On Directional Metric Regularity, Subregularity and Optimality Conditions for Nonsmooth Mathematical Programs

Gfrerer, Helmut

doi:10.1007/s11228-012-0220-5

Cited by 111 publications

(111 citation statements)

References 31 publications

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“…For the properties of the cones T A (x),N A (x) and N A (x) from Definition 2 and generalized derivatives (i), (ii) and (iii) from Definition 3 we refer the interested reader to the monographs [18] and [15]. The directional limiting normal cone and coderivative were introduced by the first author in [6] and various properties of these objects can be found also in [10] and the references therein. Note that D * F (ū,v) = D * F ((ū,v); (0, 0)) and that dom D * F ((ū,v); (d, h)) = ∅ whenever h ∈ DF (ū,v)(d).…”

Section: Problem Formulation and Preliminariesmentioning

confidence: 99%

On the Aubin property of solution maps to parameterized variational systems with implicit constraints

Gfrerer

Outrata

2019

Optimization

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In the paper a new sufficient condition for the Aubin property to a class of parameterized variational systems is derived. In these systems the constraints depend both on the parameter as well as on the decision variable itself and they include, e.g., parameterdependent quasi-variational inequalities and implicit complementarity problems. The result is based on a general condition ensuring the Aubin property of implicitly defined multifunctions which employs the recently introduced notion of the directional limiting coderivative. Our final condition can be verified, however, without an explicit computation of these coderivatives. The procedure is illustrated by an example.

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Section: Problem Formulation and Preliminariesmentioning

confidence: 99%

On the Aubin property of solution maps to parameterized variational systems with implicit constraints

Gfrerer

Outrata

2019

Optimization

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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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“…Over the last fifteen years or so, some results for characterizing metric subregularity/calmness for general set-valued maps have been obtained; see, e.g., [19,20,21,22,59]. 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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On Directional Metric Regularity, Subregularity and Optimality Conditions for Nonsmooth Mathematical Programs

Cited by 111 publications

References 31 publications

On the Aubin property of solution maps to parameterized variational systems with implicit constraints

On the Aubin property of solution maps to parameterized variational systems with implicit constraints

New Sharp Necessary Optimality Conditions for Mathematical Programs with Equilibrium Constraints

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

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