBLEM 11
1. process PrintSave[ print, save, go, cancel, layout, kbd, lpr, write
2.
3. P[ go, lpr, write, layout, kbd
4. where
5. process P[ go, lpr, write, layout, kbd
7. endproc
8. process F[ go, lpr, write, layout, kbd
9. Edit[ go, write, kbd
10. endproc
11. process Layout[ go, lpr, layout
12.
13. [] ( go; lpr ! lpdhost
14. endproc
15. process Edit[ go, write, kbd
17.
19. endproc
Figure
4. Print/Save dialog process.
The temporal structure of dialog, presentation, and application model composition
is given in the behavior specification (line 3). The behavior of P rintSave is
the behavior of the process P (defined on lines 5 through 7) with the caveat that it
may be disabled (terminated) at any time by the observation of the cancel event.
Disabling is shown with the [? operator. Process P represents which interactions
and application invocations must happen in order to send a document to a printer.
Most of this functionality is actually expressed in the sub-process Layout (defined
on lines 11 through 14). P behaves like Layout in the normal case, but it can be
disabled if the save event is observed. Recall that the save event is offered whenever
the user presses the Save to File button in the presentation model. The process
F (defined on lines 8 through 10) likewise behaves like the process Edit (defined
on lines 15 through 18) in the normal case, but is disabled if the event print is
observed. Note that F and P can disable each other, which means that the user
can switch back and forth between printing and saving as many times as he or she
likes before hitting the Go button.
\Gamma\Gamma\Gamma\Gamma! P rocess
CD
y
Component Obs
\Gamma\Gamma\Gamma\Gamma! T raceSets
Figure
5. Dialog compiler correctness.
3.4. Models, modules, and processes
like those shown in Figure 4 are useful for understanding the relationship
between models and modules. This relationship is complex, and so we describe it
first for a single model and then for the three models in composition. We now
formalize correctness conditions for the Mastermind dialog model. A similar
formalization exists for the other Mastermind models.
Figure
5 shows the relationship between dialog models (members of the set D),
runtime modules generated by dialog models (members of the set Dialog), and the
inter-model behavior of dialog models (members of the set P rocess). The relationships
between these sets are defined as functions that map members of one set into
members of another. The function CD : D ! Dialog maps dialog models to runtime
modules. Think of CD as an abstract description of the dialog-model compiler. The
rocess maps dialog models into Lotos processes describing
their inter-model behavior. Think of AD as an abstract interpretation of the dialog
model expressing its semantics in Lotos.
These sets and functions are related by the commutative diagram of Figure 5.
Externally observable model behavior is mapped into a Lotos process by AD , and
the set of traces of a module's externally observable events is recorded by Obs. We
say that a dialog model d 2 D is consistent with the module CD (d) if every trace
is in the set T r(AD (d)) and if there are no sequences
such that & 62 Obs(CD (d)). That is, the inter-model behavioral interpretation of d
agrees exactly with the observable behavior of the runtime module generated from
d. Commutativity of the diagram requires this property for any dialog model in
the set D.
3.5. Model-based synthesis
The correctness relationship between models and modules (Figure 5) can be extended
to specify the correctness of module composition. We now have functions
AP , AD , and AA that map models into Lotos processes. These processes should
compose by conjunction. We also have a runtime component Synch that combines
modules Pres, Dialog, and Appl into a single component whose actions are observable
by the Obs function. Figure 6 shows the constraints on the behavior of these
entities. Let p 2 P, d 2 D, and a 2 A. Then the code generated from these
MODEL-COMPOSITION PROBLEM 13
Figure
6. Module-composition correctness.
models is correct if and only if, for any observable behavior oe, oe is a legal trace in
the conjunction of the models. This equation defines the conditions necessary for
correct module composition without assuming any model-specific interpretation of
these actions. It serves, therefore, as a specification of design requirements. In the
next section, we present an implementation that satisfies these requirements.
4. Module-composition runtime architecture
We now turn to the designs of the run-time synchronization module and model-specific
compilers of Figure 1. The essential design problem is how to make the
generated modules compose while retaining the independence of the model-specific
compilers. The conditions of Figure 6 impose constraints on these designs. For-
tunately, these constraints do not require model-specific knowledge (e.g., graphical
concepts in the presentation model or data layout in the application model). Thus,
module-composition logic can be separated from the model-specific functionality
within a module. This separation is the key to making model-based synthesis independent
without sacrificing the correctness of module integration. The Mastermind
runtime library contains efficient primitive classes that enable independent
module synthesis and correct composition by conjunction. This library provides
a great deal of generality and flexibility for code generation. In this paper, we
describe only those aspects of the library that are relevant for supporting independent
synthesis. First, we introduce the mechanism for composing generated
modules (Section 4.1). We then describe how this mechanism implements conjunction
without sacrificing the independence of model synthesis (Section 4.2) and
demonstrate its operation through an example (Section 4.3).
4.1. Design structures to support conjunction
To facilitate the independence of model synthesis, we designed a mechanism that
enables a module to compose with other modules without directly referencing them.
As
Figure
1 suggests, generated modules compose through the aid of a special synchronization
component, called Synch. We designed the Synch interface to simplify
the generation of modules. This section describes the interface and the process of
model compilation and integration.
Figure
7 illustrates the interface between the generated modules and the Synch
component. Modules contain Action objects that link (explicitly refer to) Gate
14 STIREWALT AND RUGABER
Action Action Action Action Action Action
Gate Gate Gate
Synch
Dialog Pres
Appl
Figure
7. Structural depiction of composition according to Synch[Pres, Dialog, Appl].
objects in the Synch component. As the names suggest, an Action object reifies
a Lotos action, and a Gate object reifies a Lotos gate. At runtime, Actions
implement a unit of observable behavior in a module, and Gates implement the
synchronization of Actions by conjunction. The mathematical connection between
Lotos actions and gates is reified using explicit links between Action and Gate
objects. These links constitute the mechanism for composing generated modules
with the Synch component: A module "plugs in" to the architecture by linking its
Action objects to appropriate Gate objects in the Synch component. The dashed
lines in Figure 7 illustrate some (of many possible) links.
This architecture enables model synthesis to be treated separately from module
integration, similar to the way compilation is treated separately from linking in
traditional programming. This separation allows a module to be synthesized from
a single model, independent of the synthesis of the other models. During synthesis,
model-based compilers independently generate modules. Any behavior that must
be observed by other modules must be packaged into an instance of the class Action.
When emitting the code that creates this instance, the compiler also writes out the
name of the associated gate to an auxiliary file. Consequently the output of a
model compiler is a module and an auxiliary file listing the names of dependent
gates. During module integration, a module integrator reads in these auxiliary
files, creates the Synch component, and combines it with the generated modules to
produce an executable image.
Going back to our running example, consider the compilation of the presentation
model for the Print/Save dialog box (Figure 3). As the model is processed, the
compiler emits Action objects that interface directly with UI toolkit widgets. After
compilation, the Pres module will contain an Action for each widget in the dialog
box. For example, there will be a distinct Action object paired with the OK and
CANCEL buttons, each of the radio buttons, and Filename text-entry widget.
To integrate the Pres module with the other modules, each of these Actions must
link to Gate objects in the Synch component.
Note that when the Actions are being emitted, the corresponding Gate object
will not yet exist, as the Gate is created by the module integrator. Thus, the link
between an Action and its corresponding Gate cannot be established at compile
time. Instead, an Action object is instantiated with the name of the gate over
MODEL-COMPOSITION PROBLEM 15
void enable();
void disable();
Action {abstract}
ModuleSource
void register(Appl*);
void unregister(Appl*);
DialAppGate
void register(Pres*);
void unregister(Pres*);
PresDialGate
Gate {abstract}
void confirm(Listener*);
void synchronize();
PresDialAppGate
void unregister(Dial*);
void register(Dial*);
DialGate
void execute();
Command {abstract}
Listener {abstract}
void listen();
void ignore();
generalization
(disjoint subclasses)
generalization
(overlapping subclasses)
Legend
synchronizes
ActionRole {abstract}
Figure
8. Detailed design of action and gate classes.
which it must synchronize. At runtime, the Action uses this name to locate the
corresponding Gate. Because the module integrator creates a Gate for each named
gate, the Action object can assume that the gate will exist at runtime. This design
greatly simplifies model compilation: The presentation-model compiler need not
concern itself with locating an object in another component. Rather, the compiler
simply creates a module using Action objects and writes out the names of gates to
an auxiliary file.
4.2. Behavior of the design structures
The synthesis of one Mastermind model can proceed independently of the synthesis
of other models because the generated modules only refer to each other indirectly,
through Gate objects. The Gate objects are responsible for determining when a synchronization
should occur and dispatching control the associated Action objects in
an appropriate order once the synchronization constraints are satisfied. Conse-
quently, Action objects need not be concerned with these issues. Rather, Actions
are concerned with implementing model-specific functionality. This separation is
crucial to supporting the independence of model synthesis.
Figure
8 describes the design of classes Action and Gate. Class Gate is designed to
internalize information about the modules whose actions are required to synchronize
at the gate. Henceforth, we shall refer to this information as the synchronization
constraint of a Gate. The rules of conjunction (Figure number
of possible variations of this constraint. At runtime, a Gate determine whether or
not to synchronize by checking whether or not this constraint is satisfied. To make
this determination, a Gate must infer the location (module) of each Action that
wishes to synchronize over the Gate. We call this process of inference tabulation.
Tabulation occurs when an Action announces its readiness to synchronize. Such
announcements are made by an Action registering itself with its Gate; an Action
registers itself by passing itself to an invocation of the register operation on its Gate.
When a Gate determines that its constraint is satisfied, it invokes the synchronize
operation, which dispatches control to the registered Actions so that they may
execute.
For a Gate to tabulate the modules that request activity, the Gate must be able
to infer the module of every Action that registers. This means that an Action
must know the module in which it exists. Class Action has a subclass, called
ModuleSource, which further specializes into three subclasses, Pres, Dial,and Appl
(not shown in the figure). The concrete class of every Action must inherit from one
of these three subclasses. We implemented tabulation by specializing the register
operation so that it dispatches based on these subclasses. The subclasses of Gate
contain module variations of the register function. These subclasses embody each
of the three possible synchronization constraints that arise in Mastermind. The
constraint associated with class PresDialGate requires Pres and Dial actions to be
present at the Gate. Similarly, the constraint associated with class DialApplGate
requires Dial and Appl actions to be present at the Gate, and the constraint associated
with class PresDialogApplGate requires all actions from all three modules to be
present at the Gate. These are the only three types of synchronization constraints
required of Mastermind-generated user interfaces.
The next issue concerns dispatching control to registered actions once a Gate's
synchronization constraint is satisfied. Mastermind supports two different action-
control mechanisms (generalized by ActionRole). One mechanism is a generic interface
for executing a model-specific operation (class Command). The other mechanism
is a generic interface for reactively observing an asynchronous event, such as a user
interaction with a graphical widget (class Listener). What happens when a Gate's
synchronization constraint is satisfied depends upon the control mechanisms used
by the registered Actions. For example, if two Commands are waiting at a Gate, and
they satisfy the synchronization constraint for the Gate, then the execute method
for both Commands are invoked. If, instead, one of these actions is a Listener and
the other is a Command, then the Command is not invoked until the Listener receives
an event. Because Listeners are reactive, they need to be able to announce the
reception of an event to the Gate. This is accomplished by invoking the confirm
operation on the Gate.
A module requests the performance of an Action by invoking the operation enable.
Enabling causes an Action to register itself with its Gate. Our design abstracts
the logic for requesting the performance of an Action into the enable and disable
methods, which correctly cooperate with the corresponding Gate irrespective of the
particular synchronization constraints. Thus, the logic can be completely encapsulated
in the abstract class Action, which a model-compiler writer need never
modify. Moreover, model-compiler writers can package model-specific functionality
using one of two quite different control policies, Command and Listener. One consequence
of this design is that the module integrator must determine the type of
Gate to emit. This is a simple task, however, given the information written to the
MODEL-COMPOSITION PROBLEM 17
generalization
Legend
pointer to operation pseudocode
void listen();
void ignore();
Listener {abstract}
widget Am_Text_Input_Widget
void Do();
TextFieldAction
void listen();
void ignore();
Pres
Figure
9. Example of use.
auxiliary files by the model compilers. For example, the gate cancel that is used
in the Print/Save dialog is used in both the presentation model, where it observes
the pressing of the CANCEL button, and in the dialog model, where it observes
the completion of the dialog. Because modules compose by conjunction, the Gate
associated with cancel always synchronizes an action from the Pres module with an
action from the Dialog module. To implement this behavior, the module integrator
emits an instance of PresDialogGate, which is returned when the associated Actions
link to the named gate.
4.3. Example
We now demonstrate how these features work in the context of the Print/Save
dialog. Recall from Figure 3 the text entry field that allows a user to enter file
name in which to save a document. In the dialog model (Figure 4), the entry of
the file name is modeled as an atomic action over the gate kbd. To connect this
dialog action to the text entry widget that ultimately witnesses the action, we need
a presentation Action that knows how to attach to the text entry widget, and we
need a Gate object to represent the kbd gate.
Figure
9 illustrates how a reusable action that listens for text entry can be
created from the primitives introduced in Section 4.2. The presentation-model
compiler emits instances of this class to implement text-entry boxes. In the fig-
ure, we rendered the primitive classes in grey to distinguish them from new objects
and classes that the model-compiler writer creates. The new class is called
TextFieldAction. It inherits from class Pres because its instances will be emitted
into the Pres module. It inherits from class Listener because it is concerned with
monitoring and controlling the text-input widget. The class is associated with an
Am Text Input Widget object by an association called widget. This object is prede-
fined in the Amulet toolkit [13], which the current version of Mastermind uses for
presentation support. The TextFieldAction controls the Amulet object by invoking
the Start and Stop operations on the object, which instruct the widget to enable
and disable keyboard input. The invocations of these methods form the implementation
of listen and ignore respectively. We also need a way for the widget to
signal the Action object with the event. This is accomplished by overriding the Do
method of the widget to go find the Gate associated with the Action and invoke
the confirm operation on this Gate to signify the occurrence of the event. The Do
method can be thought of as a callback function that Amulet invokes to deliver an
event (in this case, the event is a keyboard return).
