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.

Keywords:

• augmented Lagrangian method
• Bregman distance
• Bregman projection
• cone constraint
• equilibrium problem
• proximal point method

AMS Subject Classifications::

• 90C30
• 90C48
• 65K15