ON THE BREGMAN INEXACT PROXIMAL INTERIOR POINT ALGORITHM FOR ABSTRACT PSEUDOMONOTONE EQUILIBRIUM PROBLEMS

Using a slightly modified concept of Bregman functions, we suggest a Bregman proximal interior point method (BPIPA for brevity) for solving equilibrium problems with pseudomonotone bifunctions over a convex and non-polyhedral set. We establish the weak convergence of our algorithm for pseudomonotone bifunctions satisfying a cutting plane property (the so-called pseudomonotonicity*) extending henceforth the result by N. Langenberg [41] [Pseudomonotone operators and the Bregman Proximal Point Algorithm, J. Glob. Optim. 47 (2010), 537-555] in finite dimensional Euclidian spaces to real-valued bifunctions in infinite dimensional Hilbert spaces. Our algorithm is of inexact type in the sense that exact solutions of subproblems are not needed, instead we propose a summability criterion of errors vectors to conclude the required convergence. Examples of bifunctions satisfying the assumptions of our algorithm are finally discussed in the context of quasiconvex programming and hemivariational inequalities as well as elliptic partial differential equations applicable to physics and economy.