The limiting normal cone of a complementarity set in Sobolev spaces

Harder, Felix; Wachsmuth, Gerd

doi:10.1080/02331934.2018.1484467

Cited by 12 publications

(6 citation statements)

References 17 publications

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“…In Sobolev or Lebesgue spaces, the limiting normal cone turned out to be not as effective as in finite dimensional spaces for obtaining stationarity conditions for complementaritytype optimization problems, see [Harder, Wachsmuth, 2018;Mehlitz, Wachsmuth, 2018]. Thus, it would be interesting to know whether the new elementary method from this paper can provide ideas for possible approaches for better stationarity conditions of complementarity-type optimization problems in Sobolev and Lebesgue spaces.…”

Section: Discussionmentioning

confidence: 97%

A new elementary proof for M-stationarity under MPCC-GCQ for mathematical programs with complementarity constraints

Harder

2020

Preprint

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It is known in the literature that local minimizers of mathematical programs with complementarity constraints (MPCCs) are so-called M-stationary points, if a very weak MPCC-tailored Guignard constraint qualification (called MPCC-GCQ) holds. In this paper we present a new elementary proof for this result. Our proof is significantly simpler than existing proofs and does not rely on deeper technical theory such as calculus rules for limiting normal cones. A crucial ingredient is a proof of a (to the best of our knowledge previously open) conjecture, which was formulated in a Diploma thesis by Schinabeck.

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Section: Discussionmentioning

confidence: 97%

A new elementary proof for M-stationarity under MPCC-GCQ for mathematical programs with complementarity constraints

Harder

2020

Preprint

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“…Furthermore, convergence of the method to KKT points was shown under validity of a problem-tailored version of asymptotic regularity. As soon as D becomes nonconvex, one has to face some uncomfortable properties of the appearing limiting normal cone which turns out to be comparatively large since weak- * -convergence is used for its definition as a set limit in the dual space, see [33,50]. That it why the associated M-stationarity conditions are, in general, too weak in order to yield a reasonable stationarity condition.…”

Section: Discussionmentioning

confidence: 99%

An Augmented Lagrangian Method for Optimization Problems with Structured Geometric Constraints

Jia¹,

Kanzow²,

Mehlitz³

et al. 2021

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This paper is devoted to the theoretical and numerical investigation of an augmented Lagrangian method for the solution of optimization problems with geometric constraints. Specifically, we study situations where parts of the constraints are nonconvex and possibly complicated, but allow for a fast computation of projections onto this nonconvex set. Typical problem classes which satisfy this requirement are optimization problems with disjunctive constraints (like complementarity or cardinality constraints) as well as optimization problems over sets of matrices which have to satisfy additional rank constraints. The key idea behind our method is to keep these complicated constraints explicitly in the constraints and to penalize only the remaining constraints by an augmented Lagrangian function. The resulting subproblems are then solved with the aid of a problemtailored nonmonotone projected gradient method. The corresponding convergence theory allows for an inexact solution of these subproblems. Nevertheless, the overall algorithm computes so-called Mordukhovich-stationary points of the original problem under a mild asymptotic regularity condition, which is generally weaker than most of the respective available problem-tailored constraint qualifications. Extensive numerical experiments addressing complementarity-and cardinality-constrained optimization problems as well as a semidefinite reformulation of Maxcut problems visualize the power of our approach.

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“…For limiting normals, an extension to Asplund spaces seems to be possible, directly. Nevertheless, we would like to point the reader's attention to the fact that due to certain convexification effects, limiting normals turned out to be of limited practical use in Lebesgue and Sobolev spaces which are standard in optimal control, see [24,37,38] for a detailed investigation.…”

Section: Remark 34mentioning

confidence: 99%

Calmness and Calculus: Two Basic Patterns

Benko

Mehlitz

2021

Set-Valued Var. Anal

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We establish two types of estimates for generalized derivatives of set-valued mappings which carry the essence of two basic patterns observed throughout the pile of calculus rules. These estimates also illustrate the role of the essential assumptions that accompany these two patters, namely calmness on the one hand and (fuzzy) inner calmness* on the other. Afterwards, we study the relationship between and sufficient conditions for the various notions of (inner) calmness. The aforementioned estimates are applied in order to recover several prominent calculus rules for tangents and normals as well as generalized derivatives of marginal functions and compositions as well as Cartesian products of set-valued mappings under mild conditions. We believe that our enhanced approach puts the overall generalized calculus into some other light. Some applications of our findings are presented which exemplary address necessary optimality conditions for minimax optimization problems as well as the calculus related to the recently introduced semismoothness* property.

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The limiting normal cone of a complementarity set in Sobolev spaces

Cited by 12 publications

References 17 publications

A new elementary proof for M-stationarity under MPCC-GCQ for mathematical programs with complementarity constraints

A new elementary proof for M-stationarity under MPCC-GCQ for mathematical programs with complementarity constraints

An Augmented Lagrangian Method for Optimization Problems with Structured Geometric Constraints

Calmness and Calculus: Two Basic Patterns

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