Weak convergence of explicit extragradient algorithms for solving equilibirum problems

Abstract

This paper aims to propose two new algorithms that are developed by implementing inertial and subgradient techniques to solve the problem of pseudomonotone equilibrium problems. The weak convergence of these algorithms is well established based on standard assumptions of a cost bi-function. The advantage of these algorithms was that they did not need a line search procedure or any information on Lipschitz-type bifunction constants for step-size evaluation. A practical explanation for this is that they use a sequence of step-sizes that are updated at each iteration based on some previous iterations. For numerical examples, we discuss two well-known equilibrium models that assist our well-established convergence results, and we see that the suggested algorithm has a competitive advantage over time of execution and the 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…”

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

“…The development of new iterative methods and the examination of their converging analysis are among the most effective and valuable research directions in equilibrium theory. Several numerical results for solving the problem of equilibrium in different abstract spaces have been established (for instance, see [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]).…”

Inertial Iterative Schemes with Variable Step Sizes for Variational Inequality Problem Involving Pseudomonotone Operator

“…The Variational Inequality (VI) problem Equation (1) is a fundamental problem in optimization theory which is applied in many areas of study, such as transportation problems, equilibrium, economics, engineering and so on (Refs. [1][2][3][4][5][6][7][8][9][10][11][12][13][14]). There are many approaches to the VI problem, the basic one being the regularization and projection method.…”