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835c2398f7e8c8249c224d48e7cfa05006ad2d48 | abstract | 0 | 29 | Abstract | We propose a variant of the classical augmented Lagrangian method for
constrained optimization problems in Banach spaces. Our theoretical framework
does not require any convexity or second-order assumptions and allows the
treatment of inequality constraints with infinite-dimensional image space.
Moreover, we discuss th... | {
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} | 10.1137/16M1107103 | 1807.04467 | An Augmented Lagrangian Method for Optimization Problems in Banach
Spaces | [
"Christian Kanzow",
"Daniel Steck",
"Daniel Wachsmuth"
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24f17002cb59f835a0ffe18e5c9a310152e57553 | subsection | 1 | 29 | Introduction | Let X, Y be (real) Banach spaces and let f:X\rightarrow \mathbb {R}, g:X\rightarrow Y be
given mappings. The aim of this paper is to describe an augmented Lagrangian
method for the solution of the constrained optimization problem\min \ f(x) \quad \text{subject to (s.t.)}\quad g(x)\le 0.We assume that Y\hookrightarrow L... | {
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... | 10.1137/16M1107103 | 1807.04467 | An Augmented Lagrangian Method for Optimization Problems in Banach
Spaces | [
"Christian Kanzow",
"Daniel Steck",
"Daniel Wachsmuth"
] | [
"math.OC"
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c6774eeaa31db2b0bba9c7a4ebbb16cde32a113d | subsection | 2 | 29 | Introduction | The norms on X, Y, etc. are
denoted by \Vert \cdot \Vert , where an index (as in \Vert \cdot \Vert _X) is appended if necessary.
Furthermore, we write \rightarrow , \rightharpoonup , and \rightharpoonup ^* for strong, weak, and weak-^*
convergence, respectively. Finally, we use the abbreviation lsc for a lower
semicont... | {
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} | 10.1137/16M1107103 | 1807.04467 | An Augmented Lagrangian Method for Optimization Problems in Banach
Spaces | [
"Christian Kanzow",
"Daniel Steck",
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9442428e98990dfcb9863210809d1c637a7cbad6 | subsection | 3 | 29 | Preliminaries and Assumptions | We denote by e:Y\rightarrow Z the (linear and continuous) dense embedding of Y into
Z:=L^2(\Omega ), and by K_Y, K_Z the respective nonnegative cones in
Y and Z, i.e.K_Z:=\lbrace z\in Z\mid z(t)\ge 0~\text{a.e.}\rbrace \quad \text{and}\quad K_Y:= \lbrace y\in Y \mid e(y) \in K_Z\rbrace .Note that the adjoint mapping e^... | {
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f1ab019fe369649a5766d80f63daae0213b96fdc | subsection | 4 | 29 | Preliminaries and Assumptions | Hence, if \Vert g_+\Vert is convex (which is true if g is convex with respect to the order in Y), then
the (strong) lower semicontinuity of g already implies the weak lower
semicontinuity. We conclude that (A1) holds, in particular, for every
lsc. convex function f and any mapping g\in \mathcal {L}(X,Y).On a further no... | {
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Spaces | [
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60da3cf59d9c7494d114e3685aee14fef517830e | subsection | 5 | 29 | Preliminaries and Assumptions | For instance, consider the case where \Omega =\lbrace 1\rbrace and
Z=L^2(\Omega ), which can be identified with \mathbb {R}. Then the sequences
a^k=k and b^k=1/k provide a simple counterexample. | {
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} | 10.1137/16M1107103 | 1807.04467 | An Augmented Lagrangian Method for Optimization Problems in Banach
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eb0f67bdbb7a5023f11763a778802f595b66bc66 | subsection | 6 | 29 | An Augmented Lagrangian Method | This section gives a detailed statement of our augmented Lagrangian
method for the solution of the optimization problem (REF ).
