A Self-Adaptive Extra-Gradient Methods for a Family of Pseudomonotone Equilibrium Programming with Application in Different Classes of Variational Inequality Problems

Abstract

The main objective of this article is to propose a new method that would extend Popov’s extragradient method by changing two natural projections with two convex optimization problems. We also show the weak convergence of our designed method by taking mild assumptions on a cost bifunction. The method is evaluating only one value of the bifunction per iteration and it is uses an explicit formula for identifying the appropriate stepsize parameter for each iteration. The variable stepsize is going to be effective for enhancing iterative algorithm performance. The variable stepsize is updating for each iteration based on the previous iterations. After numerical examples, we conclude that the effect of the inertial term and variable stepsize has a significant improvement over the processing time and number of iterations.

Modified Popov's Subgradient Extragradient Algorithm With Inertial Technique for Equilibrium Problems and Its Applications

“…One of the most interesting and effective areas of research in equilibrium problem theory is the development of new iterative methods, the improvement of existing methods, and the examination of their convergence analysis. Several methods have already been used in recent years to estimate the solution of the problem of equilibrium in both finite and infinite-dimensional spaces, i.e., the extragradient methods [6,7,8,9,9,10,11,12,13,14,15,16] and others in [17,18,19,20,21,22,23,24,25,26].…”

Convergence Analysis of Self-Adaptive Inertial Extra-Gradient Method for Solving a Family of Pseudomonotone Equilibrium Problems with Application

“…The construction of new iterative schemes and the modification of existing methods, as well as the study their convergence analysis, constitute an important research direction in equilibrium problem theory. Several methods have been developed in the past few years to approximate the solution of an equilibrium problem in finite and infinite dimensional real Hilbert spaces, i.e., extragradient methods [7][8][9][10][11][12][13][14][15][16], subgradient methods [17][18][19][20][21][22], inertial methods [23][24][25] and methods for particular classes of equilibrium problems [26][27][28][29][30][31][32][33][34][35].…”

A Weak Convergence Self-Adaptive Method for Solving Pseudomonotone Equilibrium Problems in a Real Hilbert Space

“…Many methods have been well-established to figure out the solution of an equilibrium problem (1) in finite and infinite-dimensional spaces. Some of these algorithms involve projection methods [5][6][7][8], the proximal point methods [9,10], the extragradient methods with or without line searches [11][12][13][14][15][16][17][18], the methods using the inertial effect [19][20][21][22] and other methods in [23][24][25][26][27][28].…”