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LECTURE SLIDES ON
CONVEX ANALYSIS AND OPTIMIZATION
BASED ON 6.253 CLASS LECTURES AT THE
MASS. INSTITUTE OF TECHNOLOGY
CAMBRIDGE, MASS
SPRING 2012
BY DIMITRI P. BERTSEKAS
http://web.mit.edu/dimitrib/www/home.html
Based on the book
“Convex Optimization Theory,” Athena Scientific,
2009, including the on-line Chapter 6 and ... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
⇤
⇤
x
|
•
•
which f and C are convex
−
−
They are continuous problems
They are nice, and have beautiful and intu-
itive structure
However, convexity permeates all of optimiza-
•
tion, including discrete problems
Principal vehicle for continuous-discrete con-
•
nection is duality:
−
−
The dual problem of a discrete prob... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
(y)
Primal Description
Values f (x)
Dual Description
Crossing points f ∗(y)
7 FENCHEL PRIMAL AND DUAL PROBLEMS
f 1 (y)
f 1 (y) +f 2 (
y)
f 2 (
y)
f1(x)
Slope y
f2(x)
x
x
Primal Problem Description
Vertical Distances
Dual Problem Description
Crossing Point Dierentials
Primal problem:
f1(x) +f 2(x)
min
x
⇤
⌅
Dual problem... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
%
Max Crossing
%#0()1*22&'3(,*&'-(4/
Point q
0!
0
u
7
0
!
u
7
Max Crossing
%#0()1*22&'3(,*&'-(4/
Point q
(b)
"5$
6
M
%
9
M
%
(a)
"#$
w
.
Min Common
%&'()*++*'(,*&'-(./
Point w
Max Crossing
Point q
%#0()1*22&'3(,*&'-(4/
0
!
(c)
"8$
u
7
All of duality theory and all of (convex/concave)
•
minimax theory can be developed/e... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
).
13 MODERN VIEW OF CONVEX OPTIMIZATION
•
•
Traditional view: Pre 1990s
LPs are solved by simplex method
NLPs are solved by gradient/Newton meth-
ods
Convex programs are special cases of NLPs
−
−
−
LP
CONVEX
NLP
Simplex
Duality
Gradient/Newton
Modern view: Post 1990s
−
−
−
LPs are often solved by nonsimplex/convex
met... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
, 2009, including the on-line
Chapter 6 and supplementary material at
http://www.athenasc.com/convexduality.html
Additional book references:
−
−
−
Rockafellar, “Convex Analysis,” 1970.
Boyd and Vanderbergue, “Convex Optimiza-
tion,” Cambridge U. Press, 2004. (On-line at
http://www.stanford.edu/~boyd/cvxbook/)
Bertseka... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
You can do your term paper on an applica-
tion area
18 A NOTE ON THESE SLIDES
These slides are a teaching aid, not a text
Don’t expect a rigorous mathematical develop-
•
•
ment
The statements of theorems are fairly precise,
•
but the proofs are not
Many proofs have been omitted or greatly ab-
•
breviated
Figures are me... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
convex if
�
C,
αx + (1
α)y
Operations that preserve convexity
−
⌘
⌘
x, y
C,
[0, 1]
α
⌘
−
Intersection, scalar multiplication, vector sum,
closure, interior, linear transformations
Special convex sets:
−
−
Polyhedral sets: Nonempty sets of the form
x
{
|
a�jx
⌥
bj, j = 1, . . . , r
}
(always convex, closed, not alwa... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
-VALUED FUNCTIONS
f (x)
!"#$
Epigraph
01.23415
f (x)
!"#$
Epigraph
01.23415
dom(f )
%&'()#*!+',-.&'
Convex function
x
#
x
#
dom(f )
/&',&'()#*!+',-.&'
Nonconvex function
The epigraph of a function f : X
•
the subset of
n+1 given by
�
[
] is
,
⇣
−⇣
◆→
epi(f ) =
(x, w)
x
|
⌘
X, w
⌘ �
, f (x)
w
⌥
The effective domain of f ... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
)
x
|
�
⇥
(ii)
•
(xk, wk)
✏
→
(iii): Let
(xk, wk)
(x, w). Then f (xk)
⌦
wk, and
epi(f ) with
⇤
⌅
⌥
w so (x, w)
epi(f )
x. Then
⌘
V⇥ and xk →
→
(x, ⇤), so (x, ⇤)
f (x)
lim inf f (xk)
k
⌥
✏
⌘
(iii)
•
(xk, ⇤)
epi(f ), and x
⌥
⌃
(i): Let
xk} ⌦
{
epi(f ) and (xk, ⇤)
V⇥.
⌘
(ii): If xk →
•
xk}K →
consider subsequence
- contr... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
ˆ have the same epigraph, and
both are not closed. But f is lower-semicon-
tinuous while fˆ is not.
Note that:
•
−
If f is lower semicontinuous at all x
it is not necessarily closed
If f is closed, dom(f ) is not necessarily closed
dom(f ),
⌘
−
Proposition: Let f : X
] be a func-
•
tion. If dom(f ) is closed and f is l... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
= f (Ax)
] given by
,
⇣
where A is an m
⇤
if f is convex (respectively, closed).
n matrix is convex (or closed)
(c) Consider fi :
⇣
is any index set. The function g :
given by
−⇣
→
�
(
,
n
], i
⌘
n
I, where I
]
(
,
−⇣
⇣
→
�
g(x) = sup fi(x)
i
⌦
is convex (or closed) if the fi are convex (respec-
tively, closed).
I
28◆
... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
−
x)
f (x)
−
⇥
f (x) + (z
f (x)
x)⇥
−
x
x + (z
x)
z
−
(b)
31 OPTIMALITY CONDITION
n
Let C be a nonempty convex subset of
•
let f :
an open set that contains C. Then a vector x⇤ ⌘
minimizes f over C if and only if
n and
be convex and differentiable over
C
→ �
�
�
f (x⇤)�(x
x⇤)
0,
≥
−
∇
x
⌘
C.
Proof: If the condition ho... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
2
x
�
C (called the projection of z on
−
�
•
⌘ �
(a) For every z
imum of
over all x
C).
⌘
(b) x⇤ is the projection of z if and only if
(x
−
x⇤)�(z
x⇤)
0,
⌥
−
x
⌘
C
Proof: (a) f is strictly convex and has compact
level sets.