The example serves to illustrate the sequence of behaviors that are enacted by
the Mastermind library primitives. Suppose an object of class TextFieldAction
is registered at the Gate associated with kbd. If the synchronization constraint for
this Gate is satisfied, the Gate invokes the listen method of the TextFieldAction.
This invocation in turn causes the Start method of the Am Text Input Widget to
be invoked, which enables user input at the widget. If the synchronization context
changes so that the constraint is no longer satisfied-either because the Pres module
disables the TextFieldAction or because another module disables an Action that
is waiting at the Gate, then the Gate invokes the ignore operation. This causes
the TextFieldAction to invoke the Stop method of the Am Text Input Widget, which
disables text input. If, on the other hand, the user enters a string and hits the
return key, the Do method of the widget is invoked. This causes the invocation of
the confirm method on the Gate, and the Gate proceeds to execute any Commands
that are waiting.
4.4.
Summary
Our design enables independent code generation because the Actions in a generated
module are insulated from Actions in other modules by the gate objects. We
compose modules by creating Gate objects that embody the synchronization requirements
of the models and by linking Actions to their Gates. The independence
that is afforded by this approach allows model-based code generators to apply deep
model-specific knowledge to the synthesis of code.
5. Results and status
We evaluated our solution to the model-composition problem with respect to power,
correctness, and efficiency. Multi-paradigm actions have proved easy to specialize
to accommodate features from disparate implementation toolkits and architectures.
For example, we have specialized Actions to represent actions in: the Amulet object
system [13], the C++ object system, and a special-purpose state-machine language.
Figure
2 summarizes the different applications and results.
MODEL-COMPOSITION PROBLEM 19
5.1. Power
We were able to express user interfaces using our modeling notations in several
case studies. We tested the quality of user interfaces on two specific examples: the
Print/Save widget described in Section 3.3 and an airspace-and-runway executive
that supports an air-traffic controller (ATC) [19]. The former demonstrates the
ability to generate common, highly reusable code for standard graphical user inter-
faces. The latter demonstrates the ability to support a complex application using
a direct-manipulation interface.
The ATC example testifies to the power of our approach. When a flight number is
keyed into a text-entry box, an airplane graphic, annotated with the flight number,
appears in the airspace. As more planes come into the airspace, the controller keys
their flight number into the text-entry box. When the controller decides to change
the position of a plane, she does so by dragging the airplane icon to a new location
on the screen. As soon as she presses and holds the mouse button, a feedback
object shaped like an airplane appears and follows the mouse to the new location.
When the mouse is released, the plane icon moves to the newly selected location.
The presentation model of the ATC example is quite rich. It specifies gridding so
that airplane graphics are always uniformly placed within the lanes, and it specifies
feedback objects that present controllers information about the planes during
operation. In a real deployment, the locations of flights change in response to asynchronous
application signals from special hardware monitors. In such a deployment,
these signals would be connected to Listener actions and would fit into the frame-work
without change. For more details on this case study and the Print/Save dialog,
see Stirewalt's dissertation [19].
5.2. Correctness
In addition to being able to generate and manage powerful user interfaces, the composition
of our modules is correct. Two aspects of our approach require justification
on these grounds. First is the design of runtime action synchronization. Second
is the synthesis of runtime dialog components (members of the set D) from dialog
models.
This paper addresses the theoretical issues involved in the design of runtime
action synchronization. The Gate and Action classes are traceable refinements of
the corresponding concepts in Lotos. In practice, we found this design to be quite
robust. One aspect of synchronization correctness, which we do not address in
this paper, is how to show that a model-specific specialization of Action does not
violate the delicate callback protocol that underlies the system. For example, say
that an Appl, which when modeled in Lotos observes a value x and offers a value
y, is to be implemented using a method invocation. The method should bind x to
its parameter and bind its return value to y. Since value offerings are evaluated in
sequence, how can we be sure that the ordering of evaluation does not interfere with
the invocation of the method or vice versa? Currently, we check this by inspection,
but we are investigating ways of packaging this problem so that a model checker
(e.g., SPIN [10], SMV [12]) can detect such anomalies. Stirewalt used the SMV
model checker to validate the inter-operation of Action and Gate objects [19].
As we mentioned earlier, the Mastermind Dialog model notation is a syntactic
sugaring for a subset of Lotos. This language is described in greater detail in [20].
We implemented a prototype dialog model code generator whose correctness was
validated as described in Stirewalt [19]. This code generator compiles dialog models
without reference to other models.
5.3. Efficiency
We measured efficiency empirically by applying our code generator on the ATC
example. We generated dialog modules and connected these with hand-coded presentation
and application modules. On the examples we tried, we observed no
time delays between interactions. We quantified these results by instrumenting the
source code to measure the use of computation resources and wall-clock time. The
maximum time taken during any interaction was 0:04 seconds. This compares well
to the de facto HCI benchmark of response time, 0:1 seconds. We believe that more
heavyweight, middle-ware solutions, such as implementing synchronization through
object-request brokers, are not competitive with these results.
5.4. Future work
We are currently completing a more robust dialog code generator. This new code
generator incorporates state-space reduction technology and will improve interaction
time, which in the prototype is a function of the depth of a dialog expression,
with interaction that executes in constant time.
6. Conclusions
How to generate code for a specialized modeling notation is a well understood
problem. Integrating code generated from multiple models is not. Integration is
much more complicated than merely linking compiled object modules. For models
to be declarative, they must assume that entities named in one model have behavior
that is elaborated in another model. Designers want to treat presentation, temporal
ordering, and effect separately because each aspect in isolation can be expressed in
a highly specialized language that would be less clear if it were required to express
the other aspects as well. For interactive systems, composition by conjunction is
essential to separating complex specifications into manageable pieces.
Unfortunately, programming languages like C++ and Java do not provide a conjunction
operator. Such an operator is difficult to implement correctly and effi-
ciently, and, in fact, we did not try to implement it. Rather, by casting model
composition into a formal framework that includes conjunction, we are able to
express a correct solution and then refine the correct solution into an efficient de-
MODEL-COMPOSITION PROBLEM 21
sign. This is a key difference between our approach and middle-ware solutions that
implement object composition by general event registry and callback.
Our results contribute to the body of automated software engineering research in
two ways. First, our framework is a practical solution that helps to automate the
engineering of interactive systems. Second, our use of formal methods to identify
design constraints and the subsequent refinement of these constraints into an object-oriented
design may serve as a model for other researchers trying to deal with model
composition in the context of code generation. The formality of the approach allows
us to minimize design constraints and is the key to arriving at a powerful, correct,
and efficient solution.
--R
Formal Aspects of Human-Computer Interaction
Structuring dialogues using CSP.
Developing Software for the User Interface.
Introduction to the ISO specification language Lotos.
Using declarative descriptions to model user interfaces with Mastermind.
Declarative models of presentation.
PAC, an object-oriented model for dialog design
On visual formalisms.
Communicating Sequential Processes.
The model checker spin.
A cookbook for using the model view controller user interface paradigm in smalltalk.
Symbolic Model Checking: An Approach to the State Explosion Problem.
The Amulet environment: New models for effective user-interface software development
Survey on user interface programming.
Knowledgeable development environments using shared design models.
Validating interactive system design through the verification of formal task and system models.
The Mecano project: Comprehensive and integrated support for model-based user interface development
Automatic Generation of Interactive Systems from Declarative Models.
Design and implementation of mdl: The mastermind dialogue language.
Declarative models for user-interface construction tools: the Mastermind approach
Beyond interface builders: Model-based interface tools
Beyond hacking: A model based approach to user interface design.
A compositional approach to multiparadigm programming.
Conjunction as composition.
--TR | model-based;user interface;multi-paradigm;code generation |
592049 | Explanation-Based Scenario Generation for Reactive System Models. | Reactive systems control many useful and complex real-world devices. Tool-supported specification modeling helps software engineers design such systems correctly. One such tool, a scenario generator, constructs an input event sequence for the spec model that reaches a state satisfying given criteria. It can uncover counterexamples to desired safety properties, explain feature interactions in concrete terms to requirements analysts, and even provide online help to end users learning how to use a system. However, while exhaustive search algorithms such as model checkers work in limited cases, the problem is highly intractable for the functionally rich models that correspond naturally to complex systems engineers wish to design. This paper describes a novel heuristic approach to the problem that is applicable to a large class of infinite state reactive systems. The key idea is to piece together scenarios that achieve subgoals into a single scenario achieving the conjunction of the subgoals. The scenarios are mined from a library captured independently during requirements acquisition. Explanation-based generalization then abstracts them so they may be coinstantiated and interleaved. The approach is implemented, and I present the results of applying the tool to 63 scenario generation problems arising from a case study of telephony feature validation. | Introduction
Reactive systems control many useful and complex
real-world devices, such as telephone switches, air and
space craft, and software agents. Such feature-rich
systems are difficult to design correctly, particularly
when distinct functional features are designed by different
people at different times over the lifecycle of
a product family. Specification modeling[11, 16] allows
engineers to apply relatively sophisticated validation
tools such as simulation, coverage analysis,
model checking[17, 5], or theorem proving[12, 20], to
relatively abstract models of the system's behavior in
order to find design errors before implementation. It
is the abstractness of the models that makes many of
the reasoning techniques tractable. The validated spec
model can be used as a starting point for code genera-
tion, as documentation of the behavior of the system,
and in support of maintenance and evolution[11].
A spec modeling tool suite benefits significantly
from a scenario generator, which constructs an input
event sequence for the spec model that reaches
a state satisfying given criteria. Such a tool can uncover
counterexamples to desired safety properties,
explain feature interactions in concrete terms to requirements
analysts, increase test coverage, and even
function as documentation, showing end users how to
achieve their goals while still learning how to use a
system. However, while some model checkers[17, 4]
are capable of generating scenarios for certain limited
classes of reactive systems, such as finite state machines
with small (or highly symmetric) state spaces,
the problem is intractable for functionally rich models
that arise as natural abstractions of systems engineers
wish to design. For example, in addition to requiring
search in an infinite state space, models incorporating
arithmetic operators can require the scenario generator
to find satisfying instances of arbitrary arithmetic
constraints, which is undecidable.
This paper describes a novel heuristic approach,
called
ization") which is applicable to a large class of infinite
state reactive systems. The key idea is to instantiate
and piece together abstracted scenarios that achieve
subsets of the conjuncts of a goal predicate into a
single scenario achieving the conjunction of the sub-
sets. The scenarios are mined from a library of concrete
scenarios captured independently during requirements
acquisition. Critically, they are then abstracted
via explanation-based generalization. The approach is
sound, but incomplete, so it will not succeed in finding
scenarios in all cases of satisfiable goal predicates;
however, it is intended to be fast, even in failing cases,
so that it can be a practical interactive tool. Moreover,
the approach's power can be increased by adding more
scenarios to the library, so, as more requirements are
uncovered and specified, the power of the tool grows
naturally. Even an incomplete generator is quite use-
ful. Typically, an engineer will discover (e.g. via static
analysis or proof attempts) descriptions of states in
which spec inconsistencies may arise, or correctness
properties may be violated; the scenario generator is
then run on these descriptions. Whenever the generator
is successful, a definite design flaw has been found,
so the engineer can focus attention there first. The
other cases, which may not even be satisfiable, can be
put off to later in the design process, after the known
problems are fixed. Fixing these first problems may
either alter or eliminate the other ones anyway. When
the generator fails, putting out a scenario coming as
close to the goal as possible can be helpful as well.
This paper can be summed up in three key ideas:
ffl Current limited-domain exhaustive search approaches
(such as model checking[17]) to scenario
generation are not enough; we need a usable scenario
generator that accommodates more expressive
logics and large state spaces, even though the
problem is highly intractable;
ffl The heuristic SGEN 2 approach, based on mining
and abstracting requirements knowledge using
explanation-based generalization, applies to
richly expressive logics and large state spaces;
ffl A moderate sized case study involving feature interactions
in telephony gives initial empirical evidence
that SGEN 2 is practical and useful.
After Section 2 defines terms and describes the tool
suite in which SGEN 2 is implemented, the next three
sections make these key points. I conclude with discussion
of related work, limitations, and future work.
Modeling
This work is performed within the Interactive
Specification Acquisition Tools (ISAT) framework.
ISAT[11, 12, 13] is a prototype tool suite for reactive
system design that is intended to support full-lifecycle
spec modeling as well as code generation. A reactive
system is a (not necessarily finite-) state machine that
reacts to parameterized input events by changing its
state and by performing acts, which can be thought
as output events. ISAT is based on two hypotheses:
ffl Functional requirements are most reliably elicited
from and validated by requirers as concrete, formal
behavior scenarios; and
ffl Specifications must be executable and amenable
to automatic analysis.
A designer constructs a reactive system model in the
executable spec language, while a requirer specifies
functional requirements as concrete scenarios. The
latter are interleaved sequences of input events and
act or state observations required to be true. Thus,
crucially to SGEN 2 , a natural part of the design lifecycle
is the acquisition of a library of validated concrete
scenarios describing the system's behavior.
