Proximal Point Algorithms for Quasiconvex Pseudomonotone Equilibrium Problems

We propose a proximal point method for quasiconvex pseudomonotone equilibrium problems. The subproblems of the method are optimization problems whose objective is the sum of a strongly quasiconvex function plus the standard quadratic regularization term for optimization problems. We prove, under suitable additional assumptions, convergence of the generated sequence to a solution of the equilibrium problem, whenever the bifunction is strongly quasiconvex in its second argument, thus extending the validity of the convergence analysis of proximal point methods for equilibrium problems beyond the standard assumption of convexity of the bifunction in the second argument.