(b) This is just the necessary and su⌅cient opti-
mality condition
f (x⇤)�(x
x⇤)
0,
≥
−
∇
x
⌘
... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
From the preceding result, f is convex.
f (x),
x)�∇
x, y
C
⌘
(b) Similar to (a), we have f (y) > f (x) + (y
f (x) for all x, y
x)�∇
the preceding result.
−
C with x = y, and we use
⌘
(c) By contradiction ... similar.
34◆
✓
CONVEX AND AFFINE HULLS
Given a set X
n:
⌦ �
•
•
vector of the form m
0, and m
i=1 αi = 1.
�
A... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
) Every x = 0 in cone(X) can be represented
as a positive combination of vectors x1, . . . , xm
from X that are linearly independent (so
m
n).
⌥
(b) Every x
/ X that belongs to conv(X) can
⌘
be represented as a convex combination of
vectors x1, . . . , xm from X with m
n + 1.
⌥
36✓
PROOF OF CARATHEODORY’S THEOREM
(a) ... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
�
xi X
�
⌘
dory, a sequence
n+1 αkxk
i
i
i=1
n+1
i=1 αk
, and
in conv(X) can
, where for all k and
i = 1. Since the
�
(αk
1, . . . , αk
n+1, x1, . . . , xn+1)
k
k
is bounded, it has a limit point
⇤
⌅
(α1, . . . , αn+1, x1, . . . , xn+1) ,
⇤
which must satisfy
X for all i.
xi
The vector
⌘
⌅
n+1
i=1 α = 1, and
i
αi
0,
≥
... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
case where x
C with xk →
xk} ⌦
•
•
{
C: See the figure.
/ C: Take sequence
⌘
x. Argue as in the figure.
40 ADDITIONAL MAJOR RESULTS
•
Let C be a nonempty convex set.
(a) ri(C) is a nonempty convex set, and has the
same a⌅ne hull as C.
(b) Prolongation Lemma: x
ri(C) if and
only if every line segment in C having x
as one ... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
)
≥
αf (x) + (1
−
α)f (x),
and since f (x) > f (x⇤), we must have f (x⇤) >
f (x) - a contradiction. Q.E.D.
Corollary: A nonconstant linear function can-
•
not attain a minimum at an interior point of a
convex set.
42◆
CALCULUS OF REL. INTERIORS: SUMMARY
The ri(C) and cl(C) of a convex set C “differ
•
very little.”
−
−
... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
(C) .
⌘
�
⇥
x
x
C
The proof of ri(C) = ri cl(C)
is similar.
�
⇥
44 LINEAR TRANSFORMATIONS
Let C be a nonempty convex subset of
n matrix.
•
let A be an m
n and
�
⇤
(a) We have A
·
ri(C) = ri(A
C).
(b) We have A
cl(A
·
if C is bounded, then A
cl(C)
⌦
C). Furthermore,
·
C).
cl(C) = cl(A
·
·
·
Proof: (a) Intuition: Spheres... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
)
⌦
If one of C1 and C2 is bounded, then
cl(C1) + cl(C2) = cl(C1 + C2)
(b) We have
ri(C1)
ri(C2)
⌫
ri(C1
⌫
⌦
C2), cl(C1
C2)
⌫
⌦
cl(C1)
⌫
cl(C2)
If ri(C1)
⌫
ri(C2) = Ø, then
ri(C1
⌫
C2) = ri(C1)
⌫
ri(C2), cl(C1
C2) = cl(C1)
cl(C2)
⌫
⌫
Proof of (a): C1 + C2 is the result of the linear
transformation (x1, x2)
x1 + x2.
◆→
... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
ri(C)
,
�
. For every x
⌘
Mx
⌫
ri(C) = ri(Mx
C) = (x, y)
y
|
⌘
ri(Cx) .
⌫
Combine the preceding two equations.
⇤
Q.E.D.⌅
47✓
CONTINUITY OF CONVEX FUNCTIONS
If f :
n
�
→ �
•
e4 = (
is convex, then it is continuous.
1, 1)
yk
e1 = (1, 1)
xk
xk+1
0
e3 = (
1,
1)
zk
e2 = (1,
1)
Proof: We will show that f is continuous at 0.... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
] is
,
⇣
−⇣
n
� →
[
,
⇣
−⇣
epi(cl f ) = cl epi(f )
The convex closure of f is� the function
⇥
epi(clˇ f ) = cl conv epi(f )
clˇ f with
Proposition: For any�f : X�
◆→
[
⇥⇥,
−⇣
]
⇣
•
•
inf f (x) = inf (cl f )(x) = inf (clˇ f )(x).
x
⌦
⌦�
⌦�
X
x
x
n
n
Also, any vector that attains the infimum of f over
X also attains the i... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
finitely along d, we never cross the
relative boundary of C to points outside C:
x + αd C,
⌘
x C,
⌘
α
≥
0
Recession Cone RC
C
d
x + d
0
x
Recession cone of C (denoted by RC): The set
•
of all directions of recession.
RC is a cone containing the origin.
•
51 RECESSION CONE THEOREM
•
Let C be a nonempty closed convex ... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
., let
zk = x + kd,
(z
dk = k −
zk −
�
x)
x �
�
d
�
We have
�
+
=
d
d
dk
d
� �
x
zk −
�
x
zk
� − � � �
d and x + dk →
so dk →
and closedness of C to conclude that x + d
x
x ⌅
�
x + d. Use the convexity
x
zk
− �
�
x ⌅
zk
� − �
x
−
zk
� − �
x
−
zk −
x
x
1,
�
,
C.
⌘
0,
53✓
LINEALITY SPACE
The lineality space of a convex ... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
��
⇤
⇤
⌅
These are the directions of recession of f .
epi(f)
!
“Slice” {(x,!) | f(x) " !}
0
Recession
Cone of f
Level Set V! = {x | f(x) " !}
55 RECESSION CONE OF LEVEL SETS
Proposition: Let f :
convex function
(
n
] be a closed
⇣
�
and consider the level sets
−⇣
→
,
f (x)
⇤ , where ⇤ is a scalar. Then:
•
proper
V⇥ = x... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
(a)-(d).
•
This behavior is independent of the starting
dom(f ).
•
point x, as long as x
⌘
57 RECESSION CONE OF A CONVEX FUNCTION
For a closed proper convex function f :
•
,
(
−⇣
level sets V⇥ = x
|
cession cone of f , and is denoted by Rf .
], the (common) recession cone of the nonempty
⇤ , ⇤
, is the re-
f (x)
⌘ �
⇣
... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
+ αd)
α
−
f (x)
Thus rf (d) is the “asymptotic slope” of f in the
•
direction d. In fact,
rf (d) = lim
α
⌃ ∇
f (x + αd)�d,
x, d
n
⌘ �
if f is differentiable.
