Strong Convergent Theorems Governed by Pseudo-Monotone Mappings

Liya Liu,Jen-Chih Yao,Xiaolong Qin

doi:10.3390/math8081256

Abstract

The purpose of this paper is to introduce two different kinds of iterative algorithms with inertial effects for solving variational inequalities. The iterative processes are based on the extragradient method, the Mann-type method and the viscosity method. Convergence theorems of strong convergence are established in Hilbert spaces under mild assumption that the associated mapping is Lipschitz continuous, pseudo-monotone and sequentially weakly continuous. Numerical experiments are performed to illustrate the behaviors of our proposed methods, as well as comparing them with the existing one in literature.

Highlights

Introduction and PreliminariesLet C be a closed, convex and nonempty subset of a Hilbert space H
An obvious disadvantage of Algorithms (2) and (3) is the assumption that the mapping A should be Lipschitz continuous and monotone. To avoid this restrictive assumption, in this paper, we show that our proposed algorithms can solve the pseudo-monotone variational inequality under suitable assumptions
We perform some computational experiments in support of the convergence properties of our proposed methods and compare our methods with Algorithm extragradient algorithm (EAI), see [20]

Summary

Introduction

Let C be a closed, convex and nonempty subset of a Hilbert space H. Let A : H → H be a nonlinear operator. We consider the following variational inequality problem find x ∗ ∈ V I (C, A) := { x ∗ ∈ C : hz − x ∗ , A( x ∗ )i ≥ 0, ∀z ∈ C }. The variational inequality, which serves as an important model in studying a wide class of real problems arising in traffic network, medical imaging, machine learning, transportation, etc. Due to its wide applications, this model unifies a number of optimization-related problems, such as, saddle problems, equilibrium problems, complementary problems, fixed point problems; see, e.g., [1,2,3,4,5]

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Conclusion