A proximal augmented Lagrangian method for equilibrium problems

Abstract

Considering a recently proposed proximal point method for equilibrium problems, we construct an augmented Lagrangian method for solving the same problem in reflexive Banach spaces with cone constraints generating a strongly convergent sequence to a certain solution of the problem. This is an inexact hybrid method meaning that at a certain iterate, a solution of an unconstrained equilibrium problem is found, allowing a proper error bound, followed by a Bregman projection of the initial iterate onto the intersection of two appropriate halfspaces. Assuming a set of reasonable hypotheses, we provide a full convergence analysis.

Abstract regularized equilibria: application to Becker’s household behavior theory

“…(1) This problem will be denoted by E P(F, K ) and its solution set by S(F, K ). Equilibrium problems have been widely studied both on the issue dealing with conditions for existence of solutions (e.g., Iusem et al 2009;Iusem and Sosa 2003;Konnov and Dyabilkin 2011;Chadli et al 2015) and algorithmic purposes (see, for instance Bento et al 2016;Burachik and Kassay 2012;CruzNeto et al 2019;Flåm and Antipin 1996;Iusem and Nasri 2007;Iusem and Sosa 2010;Konnov 2003;Langenberg 2012;Mordukhovich et al 2012;Moudafi 2007Moudafi , 1999Nguyen et al 2009;Oliveira et al 2013;Hieu 2018;Mashreghi and Nasri 2012).…”