A direct proof for M-stationarity under MPEC-GCQ for mathematical programs with equilibrium constraints

Flegel, Michael L.; Kanzow, Christian

doi:10.1007/0-387-34221-4_6

Cited by 24 publications

(24 citation statements)

References 21 publications

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“…It is interesting to note that the characterizations of strongly stationary and Mordukhovich stationary points can be obtained in such a unified way only by invoking the cones Ω 1 and Ω 2 . This fundamental idea is borrowed from [12], where a rather direct proof was given to show that any local minimizer of (MPCC) is a Mordukhovich stationary point under MPCC − GCQ. In [12], statement (v) was obtained by invoking Proposition 1 of [24] to show that the polyhedral set-valued map M defined by (12) is calm at (0, 0), while in our proof, statement (v) follows from Lemma 3.1, which can be proved without using Proposition 1 of [24]; see the appendix for a detailed proof of Lemma 3.1.…”

Section: Ii)x Is a Mordukhovich Stationary Point Of (Mpcc) If And Onlmentioning

confidence: 99%

“…, m+q+l} are assumed to be continuously differentiable. Stationarity (or first-order optimality) conditions for (MPCC) have been the subject of many recent papers and books; see [27,28,18,19,34,23,9,10,12]. Since there are several different approaches for deriving optimality conditions, various stationarity conditions arise; see a very recent thesis [8] for their definitions and connections.…”

Section: Introductionmentioning

confidence: 99%

“…As for various CQs ensuring Mordukhovich stationarity, we refer the reader to [34,10] and [8] for more details. Motivated by the work reported in [12], section 3 starts with a summary of characterizations of strong and Mordukhovich stationarities. Then, we apply the results obtained in section 2 to (MPCC) to derive sufficient conditions respectively for Mordukhovich stationarity by means of lower order exact penalty functions G p and H p , and for strong stationarity by means of lower order exact penalty functions G p (when 0 ≤ p < 1).…”

Section: Introductionmentioning

confidence: 99%

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Optimality Conditions via Exact Penalty Functions

Meng¹,

Yang²

2010

SIAM J. Optim.

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Abstract. In this paper, we study KKT optimality conditions for constrained nonlinear programming problems and strong and Mordukhovich stationarities for mathematical programs with complementarity constraints using lp penalty functions, with 0 ≤ p ≤ 1. We introduce some optimality indication sets by using contingent derivatives of penalty function terms. Some characterizations of optimality indication sets are obtained by virtue of the original problem data. We show that the KKT optimality condition holds at a feasible point if this point is a local minimizer of some lp penalty function with p belonging to the optimality indication set. Our result on constrained nonlinear programming includes some existing results from the literature as special cases.

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Section: Ii)x Is a Mordukhovich Stationary Point Of (Mpcc) If And Onlmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Optimality Conditions via Exact Penalty Functions

Meng¹,

Yang²

2010

SIAM J. Optim.

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“…i=l Using the expression of the normal cone to the sets A;, i = 1, ... , d, from [21] (cf. also [45,56] for particular cases), we get …”

Section: If In Addition (Av Is Satisfied At (X Y) Then Sc Is Lipschmentioning

confidence: 99%

Sensitivity Analysis for Two-Level Value Functions with Applications to Bilevel Programming

2012

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Abstract. This paper contributes to a deeper understanding of the link between a now conventional framework in hierarchical optimization spread under the name of the optimistic bilevel problem and its initial more difficult formulation that we call here the original optimistic bilevel optimization problem. It follows from this research that, although the process of deriving necessary optimality conditions for the latter problem is more involved, the conditions themselves do notto a large extent-differ from those known for the conventional problem. It has been already well recognized in the literature that for optimality conditions of the usual optimistic bilevel program appropriate coderivative constructions for the set-valued solution map of the lower-level problem could be used, while it is shown in this paper that for the original optimistic formulation we have to go a step further to require and justify a certain Lipschitz-like property of this map. This occurs to be related to the local Lipschitz continuity of the optimal value function of an optimization problem constrained by solutions to another optimization problem; this function is labeled here as the twolevel value function. More generally, we conduct a detailed sensitivity analysis for value functions of mathematical programs with extended complementarity constraints. The results obtained in this vein are applied to the two-level value function and then to the original optimistic formulation of the bilevel optimization problem, for which we derive verifiable stationarity conditions of various types entirely in terms of the initial data.

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“…The advantage of this concept consists, above all, in the fact that it requires only very weak constraint qualifications, cf. [5,29]. Simultaneously, starting with [13], another, stronger, stationarity notion has been investigated [18,24,27], referred to by the moniker strong stationarity in [22].…”

mentioning

confidence: 99%

Optimality Conditions for Disjunctive Programs with Application to Mathematical Programs with Equilibrium Constraints

2006

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We consider optimization problems with a disjunctive structure of the feasible set. Using Guignard-type constraint qualifications for these optimization problems and exploiting some results for the limiting normal cone by Mordukhovich, we derive different optimality conditions. Furthermore, we specialize these results to mathematical programs with equilibrium constraints. In particular, we show that a new constraint qualification, weaker than any other constraint qualification used in the literature, is enough in order to show that a local minimum results in a so-called M-stationary point. Additional assumptions are also discussed which guarantee that such an M-stationary point is in fact a strongly stationary point.Key words disjunctive programs · mathematical programs with equilibrium constraints · M-stationarity · strong stationarity · Guignard constraint qualification. Mathematics Subject Classifications (2000) 90C30 · 90C46

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A direct proof for M-stationarity under MPEC-GCQ for mathematical programs with equilibrium constraints

Cited by 24 publications

References 21 publications

Optimality Conditions via Exact Penalty Functions

Optimality Conditions via Exact Penalty Functions

Sensitivity Analysis for Two-Level Value Functions with Applications to Bilevel Programming

Optimality Conditions for Disjunctive Programs with Application to Mathematical Programs with Equilibrium Constraints

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