Iterative Schemes for Uniformly Nonconvex Equilibrium Problems

TL;DR

This paper introduces new implicit iterative methods for solving uniformly regular equilibrium problems on nonconvex sets, providing convergence analysis and descent schemes that improve upon previous results.

Contribution

It proposes novel implicit algorithms based on auxiliary and inertial proximal methods with convergence guarantees for uniformly regular equilibrium problems.

Findings

New implicit methods with proven convergence.

Construction of gap functions for descent schemes.

Results extend to variational inequalities and complementarity problems.

Abstract

Uniformly regular equilibrium problems are natural generalizations of abstract equilibrium prob lems and they are defined over the uniformly prox-regular nonconvex sets. Some new efficient implicit methods for solving uniformly regular equilibrium problems are analyzed by the aux iliary principle and inertial proximal methods. The convergence analysis of the new proposed methods is considered under some mild conditions. Gap functions are constructed to suggest some descent-type scheme for uniformly regular equilibrium problems. Our results can be viewed as significant refinements and improvements of the previously known results and they continue to hold for equilibrium problems, variational inequalities and complementarity problems as well.