Calculus of recession functions:
•
rf1+
···
+fm(d) =r f1(d) +
· · ·
+ rfm (d),
rsupi
I fi(d) = sup rfi(d)
i I
⌦
2
59◆
LOCAL AND GLOBAL MINIMA
Consider minimizin... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
•
pact if the level sets of f are compact.
(An Extension of the) Weierstrass’ Theo-
•
rem: The set of minima of f over X is nonempty
and compact if X is closed, f is lower semicontin-
uous over X, and one of the following conditions
holds:
(1) X is bounded.
(2) Some set
x
and bounded.
⌘
⇤
(3) For every sequence
, we ha... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
the
It follows
recession cone of X ⇤, when X ⇤ = Ø.
that X ⇤ is nonempty and compact if and only if
RX
. Q.E.D.
Rf =
⌫
0
⌫
{
}
62◆
✓
✓
✓
EXISTENCE OF SOLUTION, SUM OF FNS
Let fi :
n
(
,
], i = 1, . . . , m, be closed
⇣
•
proper convex functions such that the function
−⇣
→
�
f = f1 +
+ fm
· · ·
is proper. Assume that a... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
follows:
Does a function f :
minimum over a set X?
�
n
(
,
⇣
−⇣
→
] attain a
This is true if and only if
Intersection of nonempty x
X
|
⌘
f (x)
⇤k
⌥
is nonempty.
⇤
⌅
Level Sets of f
X
Optimal
Solution
65◆
ROLE OF CLOSED SET INTERSECTIONS II
If C is closed and A is a matrix,
closed?
is A C
Nk
x
Ck
C
y
yk+1
yk
AC
•
If C... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
poly-
•
hedral. To be shown later by a more refined method.
67 ROLE OF CLOSED SET INTERSECTIONS III
Let F :
n+m
(
,
] be a closed proper
⇣
•
convex function, and consider
−⇣
→
�
f (x) = inf F (x, z)
⌦�
m
z
If F (x, z) is closed, is f (x) closed?
Critical question in duality theory.
−
1st fact: If F is convex, then f is ... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
z) =
⌦�
z
✏
if x >0,
if x = 0,
if x <0,
0
1
⇣
is not closed.
69 PARTIAL MINIMIZATION THEOREM
Let F :
(
•
convex function, and consider f (x) = inf z
−⇣
→
⇣
�
n+m
,
] be a closed proper
m F (x, z).
⌦�
Every set intersection theorem yields a closed-
•
ness result. The simplest case is the following:
Preservation of Close... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
sets in the sequence
Once the appropriately refined set intersection
•
theory is developed, sharper results relating to the
three questions can be obtained
The remaining slides up to hyperplanes sum-
•
marize this development as an aid for self-study
using Sections 1.4.2, 1.4.3, and Sections 3.2, 3.3
71 ASYMPTOTIC SEQUE... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
ractive Set Sequence
%&'()*+,&-+./*
%0'(123,*+,&-+./*
(b) Nonretractive Set Sequence
A closed halfspace (viewed as a sequence with
•
identical components) is retractive.
Intersections and Cartesian products of retrac-
•
tive set sequences are retractive.
A polyhedral set is retractive. Also the vec-
•
tor sum of a conv... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
k)
{
⌫
corresponding to d
Proof: The set of common directions of recession
of C k is RX
R. For any asymptotic sequence
xk}
(1) xk −
(2) xk −
So C k
{
}
X (because X is retractive)
Ck (because d
is retractive.
⌫
L)
RX
R:
⌘
⌘
⌘
⌘
d
d
75 NEED TO ASSUME THAT X IS RETRACTIVE
X
X
Ck+1
Ck
Ck+1
Ck
Consider
⌫ k=0 C k, with C k ... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
�
Q.E.D.
77 CLOSURE UNDER LINEAR TRANSFORMATION
Let C be a nonempty closed convex, and let A
•
be a matrix with nullspace N (A).
(a) A C is closed if RC
N (A)
LC.
⌫
⌦
C) is closed if X is a retractive set
(b) A(X
and
⌫
y
�
⌅
Proof: (Outline) Let
We prove
RX
RC
N (A)
LC,
⌫
⌫
⌦
A C with yk →
{
⌫ k=0Ck = Ø, where Ck = C
⌫... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
, m. Then C1 +
set.
· · ·
· · ·
�
⌘
Special Case: If C1 and
−
C2 are closed convex
RC2 =
0
.
•
sets, then C1
C2 is closed if RC1 ⌫
−
}
Proof: The Cartesian product C = C1
Cm
is closed convex, and its recession cone is RC =
RCm. Let A be defined by
RC1 ⇤ · · · ⇤
⇤ · · · ⇤
{
A(x1, . . . , xm) = x1 +
+ xm
· · ·
A C = C1 +
... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
2
⌘
C2,
or a�x2
b
⌥
⌥
a�x1,
x1
⌘
C1,
x2
⌘
C2
If x belongs to the closure of a set C, a hyper-
•
plane that separates C and the singleton set
is said be supporting C at x.
x
}
{
81 VISUALIZATION
Separating and supporting hyperplanes:
•
a
C2
C1
(a)
a
C
x
(b)
A separating
x
a�x = b
•
|
C1 and C2 is called strictly s... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
.E.D.
→ ⇣
83 SEPARATING HYPERPLANE THEOREM
n.
Let C1 and C2 be two nonempty convex subsets
•
If C1 and C2 are disjoint, there exists a
of
hyperplane that separates them, i.e., there exists
a vector a = 0 such that
�
a�x1
⌥
a�x2,
x1
⌘
C1,
x2
⌘
C2.
Proof: Consider the convex set
C1
−
C2 =
x2
{
−
x1
|
x1
⌘
C1, x2
C2
}... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
C2
a strictly separating hyperplane without C1
being closed.
−
−
85 LECTURE 7
LECTURE OUTLINE
Review of hyperplane separation
Nonvertical hyperplanes
Convex conjugate functions
Conjugacy theorem
Examples
•
•
•
•
•
Reading: Section 1.5, 1.6
86 ADDITIONAL THEOREMS
Fundamental Characterization: The clo-
•
n is the
sure of... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
, ⇥) is
�
It intersects the (n+1)st axis at ξ = (µ/⇥)�u+w,
•
where (u, w) is any vector on the hyperplane.
w
(µ, )
(u, w)
µ
u + w
0
Nonvertical
Hyperplane
(µ, 0)
Vertical
Hyperplane
u
A nonvertical hyperplane that contains the epi-
•graph of a function in its “upper” halfspace, pro-
vides lower bounds to the function v... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
spaces as per (a).
