Optimality conditions for nonsmooth equilibrium problems via Hadamard directional derivative

In this paper, we consider a mathematical program with equilibrium constraints (MPEC) formulated as a mathematical program with complementarity constraints. We obtain necessary conditions of Fritz John (FJ) and Karush-Kuhn-Tucker (KKT) types for a nonsmooth (MPEC) problem in terms of the lower Hadamard directional derivative. In particular sufficient conditions for MPECs are given where the involved functions have pseudoconvex sublevel sets. The functions with pseudoconvex sublevel sets is a class of generalized convex functions that include quasiconvex functions.

Equilibrium problems · Optimality conditions · Constraint qualifications · Hadamard directional derivative