GAP FUNCTIONS FOR NONSMOOTH EQUILIBRIUM PROBLEMS

Abstract

We consider equilibrium problems (EP) with directionally differentiable (not necessarily ${\mathcal C}^1$) 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 $\alpha$ which allows us to overcome the trouble due to the lack of a descent direction.