90✓
CONJUGATE CONVEX FUNCTIONS
Consider a function f and its epigraph
•
Nonvertical hyperplanes supporting epi(f )
Crossing points of vertical axis
◆→
f (y) = sup x�y
x
n
⌦�
⇤
0
f x) ,
(
−
y
n.
⌘ �
⌅
y, 1)
(
f (x)
Slope = y
x
inf
x
⇥⇤
n{
f (x)
x�y
}
=
f (y)
For any f :
[
•
function is defined by
n
�
... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
,
−
y
n,
⌘ �
⌦�
⇤
⌅
note that f is convex and closed .
Reason: epi(f ) is the intersection of the epigraphs
•
of the linear functions of y
as x ranges over
x�y
−
f (x)
n.
�
Consider the conjugate of the conjugate:
•
•
f (x) = sup
n
y
⌦�
⇤
y�x
−
f (y)
,
x
n.
⌘ �
⌅
f is convex and closed.
Important fact/Conjugacy theorem... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
���
⇤
(a) We have
y�x
−
f (y)
,
x
n
⌘ �
⌅
f (x)
≥
f (x),
x
⌘ �
n
(b) If f is convex, then properness of any one
of f , f , and f implies properness of the
other two.
(c) If f is closed proper and convex, then
f (x) = f (x),
(d) If clˇ f (x) >
for all x
−⇣
clˇ f (x) = f (x),
x
⌘ �
n
n, then
⌘ �
x
n
⌘ �
95◆
PROOF ... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
are x�y
f
ing point of the strictly sep. hyperplane. Hence
(x), and lie st�rictly ab⇥ove and� below the⇥ cross-
−
f (x) and x�y
−
the fact f
x�y
f (x) > x�y
f . Q.E.D.
−
≥
f (x)
−
96 A COUNTEREXAMPLE
A counterexample (with closed convex but im-
•
proper f ) showing the need to assume properness
in order for f = f :
f ... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
�
y�x
f (y)
inf
x
⇥⇤
n{
f (x)
x�y
}
=
f (y)
99 A FEW EXAMPLES
•
•
lp and lq norm conjugacy, where 1
p + 1
q = 1
f (x) =
1 n
p
i=1
⌧
xi
p,
|
|
f (y) =
1
q
n
i=1
⌧
yi
q
|
|
Conjugate of a strictly convex quadratic
f (x) = x�Qx + a�x + b,
1
2
f (y) = (y
1
2
a)�Q−
1(y
a)
b.
−
−
−
Conjugate of a function obtained by inverti... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
A CONE - POLAR CONE
The conjugate of the indicator function ⌅C is
•
the support function, ↵C(y) = supx
C y�x.
⌦
If C is a cone,
•
↵C(y) =
0
⇣
�
y�x
0,
if
otherwise
⌥
x
⌘
C,
i.e., ↵C is the indicator function ⌅C⇤ of the cone
C ⇤ =
y
{
|
y�x
⌥
0,
x
C
}
⌘
This is called the polar cone of C.
By the Conjugacy Theorem th... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
COMMON / MAX CROSSING PROBLEMS
•
•
We introduce a pair of fundamental problems:
Let M be a nonempty subset of
n+1
�
(a) Min Common Point Problem: Consider all
vectors that are common to M and the (n +
1)st axis. Find one whose (n + 1)st compo-
nent is minimum.
(b) Max Crossing Point Problem: Consider non-
vertical hype... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
�
w + µ�u,
(u, w)
M
⌘
Max crossing problem is to maximize ξ subject to
ξ
n, or
, µ
inf (u,w)
w + µ�u
⌥
M {
⌦
}
⌘ �
maximize q(µ) =↵ inf
(u,w)
⌦
M{
w + µ�u
}
subject to µ
n.
⌘ �
105 GENERIC PROPERTIES – WEAK DUALITY
Min common problem
inf w
M
⌦
Max crossing problem
(0,w)
•
•
maximize q(µ) =↵ inf
(u,w)
⌦
M{
w + µ�u
}
s... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
inf
p(u)
|
{
w
⌅
}
w+µ�u ,
}
{
and finally
q(µ) = inf
u m
⌦�
(µ, 1)
p(u)
p(u) +µ
�u
⇤
⌅
M = epi(p)
w = p(0)
q = p(0)
0
u
µ)
q(µ) =
p(
Thus, q(µ) =
p (
−
−
µ) and
•
q⇤ = sup q(µ) = sup 0 ( µ) p ( µ) = p (0)
· − − −
n
µ
⌦�
n
µ
⌦� ⇤
⌅
107◆
GENERAL OPTIMIZATION DUALITY
Consider minimizing a function f :
n
[
−⇣
�
] be a fun... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
q(µ) =
r
⌦�
µ
inf F (0,
r
µ
⌦�
−
−
µ) =
inf F (0, µ),
r
µ
⌦�
−
and weak duality has the form
w⇤ = inf F (x, 0)
⌦�
x
n
≥ −
inf F (0, µ) = q
r
µ
⌦�
⇤
108◆
◆
CONSTRAINED OPTIMIZATION
Minimize f :
over the set
n
�
C =
→ �
X
x
⌘
n and g :
⇤
|
n
�
0 ,
⌥
r. ⌅
g(x)
→ �
where X
⌦ �
Introduce a “perturbed constraint set”
•
•
Cu... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
). Then
•
•
q(µ) = inf
u
⌥
p(u
) +
µ⇧u
r
r
⇤
, x
u
=
=
⌥
⌥
inf
x
⌥
−∞
�
inf
⌅f (x) + µ⇧u
X, g(x)
u
⇤
X L(x, µ)⇤ if µ
0, ⌅
⌥
otherwise.
110 LINEAR PROGRAMMING DUALITY
Consider the linear program
•
minimize c�x
subject to a�jx
≥
bj,
j = 1, . . . , r,
where c
For µ
•
⌘ �
≥
n, aj
n, and bj
, j = 1, . . . , r.
⌘ �
⌘ �
0, ... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
Point w
M
6
%
.
w
Min Common
Point w
%&'()*++*'(,*&'-(./
M
%
M
%
Max Crossing
%#0()1*22&'3(,*&'-(4/
Point q
0!
0
u
7
0
!
u
7
(a)
"#$
w
.
