Optimality and duality for nonsmooth mathematical programming problems with equilibrium constraints

In this paper, we construct a Wolfe and Mond-Weir types dual problem in terms of contingent epiderivatives for nonsmooth mathematical programming problems with equilibrium constraints (NMPEC) in real Banach spaces. First, we establish some strong and weak duality theorems for the original problem and its dual problem under suitable assumptions on the pseudo-convexity of objective and constraint functions at the point under consideration. We also impose a regularity condition of the (RC) type to have strong duality theorems using both the contingent epiderivative and the contingent hypoderivative. Second, we provide various types of sufficient optimality conditions for the (NMPEC) problem, where either the objective and constraint functions are pseudo-convex at the point under consideration, or the objective function is strict quasi-convex and the constraint functions are quasi-convex at the point under consideration. Some illustrative examples also provided for our findings.