2.1 Model Formalism and Backpropagation
An ISAT spec model consists of a theory definition
together with a set of event handlers. The theory defines
the types, functions, and semantic axioms of a
pure computational logic, as well as
the signatures of the state relations, events, and acts
that make up the system. In order to support model
simulation (execution), all primitive function declarations
in the model's theory must include a total operational
function capable of computing the value of
the function on inputs in its domain and some non-
error, type-compatible output value on inputs outside
its declared domain. (ISAT model theories are somewhat
similar to computational logic as described in
[3].) Thus, ISAT supports arbitrary functional rich-
ness, bounded only by the user's willingness and ability
to code implementations for the functions and provide
the logical axioms supporting the other reasoning
tools (see below). For example, models can operate on
arbitrary data structures. I have used this richness to
good advantage in my work on applying ISAT to the
specification and implementation of the Email Channels
system[13]: the ISAT model operates on message
data structures, lists of users and messages, and even
database relation objects.
Event handlers are expressed in a limited procedural
language P-EBF ("procedural event-based formal-
ism"), which is semantically related to the rule-based
EBF I described in [11]. The details of P-EBF are
not crucial to this paper, except that it supports a
predicate backpropagation operator BackProp. Note
that P-EBF need not be the input language seen by
the designer; many domain-appropriate front-end formalisms
(e.g. domain-specific languages or graphical
programming environments) may be compiled into P-
EBF. Such formalisms are beyond the scope of the
present paper, however.
Formally, the state of an ISAT model is represented
as a collection of parameterized (partial) functional
relations and each
D i are data domains (types). For example, the relation
Address 7! Call stores for an address
(i.e., phone number) the object representing its ongoing
call (if any). State values are referred to within P-
EBF expressions via the LOOKUP operator; for example,
(LOOKUP CALL "1234") returns the current call in which
extension "1234" is involved, if any. A state predicate
is a Boolean-typed ISAT expression. Predicates may
be parameterized by typed formal parameters. Here
is a state predicate of one address parameter, usr:
(and (member? usr (lookup known-addresses))
(equal IDLE (lookup mode usr))
(not (equal NO-CALL (lookup call usr))))
This predicate represents all states in which there is
an idle address that nevertheless still has a valid call
object. It is the negation of a desirable state invariant;
thus, a generated scenario reaching such a state proves
the existence of a design error.
The BackProp Operator. Formally, ISAT's
BackProp takes six arguments,
and returns three values a). P 0 is a state predicate
and a 0 is a list of actual (concrete) parameters
for P 0 such that P 0 is true when evaluated in model
M 's state s 0 ; s is a state for model M such that applying
the concrete input event e to M in s results in
the new state s 0 . Pictorially,
true in s 0 . The return value E is an event schema for
(variablization of) the concrete event e, defining fresh
formal parameters. P is a state predicate taking the
same arguments as P 0 plus the formals of E, and a
is a list of actuals for P such that P (a) is true in s.
Moreover, we specify that
a) if
and only if for all states S and actual parameters
applying E(AE ) to the model M in S results
in a state S 0 in which P 0 (AP 0
To clarify, the formals of P are just the union of the
formals of P 0 and those of the event schema E. Thus,
the actual list A will have values both for the formals
of E and the formals of P 0 . Intuitively, BackProp
computes a sufficient (not necessarily necessary) condition
on event E and the state prior to applying E,
such that P 0 is true afterward.
BackProp applies explanation-based generalization
[11, 8] to the P-EBF formalism. Others have described
similar operators, such as Dijkstra's predicate
transformers, or Igerashi et al's verification condition
generators[18]. It is beyond the scope of this paper to
explain the algorithm in detail, but here is an exam-
ple. In state 1, user "1234" has MODE IDLE. The event
results in state 2 in which the MODE
of "1234" is DIALING. Then BackProp applied to the
1-parameter predicate (EQUAL DIALING (LOOKUP MODE
returns the event schema (OFFHOOK ?y) and predicate
(AND (EQUAL :IDLE (LOOKUP MODE ?x)) (EQUAL ?x
?y)). (The actuals lists bind both ?x and ?y to "1234".)
Intuitively, this means that if we offhook any idle user,
that user will move to the dialing mode.
BackProp*. Note that if we have a succeeding
scenario trace involving a sequence of input events, we
can iteratively apply BackProp to get an entire generalized
scenario, where the initial predicate will not
depend on the state at all (because ISAT scenarios are
defined never to succeed if they depend on uninitialized
state values). The rest of the paper will refer to
this operation as BackProp*; it takes in a model, a
scenario trace, and a predicate to be backpropagated
together with its satisfying actuals list, and returns
this fully backpropagated predicate, its actuals list,
and the list of event schemas making up the generalized
scenario.
2.2 ISAT Tools Overview
ISAT exploits the two hypotheses above to provide
a suite of analysis tools to help the designer produce a
specification that meets the true needs of the requirer.
ISAT includes the following tools:
ffl Scenario simulation takes a scenario and a model
and executes the model to determine whether the
scenario represents correct behavior of the model.
Thus, requirements scenarios can be directly validated
ffl Coverage analysis reports states never reached by,
and statements of the model that are not executed
by, any of the requirement scenarios. This helps
the designer elicit adequate requirements from the
requirer.
ffl Layered theorem proving[12, 20] is a technique for
proving arbitrary correctness properties, such as
state invariants and pseudo-state diagrams[12].
ffl Conflict detection[14] returns predicates describing
states in which the model, if it reaches them,
will derive an inconsistent next state (potentially
causing either a crash of the simulator or, worse,
the implemented system). Inconsistencies can result
from setting state relations to two inconsistent
values or raising conflicting output events,
such as playing both the ringback tone and the
busy tone at the same time to the same phone.
Coverage analysis, conflict detection, and proof attempts
produce state predicates to which we can apply
a scenario generator in order to discover whether they
represent reachable states of the model.
3 The Scenario Generation Problem
Formally, the scenario generation problem is to take
a model M and state predicate P 0 and find a sequence
L of concrete input events and a list of actual parameters
executing L in M starting
from the undefined initial state results in a state s 0 satisfying
I have concentrated
on conjunctive state predicates, i.e. those whose expression
consists of the logical AND of a collection of
predicates. The method can be applied to disjunctions
of conjunctive state predicates by applying it concurrently
to each of the disjuncts, but that requires engineering
for efficiency that is beyond the scope of this
paper. Sections 1 and 2 discussed some ways a tool
suite can benefit from solving this problem.
Why Rich Formalisms? Model checkers[17] and
symbolic model checkers[5] guarantee that when they
find a property not valid in a model, they return a
concrete counterexample (scenario) illustrating the vi-
olation. Thus, we should explore under what circumstances
these tools solve the scenario generation problem
before inventing different ways to solve it.
Model checkers exhaustively search the state space
of the system, testing the property in each state.
Thus, they are limited by the size of the state space
they can handle. Some model checkers exploit limited
forms of state space symmetry to handle systems with
larger spaces, but all eventually run into this "state explosion
problem". And while symbolic model checkers
have checked properties in impressively large (10 120
spaces, it is not clear if the technique
can be extended to handle nonboolean logics.
For a survey of model checking and its relation to theorem
proving for verification, see[6].
Should we simply avoid models with large state
spaces? I believe the answer is "no." Several common
types of design problems are only manifest in more
complex (large or unbounded state space) models of a
system. For example, complex systems are frequently
designed in a modular fashion by designing functional
"features" independently and then combining feature
sets to meet customization or market needs. Telephone
switching systems are a good example of this
approach, yet many other systems are built this way.
The problem is that even though individual features
are valid in isolation, their combination may lead to
undesirable interactions that lead to faulty behavior.
The only way for a tool to discover these interactions is
to model the feature combinations; it follows that the
more features a system has, the more complex must
its model be in order to detect interactions.
Another reason limited-space approaches are not
the final answer is that it is difficult both to do enough
abstraction to make the problem tractable and yet
to retain enough detail to manifest the problems of
interest. In particular, each property to be checked
may require a different, hand-constructed model ab-
straction. And since designers don't know in advance
which problems the system has, there could be a lot
of wasted effort and/or false confidence in results. By
dealing with more complex models, the abstraction
can be relatively straight-forward, and a single one
can be used for all properties.
Finally, another reason to prefer a single, easily produced
abstraction that is clearly faithful to the system,
is that there is the possibility of generating implementations
directly from the models, either through code
synthesis or by direct manual implementation. Of-
ten, abstractions that are necessary for tractability are
missing too much detail to allow any direct mapping
to implementations. For example, Alur et al[1] report
on a model checking effort for a phone switch in which
it was necessary to model queue data structures by 7
bit integers (representing the number of items in the
queue). An implementation must supply all details of
queue implementation, as well as any system behavior
depending on the actual contents of the queues.
Why is scenario generation hard? As soon
as our representation language allows event and state
parameterization and functions, we have added an
uncomputable constraint satisfaction problem to the
problem of combinatorial search in large state spaces.
For example, designers commonly need models with
arithmetic, lists and other data structures, text manipulation
functions such as pattern matching, etc.
But then it is possible to define systems and properties
that are only satisfied when the system reaches
a state satisfying an arbitrary sentence of this rich
theory. Proving such a state reachable is undecidable,
by G-odel's Incompleteness Theorem; generating a scenario
that actually reaches it is even harder because
of the combinatorial search.
Thus, in summary, we want to be able to apply
scenario generation to complex modeling formalisms,
and yet the problem goes from merely search to un-
computable. Our only hope in these cases is to find an
approach that can solve the problem in usefully many
cases, and not take too long doing it. We also require
that whenever the tool returns a scenario, it actually
satisfies the goal predicate (soundness). These are the
goals of the SGEN 2 approach.
4 The SGEN 2 Approach
Let us term the overall conjunctive state predicate
the "goal predicate" and the individual conjuncts
making it up the "conjunct predicates" or simply the
"conjuncts." There are two key insights behind the
algorithm. First, the library of requirement
scenarios, while unlikely to have a scenario which
reaches a state satisfying the goal predicate, nevertheless
is likely to have scenarios that reach states
satisfying sets of the conjuncts. Thus, we might find
such scenarios and somehow paste them together into
a single scenario that achieves the full conjunction.
typically two such scenarios will operate on different
data items; for example, scenario 1 may achieve
set 1 of the conjuncts for address "1234", while scenario
2 achieves set 2 for address "5678". Thus, these
two concrete scenarios cannot be interleaved to form
a scenario that achieves the union of the sets for a
single address. However, the second key insight is
that we can solve this subproblem by abstracting the
two scenarios, using BackProp*, and finding a common
instantiation of them (binding of their variables
to data values) such that the union of the two predicate
subsets is satisfied. Once such a common instantiation
is found, a heuristic search merges the two
event sequences into one, achieving the union of the
conjunct sets. Appendix A gives a precise high-level
pseudocode description of the SGEN 2 algorithm.
The following illustrative example is taken from the
case study. Consider the goal predicate
(and (member ?y (lookup known-addresses))
(lookup fpr-active ?y)
(equal dialing (lookup mode ?x))
(lookup tcs-active ?y)
(member ?x (lookup tcs-screened-list ?y)))
This describes states in which known address ?y has
two features, "fpr" and "tcs" both active, with ?x on
its tcs-screened-list, and in which ?x is dialing.
Initialization. SGEN 2 first mines its library and
discovers scenario
(init)
(init-address "1234")
(activate-tcs "1234" "1234")
(offhook "1234")
which results in a state satisfying 4 of the 5 conjuncts:
(and (member ?y (lookup known-addresses))
(equal dialing (lookup mode ?x))
(lookup tcs-active ?y)
(member ?x (lookup tcs-screened-list ?y)))
when we bind both ?x and ?y to "1234". Since it is
unlikely we will find another scenario that fortuitously
achieves the rest of the goal for the constant "1234",
we apply BackProp* to the above predicate and the
trace of scenario S 1 to get the generalized scenario
(init)
(init-address ?x)
(activate-tcs ?y ?x)
(offhook ?x)
subject to the backpropagated condition (equal ?x
?y). SGEN 2 also records the actual bindings
recursive step. SGEN 2 -REC continues
by searching the mined library information for satisfiers
of the remaining conjunct(s) of the goal. In this
case, it discovers (among others) that the scenario S 2
(activate-fpr
achieves the remaining conjunct (lookup fpr-active
?y) when ?y is bound to "5678". Note that since S 1
and S 2 operate on different constants, they cannot be
directly interleaved to get a scenario reaching the desired
conjunction. Applying BackProp* to the remaining
conjunct and the trace of S 2 , we get the generalized
(activate-fpr ?y ?t1 ?t2 ?w)
subject to no constraints (other than implicit type
constraints), with actual bindings:
"1357"g.
calls the Coinstantiate routine
which attempts to find a common instantiation
of G 1 and G 2 obeying both sets of constraints. In
this case, since the constraint set for G 2 is empty,
Coinstantiate quickly finds that the common instantiation
sets.
finally calls MergeScenarios on
the two scenarios G 1 (I) and G 2 (I), which denote
the instances of G 1 and G 2 obtained by applying
I . MergeScenarios also takes the two predicates
are satisfied by G 1 (I) and
respectively, so that it can check whether its result
satisfies both simultaneously. In the case above,
MergeScenarios finds the following interleaving
which does, indeed, satisfy the conjunct sets.
(init)
(init-address "1234")
(activate-tcs "1234" "1234")
(activate-fpr
(offhook "1234")
If at this point there were still unsatisfied conjuncts
of the goal, SGEN 2 -REC would call BackProp* to
generalize this result scenario and then recur to search
for yet another scenario to satisfy the next subset.
If Coinstantiate or MergeScenarios fails, then
move on to the next candidates
in the search (cf Appendix A).