Min Common
%&'()*++*'(,*&'-(./
Point w
Max Crossing
Point q
%#0()1*22&'3(,*&'-(4/
Max Crossing
%#0()1*22&'3(,*&'-(4/
Point q
(b)
"5$
6
M
%
9
M
%
0
!
(c)
"8$
u
7
112 REVIEW OF THE MC/M... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
113 MINIMAX PROBLEMS
Given φ : X
consider
Z
◆→ �
⇤
minimize
, where X
n, Z
m
⌦ �
⌦ �
sup φ(x, z)
z
⌦
subject to x
X
Z
⌘
or
•
•
•
maximize
inf φ(x, z)
x
subject to z
Z.
X
⌦
⌘
Some important contexts:
Constrained optimization duality theory
Zero sum game theory
−
−
We always have
sup inf φ(x, z)
z
x X
Z ⌦
⌦
⌥
inf
x
⌦
X z... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
gives aij
•
to the 2nd.
Mixed strategies are allowed: The two players
•
select probability distributions
x = (x1, . . . , xn),
z = (z1, . . . , zm)
over their possible choices.
Probability of (i, j) is xizj, so the expected
•
amount to be paid by the 1st player
x�Az =
aijxizj
i,j
⌧
where A is the n
m matrix with elemen... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
, z)
z
x X
Z ⌥
⌥
By the minimax inequality, the above holds as an
equality throughout, so the minimax equality and
Eq. (*) hold.
Conversely, if Eq. (*) holds, then
sup inf ⌅(x, z) = inf ⌅(x, z⇥)
z Z
⌥
x X
⌥
⇤
⌅(x⇥, z⇥)
x X
⌥
sup ⌅(x⇥, z) = inf sup ⌅(x, z)
x
z
X z
Z
Z
⌥
⌥
⇤
⌥
Using the minimax equ., (x⇤, z⇤) is a saddle... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
, z),
z
z
⌦
⌦�
Z
m
w⇤ = p(0) = inf
⌦
The dual function can be shown to be
sup (clˆ φ)(x, z).
⌦�
X z
x
m
•
so
•
q(µ) = inf (clˆ φ)(x, µ),
X
x
⌦
µ
⌘ �
m
so if φ(x,
·
) is concave and closed,
w⇤ = inf
⌦
x
X z
m
⌦�
sup φ(x, z),
q⇤ = sup inf φ(x, z)
m x
X
z
⌦�
⌦
119◆
PROOF OF FORM OF DUAL FUNCTION
Write p(u) = inf
x
p
xX... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
X
p⇤
x(
µ)
⌅
m
X u
⌦�
−
= inf (clˆ φ)(x, µ) ⌅
−
⇤
⌦
m
m
u
⌦�
= inf
u
⌦�
inf
x
⌦
= inf
X
x
=
x X
⌦
120
DUALITY THEOREMS
Assume that w⇤ <
⇣
•
and that the set
M =
(u, w)
|
⇤
is convex.
there exists w with w
w and (u, w)
M
⌃
⇤
⌅
Min Common/Max Crossing Theorem I:
•
We have q⇤ = w⇤ if and only if for every sequence
(uk, ... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
⌘
−⇣
⌘ �
D =
u
|
there exists w
⇤
contains the origin in its relative interior. Then
q⇤ = w⇤ and there exists µ such that q(µ) = q⇤.
w
(µ, 1)
w∗ = q∗
M
M
0
D
u
w
w
q∗
0
M
M
u
D
Furthermore, the set
is nonempty
µ
•
and compact if and only if D contains the origin
in its interior.
q(µ) = q⇤}
{
|
Min Common/Max Crossing T... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
w⇤ −
⇧)
•
/ cl(M ) for any ⇧ > 0.
⌘
w
w
w∗ −
⇥
wk
lim inf
k
⇥⇤
0
M
(uk, wk)
(uk+1, wk+1)
(uk, wk)
(uk+1, wk+1)
M
u
123 PROOF OF THEOREM I (CONTINUED)
⇧
⌘
Step 2: M does not contain any vertical lines.
1) would be a direction
cl(M ),
⇤)
0 belongs to
•
If this were not so, (0,
−
). Because (0, w
of recession of cl(M
⇧)
(... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
through
(0, w⇤), contains M in one of its closed halfspaces,
but does not fully contain M , i.e., for some (µ, ⇥) =
(0, 0)
⇥w⇤ ⌥
µ�u + ⇥w,
⇥w⇤ < sup
(u,w)
⌦
{
M
M ,
(u, w)
µ�u + ⇥w
⌘
}
Will show that the hyperplane is nonvertical.
any (u, w)
M , the set M contains the
Since for
•
0. If
w
halfline (u, w)
|
≥
ri(D)
µ�u ... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
≥
X .
⇤
Then Q⇤ is nonempty and compact if and only⌅ if
X such that gj(x) < 0
there exists a vector x
for all j = 1, . . . , r.
⌘
(g(x), f (x))
x
|
⌅
X
(g(x), f (x))
x
|
⌅
X
(g(x), f (x))
x
|
⌅
X
�
g(x), f (x)
�
⇥
0
⇥
�
⇥
�
⇥
(µ, 1)
0
(µ, 1)
0
0
}
(a)
(b)
(c)
The lemma asserts the existence of a nonverti-
r+1, with nor... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
/MC Theorem II applies: we have
•
D =
u
|
there exists w
with (u, w)
M
⌘
⌘ �
and 0
⇤
⌘
int(D), because g(x), f (x) M .
⌘
⌅
�
⇥
127 LECTURE 10
LECTURE OUTLINE
Min Common/Max Crossing Th. III
Nonlinear Farkas Lemma/Linear Constraints
Linear Programming Duality
Convex Programming Duality
•
•
•
•
Optimality Conditions
•
Re... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
P = Ø, where
⌫
D˜ =
u
|
there exists w
with (u, w)
⌘ �
˜
M
}
⌘
⇤
⌘
Then q⇤ = w⇤, there is a max crossing solution,
0
and all max crossing solutions µ satisfy µ�d
for all d
⌥
RP .
