The Inertial Sub-Gradient Extra-Gradient Method for a Class of Pseudo-Monotone Equilibrium Problems

Abstract

In this article, we focus on improving the sub-gradient extra-gradient method to find a solution to the problems of pseudo-monotone equilibrium in a real Hilbert space. The weak convergence of our method is well-established based on the standard assumptions on a bifunction. We also present the application of our results that enable to solve numerically the pseudo-monotone and monotone variational inequality problems, in addition to the particular presumptions required by the operator. We have used various numerical examples to support our well-proved convergence results, and we can show that the proposed method involves a considerable influence over-running time and the total number of iterations.

Inertial Extra-Gradient Method for Solving a Family of Strongly Pseudomonotone Equilibrium Problems in Real Hilbert Spaces with Application in Variational Inequality Problem

“…From the expression (23), (25) and (29) implies that the sequences {u n }, {w n } and {v n } are bounded, and for each u * ∈ EP( f , C), the lim n→∞ u n − u * 2 , lim n→∞ v n − u * 2 , lim n→∞ w n − u * 2 exists. Next, we are going to prove that the sequence {u n } strongly converges to u * .…”

“…Furthermore, Konnov [12] also provides a different variant of the proximal point method with weaker assumptions in the case of equilibrium problems. Several numerical methods based on these techniques have been developed to solve different classes of equilibrium problems in finite and infinite-dimensional abstract spaces (for more details see, [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]). More specifically, Hieu et al in [30] developed an iterative sequence sequence {u n } recursively as…”

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

“…Many researchers have provided and generalized many results corresponding to the existence of a solution for the equilibrium problem (see [7][8][9][10]). A considerable number of methods are the earliest set up over the last few years concentrating on the different equilibrium problem classes and other particular forms of an equilibrium problem in abstract spaces (see [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]).…”

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].…”