Enhanced Karush-Kuhn-Tucker Condition for Mathematical Programs with Equilibrium Constraints

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 argue that the strong stationary condition is unlikely for a mathematical program with equilibrium constraints with a nonsmooth objective function to hold at a local minimizer. We then focus on the study of the enhanced version of the Mordukhovich stationary condition, which is the next strongest optimality condition. We introduce the Mathematical Program with Equilibrium Constraints Generalized Pseudonormality, the Mathematical Program with Equilibrium Constraints Generalized Quasinormality, and the Mathematical Program with Equilibrium Constraints Constant Positive Linear Dependence. We show that the enhanced Mordukhovich stationary condition holds under these new constraint qualifications. Finally, we prove that either the Mathematical Program with Equilibrium Constrains Generalized Pseudonormality or the Mathematical Program with Equilibrium Constraints Generalized Quasinormality with regularity on the constraint functions and the set constraint implies the existence of a local error bound.