Reducing the complexity of equilibrium problems and applications to best approximation problems

TL;DR

This paper simplifies the analysis of scalar equilibrium problems using convex analysis, enabling better understanding and solutions for related variational inequalities and approximation tasks.

Contribution

It introduces a new characterization of equilibrium solutions via extreme points under generalized convexity assumptions.

Findings

Solutions can be characterized by extreme or exposed points.

Applicable to variational inequalities and optimization problems.

Enhances understanding of best approximation problems.

Abstract

We consider scalar equilibrium problems governed by a bifunction in a finite-dimensional framework. By using classical arguments in Convex Analysis, we show that under suitable generalized convexity assumptions imposed on the bifunction, the solutions of the equilibrium problem can be characterized by means of extreme or exposed points of the feasible domain. Our results are relevant for different particular instances, such as variational inequalities and optimization problems, especially for best approximation problems.