It is motivated by the finite-dimensional discussion in, e.g.,
and differs from the traditional augmented
Lagrangian method as applied, e.g., in , to a class
of infinite-dimensional problems,... | {
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Spaces | [
"Christian Kanzow",
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047382410681af3ef9dc5009574dd699071d0e5c | subsection | 7 | 29 | An Augmented Lagrangian Method | Going a
little further, our method also includes the Moreau-Yosida regularization scheme
(see , and Section ) as a special
case, which arises if (w^k) is chosen as a constant sequence. However, the
most natural choice, which also brings the method closer to traditional augmented
Lagrangian schemes, is w^k:=\min \lbrace... | {
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7654f86924960682501f2cfe947ba575ad46a01c | subsection | 8 | 29 | Global Minimization | We begin by considering Algorithm REF from a global optimization
perspective. Note that most of the analysis in this section can be carried
out in the more general case where f is an extended real-valued function,
i.e. f maps to \mathbb {R}\cup \lbrace +\infty \rbrace .The global optimization perspective is particularl... | {
"cite_spans": []
} | 10.1137/16M1107103 | 1807.04467 | An Augmented Lagrangian Method for Optimization Problems in Banach
Spaces | [
"Christian Kanzow",
"Daniel Steck",
"Daniel Wachsmuth"
] | [
"math.OC"
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9cfbf62dd0ef0291621a5f640f345a0db0d9bc07 | subsection | 9 | 29 | Global Minimization | Let \mathcal {K}\subset \mathbb {N}
be such that x^{k+1}\rightharpoonup _{\mathcal {K}}\bar{x} and assume that there is an
x\in X with \Vert g_+(x)\Vert _Z^2<\Vert g_+(\bar{x})\Vert _Z^2.
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} | 10.1137/16M1107103 | 1807.04467 | An Augmented Lagrangian Method for Optimization Problems in Banach
Spaces | [
"Christian Kanzow",
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"math.OC"
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a3b6a38612babdc3acd23dcc58a29ac6e8b6c2d7 | subsection | 10 | 29 | Global Minimization | Using the feasibility
of x and a similar inequality to above, it follows thatf(x^{k+1})+\frac{\rho _k}{2} \left\Vert \left( g(x^{k+1})+
\frac{w^k}{\rho _k} \right)_+ \right\Vert _Z^2 \le f(x)+\frac{\rho _k}{2}\left\Vert \frac{w^k}{\rho _k}\right\Vert _Z^2+\varepsilon _k.But\left( g(x^{k+1})+\frac{w^k}{\rho _k} \right)_... | {
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Spaces | [
"Christian Kanzow",
"Daniel Steck",
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"math.OC"
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961a48f847780f0ec23d735d01cee6ad1b9a4562 | subsection | 11 | 29 | Global Minimization | Therefore, existence and uniqueness of the solution \bar{x} follow from
standard arguments.Now, denoting by c>0 the modulus of convexity of f, it follows that\frac{c}{8} \Vert x^{k+1}-\bar{x}\Vert _X^2 \le \frac{f(x^{k+1})+f(\bar{x})}{2}-
f( \frac{x^{k+1}+\bar{x}}{2} )for all k. By the proof of Theorem REF (b), it is e... | {
"cite_spans": []
} | 10.1137/16M1107103 | 1807.04467 | An Augmented Lagrangian Method for Optimization Problems in Banach
Spaces | [
"Christian Kanzow",
"Daniel Steck",
"Daniel Wachsmuth"
] | [
"math.OC"
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f64ec1d39d0355868b31bd34b96d58ca0438f9a4 | subsection | 12 | 29 | Sequential KKT conditions | Throughout this section, we assume that f and g are continuously
Fréchet-differentiable on X, and discuss the KKT conditions of the
optimization problem (REF ). Recalling that K_Y is the nonnegative
cone in Y, we denote byK_Y^+ := \lbrace f\in Y^* \mid \left\langle f,y \right\rangle \ge 0~
\forall y\in K_Y \rbraceits d... | {
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Spaces | [
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fcaa62f7335da7b0af90efd5e4820c57df9e8722 | subsection | 13 | 29 | Sequential KKT conditions | Due to (x^k)\subset B_r(\bar{x}),
there is a \mathcal {K}\subset \mathbb {N} such that x^k\rightharpoonup _{\mathcal {K}}\bar{y}
for some \bar{y}\in B_r(\bar{x}). Since x^k is a solution of
(REF ), we havef(x^k)+k\Vert g_+(x^k)\Vert _Z^2+\Vert x^k-\bar{x}\Vert _X^2 \le f(\bar{x})for every k. Dividing by k and taking th... | {
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6a1748eac7260116154f1fbdec4d1004befb1205 | subsection | 14 | 29 | Sequential KKT conditions | However, in the infinite-dimensional
setting, our choice of constraint qualification is much more restricted. For
instance, we are not aware of any infinite-dimensional analogues of the (very
