Inertial Iterative Self-Adaptive Step Size Extragradient-Like Method for Solving Equilibrium Problems in Real Hilbert Space with Applications

Wiyada Kumam,Kanikar Muangchoo

doi:10.3390/axioms9040127

Wiyada Kumam, Kanikar Muangchoo

Open Access

https://doi.org/10.3390/axioms9040127

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Abstract

A number of applications from mathematical programmings, such as minimization problems, variational inequality problems and fixed point problems, can be written as equilibrium problems. Most of the schemes being used to solve this problem involve iterative methods, and for that reason, in this paper, we introduce a modified iterative method to solve equilibrium problems in real Hilbert space. This method can be seen as a modification of the paper titled “A new two-step proximal algorithm of solving the problem of equilibrium programming” by Lyashko et al. (Optimization and its applications in control and data sciences, Springer book pp. 315–325, 2016). A weak convergence result has been proven by considering the mild conditions on the cost bifunction. We have given the application of our results to solve variational inequality problems. A detailed numerical study on the Nash–Cournot electricity equilibrium model and other test problems is considered to verify the convergence result and its performance.

Highlights

An equilibrium problem (EP) is a generalized concept that unifies several mathematical problems, such as the variational inequality problems, minimization problems, complementarity problems, the fixed point problems, non-cooperative games of Nash equilibrium, the saddle point problems and scalar and vector minimization problems
We consider the concept of an equilibrium problem introduced by Blum and Oettli in [1]
A equilibrium problem regarding f on the set C is defined in the following way: Find p ∈ C such that f ( p, v) ≥ 0, for all v ∈ C

Summary

Introduction

An equilibrium problem (EP) is a generalized concept that unifies several mathematical problems, such as the variational inequality problems, minimization problems, complementarity problems, the fixed point problems, non-cooperative games of Nash equilibrium, the saddle point problems and scalar and vector minimization problems (see e.g., [1,2,3]). A equilibrium problem regarding f on the set C is defined in the following way: Find p ∈ C such that f ( p, v) ≥ 0, for all v ∈ C. Lyashko et al [25] in 2016 introduced an improvement of the method (2) to solve equilibrium problem and sequence {un } was generated in the following way: u0 , v0 ∈ C, vn+1 = arg min{λ f (vn , y) + 12 kun+1 − yk : y ∈ C },. A few other formulations of the problem of variational inequalities are discussed, and many computational examples in finite and infinite dimensions spaces are presented to demonstrate the applicability of our proposed results.

Preliminaries

Main Results

Applications

Computational Illustration

Conclusions