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

Thanatporn Bantaojai,Habib Ur Rehman,Poom Kumam,Nuttapol Pakkaranang,Wiyada Kumam

doi:10.3390/sym12081332

Abstract

In this article, we propose a new modified extragradient-like method to solve pseudomonotone equilibrium problems in real Hilbert space with a Lipschitz-type condition on a bifunction. This method uses a variable stepsize formula that is updated at each iteration based on the previous iterations. The advantage of the method is that it operates without prior knowledge of Lipschitz-type constants and any line search method. The weak convergence of the method is established by taking mild conditions on a bifunction. In the context of an application, fixed-point theorems involving strict pseudo-contraction and results for pseudomonotone variational inequalities are considered. Many numerical results have been reported to explain the numerical behavior of the proposed method.

Highlights

Let C be a nonempty, closed and convex subset of a real Hilbert space H and R, N be the sets of real numbers and natural numbers, respectively
A bifunction f : H × H → R is said to be Lipschitz-type continuous on C if there exist two positive constants c1, c2 such that f ( z1, z3 ) ≤ f ( z1, z2 ) + f ( z2, z3 ) + c1 k z1 − z2 k2 + c2 k z2 − z3 k2, ∀ z1, z2, z3 ∈ C
Let C be a nonempty, closed and convex subset of a real Hilbert space H and a normal cone of C at z1 ∈ C is defined by: NC (z1 ) = {z3 ∈ H : hz3, z2 − z1 i ≤ 0, ∀z2 ∈ C}

Summary

Introduction

Let C be a nonempty, closed and convex subset of a real Hilbert space H and R, N be the sets of real numbers and natural numbers, respectively. The iterative sequence generated from the above-mentioned method provides a weak convergent iterative sequence and in order to operate it, prior information regarding the Lipschitz-type constants is required. These Lipschitz-type constants are mostly unknown or hard to compute. Vinh and Muu proposed an inertial iterative algorithm in [39] to solve a pseudomonotone equilibrium problem. Their main contribution is the availability of an inertial effect in the algorithm that is used to improve the convergence rate of the iterative sequence.

Background

Convergence Analysis for an Algorithm

Applications to Solve Fixed Point Problems

Application to Solve Variational Inequality Problems

Numerical Experiments