Proximal point method with Bregman distance for nonconvex quasi-equilibrium problems

Abstract

We introduce a Bregman proximal point method for solving pseudomonotone quasi-equilibrium problems. The algorithm is formulated using Bregman distances generated by Legendre functions and allows the bifunction to be nonconvex. Unlike existing Euclidean proximal approaches, our method is formulated using Bregman distances, allowing the algorithm to exploit problem-dependent geometries and ensuring well-posedness and convergence under weaker assumptions. In addition, we analyse a regularized Bregman proximal point method and show that it converges to the unique solution of the quasi-equilibrium problem. This analysis/approach significantly broadens the applicability of proximal methods to nonconvex quasi-equilibrium problems and is illustrated through applications to quasi-optimization problems and generalized Nash equilibrium models, including Nash-Cournot type models.