Proximal point method with Bregman distance for quasiconvex pseudomonotone equilibrium problems

Abstract

In this paper, we propose a proximal point method by using Bregman distance to solve the quasiconvex pseudomonotone equilibrium problems. Under suitable assumptions, we prove that the proposed algorithm is well defined and the sequence generated by it converges to a solution of the equilibrium problem, whenever the bifunction is strongly quasiconvex in its second argument. Our method goes beyond the usual assumption of the bifunction's convexity in the second argument, extending the validity of the convergence analysis of proximal point methods for equilibrium problems. For a particular choice of the Bregman function, our method reduces to the traditional proximal point method. At the end, we illustrate our algorithm and convergence result by numerical examples.