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A relaxed proximal point algorithm with double-inertial effects for nonconvex equilibrium problems

Nam Van Tran

TL;DR

This work addresses nonconvex equilibrium problems defined by a bifunction $f$ with $f(x,x)=0$ and seeks solutions in $S(C;f)$. It introduces a relaxed proximal point algorithm with double inertial extrapolation (RTIPPA-EP) that leverages strong quasiconvexity to accelerate convergence beyond convex settings. The authors establish global convergence of the generated sequence to a solution of the equilibrium problem under standard regularity assumptions and demonstrate practical speedups through numerical experiments against existing one-step inertial methods. The results extend proximal point methods to nonconvex equilibrium problems and provide a computationally efficient approach for broader applications in optimization and game theory.

Abstract

In this paper, we present a relaxation proximal point method with double inertial effects to approximate a solution of a non-convex equilibrium problem. We give global convergence results of the iterative sequence generated by our algorithm. Some known results are recovered as special cases of our results. Numerical test is given to support the theoretical findings.

A relaxed proximal point algorithm with double-inertial effects for nonconvex equilibrium problems

TL;DR

This work addresses nonconvex equilibrium problems defined by a bifunction with and seeks solutions in . It introduces a relaxed proximal point algorithm with double inertial extrapolation (RTIPPA-EP) that leverages strong quasiconvexity to accelerate convergence beyond convex settings. The authors establish global convergence of the generated sequence to a solution of the equilibrium problem under standard regularity assumptions and demonstrate practical speedups through numerical experiments against existing one-step inertial methods. The results extend proximal point methods to nonconvex equilibrium problems and provide a computationally efficient approach for broader applications in optimization and game theory.

Abstract

In this paper, we present a relaxation proximal point method with double inertial effects to approximate a solution of a non-convex equilibrium problem. We give global convergence results of the iterative sequence generated by our algorithm. Some known results are recovered as special cases of our results. Numerical test is given to support the theoretical findings.

Paper Structure

This paper contains 7 sections, 7 theorems, 54 equations, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Algorithm and Convergence Analysis
  4. Proposed Method
  5. Convergence Analysis
  6. Numerical Experiment
  7. Conclusion

Key Result

Lemma 2.1

(JOV) Let $C$ be a closed and convex set in $\mathbb{R}^n$, $g:~\mathbb{R}^n \longrightarrow \overline{\mathbb{R}}$ be a proper, lower semicontinuous, strongly quasiconvex function with modulus $\gamma>0$ and such that $C\subseteq \hbox{\rm dom}\ g$, $\lambda>0$ and $x\in C$. If $\hat{x}\in \hbox{\r

Theorems & Definitions (11)

  • Lemma 2.1
  • Lemma 2.2
  • Remark 3.1
  • Proposition 3.3
  • Proposition 3.4
  • Corollary 3.5
  • Lemma 3.6
  • proof
  • Theorem 3.7
  • proof
  • ...and 1 more