4.1 Library Mining
The first step of SGEN 2 is to search the library of
execution traces of requirement scenarios for states in
which sets of conjuncts are satisfied. The subroutine
MineLibrary accomplishes this as follows. For each
scenario in the requirements library, it first generates
an execution trace by calling the simulator. It then
extracts from the trace sets of data values (grouped
by type) appearing in the trace. Then, for each possible
well-typed assignment of these data values to the
formal parameters of the goal predicate, it searches
the states of the execution trace for those in which
a conjunct first becomes true (for that parameter as-
signment). It creates a predicate group satisfier (pgs)
for that state, which records the assignment and which
set of conjuncts are satisfied. This set of satisfied conjuncts
is termed the satset of the pgs. MineLibrary
returns the entire collection of PGSs found in this way
in all traces. It sorts the list in decreasing order of the
size of the satset so that SGEN 2 will consider earlier
those PGSs that satisfy the most predicates at once.
MineLibrary is linear in the total number of
states in all traces in the library. More importantly,
however, it is proportional to the number of parameter
assignments, which is exponential in the number of
goal predicate parameters. While the current implementation
seems to work adequately fast on the case
study examples (- 5 parameters each), it may be necessary
to limit the number of assignments considered
when the goal predicate has many parameters.
4.2 Coinstantiation
Coinstantiate heuristically attacks the (in gen-
eral) uncomputable problem of coinstantiation by simply
trying out all possible well-typed assignments of
constants to the parameters of G 1 and G 2 , where the
constant pool is simply the union of all constants in
the actual-bindings of the PGSs from which G 1 and G 2
were generalized. This has proven effective in the case
study, and takes negligible time (see statistics below).
If necessary, Coinstantiate can be made to consider
larger constant pools, such as those in all scenarios.
4.3 Scenario Merging
MergeScenarios takes in two scenario/predicate
pairs, where each scenario results in a state satisfying
its predicate. The goal is to return an interleaving
of the two scenarios that satisfies both predicates.
MergeScenarios does not attempt to check all possible
interleavings, as this would require checking exponentially
many (in the sum of the lengths of the two
input scenarios) interleavings in the worst case. And
note that the worst case occurs any time no interleaving
exists, so it is fairly common. Designate the input
scenario/predicate pairs as the "left" scenario and
predicate and the "right" scenario and predicate. Our
approach is to sequentially select the front event off of
either the left or right scenario and add it to the end of
the result scenario until both left and right are empty.
Doing this in all possible ways, waiting until left and
right are empty before checking the predicates, would
result in the exponential worst case mentioned above.
Instead, MergeScenarios heuristically limits the
search as follows. Each time it selects an event e l
from the left scenario, it checks to see whether, if the
result scenario were extended from that point with
the remainder of the right scenario, the right predicate
would be satisfied. If not, e l is vetoed; otherwise,
it proceeds to the next choice. (By induction, one
can show that if instead we extended the result with
the remainder of the left scenario, the left predicate
would also be satisfied.) The dual check is done when
the event is selected from the right. When the front
events on left and right are identical, the algorithm
also attempts the third option of adding one event
and discarding the other.
Note that since there can be interleavings that satisfy
both predicates at the end but which contain intermediate
points at which the check would fail, this
approach is less powerful than brute force search; how-
ever, in the case study, MergeScenarios only failed
Total Scenario Satisfiable/ Not
Attempts Generated No scenario Satisfiable
Table
1: SGEN 2 success on case study
once when a brute force search would have succeeded,
and yet was as much as 12 times faster (average: 2x).
5 Case Study
I ran SGEN 2 on 63 distinct scenario generation
problems that arose in a larger case study of feature
interactions in a telephone switch specification. (The
study actually produced 66 problems, but three were
duplicates, so were discarded for this paper.) The
larger study is actually a tool contest associated with
the 1998 Feature Interactions Workshop[7]. The system
being modeled is a telephone switch implementing
Plain Old Telephone Service (POTS), plus 12 functional
features such as Call Forwarding (CF), Terminating
Call Screening (TCS), FreePhone Routing
and nine others. This SGEN 2 case study was
performed before four of the twelve were modeled, so
only POTS and eight features are included here. In a
related paper[14], I explain how I used the ISAT tool
set to model these specs and to detect various types of
feature interactions among them, many of which are
predicates describing states in which undesired things
may happen, such as feature inconsistencies becoming
manifest (conflicts) or feature correctness properties
being violated. In the absence of a scenario genera-
tor, it is left to the user to determine whether those
state predicates describe reachable states of the model.
Thus, these 63 problems provide a moderately complex
test of the power and usefulness of a scenario
generator, and are representative of the problems that
may be encountered by such a tool. The full data is
available at [15].
Results. The 63 predicates averaged 1.72 parameters
and 5.98 conjuncts each. Table 1 shows the results
of running the generator. "Scenario Generated"
refers to trials in which SGEN 2 succeeded in finding
a scenario; "Satisfiable/No Scenario" refers to the
cases when it failed to find a scenario, even though the
predicate is satisfiable; and "Not satisfiable" refers to
those cases determined (through external means) to
be unsatisfiable and, hence, there exists no scenario
to generate.
Scen.Gen No Scen.Gen
All Only Only
Total 8938 603 8335
Library Mining 658 510 148
BackProp 1766
Coinstantiation
Merge
Table
2: SGEN 2 aggregate run times (rounded to
nearest second).
Table
shows run time statistics for the 63 tri-
als. All times are measured on a 225 MHz Macintosh
clone (144 MB memory) running the ISAT system under
Macintosh Common Lisp 4.2. For this table, the
"no.scen.gen only" condition includes all cases where
the tool did not find a scenario, whether or not the
goal predicate was satisfiable (since to the user these
are equivalent when waiting for the tool to finish).
Discussion. Of the 60 cases in which it was possible
to generate a scenario, SGEN 2 succeeded 40%
of the time. Thus, the user can be sure that at least
those cases illustrate real design errors and therefore
concentrate first on fixing them. Note that one error
can cause scenarios to fail (due to conflicts) that
would otherwise succeed far enough to reach a second
error state. Thus, fixing an error can cause SGEN 2
to succeed when it failed previously. I know of one
definite case (and some others suspected) where this
sort of error interference occurred in the case study.
When I first ran the study, a few cases failed because
individual conjuncts were not covered by the
scenario library. Of course, if there is no known way
to satisfy a single conjunct, the goal predicate won't
be satisfied either. Fortunately, it is relatively easy to
discover a scenario covering a single conjunct, such as
(member ?x (lookup tcs-screened-list ?x)). I easily
created three scenarios to cover these cases, resulting
in one more success and several failures. The results
above reflect these additional scenarios.
Turning to time, we see that the average time per
trial is 142 seconds overall, with succeeding cases taking
25 seconds on average (101 sec maximum) and
failing cases requiring 214 seconds (1054 sec maxi-
mum). Note that the distribution of time is radically
different between succeeding and failing cases,
with MergeScenarios dominating for failing cases
and MineLibrary dominating for succeeding cases.
Coinstantiate was never significant, suggesting that
there is room to improve its power (by checking
larger constant pools, for example) without significantly
harming the overall run time. On the other
hand, we must be extremely careful in increasing the
power of MergeScenarios since that is the bottle-neck
in failing cases. These results are only intended
to be suggestive of future algorithmic improvements; I
believe they can be significantly reduced by a careful
re-engineering effort. (The current ISAT system is an
exploratory prototype.) Note also that these results
depend on the model and scenario library as well.
In summary, it seems that at least for validation
purposes an imperfect scenario generator can still be
quite useful as long as it doesn't take too long. Of
course, we can always hope for a better success av-
erage, and future work will go into improving the
heuristics. However, it is desirable to keep the times
relatively low in all cases, including failure cases, so
the tool is still usable. Thus, we must engineer the
power/speed tradeoff carefully.
6 Related Work
Having discussed model checkers above, I will only
summarize here. Model checkers are useful solutions
to the problem of scenario generation as long as one
can effectively generate models in the limited formalism
necessary to run the tool tractably. However,
there is reason to believe that we need to handle the
more complex formalisms addressed in this work for
at least the reasons discussed in Section 3. In addi-
tion, we may wish to use scenario generation in ways
beyond validation, such as online help systems. For
comparison, it is amusing to estimate the state space
size necessary to model the telephony case study specs
in a finite state formalism. If we model all 12 features
for n users, I estimate there are at least
reachable states (logs are to base 2). If we consider
call-waiting and similar features, we need at least 3
users, but if we add forwarding and other multi-user
features, one can easily imagine properties referring to
6 or more users, leading to
which would challenge even the best model checkers.
Note that even infinite-state model checkers, such
as that of Bultan et al[4], are highly restrictive. That
system is restricted to state spaces that are the cross
product of a boolean state space and one representing
integer inequalities (higher dimensional polyhedra).
While increasing model checking power by adding specialized
constraint reasoners shows promise, it is not
even clear that most reactive systems people design
will be expressible within such restricted formalisms,
due to the common occurrence of functions mixing arguments
of several different types.
Another class of approaches to the problem that
may seem applicable are AI-based planners, such as
STRIPS[9] or Prodigy[19]. The problem with applying
these systems, where the spec model provides the planning
operators, is that planning operators must explicitly
list their consequences; for example, STRIPS
operators must have ADD and DELETE lists. Simi-
larly, the macro operators learned by the EBG-based
PRODIGY system must explicitly include the goal(s)
they achieve. This is too limiting, because users of
scenario generation may give any goal statement they
wish in terms of functions defined in the logic. Any
planning operator derived from the spec model potentially
achieves too many (infinitely many, in fact)
different goals; far too many to be stored explicitly
even if we could bound the vocabulary. SGEN 2 avoids
this problem by doing its abstraction and reasoning
on the fly in MineLibrary and BackProp*. The
only knowledge stored is the "raw" scenario traces,
unadorned with any goal information.
There is also work in the traditional testing literature
on generating test inputs to cover a given path
in a program. For example, Gotlieb et al[10] describe
a constraint-based approach, which essentially
reduces to trying to find a satisfying assignment for a
boolean functional expression which is, of course, uncomputable
once we enrich the formalism to include
(e.g.) arithmetic. However, the constraint based approach
may prove useful in improving Coinstantiate
and MineLibrary; it does not address the state space
search needed to handle reactive systems.
Finally, there are other spec modeling tool suites
providing some of the same (and many contrasting)
tools as ISAT, such as the SCR tool suite[16]. Such environments
may incorporate model checking, but none
capable of dealing with rich formalisms have scenario
generators, to my knowledge.
7 Limitations and Future Work
The most basic limitation of SGEN 2 is that it is
fundamentally a hill-climbing algorithm. In particu-
lar, there are examples in the case study which are
easily solved by merging two scenarios from the li-
brary, but which SGEN 2 cannot find. As an example,
one scenario achieves a particular conjunct set halfway
through its event sequence, but then has several more
steps that are removed by BackProp* as irrelevant
to achieving that set. It turns out they are necessary,
however, if one wishes to later merge it with a second
scenario achieving the rest of the goal. (These
"extra" steps are things like hanging up a phone after
activating a feature, because a subsequent scenario
must start from the idle state.)
power comes from the richness of the scenario
library; it is therefore likely to be more useful in
development processes and environments that encourage
the formalization of such scenarios. SGEN 2 pro-
vides, perhaps, a new argument in favor of integrating
formal scenarios into the software process.
SGEN 2 is still in its early youth, and there are
many ways it can be improved. For example, in its
search, SGEN 2 only considers the first PGS having
a given sat-set. A better, but more expensive, approach
is to try a PGS if and only if its BackProp*-
generalization is not isomorphic to one seen previously.
The effect of this on run-time must be monitored, how-
ever. MergeScenarios, being the time bottleneck
on failing cases, may profit from more work on limiting
its search. MineLibrary needs to search fewer
cases when the predicate takes many parameters.
Of course, results from one case study are not con-
clusive, so future work should investigate SGEN 2 's
effectiveness on other domains and systems.
Conclusions
A scenario generation tool can be useful in a specification
modeling tool suite, in focusing attention on
design errors demonstrably present, in helping communicate
errors in the requirements, and even in implementing
online help systems. Exhaustive-search
approaches, such as model checking, while useful, are
not tractable in rich formalisms allowing more direct
system models to be expressed. SGEN 2 is a heuristic
approach to a highly uncomputable problem, based
on the simple idea of piecing together partially satisfying
scenarios from the requirements library, using
explanation-based generalization to abstract them
in order to be able to coinstantiate them. Results
from the case study are encouraging; SGEN 2 seems
to succeed often enough to be useful and yet be efficient
enough to be engineered into an interactive tool.
While the work needs further empirical validation, it
seems promising and should be pursued.
--R
Model checking of real-time systems: a telecommunications application
A computational logic hand- book
Verifying systems with integer constraints and boolean predicates: a composite approach
states and beyond
Formal methods: state of the art and future directions
Feature Interaction Detection Tool Contest
Learning and executing generalized robot plans
Automatic test data generation using constraint solving techniques
Systematic incremental validation of reactive systems via sound scenario generalization
Reactive system validation using automated reasoning over a fragment library
How to avoid unwanted email
Feature combination and interaction detection via foreground/background models
Complete case study data for this paper
Automated consistency checking of requirements specifi- cations
Design and validation of computer protocols
Automatic program verification I: a logical basis and its imple- mentation
Quantitative results concerning the utility of explanation-based learning
Seven layers of knowledge representation and reasoning in support of software development
--TR | scenario generation;validation;reactive system;explanation-based generalization |
592050 | Specification-Based Browsing of Software Component Libraries. | Specification-based retrieval provides exact content-oriented access to component libraries but requires too much deductive power. Specification-based browsing evades this bottleneck by moving any deduction into an off-line indexing phase. In this paper, we show how match relations are used to build an appropriate index and how formal concept analysis is used to build a suitable navigation structure. This structure has the single-focus property (i.e., any sensible subset of a library is represented by a single node) and supports attribute-based (via explicit component properties) and object-based (via implicit component similarities) navigation styles. It thus combines the exact semantics of formal methods with the interactive navigation possibilities of informal methods. Experiments show that current theorem provers can solve enough of the emerging proof problems to make browsing feasible. The navigation structure also indicates situations where additional abstractions are required to build a better index and thus helps to understand and to re-engineer component libraries. | Introduction
Large software libraries represent valuable assets but the larger they grow, the
harder it becomes to capitalize them for reuse purposes. The main problems are
to keep the overview over the library and to extract appropriate components.
