Optimality conditions for mathematical programs with equilibrium constraints using directional convexificators

Abstract

We consider a nonsmooth and nonconvex mathematical program with equilibrium constraints (MPEC) where its functions are not necessarily locally Lipschitz/continuous. In exploiting the idea of directional convexificators introduced by Dempe and Pelicka [Necessary optimality conditions for optimistic bilevel programming problems using set-valued programming. J Global Optim. 2015;61:769–788], we define two forms of nonsmooth versions of the classical Abadie constraint qualifications, and introduce nonsmooth stationarity conditions, which are based on directional convexificators. Then, we employ these conditions to derive first-order optimality conditions subject to generalized Abadie constraint qualifications. Finally, we establish sufficient optimality conditions for MPEC under a new assumption of generalized convexity.