Bilevel Optimization and Variational Analysis

Mordukhovich, Boris S.

doi:10.48550/arxiv.1907.06140

Cited by 1 publication

(1 citation statement)

References 29 publications

(62 reference statements)

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“…Instead of using y ∈ S(x) as a constraint in (BP), Outrata [25] proposed to replace it with f (x, y) − V (x) = 0, y ∈ Y (x) where V (x) := inf y∈Y (x) f (x, y) is the value function of the lower level program for a numerical consideration. This so-called value function approach was further used in Ye and Zhu [32] and later in other papers such as [7,8,21,22,23,24] to derive various necessary optimality conditions under the partial calmness condition. The partial calmness condition for the value function reformulation of the bilevel program, however, is a very strong assumption.…”

Section: Introductionmentioning

confidence: 99%

Relaxed constant positive linear dependence constraint qualification and its application to bilevel programs

Xu¹,

Ye²

2019

Preprint

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Relaxed constant positive linear dependence constraint qualification (RCPLD) for a system of smooth equalities and inequalities is a constraint qualification that is weaker than the usual constraint qualifications such as Mangasarian Fromovitz constraint qualification and the linear constraint qualification. Moreover RCPLD is known to induce an error bound property. In this paper we extend RCPLD to a very general feasibility system which may include Lipschitz continuous inequality constraints, complementarity constraints and abstract constraints. We show that this RCPLD for the general system is a constraint qualification for the optimality condition in terms of limiting subdifferential and limiting normal cone and it is a sufficient condition for the error bound property under the strict complementarity condition for the complementarity system and Clarke regularity conditions for the inequality constraints and the abstract constraint set. Moreover we introduce and study some sufficient conditions for RCPLD including the relaxed constant rank constraint qualification (RCRCQ). Finally we apply our results to the bilevel program.

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