This requires better library organizations and retrieval algorithms than a linear
search through a
at list of components.
Libraries are thus often structured by syntactic means, e.g., inheritance
hierarchies. But this is misleading because it need not to express any semantic
relation between components. Information science oers semantic methods for
library organization and component retrieval e.g., [17, 24], but these methods
are informal because they rely only on the meaning conveyed by words.
As a more exact alternative, the application of formal specication methods
to software libraries has been investigated, starting with [10, 23, 25]. The general
idea is quite simple. Each component is indexed with a formal specication
which captures its relevant behavior. Any desired relation between two components
(e.g., renement or matching) is expressed by a logical formula composed
This work is supported by the DFG within the Schwerpunkt \Deduktion", grant Sn11/2-3.
from the indices. An automated theorem prover is used to check the validity of
the formula. If (and only if) the prover succeeds the relation is considered to
be established. The most ambitious of these approaches is specication-based
retrieval [21, 22, 18, 26, 6]. It allows arbitrary specications as search keys and
retrieves all components from a library whose indexes satisfy a given match
relation with respect to the key.
However, in spite of all research eorts (cf. [19] for a detailed survey),
it is still far away from being practicable. Notwithstanding all progress in
automated deduction, the required theorem proving capabilities remain the
main bottleneck. Here, we investigate a more practical approach, specication-
based browsing of component libraries. Its crucial success factor is that any
time-consuming deduction can be moved into an o-line indexing phase (\pre-
processing") and can thus be separated from navigation. The user works only
on the pre-processed, xed navigation structure which re
ects the semantic
properties of the components with respect to the index.
We show that dierent match relations must be used to build an appropriate
index and how formal concept analysis can be used to build a concept lattice
which serves as navigation structure. Both techniques|specication-based
library organization [9, 18] and concept-based browsing [7, 12]|have been proposed
before, but their combination is new and unique to this research. It thus
combines the exact semantics of formal methods with the interactive navigation
of informal methods.
Experiments show that this approach is feasible. Apart from writing the
specications in the rst place, indexing can be fully automated. Current theorem
provers can solve enough of the emerging problems, even with modest
timeouts. Calculation of the concept lattice is fast enough and navigation works
without delay.
Specication-based browsing is not only useful for reuse but also for analyz-
ing, understanding, and re-engineering component libraries. Although browsing
is dened via specications, they are not actually required for navigation.
Instead, symbolic names can be used which \hide" the actual formulas. An
intelligent choice of such abstractions can thus speed-up and improve under-
standing. The lattice even indicates situations where additional abstractions
are required to build a better index.
Browsing vs. Retrieval
Library browsing and retrieval are closely related but following [19] a clear distinction
can be made. Retrieval consists in extracting components which satisfy
a predened matching criterion. Its main operation is thus the satisfaction check
or matching. The criterion is usually given by an arbitrary user-dened search
or query which is matched against the candidates' indices. Browsing consists
in inspecting candidates for possible extraction, but without a predened
criterion. Its main operation is thus navigation which determines in what order
the components are visited and whether they are visited at all. Browsing
usually works stepwise and we denote the set of all components which can be
visited in the next step as the focus. In contrast to retrieval, it requires no
search key but works on a pre-processed, usually hierarchical navigation struc-
ture. The obvious although not optimal way to compute such a structure is
to order the components by inclusion on their retrieval results using their own
index as query.
In the specication-based case, these dierences prove to be crucial for the
greater practicability of browsing. The pre-processing of the navigation structure
allows us to resort to o-line proving and thus to evade the deductive
bottleneck. Less obvious but equally important, the construction of the hierarchy
via a cross-match of the component library against itself benets the
proof problems. Since no arbitrary user specications are involved, the spec-
ications are much more uniform in style. This allows some obvious prover
tuning; however, the real gain comes from the absence of data mismatches.
Consider for example a graph library where the graphs are represented as map
from nodes to node sets and a query using a representation as a list of node
pairs. Then, the prover must repeatedly, for each candidate, show that both
data representations are equivalent. Although signature matching can mitigate
the data mismatch problem [6], it is still the major source of complexity in
deduction-based retrieval.
Renement Lattices Reconsidered
Formal specications can be used to order components and hence to organize
libraries hierarchically. These hierarchies can then be exploited to optimize
retrieval or to compute a navigation structure. The obvious question is how
to order the components and the obvious answer is by renement or plug-in-
compatibility [21, 6]. Given two components G and S with respective axiomatic
specications (pre G ; post G ) and (pre S ; post S ), S is said to rene G (or to be
more specic than G; S w G, or G to subsume S), i
(pre G ) pre
holds. 1 Intuitively, (1) expresses the fact that S can be plugged into any place
where G is used because it has a wider domain and produces more specic results
than G. Using a relational view (i.e., specications are considered as sets of
valid (input, output)-pairs), [18] show that (1) denes a partial order which
induces a lattice-like structure on the set of all specications. This structure
is generally known as the renement lattice although it is strictly spoken no
lattice.
Turning the renement lattice into a navigation structure for library browsing
exposes, however, some unexpected problems. First of all, libraries do not
oer enough structure, i.e., the renement hierarchies they induce are too shal-
low. While this is a good thing from a design point of view|it simply says
that the library contains only little redundancy|it is a bad thing for browsing.
It can be overcome by the introduction of meta-nodes or abstractions. Such
specications do not represent real, existing components but just factor out
1 For the sake of brevity, we shall omit the quantication over the respective argument and
return variables and their identication via type compatibility predicates. For arguments ~x
and result variables ~y, the full form is
pre S (~x S post G (~x G ~y G ))).
similarities between some of them. As an example, consider the specication of
an abstract element lter: 2
lter some (l : list) r : list
pre l
post exists l1,
lter some species only that a singleton element is removed from the list (hence
it cannot be empty) but not which one. It is thus via (1) rened by both
components tail and lead:
list lead (l list
pre l pre l
post exists post exists
However, a nave introduction of meta-nodes yields unexpected results. If we
introduce another meta-node segment
segment (l list
pre true
post exists l1, l2 : list &
to capture the property that both components return continuous sublists of
their argument, this does not work: neither tail nor lead rene segment. The
reason for this at rst glance counterintuitive behavior is that segment is specied
as a total function (pre segment
both tail and lead are partial. And
while we can x this particular
aw by setting segment's precondition also to
becomes increasingly infeasible. If the library also contains
components which work on sorted lists only, we have to integrate this property
into the precondition, too. In eect, if we want an abstraction which captures
all segment-like components we have to adjoin all occurring preconditions con-
junctively. If, however, two of them are contradictory the result becomes false
and segment subsumes the entire library.
The solution to this dilemma is easy. While we can use renement to index
components with abstractions, we additionally need a second relation to model
the above situation. Since we are only interested in the eect of the calculation
(i.e., the postcondition post G ) we can drop pre G . We want post G to hold on
the appropriate domain only, hence
pre post G (2)
which is also known as conditional compatibility [6] or weak post match [21]
in deduction-based retrieval. We can then consider G as derived attribute or
feature [22] of holds whenever the execution of S was legal
(pre S holds) and terminated (post S holds.) In our example, segment is a feature
of tail and lead, as expected. It is easy to verify that features are inherited along
with the renement relation, i.e., if R renes S and G is a feature of S, then G
is a feature of R, too.
We use VDM-SL for our example specications. Here, ^ means concatenation of lists, [
the empty list, [i] a singleton list with item i. & reads as \such that".
A similar problem arises when we want to consider preconditions only. While
we can use the abstraction total
list
pre true
post true
to subsume all total functions, it is much harder to index partial functions
properly. The meta-node
requires non empty (l list
pre l
post true
correctly subsumes all functions which work on non-empty lists only but it
is not really appropriate: it also subsumes all total functions and is thus not
discriminative.
Hence, we need a third relation. Since we are now only interested in the
properties of the legal domains, we can drop the postconditions. But in contrast
to renement we want the domain of S now to be more restricted, hence
pre S ) pre G (3)
Again, G is a derived attribute of S|it is a requisite, S w r G|and using (3),
requires non empty now works as index. Requisites are also compatible with
renement but in contrast to features their absence is propagated. If R renes
S and G is no requisite of S, then G cannot be a requisite of R.
top
run lead tail
duplicate
rst
requires
non
empty
segment front
segment
works
on
empty
total lter
some
top
run w w f w f w w
lead
tail w w r w f w
duplicate rst w w r
Figure
1: Example index
Figure
1 shows the index for the examples in this paper. The components
are represented as rows, the attributes as columns; the symbols indicate which
relations have been used to index the components with the respective attributes.
We also see that the library is indeed shallow: each component indexes only
itself.
However, (1-3) are not the only sensible relations we could use. Instead
of indexing a component S with its requisites, we could also index S with all
requisites it does not require, i.e., with all its valid border conditions. In terms
of preconditions, this is formalized by
:pre G ) pre S (4)
and denoted by S w r G: G is not a requisite for S, or S also works on G.
Hence, we have of course tail 6w
r requires non empty but for a topological sort
function
top list
pre acyclic(l)
post top
we have top sort w
r requires non empty as expected. However, in principle, (4)
is not necessary. We can achieve the same eect using a modied version
works on empty (l list
pre
post true
of requires non empty and renement: top sort w works on empty. 3 However,
this hides the fact that requires non empty and works on empty are complementary
to each other.
We now use (1-3) 4 to compute an appropriately modied version of the
renement lattice but even this variant is not yet adequate for browsing. It
still lacks the single-focus property, i.e., it does not contain enough structure to
represent the focus by a single node. Consider for example lead and tail. Apart
from further renements, they are the only two components which have the
feature segment and are subsumed by lter some at the same time. 5 Yet there
is no meta-node to represent this and a user has to keep his focus on both distinguishing
properties to capture the conceptual similarity of the components.
The deeper reason for this is that even the modied renement lattice has
lattice-like properties only on the set of all possible specications, not on arbitrary
subsets or libraries. True lattices, on the other hand, have the single-focus
property by denition and we will show how to embed the renement lattice
into a true lattice using formal concept analysis.
4 Concept Lattices
4.1 Formal Concept Analysis
Formal concept analysis [30, 8, 3] applies lattice-theoretic methods to investigate
abstract relations between objects and their attributes. A concept lattice
is a structure with strong mathematical properties which reveals hidden structural
and hierarchical properties of the original relation. It can be computed
automatically from any given relation.
3 Notice that this relies on the fact that post works on empty = true|otherwise, the postcondition
part of (1) would not be valid.
4 We still need renement to represent all information of interest. E.g., we cannot split total
into a requisite and a feature which have both the value true because both of them index the
library.
5 By lter some we have to remove an element, but by segment we are not allowed to split
the list. Hence, there are only the two choices to remove the element at either end of the list.
Denition 1 A formal context is a triple (O; A; R) where O and A are sets
of objects and attributes, respectively, and R O A is an arbitrary relation.
Contexts can be imagined as cross tables where the rows are objects and
the columns are attributes. Hence, the index shown in Figure 1 can also be
considered as a formal context, provided that the dierent relations (i.e., w; w r
and w f ) are merged.
Denition 2 Let (O; A; R) be a context, O O and A A. The common
attributes of O are dened by (O) def
Rg, the
common objects of A by !(A) def
Rg.
Objects from a context share a set of common attributes and vice versa.
Concepts are pairs of objects and attributes which are synonymous and thus
characterize each other.
Denition 3 Let C be a context. is called a concept of C
A and
O and A (c) def
are called c's extent and intent,
respectively. The set of all concepts of C is denoted by B(C).
Concepts can be imagined as maximal rectangles (modulo permutation
of rows and columns) in the context table, e.g., (flead, tailg, fsegment, requires
non empty, lter someg). They are partially ordered by inclusion of extents
(and intents) such that a concept's extent includes the extent of all of its
subconcepts (and its intent includes the intent of all of its superconcepts).
Denition 4 Let C be a context, c 1
and
c 2 are ordered by the subconcept relation, c 1 c 2 , i O 1 O 2 . The structure
of B and is denoted by B(C).
The intent-part follows by duality. As an immediate consequence of the
preceding denitions we get that the strict order corresponds to strict inclusion
of extents and intents, i.e., c 1
and A 1
The following basic theorem of formal concept analysis states that the structure
induced by the concepts of a formal context and their ordering is always a
complete lattice. (Cf. Figure 2 for an example lattice.)
Theorem 5 ([30]) Let C be a context. Then B(C) is a complete lattice, the
concept lattice of C. Its inmum and supremum operation are given by
i2I
i2I
O
i2I
_
i2I
i2I
O i ));
i2I
Each attribute and object has a uniquely determined dening concept in
the lattice. This can be calculated from the attribute or object, respectively,
alone.
Denition 6 Let B(O; A; R) be a concept lattice. The dening concept of an
attribute a 2 A (object is the greatest (smallest) concept c such that
a 2 A (c) (o 2 O (c)) holds. It is denoted by (a) ((o)).
Theorem 7 ([3]) In any concept lattice we have
and
4.2 From Renement Lattices to Concept Lattices
[12] has shown that keyword-indexed components can be considered as a formal
context with the components as objects and the (informal) keywords as
attributes. We now lift this idea to formal specications.