Comparison with Th. II: Since D = D˜
•
the condition 0
ri(D) of Theorem II is
P ,
−
⌘
ri(D˜ )
ri(P ) = Ø
⌫
129✓
✓
PROOF OF MC/MC TH. III
Con... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
assumption, ⇥ = 0, so we may assume that ⇥ = 1.
a direction of recession
of C
≥
⇤
130✓
✓
PROOF (CONTINUED)
Hence,
•
w⇤ + µ�z
so that
⌥
inf
(u,v)
⌦
C1
{
v + µ�u
,
}
z P,
⌘
w⇤ ⌥
=
(u,v)
inf
C1, z
⌦
inf
˜M
⌦
P {
v + µ�(u
z)
−
P
⇤
v + µ�u
⌅
}
(u,v)
⌦
= inf
(u,v)
⌦
= q(µ)
−
v + µ�u
}
{
M
Using q⇤ ⌥
q⇤ = w⇤.
w⇤ (weak dual... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
= M˜ + Positive Orthant, where
⌅
M˜ = (Ax
b, w)
|
−
(x, w)
⌘
epi(f )
⇤
w
w
epi(f )
(x⇥, w⇥)
(x, w)
0
}
Ax
b
⇥
(Ax
−
b, w)
⇧⌅
x
˜M
w⇥
0
}
p(u) = inf
b
⇤
−
Ax
⌅
u
f (x)
(µ, 1)
M
w⇥
q(µ)
0
}
p(u) < w
u
(u, w)
|
�
M
⇤
⇥
u
⇤
epi(p)
(2) There is an x
ri(dom(f )) s. t. Ax
b
0.
Then q⇤ = w⇤ and there is a µ
⌘
−
0 with q(µ) = q... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
.
⇤
Assume that there exists a vector x
that Ax
0. Then Q⇤ is nonempty.
⌘
b
−
⌥
⌅
ri(X) such
Proof: As before, apply special case of MC/MC
Th. III of preceding slide, using the fact w⇤ ≥
0,
implied by the assumption.
w
(0, w∗)
0
M =
(u, w)
Ax
b
−
⇥
|
u, for some (x, w)
epi(f )
⌅
⇤
(Ax
−
b, f (x))
x
|
⌅
X
⇤
⌅
⌅
u
(µ, 1)... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
,
≥
−
x
⌘ �
m,
or Aµ = c.
134 LINEAR PROGRAMMING DUALITY
Consider the linear program
•
minimize c�x
subject to a�jx
≥
bj,
j = 1, . . . , r,
where c
n, aj
n, and bj
⌘ �
⌘ �
The dual problem is
, j = 1, . . . , r.
⌘ �
•
•
maximize b�µ
r
subject to
ajµj = c, µ
0.
≥
j=1
⌧
Linear Programming Duality Theorem:
(a) If either... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
}
0, c can
c =
r
j=1
⌧
µ⇤j aj, µ⇤j ≥
0,
j
⌘
J, µ⇤j = 0,
j /
⌘
J.
Taking inner product with x⇤, we obtain c�x⇤ =
f ⇤, shows that q⇤ = f ⇤
b�µ⇤, which in view of q⇤ ⌥
and that µ⇤ is optimal.
136 LINEAR PROGRAMMING OPT. CONDITIONS
A pair of vectors (x⇤, µ⇤) form a primal and dual
optimal solution pair if and only if... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
b�µ⇤ = c�x⇤. From Eq. (**), we obtain Eq.
(*).
137 CONVEX PROGRAMMING
Consider the problem
minimize f (x)
subject to x
X, gj(x)
⌘
⌥
0, j = 1, . . . , r,
⌦ �
◆→ �
where X
gj : X
n is convex, and f : X
are convex. Assume f ⇤: finite.
Recall the connection with the max crossing
•
problem in the MC/MC framework where M =
ep... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
X w/ g(x)
−
⌥
r
f ⇤ ⌥
f (x) +
µ⇤j gj(x),
x
⌘
X
j=1
⌧
It follows that
•
f ⇤ ⌥
inf
X
x
⌦
⇤
f (x)+µ⇤�g(x)
⌅
inf
X, g(x)
0
⌅
⌥
x
⌦
f (x) = f ⇤.
Thus equality
holds throughout, and we have
f ⇤ = inf
◆
x X
⌦ ⌫
f (x) +
r
j=1
⌧
µ⇤j gj(x)
⇠
= q(µ⇤)
139 QUADRATIC PROGRAMMING DUALITY
•
Consider the quadratic program
2 x�Q... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
have q⇤ = f ⇤, and the vectors x⇤ and µ⇤ are
•
optimal solutions of the primal and dual problems,
respectively, iff x⇤ is feasible, µ⇤ ≥
x⇤ ⌘
arg min L(x, µ⇤),
µ⇤j gj(x⇤) = 0,
0, and
(1)
Proof: If q⇤ = f ⇤, and x⇤, µ⇤ are optimal, then
j.
X
⌦
x
f ⇤ = q⇤ = q(µ⇤) = inf L(x, µ⇤)
x
X
⌦
r
L(x⇤, µ⇤)
⌥
= f (x⇤) +
µ⇤j gj(x⇤)
... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
:
•
Ax⇤ ⌥
b,
µ⇤ ≥
0
Lagrangian optimality holds [x⇤ minimizes L(x, µ⇤)
•
over x
n]. This yields
⌘ �
x⇤ =
−
Q−
1(c + A�µ⇤)
Complementary slackness holds [(Ax⇤ −
•
0]. It can be written as
b)�µ⇤ =
µ⇤j > 0
✏
a�jx⇤ = bj,
j = 1, . . . , r,
where a�j is the jth row of A, and bj is the jth
component of b.
142 LINEAR EQUALIT... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
m.
⌘ �
143 DUALITY AND OPTIMALITY COND.
•
Pure equality constraints:
(a) Assume that f ⇤: finite and there exists x
⌘
ri(X) such that Ax = b. Then f ⇤ = q⇤ and
there exists a dual optimal solution.
(b) f ⇤ = q⇤, and (x⇤, ⌃⇤) are a primal and dual
optimal solution pair if and only if x⇤ is fea-
sible, and
x⇤ ⌘
arg min L(... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
Farkas’ Lemma
Linear programming (duality, opt. conditions)
Convex programming
•
•
•
minimize
subject to x
f (x)
X, g(x)
⌘
⌥
0, Ax = b,
g1(x), . . . , gr(x) �, f :
where X is convex, g(x) =
vex.
and gj : X
X
(Nonlin. Farkas’ Lemma, duality, opt. conditions)
, j = 1, . . . , r, are con
⇥
◆→ �
◆→ �
�
145 DUALITY AND OPTI... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
��
j
146 COUNTEREXAMPLE I
Strong Duality Counterexample: Consider
•
minimize f (x) = e−
subject to x1 = 0,
x1x2
X =
x
⌘
x
x
{
|
≥
0
}
Here f ⇤ = 1 and f is convex (its Hessian is > 0 in
the interior of X). The dual function is
q(⌃) = inf
0
⇧
x
e−
x1x2 + ⌃x1
=
0
0,
if ⌃
≥
otherwise,
�
−⇣
(when ⌃
≥
ative for x
x1
≥... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
Solutions Counterexample:
, f (x) = x, g(x) = x2. Then x⇤ = 0 is
•
Let X =
the only feasible/optimal solution, and we have
�
q(µ) = inf
{
⌦�
x
x + µx2
=
}
−
1
4µ
,
µ > 0,
for µ
and q(µ) =
−⇣
However, there is no µ⇤ ≥
q⇤ = 0.
