A Class of Novel Mann-Type Subgradient Extragradient Algorithms for Solving Quasimonotone Variational Inequalities

Nopparat Wairojjana,Ioannis K Argyros,Meshal Shutaywi,Wejdan Deebani,Christopher I Argyros

doi:10.3390/sym13071108

Abstract

Symmetries play an important role in the dynamics of physical systems. As an example, quantum physics and microworld are the basis of symmetry principles. These problems are reduced to solving inequalities in general. That is why in this article, we study the numerical approximation of solutions to variational inequality problems involving quasimonotone operators in an infinite-dimensional real Hilbert space. We prove that the iterative sequences generated by the proposed iterative schemes for solving variational inequalities with quasimonotone mapping converge strongly to some solution. The main advantage of the proposed iterative schemes is that they use a monotone and non-monotone step size rule based on operator knowledge rather than a Lipschitz constant or some line search method. We present a number of numerical experiments for the proposed algorithms.

Highlights

Our main concern here is to study the different iterative algorithms that are used to evaluate the numerical solution of the variational inequality problem (shortly, (1)) involving quasimonotone operators in any real Hilbert space E
In order to prove the strong convergence, it is assumed that the following conditions are satisfied: (T 1) The solution set for the problem (1) is denoted by V I(K, T ) and it is nonempty; (T 2) A mapping T : E → E is said to be quasimonotone if
We prove that the proposed iterative sequence generated by the subgradient extragradient algorithms for solving variational inequalities involving quasimonotone operators converges strongly to a solution

Summary

Introduction

Our main concern here is to study the different iterative algorithms that are used to evaluate the numerical solution of the variational inequality problem (shortly, (1)) involving quasimonotone operators in any real Hilbert space E. (T 4) A mapping T : E → E is said to be weakly sequentially continuous if {T (un)} weakly converges to T (u) for each sequence {un} that weakly converges to u It is well-established that the problem (1) is a key problem in nonlinear analysis. It is an important mathematical model that incorporates many important topics in pure and applied mathematics, such as a nonlinear system of equations, optimization conditions for problems with the optimization process, complementarity problems, network equilibrium problems and finance (see [3–13] and others in [14–19]) As a result, this problem has a number of applications in engineering, mathematical programming, network economics, transportation analysis, game theory and software engineering. The basis of symmetry principles, for example, is quantum physics and the microworld These problems are reduced to solving inequalities in general

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