Denition 8 Let be a formally specied library with components
L, requisites R, features F , and abstractions A. Its induced context is
dened by
Again, we consider the components as objects, and, of course, the keywords
are replaced by (the names of) the specications 6 but the context table is
slightly more complicated. To prevent dierent components from \collapsing"
into a single concept if the index is insu-cient, the component specications L
double as objects and attributes. The relations, however, remain original.
works on empty
(i)
segment
@
@
@
@
@
(ii)
requires non empty
total
front segment
(iii)
lter some
top sort
top sort
run
run
(iv)
lead
lead
tail
tail
duplicate rst
duplicate rst
Figure
2: Example lattice
We then calculate the concept lattice from this context. Figure 2 shows the
result for the example context. Each bullet represents a concept. The labels
6 Wlog. we assume that L; R; F , and A are pairwise disjoint.
over the bullet are the attributes dened at this concept. E.g., the concept (iii)
denes the attribute lter some. However, since attributes in this representation
are inherited downwards, its intent A is the set fsegment, requires non empty,
lter someg. None of the attributes are equivalent in the sense that they index
the same set of components. Hence, each concept introduces only one attribute.
The labels under the bullet denote the objects dened at this concept, e.g., lead
at (iv). Since none of the actual components subsumes an other, each concept
introduces at most one object and is atomic if it introduces an object at all.
The concept lattice is not an \extension" of the renement lattice: for two
attributes a 1 ; a 2 with (a 1 ) (a 2 ) it is possible to be completely unrelated,
i.e., neither of the relations (1-3) holds. However, for two reasons, it is an
adequate representation of the index. First, subconcepts preserve renement
of the original components. Second, a superconcept can be distinguished from
any subconcept by an attribute which is not valid for at least one component
in the extent of the superconcept but is valid for all components in the extent
of the subconcept. Formally:
Theorem 9 Let B(CL ) be the concept lattice of the context CL induced by a
library L and c 1
) such that either
1. 9m 2 O
2. 9 a 2 R a 2 A (c 1
9 a 2 F a 2 A (c 1
9 a 2 A a 2 A (c 1
This theorem, which follows from denitions 3 and 4, makes the concept
lattice already suitable for specication-based navigation. However, we can
impose even more structure if we double R and use w r in addition to dene
the induced context. Then, Theorem 9 holds appropriately and, additionally,
we get
Theorem 10 Let B(CL ) be the concept lattice of the context CL induced by
a library L. Then, for any two complementary requisites a; a 2 R we have
a and consequently (a) u
Hence, the dening concepts of two complementary requisites are complementary
to each other in the lattice. Moreover, their extents divide the entire
library into two partitions which is not the case for two arbitrary complementary
nodes in the lattice.
5 Navigation in Concept Lattices
[12] has also shown how concept lattices can be used as navigation structure for
interactive and incremental retrieval (i.e., browsing in our terminology). The
focus is represented by (the extent of) a concept. Narrowing the focus is a
downward movement in the lattice and is done in two steps:
1. The user selects an additional attribute. As a consequence of the lattice
structure, the system can support this selection by calculating all
attributes which actually narrow the focus but do not sweep it entirely.
It can thus prevent navigation into dead ends (i.e., an empty focus.)
2. The system calculates the new focus in the lattice as the meet (which
exists due to Theorem 5) of the actual focus and the dening concept of
the selected attribute (obtained by Theorem 7.)
Similarly, the focus can also be widened again by de-selecting an attribute. The
system then calculates the new focus using the join operation.
In the specication-based case, navigation works quite similar. We use the
derived properties (i.e., R; F , and A) as navigation attributes. Since the property
sets are pairwise disjoint, we can even split the set of navigation attributes
into three dimensions. These dimensions are not independent of each other
but can be selected independently because all interdependencies are contained
in the concepts of the lattice. If we use the modied context (i.e., double R
and use (1-4)), we get a fourth dimension. This is still independent but due to
Theorem 10, independent selection from R and
R is not benecial. Instead, we
can toggle between them, in addition to selection/de-selection.
Initially, all attributes are de-selected and the focus concept is >: the focus
is the entire library. Now, for an example, assume that we select segment. This
reduces the focus to O tailg. Further renement is possible
by attributes whose dening concepts have a strictly smaller but non-bottom
meet with the current focus concept. Thus, for (i), any navigation attribute
is possible. If we select requires non empty, the new focus concept is (i) u
the choice of requires non empty eliminates run from the focus.
Moreover, it leaves front segment as the only possible further renement.
This navigation style is attribute-based : the focus is essentially a function
of the selected attributes. Due to their dual nature, concept-lattices also allow
object-based navigation. Here, the user selects or de-selects a single component
and the system calculates the new focus similarly. However, selecting an additional
component widens the focus and is thus realized by the join operation.
While attribute-based navigation depends on the explicit and learned choice
of functional properties and thus is more suited for reuse purposes, object-based
navigation exposes implicit conceptual similarities of components: the intent
of the focus concept contains all properties which are common to all selected
components; its extent also contains all other components which share these
properties, even if they have not been selected explicitly. Hence, it is more
appropriate for library understanding and re-engineering.
6 Practical Aspects
We made a series of experiments to support the claim that browsing is more
practical in the specication-based case than retrieval. For these, we used a variant
of the list processing library which we also used in our retrieval experiments
[6]. It comprises 5 requisites, 31 features, and 86 components and abstractions.
All example specications in this paper are taken from that library.
6.1 Calculation of the Renement Lattice
Even if the calculation of the renement lattice is done in advance and is thus
not time-criticle in principle, it is not obvious that it is feasible at all. Two
questions are of main concern:
1. How high is the computational eort in practice?
2. How di-cult are the proof problems in practice? Are current theorem
provers powerful enough?
The answer to both questions depends on the number and structure of the
arising proof problems.
At rst glance, it seems that we have to check each requisite, feature, ab-
straction, and component against each other to calculate the modied rene-
ment lattice. However, in practice this can be optimized due to three obser-
vations. First, we do not need to compare the components and abstractions
pairwise but can use recursive comparison as in [9] because renement is tran-
sitive. Then, we do not need to check requisites and features against each other
but only against the components and abstractions. Finally, since the former are
compatible with renement, we can \sink them in" once we have the renement
lattice on the other nodes ready. In the worst case, the number of problems is
thus O(jR [ F j jA [ Lj 2 ). Nevertheless, still too many problems arise to be
handled manually. As in other software engineering applications, a fully automated
system is required which feeds and controls the prover. However, the
sheer numbers become a problem only because most of the proof problems (ap-
proximately 85% in our experiments) are logically invalid and thus not provable
at all. But theorem provers do usually not check for unprovability and are thus
stopped by time-out only. Hence, dedicated disproving lters are required.
Nevertheless, the computation is practically feasible. Using techniques from
[6] we generated the full set of more than 14.000 proof tasks (i.e., \ready-to-
run" versions of the problems which also contain appropriate axioms and prover
control information) and ltered out approximately 6.600 as unprovable. This
took approximately 8 hours on a Sun SparcStation 20. For simplicity, we did not
use the optimizations explained above. This would have reduced the original
number of tasks to about 11.000.
We then used the automated theorem prover SPASS [29] on a network of
PCs to check the surviving tasks. With a time-out of 60 seconds, SPASS was
able to solve 1.250 tasks. For the remaining 6.250 problems, we re-generated
the tasks, using a dierent subset of the axioms. After a third iteration, SPASS
had solved a total of 1.460 or almost 80% of the valid problems. This required
a total of approximately 340 hours runtime, or equivalently, a weekend of real
time.
6.2 Calculation of the Concept Lattice
Concept lattices can grow exponentially in the number of attributes and objects.
In practice, however, the worst case rarely occurs and a polynomial behavior is
usual. [12] contains more experimental evidence for this.
For our example library, the concept lattices from the full (i.e., manually
computed) and the approximated (i.e., computed using SPASS) renement lattices
contained 153 and 180 concepts, respectively. Their computation took
approximately a second and is thus negligible compared to the time required
for proving.
6.3 Navigation
During our experiments it became quickly obvious that neither the modied
renement lattice nor the concept lattice are suitable for presentation because
they are too big and complex. [12] makes the same observation and describes
a simple text-based interface which works on the attribute and object names
only. We are currently adapting this system. The navigation process itself,
however, is very fast: the system responds without noticeable delay, even for
much larger concept lattices than we are currently investigating.
6.4 Knowledge Acquisition
The formal specications of the library components and some initial abstractions
7 must be supplied. Once this seed is available, specication-based browsing
can already support further knowledge acquisition.
Consider for example a seed comprising lter some, segment, tail, lead, and
list
pre true
post exists l1 : list &
list
which computes the longest ordered initial subsegment (i.e., run) of a list. From
this seed, an initial concept lattice is calculated. Object-based navigation conrms
that both tail and lead have a common superconcept, which has the attributes
lter some and segment, and, as expected, no other objects. But it also
reveals that there is no concept which has the extent of lead and run only|
selecting both also causes tail to appear. To disambiguate tail, a feature
front segment (l list
pre true
post exists l1 : list &
must be introduced which factors out the common property of lead and run.
7 Related Work
Most work on applying specication-based techniques to software libraries examines
retrieval only. Relevant for browsing are the investigation of dierent
match relations [21] and their eect on software reuse [5, 6]. [22] introduced
features as indexes to speed up retrieval.
7 Initial requisites and features can be derived automatically by splitting of the supplied
specications. Any resulting indiscriminate attributes are merged into a single concept by
construction of the lattice.
[9] builds a two-tiered hierarchy from the library. The lower level is based on
a modied denition of subsumption which works modulo arbitrary user-dened
congruences on literals and is thus unsound in general. The upper level uses
a similarity metric derived from the normal forms of the specications. This
hierarchy is then visualized to support browsing. [18] only use subsumption to
build a hierarchical representation of a library and exploit that only to optimize
retrieval.
In programming language research, [15] and [16] apply formal methods to
the specication and verication of object-oriented class libraries. There, behavioral
subtyping corresponds to subsumption.
Concept lattices or Galois lattices have been developed as a means to structure
arbitrary observations. They have already been applied to various problems
in software engineering, e.g., inference of conguration structures [11] or
identication of modules [14, 28] and objects [27] in legacy programs. Their
application to software component libraries, however, seems to be obvious only
in retrospect, and there is only little related work. [7] also uses concept lattices
for navigation but presents the entire lattice to the user and oers only a subset
of all possible attributes for selection. As far as navigation is concerned, [12] is
thus most closely related to our own work. But there, object-based navigation,
which is instrumental in knowledge acquisition, is not supported.
Conclusions
Only specication-based methods can provide exact content-oriented access to
software components. Retrieval, however, still requires more deductive power
than current theorem provers and hardware can oer. Browsing can evade this
bottleneck by moving any time-consuming deduction into an o-line indexing
phase.
In this paper, we have shown that dierent match relations must be used to
index a library properly and how this index is turned into a navigation structure
using formal concept analysis. Experiments show that it is feasible to calculate
an approximation of the index which is accurate enough for browsing purposes,
using current theorem provers and hardware (e.g., SPASS on a small network
of PCs.) The computational eort, however, is still high.
The concept lattice reveals the implicit structure of a library as it follows
from the index. It can even indicate situations where a ner index is required.
Due to its dual nature, the lattice allows two complementary navigation styles
which are based either on attributes or on objects. Due to the lattice nature,
both navigation styles automatically have the single-focus property and refrain
the user from dead ends.
In our approach, theorem provers are used to derive formally dened properties
of components. For navigation, these formal denitions are still available
but not actually required|symbolic property names su-ce. However, since
informally dened and derived properties (e.g., reliability) are usually also represented
by symbolic names (e.g., trusthworty), concept-based browsing allows
a smooth integration of formal and informal attributes and thus refutes a conjecture
of [1] that formal and informal methods are incompatible.
Future work especially concerns scale-up. We expect the fraction of non-
theorems to grow further with increasing library size; dedicated disproving techniques
are thus one area of interest. Since the remaining tasks are homogeneous
in style, learning theorem provers [4, 2] can be expected to perform well on them.
Acknowledgments
Christian Lindig's work on concept-based retrieval also triggered this research; discussions
with him greatly improved my own understanding of formal concept analysis.
Comments by Christian, Jens Krinke, and Gregor Snelting improved the presentation
of this paper. Christoph Weidenbach did the actual theorem proving at the MPII.
--R
"Semantic-Based Software Retrieval to Support Rapid Prototyping"
"DISCOUNT: A Distributed and Learning Equational Prover"
Introduction to Lattices and Order.