⌥
0, so that q⇤ = f ⇤ = 0.
0 such that q(µ⇤) =
The perturbation function is
•
p(u) = inf x =
... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
�
Dual problem: max
⌅
{−
min⌅{−
q(⌃)
or
}
•
−
f (⌃)
1
⇤
−
f
2 (
⌃)
}
−
⌅
=
1 (⌃) +f
f
2 (
minimize
n,
subject to ⌃
⌘ �
⌃)
−
where f
1 and f
2 are the conjugates.
150◆
◆
•
FENCHEL DUALITY THEOREM
Consider the Fenchel framework:
(a) If f ⇤ is finite and ri
=
dom(f1)
, then f ⇤ = q⇤ and there exists at least one
dom(... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
RIC INTERPRETATION
f
1 ()
q()
f
2 (
)
f = q
f1(x)
Slope
Slope
f2(x)
x
x
When dom(f1) = dom(f2) =
n, and f1 and
•
f2 are differentiable, the optimality condition is
equivalent to
�
⌃⇤ =
∇
f1(x⇤) =
−∇
f2(x⇤)
By reversing the roles of the (symmetric) primal
•
and dual problems, we obtain alternative criteria
for strong... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
f ⌥(⇤) = sup
1
n
x
⌥
⇤
where C ⇤ =
⇤⇧x
−
f (x)
⌅
, f ⌥
2 (⇤) = sup ⇤⇧x =
x
C
⌥
0
⇧
if ⇤
if ⇤
C⇥,
⌃
/ C ,
⇥
⌃
�
⌃
⌃�x
⌥
{
The dual problem is
|
0,
x
C
.
}
⌘
•
•
minimize
subject to ⌃
f (⌃)
ˆ
C,
⌘
where f is the conjugate of f and
Cˆ =
⌃
{
|
⌃�x
0,
x
C
.
}
⌘
≥
Cˆ and
−
ˆ
C are called the dual and polar cones.
154◆
... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
•
•
•
Primal problem is
minimize
c�x
subject to x
We have
b
−
⌘
S,
x
C.
⌘
f (⌃) = sup (⌃
x
S −
b
c)�x = sup(⌃
c)�(y + b)
=
⌦
−
(⌃
�
⇣
c)�b
−
if ⌃
if ⌃
y S −
⌦
c
⌘
c /
⌘
S⊥,
S.
−
−
Dual problem is equivalent to
minimize
b�⌃
subject to ⌃
c S ,
− ⌘ ⊥
⌃ C.ˆ
⌘
If X
ri(C) =Ø, there is no dualit y gap an
d
•
there exists a du... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
ization:
⇣
�
p(u) = inf F (x, u).
⌦�
Under the given assumption, p is closed convex.
x
n
157 LECTURE 12
LECTURE OUTLINE
Subgradients
Fenchel inequality
Sensitivity in constrained optimization
Subdifferential calculus
•
•
•
•
Optimality conditions
•
Reading: Section 5.4
158 SUBGRADIENTS
f (z)
0
g, 1)
(
x, f (x)
�
⇥z
Let ... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
:3)(cid:1)(cid:10)(cid:1)(cid:13)(cid:11)(cid:14)(cid:15)(cid:7)(cid:4)(cid:1)(cid:2)(cid:8)(cid:6)(cid:9)(cid:3)(cid:2)(cid:14)(cid:9)(cid:1)(cid:5)(cid:1)(cid:8)(cid:3)(cid:17)
1)
f (x) = max
0, (1/2)(x2
�
-1
(cid:5)(cid:1)(cid:8)
0
(cid:7)
1
(cid:8)
⇥
x
(cid:14)
(cid:2)(cid:12)(cid:2)(cid:14)(cid:3)
f (x)
(cid:8)
1
... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
proper convex.
•
•
M = epi(fx),
fx(z) = f (x + z)
f (x)
−
f (z)
0
Epigraph of f
g, 1)
(
x, f (x)
�
⇥z
fx(z)
Translated
Epigraph of f
g, 1)
(
0
z
By 2nd MC/MC Duality Theorem, ◆f (x) is
•
nonempty and compact if and only if x is in the
interior of dom(f ).
More generally: for every x
ri dom(f )),
⌘
•
◆f (x) = S⊥ + G, �
... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
}
,
⌥
we have
NC(x) =
0
{
}
cone
�
ai
{
|
a�ix = bi
}
if x
if x
int(C),
⌘
/ int(C).
⌘
�
⇥
Proof: Given x, disregard inequalities with
•
a�ix < bi, and translate C to move x to 0, so it
becomes a cone. The polar cone is NC(x).
163 FENCHEL INEQUALITY
n
�
Let f :
] be proper convex and
⇣
•
let f be its conjugate. Using th... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
) We have x⇥
�
n. So
all x
⌘ �
⇤ ⌘
X ⇤ iff f (x)
f (x⇤) for
≥
X ⇤
x⇤ ⌘
where:
iff
0
⌘
◆f (x⇤)
iff
x⇤ ⌘
◆f (0)
−
−
1st relation follows from the subgradient in-
equality
2nd relation follows from the conjugate sub-
gradient theorem
(b) ◆f (0) is nonempty if 0
ri dom(f ) .
⌘
(c) ◆f (0) is nonempty and compact
int dom(f ) . ... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
∇
−
p(0) is
•
⌅
Q⇤
p(u) =
x
inf
X, g(x)
f (x),
u
⌦
⌅
If p is convex and differentiable,
µ⇤j =
◆p(0)
◆uj
,
−
j = 1, . . . , r.
166 EXAMPLE: SUBDIFF. OF SUPPORT FUNCTION
Consider the support function ↵X (y) of a set
•
X. To calculate ◆↵X (y) at some y, we introduce
r(y) = ↵X (y + y),
y
n.
