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

Valerian-Alin Fodor, Nicolae Popovici

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.

Reducing the complexity of equilibrium problems and applications to 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.
Paper Structure (3 sections, 15 theorems, 79 equations, 1 figure)

This paper contains 3 sections, 15 theorems, 79 equations, 1 figure.

Table of Contents

  1. Introduction and preliminaries
  2. The inverse images of the metric projection
  3. Equilibrium problems

Key Result

Theorem 1.1

Every compact convex set in $\mathbb{R}^n$ is the convex hull of its extreme points.

Figures (1)

  • Figure 1: Proposition \ref{['partitionofR^nbynormalcones']} applied for the particular case of a square in $\mathbb{R}^2$

Theorems & Definitions (37)

  • Theorem 1.1: Minkowski (Krein-Milman)
  • Theorem 1.2: Straszewicz
  • Definition 1.1
  • Remark 1.1
  • Definition 1.2
  • Proposition 1.1
  • Theorem 1.3: existence of elements of best approximation
  • Theorem 1.4: unicity of the element of best approximation
  • Theorem 1.5: characterization of elements of best approximation
  • Corollary 1.1
  • ...and 27 more