"Learning Domain Knowledge to Improve Theorem Proving"
"Reuse by Contract"
"Deduction-Based Software Component Retrieval"
"Design of a Browsing Interface for Information Retrieval"
Formale Begri
"Using formal methods to construct a software component library"
"PARIS: A System for Reusing Partially Interpreted Schemas"
"On The Inference of Con guration Structures from Source Code"
"Concept-Based Component Retrieval"
"Assessing modular structure of legacy code based on mathematical concept analysis"
"A Behavioral Notion of Subtyping"
"Speci cation and Veri cation of Object-Oriented Programs Using Supertype Abstraction"
"An Information Retrieval Approach For Automatically Constructing Software Libraries"
"Storing and Retrieving Software Components: A Re nement-Based System"
"A Survey of Software Reuse Li- braries"
"Speci cation Matching of Software Components"
"Classi cation and Retrieval of Reusable Components Using Semantic Features"
"The Inscape Environment"
"Implementing Faceted Classi cation for Software Reuse"
"Speci cations as Search Keys for Software Libraries"
"NORA/HAMMR: Making Deduction- Based Software Component Retrieval Practical"
"Applying Concept Formation Methods to Object caton in Procedural Code"
"Identifying Modules Via Concept Analysis"
"Spass and Flotter version 0.42"
"Restructuring Lattice Theory: An Approach Based on Hierarchies of Concepts"
--TR
--CTR
Katsuro Inoue , Reishi Yokomori , Tetsuo Yamamoto , Makoto Matsushita , Shinji Kusumoto, Ranking Significance of Software Components Based on Use Relations, IEEE Transactions on Software Engineering, v.31 n.3, p.213-225, March 2005
Benedikt Kratz , Ralf Reussner , Willem-Jan van den Heuvel, Empirical Research Similarity Metrics For Software Component Interfaces, Journal of Integrated Design & Process Science, v.8 n.4, p.1-17, December 2004
Ge Li , Lu Zhang , Yan Li , Bing Xie , Weizhong Shao, Shortening retrieval sequences in browsing-based component retrieval using information entropy, Journal of Systems and Software, v.79 n.2, p.216-230, February 2006
Balaji Padmanabhan , Alexander Tuzhilin, On the Use of Optimization for Data Mining: Theoretical Interactions and eCRM Opportunities, Management Science, v.49 n.10, p.1327-1343, October | browsing;software reuse;formal specifications;retrieval;software component libraries |
592961 | QoS and Contention-Aware Multi-Resource Reservation. | To provide Quality of Service (QoS) guarantee in distributed services, it is necessary to reserve multiple computing and communication resources for each service session. Meanwhile, techniques have been available for the reservation and enforcement of various types of resources. Therefore, there is a need to create an integrated framework for coordinated multi-resource reservation. One challenge in creating such a framework is the complex relation between the end-to-end application-level QoS and the corresponding end-to-end resource requirement. Furthermore, the goals of (1) providing the best end-to-end QoS for each distributed service session and (2) increasing the overall reservation success rate of all service sessions are in conflict with each other. In this paper, we present a QoS and contention-aware framework of end-to-end multi-resource reservation for distributed services. The framework assumes a reservation-enabled environment, where each type of resource can be reserved. The framework consists of (1) a component-based QoS-Resource Model, (2) a runtime system architecture for coordinated reservation, and (3) a runtime algorithm for the computation of end-to-end multi-resource reservation plans. The algorithm provides a solution to alleviating the conflict between the QoS of an individual service session and the success rate of all service sessions. More specifically, for each service session, the algorithm computes an end-to-end reservation plan, such that it guarantees the highest possible end-to-end QoS level under the current end-to-end resource availability, and requires the lowest percentage of bottleneck resource(s) among all feasible reservation plans. Our simulation results show excellent performance of this algorithm. | Introduction
With the advances in resource reservation and scheduling
techniques, it is possible to provide end-to-end Quality
of Service (QoS) guarantees for distributed applications
and services. Various resource reservation and scheduling
frameworks have been proposed for individual system resources
such as CPU [7, 5], network bandwidth [11], disk
I/O bandwidth [9], and memory [7]. Now it becomes necessary
to create an environment where all these resources can
be reserved and scheduled in an integrated manner. In such
an environment, an end-to-end multi-resource reservation
will be performed for each client requesting a distributed
This work was supported by the National Science Foundation under
contract number 9870736, the Air Force Grant under contract number
F30602-97-2-0121, National Science Foundation Career Grant under contract
number NSF CCR 96-23867, NSF PACI grant under contract number
Infrastructure grant under contract
number NSF EIA 99-72884, NSF CISE Infrastructure grant under contract
number NSF CDA 96-24396, and NASA grant under contract number
NASA NAG 2-1250.
service, so that it can be guaranteed a certain level of end-
to-end QoS. A key question in creating such an environment
is: for a distributed service, how to determine the best end-
to-end QoS level and the corresponding multi-resource re-
quirement, under the constraint of current end-to-end multi-resource
availability.
One difficulty in answering the above question is: the relation
between an end-to-end QoS level and the corresponding
end-to-end resource requirement can be very complex.
Both the QoS level and the multi-resource requirement are
generally expressed as partial-ordered multi-dimensional
vectors. Every resource contributes to the end-to-end QoS,
and there may exist trade-offs between different resources
for the same end-to-end QoS level. In this case, the multi-resource
requirement can not be determined by looking at
these resources separately. Instead, it must be determined
by a coordinating entity placed on top of these resources.
Another difficulty is: even in a reservation-based envi-
ronment, there is still resource contention. Different applications
and services need to reserve from the same pool of
resources, inevitably causing the reservations for some ap-
plications/services to fail. In fact, the goals of (1) increasing
the overall success rate of multi-resource reservations for
different service requests, and (2) achieving the best end-to-
end QoS for each service request, are in conflict with each
other.
In this paper, we propose a solution to the difficulties
discussed above. We present a QoS and contention-
aware multi-resource reservation algorithm for distributed
and component-based services. The algorithm computes an
end-to-end multi-resource reservation plan, which achieves
the highest possible end-to-end QoS level under the constraint
of current resource availability. In the meantime, the
multi-resource reservation plan tends to cause low bottle-neck
resource contention among all feasible resource reservation
plans which lead to the same level of end-to-end
QoS.
The rest of the paper is organized as follows. In Section
2, we describe an enabling system architecture for multi-resource
reservation, and a QoS-resource model on which
our algorithm is based. In Section 3, we present the QoS
and contention aware multi-resource reservation algorithm.
In Section 4, we show the performance of this algorithm by
simulation. Finally, we conclude this paper in Section 6.
System Architecture and QoS-Resource
Model
2.1 Distributed and Component-Based Services
The distributed services studied in this paper are
component-based. With distributed object programming
techniques, a distributed service can be implemented as a
set of collaborating service components. A service component
is a functional unit participating in the service delivery.
For example, in a distributed video streaming service with
object tracking functionality, besides streaming a video to a
client, the service can also track an object of interest in the
video for the client. The client host will be able to playback
the video, and there will be a rectangle around the object being
tracked. In this service, the service components include
a VideoSender service component running on a video server,
an ObjectTracking service component running on a tracking
server, and a VideoPlayer service component running
on each client host. Each service component in a distributed
service is able to achieve one or more levels of service qual-
ity, depending on the amounts of resources reserved for this
component. The service quality achieved by each individual
service component finally leads to the end-to-end QoS
provided for the client.
2.2 An Architecture for Multi-Resource Reserva-
tion
In order to deploy such a distributed and component-based
service in a reservation-enabled environment, we introduce
an enabling system architecture. The architecture
is shown in Figure 1. It involves the following entities:
Resource Brokers (RBs), QoSProxies, and service compo-
nents. On each host in the environment, there is one or
more RBs managing individual resources, one for each type
of resource. A QoSProxy runs on each host, coordinating
the reservation activities of local RBs. The multi-resource
reservation algorithm will be executed by the QoSProxies
on the hosts involved in a distributed service.
A RB is responsible for the reservation and scheduling
of a resource 1 . The basic functions of a RB includes:
(1) reporting current resource availability, (2) making
1 For end-to-end network bandwidth, we look at it as one resource - a
pipe from the sender to the receiver. To be compatible with RSVP, we
assume that the network RB of the receiver is always responsible for initiating
an end-to-end bandwidth reservation.
component
Service
component
Service
QoSProxy
component
Service
QoSProxy
Translation function
QoSProxy
Figure
1. An Architecture for Multi-Resource
Reservation
and enforcing reservations, (3) releasing reservations,
and (4) reporting possible reservation degradations.
A QoSProxy is responsible for coordinating the reservation
activities of individual RBs on the same host. Its
basic functions include: (1) collecting resource availability
information from individual RBs, (2) executing
the multi-resource reservation algorithm and dispatching
the resultant reservation plan to individual RBs,
and (3) starting the service component on the same
host when the multi-resource reservation is completed.
A QoSProxy has to understand the relation between the
QoS levels and the corresponding resource requirements of
a service component, in order to compute a resource reservation
plan. However, this relation is highly application-
specific. For this reason, our architecture allows service developers
to provide translation functions as plug-ins for the
QoSProxies, as shown in Figure 1. Each translation function
the relation between the multiple QoS levels
and their resource requirements of a service component (the
formal definition of a translation function will be given in
Section 2.3). Therefore, a QoSProxy can call the translation
function during the execution of the multi-resource reservation
algorithm.
2.3 QoS-Resource Model
To express the relation between a service component's
QoS and its resource requirement, we adopt a QoS-resource
model. This model was originally proposed in [8]. Each service
component c is associated with both the input quality
Q in and the output quality Q out . Q in and Q out are both
represented as vectors of multiple QoS parameters. Their
QoS parameter sets may not be identical. For simplicity, we
assume that each parameter takes discrete values. There-
fore, Q in and Q out of each service component are enumer-
able. To compare two QoS vectors, they must have the same
parameter set. For two QoS vectors Q a and Q b , Q a Q b
holds if and only if for each QoS parameter, the corresponding
value of Q a is not larger than that of Q b .
For a service component c, the resource requirement
to achieve a certain output quality Q out , given an input
quality Q in , is computed by the translation function T c
R). The resource requirement is
formally represented as a resource requirement vector R.
Therefore, given a pair (Q in ; Q out ), we have:
The resource requirement vector
M) is the required amount of the mth re-
source. To compare two resource requirement vectors, they
must have the same set of resources. For two resource requirement
vectors R a and R b , R a R b holds if and only if
for each type of resource, the corresponding value of R a is
not larger than that of R b .
For a distributed service, the participating service components
organize themselves into a dependency graph. In
general, the dependency graph is a Direct Acyclic Graph
(DAG). Nodes of a dependency graph represent service
components. Edges of a dependency graph represent the dependencies
among the service components. Figure 2 shows
an example of the dependency graph. An edge from service
components c to c 0 indicates that the output of c is the input
of c 0 ; and the Q out of c is equivalent to the Q in of c 0 . In
addition, for a node with no in-coming edges (for example,
c 1 in
Figure
2), its Q in represents the original quality of the
source data involved in this service (for example, a stored
video clip or a live video source involved in a multimedia
service); for a node with no out-going edges (for example,
c 3 in
Figure
2), its Q out represents the resultant end-to-end
QoS.
out in out in
in
Figure
2. Example Dependency Graph of a
Distributed Service
3 Multi-Resource Reservation Algorithm
After introducing the system architecture for multi-resource
reservation and the QoS-resource model, we now
present the QoS and contention-aware multi-resource reservation
algorithm. Given a service request, the algorithm
computes an end-to-end resource reservation plan for service
components participating in this distributed service, so
that the best end-to-end QoS can be delivered to the client,
under the constraint of current end-to-end resource availability
observed by the client. The goals of our algorithm
involve both QoS-awareness and contention-awareness:
QoS-awareness Each service component may accept
multiple levels of Q in , and achieve multiple levels of
Q out . The algorithm must compute a resource reservation
plan by selecting appropriate levels of Q in and
Q out for each service component, so that it will lead
to the best possible end-to-end QoS for the client-side
service component according to the dependency graph.
Contention-awareness Resources may be shared by
other services and applications on a competitive basis.
Therefore, resource contention may exist during resource
reservation. The degree of resource contention
varies from time to time, from resource to resource. It
may affect the overall success rate of resource reservations
in the environment 2 . The algorithm must find a
resource reservation plan among all possible reservation
plans, such that it will reserve only the minimum
amount of bottleneck resource(s). Therefore, if every
multi-resource reservation is disciplined by this algo-
rithm, the overall resource contention in the environment
will be alleviated.
In the following subsections, we first define QoS-
Resource Graph (QRG) - a key data structure to study the
multi-resource reservation problem. We then study the special
case in which the dependency graph of a distributed
service is a chain. Finally, we extend the algorithm to deal
with the general case in which the dependency graph of a
distributed service is a DAG.
3.1 QoS-Resource Graph
We formally define the multi-resource reservation problem
using a QoS-Resource Graph (QRG). For a distributed
service, a QRG is generated for each service request at run-
time, based on the dependency graph of the requested ser-
vice. However, the definition of a QRG is different from
that of a dependency graph. A node in a QRG represents
a QoS level for the Q in or Q out of a service component
c. An edge in a QRG from a node Q in to a node Q out
represents the corresponding resource requirement vector
computed by the translation function T c . However, such
an edge exists if and only if the current resource availability
(also represented as a vector) is no less than the resource
requirement vector. Figure 3 shows an example QRG generated
from the dependency graph in Figure 2. The dotted
rectangles in the QRG represent the corresponding service
components in the dependency graph.
We assume that a multi-resource reservation is not successful, if at
least one resource can not be reserved.
For simplicity (without lowering the problem's complex-
ity) , we further assume that the original quality of the
source data associated with a service request has a single
QoS level. We define the node representing this QoS level
as the source node of a QRG (for example, Q a in Figure 3).000.50.60.50.6
Figure
3. Example QRG (weights of the edges
are shown)
For the client-side service component, whose Q out
nodes represent the end-to-end QoS levels (for example,
service component c 3 in Figure 3), we define its Q out nodes
as the sink nodes of a QRG (for example, Q l and Qm in
Figure
3). We also assume that the sink nodes (i.e. the
end-to-end QoS levels) can be ranked in a linear order. The
linear ranking can be determined by a client's preferences,
and may be subjective. For example, when two end-to-end
QoS levels are not comparable, the client requesting the
distributed service can arbitrate that the QoS level with a
smaller delay parameter value is better than the one with a
larger value.
We now define the weight of each edge in a QRG. The
weight will reflect the degree of resource contention caused
by the resource requirement represented by the edge. For
an edge from a node Q in to a node Q out , let R
[r req
M ] be the corresponding resource requirement
vector (computed by calling T c (Q in ; Q out )). On the
other hand, let
M ] be the current
resource availability vector (collected by querying the
RBs). We first define a contention index i to evaluate how
'competitive' it is to reserve r req
i amount of resource r i , under
the constraint of availability r avail
i . In this paper, we
choose a simple definition of as follows:
r req
r avail
Intuitively, the larger the percentage of a resource one tries
to reserve under the current availability constraint, the less
likely the reservation will succeed 3 . Now, we can further
3 In fact, there are other definitions for which also exhibit this charac-
define the weight for the edge as:
r req
r avail
For an edge from a node Q out to a node Q in , it only
represents their equivalence - the output quality of a service
component is the input quality of its dependent service com-
ponent. Therefore, the weight of such an edges is defined
as zero (shown in Figure 3).
