A Bundle Method for Solving Equilibrium Problems

We present a bundle method for solving nonsmooth convex equilibrium problems based on the auxiliary problem principle. First, we consider a general algorithm that we prove to be convergent. Then we explain how to make this algorithm implementable. The strategy is to approximate the nonsmooth convex functions by piecewise linear convex functions in such a way that the subproblems are easy to solve and the convergence is preserved. In particular, we introduce a stopping criterion which is satisfied after finitely many iterations and which gives rise to delta-stationary points. Finally, we apply our implementable algorithm for solving the particular case of singlevalued and multivalued variational inequalities and we find again the results obtained recently by Salmon