⌘ �
We have ◆↵X (y) = ◆r(0) = arg... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
xed x
n, consider
⌘ �
•
Ax =
j
|
a�jx + bj = f (x)
and the function r(x) = max a�
⇤
jx
j
|
⌅
⌘
Ax .
It can be seen that ◆f (x) =⇤ ◆r(0).
Since r is the support function of the finite set
⌅
•
•
aj
{
j
Ax}
⌘
|
, we see that
◆f (x) = ◆r(0) = conv
aj
{
|
j
⌘
Ax}
⇥
�
168 CHAIN RULE
m
(
�
Let f :
] be convex, and A be
,
⇣
•
a... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
⌃
�(Az
y)
−
Since the min over z is unconstrained, we hav⌅e
⇤
d = A�⌃, so Ax
arg miny
f (y)
⌃�y
, or
m
⌘
⌦�
−
f (y)
≥
⇤
f (Ax) +⌃ �(y Ax),
−
y
⌅
m.
⌘ �
◆f (Ax), so that d = A�⌃
Hence ⌃
⌘
It follows that ◆F (x)
dral case, dom(f ) is polyhedral. Q.E.D.
A�◆f (Ax).
⌘
A�◆f (Ax). In the polyhe-
⌦
169◆
✓
✓
SUM OF FUNCTIONS... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
��
⌥
⌫
⌫
�
⌥
m
i=k+1 ri
dom(fi)
= Ø.
�
⇥�
170◆
✓
◆
✓
CONSTRAINED OPTIMALITY CONDITION
n
(
Let f :
,
−⇣
•
�
be a convex subset of
the following four conditions holds:
] be proper convex, let X
n, and assume that one of
⇣
�
→
(i) ri
dom(f )
ri(X) = Ø.
⌫
f
(ii) f is polyhedral and dom( )
⇥
�
ri(
X
) = Ø
.
⌫
⌫
(iii) X is ... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
�)
⌘
−∇
NC(x⇤),
which is equivalent to
f (x⇤)�(x
x⇤)
0,
≥
−
∇
x
⌘
X.
In the figure on the right, f is nondifferentiable,
•
and the condition is that
g
−
⌘
NC(x⇤)
for some g
◆f (x⇤).
⌘
172 LECTURE 13
LECTURE OUTLINE
•
•
Problem Structures
Separable problems
Integer/discrete problems – Branch-and-bound
Large sum problems... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
It is
still useful in a branch-and-bound scheme.
0, 1
{
174◆
◆
LARGE SUM PROBLEMS
Consider cost function of the form
m
•
f (x) =
fi(x),
m is very large,
i=1
⌧
→ �
n
�
where fi :
are convex. Some examples:
Dual cost of a separable problem.
•
•
•
Data analysis/machine learning: x is pa-
•
rameter vector of a model; each... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
0, and c is a
Examples:
•
0, t
The quadratic penalty P (t) = max
−
}
The nondifferentiable penalty P�(t) = max⇥{
−
Another possibility: Initially discard some of
•
the constraints, solve a less constrained problem,
and later reintroduce constraints that seem to be
violated at the optimum (outer approximation).
{
2
.
0, ... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
uality with the definitions
•
f1(x) = f (x),
f2(x) =
0
⇣
�
if x
if x
C,
⌘
/ C.
⌘
The conjugates are
f ⌥
1 (⇤) = sup
n
⌥
x
⇤⇧x
−
)
f (x
⇤
⌅
, f ⌥
2 (⇤) = sup ⇤⇧x =
x
C
⌥
0
⇧
if ⇤
if ⇤
C⇥
,
⌃
/ C⇥,
⌃
�
where C ⇤ =
cone of C.
⌃
{
|
⌃�x
0,
x
C
}
⌘
⌥
is the polar
The dual problem is
•
minimize
subject to ⌃
f (⌃)
ˆ
C,
⌘
wh... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
⌃
y S −
⌦
⌘ ⊥,
c S
S⊥,
c /
⌘
−
−
so the dual problem can be written as
minimize
b�⌃
subject to ⌃
c
−
⌘
S⊥,
ˆ
C.
⌃
⌘
The primal and dual have the same form.
If C is closed, the dual of the dual yields the
•
•
primal.
180 SPECIAL LINEAR-CONIC FORMS
min
Ax=b, x
C
⌦
c�x
⇐✏
min c�x
C
⌦
−
b
Ax
⇐✏
A
c
max b�⌃,
ˆ
C
A0⌅
−
⌦
max... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
the dual as
minimize x�(c
subject to c
−
A�⌃)
ˆ
C
−
A�⌃
⌘
discard the constant x�c, use the fact Ax = b, and
change from min to max.
181 SOME EXAMPLES
Nonnegative Orthant: C =
{
The Second Order Cone: Let
x
x
0
.
}
≥
|
•
•
C =
(x1, . . . , xn)
�
xn
|
≥
x3
!
x2
1 +
· · ·
+ x2
n
1
−
�
x1
x2
The Positive Semidefinite Cone:... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
m
i=1
⌧
m
i=1
⌧
b�i⌃i
A�i⌃i = c,
⌃i
⌘
Ci, i = 1, . . . , m,
where ⌃ = (⌃1, . . . , ⌃m).
The duality theory is no more favorable than
•
the one for linear-conic problems.
There is no duality gap if there exists a feasible
•
solution in the interior of the 2nd order cones Ci.
Generally, 2nd order cone problems can be
•
r... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
� · · · ⇤
•
•
maximize
subject to
m
i=1
⌧
m
i=1
⌧
b�i⌃i
A�i⌃i = c,
⌃i
⌘
Ci, i = 1, . . . , m,
where ⌃ = (⌃1, . . . , ⌃m).
186 EXAMPLE: ROBUST LINEAR PROGRAMMING
minimize c�x
subject to a�jx
bj,
⌥
(aj, bj)
⌘
Tj,
j = 1, . . . , r,
where c
n, and Tj is a given subset of
n+1.
⌘ �
�
We convert the problem to the equivalen... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
−
⌘
Cj, where
187 SEMIDEFINITE PROGRAMMING
Consider the symmetric n
•
product < X, Y >= trace(XY ) =
⇤
n matrices. Inner
n
i,j=1 x y .
ij ij
•
Let C be the cone of pos. semidefinite matrices.
�
C is self-dual, and its interior is the set of pos-
•
itive definite matrices.
Fix symmetric matrices D, A1, . . . , Am, and
•
v... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/6c63c6219c60378bc27d5b4a9167f1bc_MIT6_253S12_lec_comp.pdf |
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