Two strongly convergent self-adaptive iterative schemes for solving pseudo-monotone equilibrium problems with applications

Nuttapol Pakkaranang,Wiyada Kumam,Habib Ur Rehman

doi:10.1515/dema-2021-0030

Abstract

Abstract The aim of this paper is to propose two new modified extragradient methods to solve the pseudo-monotone equilibrium problem in a real Hilbert space with the Lipschitz-type condition. The iterative schemes use a new step size rule that is updated on each iteration based on the value of previous iterations. By using mild conditions on a bi-function, two strong convergence theorems are established. The applications of proposed results are studied to solve variational inequalities and fixed point problems in the setting of real Hilbert spaces. Many numerical experiments have been provided in order to show the algorithmic performance of the proposed methods and compare them with the existing ones.

Highlights

Assume that is a convex subset of a real Hilbert space
We provide a positive answer to the above question, that is, the gradient method provides a strong convergence sequence by using a non-monotonic step size rule for solving equilibrium problem (EP) accompanied with pseudo-monotone bi-functions
Motivated by the works of Censor et al [32] and Wang et al [31] we introduce a new gradient-type method to figure out the problem (EP) in the setting of an infinite-dimensional real Hilbert space

Summary

Introduction

Assume that is a convex subset of a real Hilbert space. Suppose that f : × → satisfying f (y1, y1) = 0 for each y1 ∈ and the equilibrium problem (EP) [1,2] for f on is defined in the following manner: Find x∗ ∈ such that f (x∗, y1) ≥ 0, for all y1 ∈. The proximal method [28] is an effective method to solve EPs and need to solve minimization problems on each iterative step This method was known as the two-step extragradient method in [29] due to the previous contribution of Korpelevich extragradient method [30] to solve the saddle point problems. 2c1 2c2 Recently, Wang et al in [31] introduced a non-convex combination iterative method to solve pseudomonotone EPs. Strong convergence of iterative sequences is the main contribution of the proposed method. We provide a positive answer to the above question, that is, the gradient method provides a strong convergence sequence by using a non-monotonic step size rule for solving EPs accompanied with pseudo-monotone bi-functions.

Preliminaries

Explicit subgradient extragradient method and its convergence analysis

Applications

Numerical illustration

Conclusion