Lipschitz and Hölder stability of optimization problems and generalized equations

Gfrerer, Helmut; Klatte, Diethard

doi:10.1007/s10107-015-0914-1

Cited by 54 publications

(48 citation statements)

References 28 publications

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“…For verifying the property of metric subregularity there are some sufficient conditions known, see e.g. [8,9,10,11,13]. In this paper polyhedrality will play an important role.…”

Section: Preliminaries From Variational Geometry and Variational Analmentioning

confidence: 99%

New Sharp Necessary Optimality Conditions for Mathematical Programs with Equilibrium Constraints

Gfrerer

2019

Set-Valued Var. Anal

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In this paper, we study the mathematical program with equilibrium constraints (MPEC) formulated as a mathematical program with a parametric generalized equation involving the regular normal cone. We derive a new necessary optimality condition which is sharper than the usual M-stationary condition and is applicable even when no constraint qualifications hold for the corresponding mathematical program with complementarity constraints (MPCC) reformulation.

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“…For verifying the property of metric subregularity there are some sufficient conditions known, see e.g. [8,9,10,11,13]. In this paper polyhedrality will play an important role.…”

Section: Preliminaries From Variational Geometry and Variational Analmentioning

confidence: 99%

New Sharp Necessary Optimality Conditions for Mathematical Programs with Equilibrium Constraints

Gfrerer

2019

Set-Valued Var. Anal

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“…Γ ∞ (a k , t k ) := {ω ∈ S | t k / a k → 0, ∃ a subsequence K of N : a k / a k → ω when k ∈ K}. (7) Note that exactly one of these sets is not empty, since Γ(a k , t k ) = ∅ is equivalent to t k / a k → 0. In the situation considered above sequences a k appear in form a k ∈ S(ȳ + t k v k ) − S(ȳ).…”

Section: Preliminariesmentioning

confidence: 99%

“…Thus, if Γ(a k , t k ) = ∅, one can clearly take a suitable direction h ∈ Γ(a k , t k ), while in the other case one can still proceed with h ∈ Γ ∞ (a k , t k ) to obtain different (but rather rough) estimates. Notation (6), (7) will be extensively used throughout the whole sequel.…”

Section: Preliminariesmentioning

confidence: 99%

Calculus for Directional Limiting Normal Cones and Subdifferentials

Benko

Gfrerer

Outrata

2018

Set-Valued Var. Anal

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The paper is devoted to the development of a comprehensive calculus for directional limiting normal cones, subdifferentials and coderivatives in finite dimensions. This calculus encompasses the whole range of the standard generalized differential calculus for (non-directional) limiting notions and relies on very weak (non-restrictive) qualification conditions having also a directional character. The derived rules facilitate the application of tools exploiting the directional limiting notions to difficult problems of variational analysis including, for instance, various stability and sensitivity issues. This is illustrated by some selected applications in the last part of the paper.

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“…Recently the concept of a directional limiting normal cone which is in general a smaller set than the limiting normal cone was introduced [16,10]. Based on the result for general set-valued maps in [10], Gfrerer and Klatte [14,Corollary 1] showed that metric subregularity holds for system (1) atx under the first-order sufficient condition for metric subregularity (FOSCMS): assuming P (x) is C 1 , if for each nonzero direction u satisfying ∇P (x)u ∈ T Λ (P (x)), there is no nonzero ζ such that…”

Section: Introductionmentioning

confidence: 99%

Directional Quasi-/Pseudo-Normality as Sufficient Conditions for Metric Subregularity

Bai¹,

Ye²,

Jin³

2019

SIAM J. Optim.

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In this paper we study sufficient conditions for metric subregularity of a set-valued map which is the sum of a single-valued continuous map and a locally closed subset. First we derive a sufficient condition for metric subregularity which is weaker than the so-called first-order sufficient condition for metric subregularity (FOSCMS) by adding an extra sequential condition. Then we introduce directional versions of quasi-normality and pseudo-normality which are stronger than the new weak sufficient condition for metric subregularity but weaker than the classical quasi-normality and pseudo-normality respectively. Moreover we introduce a nonsmooth version of the second-order sufficient condition for metric subregularity and show that it is a sufficient condition for the new sufficient condition for metric subregularity to hold. An example is used to illustrate that directional pseduo-normality can be weaker than FOSCMS. For the class of setvalued maps where the single-valued mapping is affine and the abstract set is the union of finitely many convex polyhedral sets, we show that pseudo-normality and hence directional pseudo-normality holds automatically at each point of the graph. Finally we apply our results to complementarity and the Karush-Kuhn-Tucker systems.

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Lipschitz and Hölder stability of optimization problems and generalized equations

Cited by 54 publications

References 28 publications

New Sharp Necessary Optimality Conditions for Mathematical Programs with Equilibrium Constraints

New Sharp Necessary Optimality Conditions for Mathematical Programs with Equilibrium Constraints

Calculus for Directional Limiting Normal Cones and Subdifferentials

Directional Quasi-/Pseudo-Normality as Sufficient Conditions for Metric Subregularity

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