A proximal point method for quasi-equilibrium problems in Hilbert spaces

Abstract

In this paper, we study the convergence of a proximal point method for solving quasi-equilibrium problems (QEP) in Hilbert spaces. We extent the method proposed by Moudafi [Proximal point algorithm extended to equilibrium problems. J Nat Geom. 1999;15(1-2):91–100] and Iusem and Sosa [Iterative algorithms for equilibrium problems. Optimization. 2003;52(3):301–316] to the more general context of quasi-equilibrium problems. In our method a quasi-equilibrium problem is solved by computing a solution of an equilibrium problem at each iteration. We obtain weak convergence of the sequence to a solution of the QEP under some mild assumptions. Some encouraging numerical experiments are presented to show the performance of the method.