Enhanced Karush–Kuhn–Tucker Conditions for Mathematical Programs with Equilibrium Constraints
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In this paper, we study necessary optimality conditions for nonsmooth mathematical programs with equilibrium constraints. We first show that, unlike the smooth case, the mathematical program with equilibrium constraints linear independent constraint qualification is not a constraint qualification for the strong stationary condition when the objective function is nonsmooth. We then focus on the study of the enhanced version of the Mordukhovich stationary condition, which is a weaker optimality condition than the strong stationary condition. We introduce the quasi-normality and several other new constraint qualifications and show that the enhanced Mordukhovich stationary condition holds under them. Finally, we prove that quasi-normality with regularity implies the existence of a local error bound.
KeywordsEnhanced Karush–Kuhn–Tucker conditions Constraint qualification Mathematical program with equilibrium constraints Local error bound
The first author’s work was supported in part by NSERC. The authors are grateful to the four anonymous referees and Professor Franco Giannessi for their helpful suggestions and comments.
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