amenable) CPLD condition. Hence, we have decided to employ the Zowe-Kurcyusz
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8801bd7a1b8f4262037c4a105bf087996fac3a1e | subsection | 15 | 29 | Sequential KKT conditions | By the AKKT conditions and REF , there is a
k_0\in \mathbb {N} such that\Vert g(x^k)-g(x)\Vert _Y\le \frac{r}{4}
\quad \text{and}\quad \Vert g^{\prime }(x^k)-g^{\prime }(x)\Vert _{\mathcal {L}(X,Y)}\le \frac{r}{4}for every k\ge k_0. Now, let u\in B_r^Y and k\ge k_0. It follows that
-u=g^{\prime }(x)w+z with \Vert w\Ver... | {
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} | 10.1137/16M1107103 | 1807.04467 | An Augmented Lagrangian Method for Optimization Problems in Banach
Spaces | [
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f4522f2b5faeebf6f239aced890383513dd77e2a | subsection | 16 | 29 | Sequential KKT conditions | We conclude that\Vert \lambda ^k\Vert _{Y^*}=\sup _{\Vert u\Vert \le r}\left\langle \lambda ^k,\frac{1}{r}u \right\rangle \le \frac{1}{r}
\left(C+\frac{r}{2}\Vert \lambda ^k\Vert _{Y^*}\right)and, hence, \Vert \lambda ^k\Vert _{Y^*}\le 2C/r.(b): Since (\lambda ^k) is bounded in Y^* and the unit ball in Y^* is
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bba8e39c0fc1133726f55bc5a6c01f70f88ecb76 | subsection | 17 | 29 | Convergence to KKT Points | We now discuss the convergence properties of Algorithm REF
from the perspective of KKT points. To this end, we make the following
assumption.Assumption 6.1
In Step 2 of Algorithm REF , we obtain x^{k+1} such that
L_{\rho _k}^{\prime }(x^{k+1},w^k)\rightarrow 0 as k\rightarrow \infty .The above is a very natural assum... | {
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Spaces | [
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b2aeb732a4ee750ade0d528cfdc5e7e470e8b827 | subsection | 18 | 29 | Convergence to KKT Points | To this end, recall that Assumption REF implies thatf^{\prime }(x^{k+1})+g^{\prime }(x^{k+1})^*\lambda ^{k+1}\rightarrow 0,which already suggests that the sequence of tuples (x^k,\lambda ^k) satisfies
AKKT for the optimization problem (REF ). In fact, the only missing
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} | 10.1137/16M1107103 | 1807.04467 | An Augmented Lagrangian Method for Optimization Problems in Banach
Spaces | [
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7bb8f4896b334fda7301d111792f09f6435a11b2 | subsection | 19 | 29 | Convergence to KKT Points | Now, the claim essentially follows from Theorem REF (b),
the only difference here is that we are working in the Hilbert space
Z instead of Y or Y^* , hence the two conditions REF and
REF formally required in Theorem REF (b) are
automatically satisfied in the current Hilbert space situation.Some further remarks about t... | {
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033b1f4fb37c0c9edcc52d50380cbcf584de6a8f | subsection | 20 | 29 | Convergence to KKT Points | In this case, the pointwise convergence
implies that w^k(t)+\rho _k g(x^{k+1})(t)<0 for sufficiently large k and,
hence, v^k(t)=0 for all such k.Case 2. g(\bar{x})(t)=0. Then consider a fixed k \in \mathcal {K} . If g(x^{k+1})(t)\ge 0, it follows again from the definition
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} | 10.1137/16M1107103 | 1807.04467 | An Augmented Lagrangian Method for Optimization Problems in Banach
Spaces | [
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70af1a901cdb4cafee0cfcac519981af64dffeec | subsection | 21 | 29 | Applications | We now give some applications and numerical results for Algorithm REF . To this end, we consider some standard problems from the literature. Apart from the first example, we place special emphasis on nonlinear and nonconvex problems since the appropriate treatment of these is one of the focal points of our method.All o... | {
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"Christian Kanzow",
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a73de8f9aedd471f31e69207c3fc86d18c2c9bca | subsection | 22 | 29 | The Obstacle Problem | We consider the well-known obstacle problem , . To this end, let \Omega \subseteq \mathbb {R}^d be a bounded domain, and let X:=Y:=H_0^1(\Omega ), Z:=L^2(\Omega ). The obstacle problem considers the minimization problem\min \ f(u) \quad \text{s.t.}\quad u\ge \psi ,where f(u):=\Vert \nabla u\Vert _{L^2(\Omega )}^2 and \... | {
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"Christian Kanzow",
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ffe856df2a17cb446a3467092265670dd719c894 | subsection | 23 | 29 | The Obstacle Problem | The subproblems occurring in Algorithm REF are unconstrained minimization problems which we solve by means of a standard semismooth Newton method.