3.2 Algorithm: the Chain Case
After defining the QRG, we are now ready to present the
algorithm. We first consider the special case in which the
dependency graph of a distributed service is a chain.
In a QRG, an edge with non-zero weight exists if and
only if the corresponding R req R avail , i.e. the reservation
of resources according to R req is feasible. There-
fore, we have the following observation: every path from
the source node to one of the sink nodes represents a feasible
end-to-end resource reservation plan - in other words, if
we reserve resources according to the resource requirement
vectors represented by the non-zero-weight edges on the
path, the end-to-end QoS represented by the sink node will
be guaranteed. Furthermore, the best achievable end-to-end
QoS under the current resource availability is represented
by the sink node which has the highest ranking among all
'reachable' sink nodes from the source node. For example,
in
Figure
3, if we assume that Q l ranks higher than Qm ,
then Q l is the best achievable end-to-end QoS level.
However, there are multiple paths from Q a to Q l , i.e.
there exist more than one feasible resource reservation plans
to achieve Q l . To minimize resource contention, our algorithm
will select a path such that the value of P is the
smallest among these paths - P is defined as:
(each edge e on path P )
e (4)
By definition, it is easy to see that P represents the
contention index of the bottleneck resource on the path (note
that the bottleneck resource on each path may be different).
To find a path from Q a to Q l whose bottleneck resource
has the smallest contention index, our algorithm finds the
shortest path from Q a to Q l , with operator '+' re-defined
as 'max'. This is done by running Dijkstra's algorithm on
the QRG. Figure 4 shows such a shortest path (shown by
the thicker edges). The value inside each node is generated
during the execution of Dijkstra's algorithm.
The computation complexity of the reservation algorithm
in the chain case is O(KQ 2 ). K is the number of service
components in the dependency graph of a distributed
teristics. Fortunately, it is easy for our algorithm to adopt a different (and
more accurate) definition in the future.
f
Qm000.50.60.50.6
Figure
4. The Shortest Path from Q a to Q l
(representing the end-to-end reservation plan
computed by the algorithm)
service. Q is the maximum number of Q out levels (nodes)
among the service components. For example, in Figure 4,
(which is the number of Q out nodes of
Fortunately, K and Q usually have fairly small values
in practice (for example, K, Q 10). Therefore, scalability
is not a major concern for the multi-resource reservation
algorithm.
The number of Q out levels of a service component is set
by the service developer in the translating function of the
service component. Our experience shows that this number
can be effectively controlled by limiting the number of
possible values for each QoS parameter.
3.3 Algorithm: the DAG Case
We now consider the more general case in which the dependency
graph of a distributed service is a DAG. For a
DAG dependency graph, we first extend the definition of a
service component's QoS levels:
For a service component with more than one out-going
edge, its Q out will become the Q in of each service
component on the other end of the out-going edge (as
shown in Figure 5). We call the service component
with more than one out-going edges a fan-out service
component. For example, c 2 in Figure 5 is a fan-out
service component.
For a service component with more than one in-coming
edge, its Q in is defined as the concatenation
of the Q out of service components on the other end of
the in-coming edges. We call the service component
with more than one in-coming edges a fan-in service
component. For example, c 5 in Figure 5 is a fan-in
service component.
An example QRG generated from this DAG dependency
graph is shown in Figure 6. A feasible end-to-end reser-
out in C3
out
out in
out in
out
q 33, . ]
[q q . ]
out
41, 43,
q 33, .
C1in
Figure
5. Example DAG Dependency Graph of
a Distributed Service
vation plan is now represented by an embedded graph in
the QRG such that: (1) for each service component, among
the edges from a Q in node to a Q out node, there is one -
and only one edge that belongs to the embedded graph; (2)
within the embedded graph, the sink node (representing the
end-to-end QoS level achieved by this reservation plan) is
reachable from each node in the embedded graph. The goal
of our reservation algorithm is to compute a feasible end-to-
reservation plan, represented by an embedded graph G,
such that (1) the sink node in the embedded graph has the
highest QoS ranking; and (2) the value of G is the smallest
both among all feasible end-to-end reservation plans. G
is defined as follows:
(each edge e in G) e (5)00.530.30.20.60000000.60.80.70.40.450.5Q a
c cc
c
c
Figure
6. Example QRG Based on a DAG Dependency
Graph
It can be shown that such a problem is NP-complete.
Therefore, we focus on providing an efficient and effective
heuristics to compute an end-to-end reservation plan that
achieves the best end-to-end QoS, while trying to maintain
a low value of G . Our heuristics is based on the following
two-pass procedure on the QRG.
Pass I on the QRG is similar to the reservation algorithm
in the case of chain dependency graph. It also runs
Dijkstra's algorithm to explore the 'shortest path' from the
source node to the sink nodes of the QRG. As an example,
Figure
7 shows the result of pass I on the QRG in Figure
6. Notice that when generating the value in a Q in node of
a fan-in service component (for example, node Q r of c 5 ),
we set the value as the maximum of the values in Q out
nodes on the other end of the in-coming edges (for exam-
ple, nodes Q n and Q p ). This is different from Dijkstra's
algorithm. By our definition, Q r is the concatenation of Q n
and Q p . Therefore, the resource contention to reach Q r is
the maximum of resource contention to reach Q n and Q p ,
respectively.
s
c
c
c
Figure
7. Running the Heuristics: Result of
Pass I
Pass II on the QRG proceeds in the reversed direction of
pass I. Starting from the reachable sink node with the highest
QoS ranking (for example, Q v ), we backtrack the edges
toward the source node, according to the result of pass I.
This is to determine the embedded graph that represents the
resultant end-to-end reservation plan. However, the back-tracking
may encounter the following problem: when arriving
at a fan-out service component, the backtracked edges
do not converge at the same Q out node. For example, in
Figure
7, the backtracked edges (the thicker ones in the
Figure) lead to different Q out nodes Q h and Q i . In our
heuristics, we use the following method to resolve this non-convergence
locally:
For the service components dependent on the fan-out
component (for example, c 3 and c 4 ), fix their Q out nodes
that have been backtracked (for example, nodes Q n and
select such a Q out node of the fan-out service
component: it causes the lowest resource contention to
reach the fixed Q out nodes of the dependent service com-
ponents. For example, in Figure 7, Q i will be selected (in-
stead of Q h ), because for Q i to reach Q n and Q p , the resource
contention is 0.6, while for Q h to reach Q n and Q p ,
the resource contention is 0.7.
e
c c
c
c
c
Figure
8. Running the Heuristics: the Embedded
Graph Representing the Resultant Reservation
Plan
By using this two-pass heuristics, an end-to-end reservation
plan can be computed for the QRG in Figure 7. The
embedded graph representing the reservation plan is shown
in
Figure
8. The limitation of this heuristics is: for a sink
node of the QRG which is reachable in pass I, the heuristics
may not necessarily find a feasible reservation plan in
pass II to guarantee the end-to-end QoS level represented
by the sink node. Furthermore, due to the local (instead
of global) nature of the non-convergence resolution in pass
II, the reservation plan computed by the heuristics may not
incur the lowest bottleneck resource contention among all
feasible reservation plans.
4 Simulation Results
In this section, we evaluate the success rate of multi-resource
reservations achieved by the proposed reservation
algorithm. The results in this section are initial, and obtained
by simulation. We simulate a simple scenario: there
is a distributed service which involves three service components
In our simulated environment, c 1
runs on one host. c 2 runs on another host. c 3 is the client-side
service component, and runs on each client host. Resource
contention exists on the hosts where c 1 and c 2 ex-
ecute. In addition, we introduce background computation
task on each client host, so that resource contention also
exists between the execution of c 3 and the background task.
For simplicity, we assume that each service component
only requires one type of resource. The QoS levels and the
corresponding resource requirements are shown in figure 9.
Each value in the brackets denotes the required amount of
resource for the corresponding (Q in ; Q out ) pair. We also
assume that Q l has a higher QoS ranking than Qm . Notice
that
Figure
9 is not a QRG.
The total amount of resource on the host where c 1 executes
is 800 units. The total amount of resource on the
host where c 2 executes is 400 units. The total amount of
resource on each client host where c 3 executes is 1 unit. We
assume that at the beginning of the simulation, all these resources
are free. We also assume that for each client host,
right before it makes a service request, a background computation
task will begin to run with a 0.5 probability, and
the amount of resource it consumes is uniformly distributed
between 0.25 and 0.75 unit.
[1.5]
[2.5]
[1.0]
[0.3]
[0.3]
[x]: Amount of Resource Required
Figure
9. QoS Levels and Resource Requirements
of the Simulated Service Components
In the first experiment, we simulate multi-resource reservations
made for 16000 service requests spreading over 400
minutes. The duration (i.e. resource holding time) of each
service session varies uniformly between 5 and 50 minutes.
Service requests from different clients arrive at an average
rate of 40 requests per minute. The success rates of multi-resource
reservations are shown in Figure 10 - each point
represents the success rate in a 5-minute interval. Here,
we compare our algorithm with a random algorithm, which
randomly selects a feasible multi-resource reservation plan
represented by a path from Q a to Q l in Figure 9. During
the 400-minute period, the overall success rate using our algorithm
is 96.33%, while the random algorithm achieves an
overall success rate of only 78.06%.
In the second experiment, we simulate different average
request arrival rates for the same distributed service. For10305070900 50 100 150 200 250 300 350 400
Success
Rate
of
Multi-Resource
Reservations
(%,
Time (minutes)
'Our Algorithm'
'Random Algorithm'
Figure
10. Multi-Resource Reservation Success
Rate over the 400-Minute Period
each average arrival rate, we measure the overall success
rate over a 400 minute period using our algorithm and using
the random algorithm. Figure 11 shows the overall multi-resource
reservation success rate under different service request
arrival rates. The results show that our algorithm constantly
achieves higher overall success rate than the random
algorithm.
5 Related Work
The problem of multi-resource reservation has been addressed
from different angles. In [3] and [4], a resource
co-allocation architecture and its mechanisms for alloca-
tion, configuration, monitoring, and control are presented.
It is suggested that resource co-allocation should be an integral
part of the resource management architecture for Grid
environments. In addition, an advance reservation mechanism
is also proposed. One of our next steps is to extend
our algorithm to accommodate advance reservation. In
[6], the problem of apportioning multiple finite resources to
satisfy the QoS needs of multiple applications along multiple
QoS dimensions is studied. However, their solution
is based on a static set of applications to be executed at
the same time, and they do not consider the dynamic arrival
and completion of applications. Therefore, their solution
is not contention-aware. In the Darwin Project [1], a
hierarchical service and resource brokerage architecture is
Success
Rate
of
Multi-Resource
Reservations
(%,
overminutes)
Arrival Rate of Multi-Resource Reservation Requests (requests/min)
'Our Algorithm'
'Random Algorithm'
Figure
11. Multi-Resource Reservation Success
Rate under Different Request Arrival
Rates
introduced. In order to compose value-added services, allocation
of multiple resources is needed. The signaling protocol
during multi-resource allocation is the Beagle signaling
protocol [2]. However, this protocol is not contention-aware
either. In our earlier work of Qualman system [7], different
QoS-aware resource brokers are proposed. They are responsible
for the reservation and enforcement of CPU, network
bandwidth, and memory resources, respectively. However,
there is no coordination among these resource brokers, and
no algorithm is proposed to compute multi-resource reservation
plans to guarantee end-to-end application level QoS.
Finally, in [10], we study the multi-resource reservation
problem only in the case of chain dependency graph. In this
paper, we extend our solution to deal with the more general
case of DAG dependency graph.
6 Conclusion
In a reservation-based environment where every type
of resource can be reserved, we need system support to
compute end-to-end multi-resource reservation plans and to
make corresponding reservations in an integrated and systematic
manner. In this paper, we first propose a system
architecture that enables such an integrated multi-resource
reservation for distributed and component-based services.
We then present a QoS and contention-aware multi-resource
reservation algorithm that computes a reservation plan for
each distributed service request, such that (1) it achieves
the highest level of end-to-end QoS under the constraint of
current resource availability, and (2) it causes the least bottleneck
resource contention in the case of chain dependency
while it tends to cause low bottleneck resource contention
in the case of DAG dependency graph. Our future
work includes the extension to support advance reservation,
and the study of reservation fairness among service requests
with highly heterogeneous resource requirements and service
durations.
--R
Resource management for value-added customizable network service
A signaling protocol for structured resource allocation.
Resource co-allocation in computational grids
A distributed resource management architecture that supports advance reservations and co-allocation
CPU reservation and time constraints: efficient
A scalable solution to the multi-resource QoS problem
A disk scheduling framework for next generation operating systems.
Multimedia service configuration and reservation in heterogeneous environ- ments
RSVP: A new resource reservation protocol.
--TR
--CTR
Ashish M. Mehta , Jay Smith , H. J. Siegel , Anthony A. Maciejewski , Arun Jayaseelan , Bin Ye, Dynamic resource allocation heuristics that manage tradeoff between makespan and robustness, The Journal of Supercomputing, v.42 n.1, p.33-58, October 2007
Jong-Kook Kim , Sameer Shivle , Howard Jay Siegel , Anthony A. Maciejewski , Tracy D. Braun , Myron Schneider , Sonja Tideman , Ramakrishna Chitta , Raheleh B. Dilmaghani , Rohit Joshi , Aditya Kaul , Ashish Sharma , Siddhartha Sripada , Praveen Vangari , Siva Sankar Yellampalli, Dynamically mapping tasks with priorities and multiple deadlines in a heterogeneous environment, Journal of Parallel and Distributed Computing, v.67 n.2, p.154-169, February, 2007 | resource contention;distributed service;resource reservation |
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