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

In this paper we study necessary optimality conditions for nonsmooth mathematical programs with equilibrium constraints (MPECs). We first show that MPEC-LICQ is not a constraint qualification for the strong (S-) stationary condition when the objective function is nonsmooth. Enhanced Fritz John conditions provide stronger necessary optimality conditions under weaker constraint qualifications. In this paper we derive the enhanced Fritz John Mordukhovich (M-) stationary condition for nonsmooth MPECs. From this enhanced Fritz John M-stationary condition we introduce the associated MPEC generalized pseudonormality and quasinormality condition and build the relations between them and some other widely used MPEC constraint qualifications. At last we prove that either MPEC generalized pseudonormality or quasinormality with regularity on the constraint functions and the set constraint implies the existence of a local error bound.