[Table: Numerical results for the obstacle problem.]Table REF contains the inner and outer iteration numbers together with the final penalty parameter for different values o... | {
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} | 10.1137/16M1107103 | 1807.04467 | An Augmented Lagrangian Method for Optimization Problems in Banach
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1119e6d5f35b84a1d3c8ffd6105c1261f7611cf7 | subsection | 24 | 29 | The Obstacle Bratu Problem | Let us briefly consider the obstacle Bratu problem , , which we simply refer to as Bratu problem. This is a non-quadratic and nonconvex problem which differs from (REF ) in the choice of objective function. To this end, letf(u):=\Vert \nabla u\Vert _{L^2(\Omega )}^2 - \alpha \int _{\Omega } e^{-u(x)} dxfor some fixed \... | {
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97e93280043e0fde1926042c7051ce875f8d7b2c | subsection | 25 | 29 | The Obstacle Bratu Problem | Let us writee^{u+h}-e^u-e^uh = \int _0^1 \int _0^1 e^{u+sth}sh^2 \ dt \ ds,which implies\Vert e^{u+h}-e^u-e^uh\Vert _{L^1(\Omega )} \le \frac{1}{2}\Vert e^{u+|h|}\Vert _{L^2(\Omega )} \Vert h\Vert _{L^4(\Omega )}^2
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1baafd76f0eddb44db7af031439bb357afb9a074 | subsection | 26 | 29 | Optimal Control Problems | We now turn to a class of optimal control problems subject to a semilinear elliptic equation. Let \Omega \subseteq \mathbb {R}^d, d=2,3, be a bounded Lipschitz domain.
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fb241414526e157ac87934060856fcc3f3e603f9 | subsection | 27 | 29 | Optimal Control Problems | By reintroducing the state variable y, we can write these subproblems as\min \ J(y,u)+\frac{\rho _k}{2}\left\Vert \left( y_c-y+\frac{w^k}{\rho _k} \right)_+ \right\Vert ^2
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11a9fd845fa22e5520c608139fcf032888497abc | subsection | 28 | 29 | Final Remarks | We have presented an augmented Lagrangian method for the solution of optimization problems in Banach spaces, which is essentially a generalization of the modified augmented Lagrangian method from . Furthermore, we have shown how the method can be applied to well-known problem classes, and the corresponding numerical re... | {
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8e2a35eef5f22c41da49cbdd7b273937fbf6b21a | abstract | 0 | 126 | Abstract | This article introduces the DPG-star (from now on, denoted DPG$^*$) finite
element method. It is a method that is in some sense dual to the discontinuous
Petrov-Galerkin (DPG) method. The DPG methodology can be viewed as a means to
solve an overdetermined discretization of a boundary value problem. In the same
vein, th... | {
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} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
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End of preview. Expand in Data Studio
EviGraph-R Sparse Index
This dataset contains the sparse retrieval index generated by the EviGraph-R indexing pipeline. It is exported from the finalized shard records after the collection has been written to Qdrant, so the Hub copy matches the indexed corpus that was prepared for retrieval.
What is inside
- One row per indexed chunk.
- Original chunk payload metadata used by retrieval and analysis.
- Vector columns:
sparse_indices / sparse_values. - Source collection:
unarxive_chunks. - Embedding model key:
bge-m3. - Runtime profile:
hpc.
Build summary
- Repository:
lostelf/unarxive_sparse - Split:
train - Shards exported:
15 - Rows exported:
127353 - Generated at:
2026-04-12T19:26:45.835499+00:00
Suggested use
Use this dataset as a portable snapshot of the EviGraph-R retrieval index for reproducible experiments, offline analysis, or mirroring the vector store outside Qdrant.
- Downloads last month
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