Gap functions for nonsmooth equilibrium problems

Abstract

We consider equilibrium problems (EP) with directionally differentiable (not necessarily C1) bifunctions which are convex with respect to the second variable and we use a gap function approach to solve them. In the first part of the paper we establish a condition under which any stationary point of the gap function solves (EP) and we propose a solution method which uses descent directions of the gap function. In the final section we study the problem when this condition is not satisfied. In this case we use a family of gap functions depending on a parameter α which allows us to overcome the trouble due to the lack of a descent direction.