Bilevel Optimization as a Regularization Approach to Pseudomonotone Equilibrium Problems

Abstract

We study properties of an inexact proximal point method for pseudomonotone equilibrium problems in real Hilbert spaces. Unlike monotone problems, in pseudomonotone problems, the regularized subproblems may not be strongly monotone, even not pseudomonotone. However, we show that every inexact proximal trajectory weakly converges to the same limit. We use these properties to extend a viscosity-proximal point algorithm developed in [28 A. Tada and W. Takahashi ( 2007 ). Weak and strong convergence theorem for nonexpansive mapping and equilibrium problem . J. Optim. Theory and Appl. 133 : 359 – 370 .[Crossref], [Web of Science ®] , [Google Scholar]] to pseudomonotone equilibrium problems. Then we propose a hybrid extragradient-cutting plane algorithm for approximating the limit point by solving a bilevel strongly convex optimization problem. Finally, we show that by using this bilevel convex optimization, the proximal point method can be used for handling ill-possed pseudomonotone equilibrium problems.