Composite inertial subgradient extragradient methods for variational inequalities and fixed point problems

Lu-Chuan Ceng,Qing Yuan

doi:10.1186/s13660-019-2229-x

Abstract

In this paper, we introduce and investigate composite inertial gradient-based algorithms with a line-search process for solving a variational inequality problem (VIP) with a pseudomonotone and Lipschitz continuous mapping and a common fixed-point problem (CFPP) of finitely many nonexpansive mappings and a strictly pseudocontractive mapping in the framework infinite-dimensional Hilbert spaces. The proposed algorithms are based on an inertial subgradient–extragradient method with the line-search process, hybrid steepest-descent methods, viscosity approximation methods and Mann iteration methods. Under weak conditions, we prove strong convergence of the proposed algorithms to the element in the common solution set of the VIP and CFPP, which solves a certain hierarchical VIP defined on this common solution set.

Highlights

Throughout this work, H is assumed to be a real Hilbert space with norm · and inner product ·, ·
Inspired by the research work of [23,24,25,26,27,28], we introduce two composite inertial subgradient–extragradient algorithms with a line-search process for solving the variational inequality problem (VIP) with pseudomonotone and Lipschitz continuous mapping and the common fixed-point problem (CFPP) of finitely many nonexpansive mappings and a strictly pseudocontractive mapping in a real Hilbert space
We prove strong convergence of the proposed algorithms to an element in the common solution set of the VIP and CFPP, which solves a certain hierarchical VIP defined on this common solution set

Summary

Introduction

Throughout this work, H is assumed to be a real Hilbert space with norm · and inner product ·, ·. For the well-known extragradient method, one needs to compute two nearest point projections for every iterative step/process. Thong and Hieu [25] introduced two inertial subgradient–extragradient algorithms with linear-search process for solving the VIP with monotone and Lipschitz continuous mapping A and the fixedpoint problem (FPP) of a quasi-nonexpansive mapping T with a demiclosedness property in a real Hilbert space. Inspired by the research work of [23,24,25,26,27,28], we introduce two composite inertial subgradient–extragradient algorithms with a line-search process for solving the VIP with pseudomonotone and Lipschitz continuous mapping and the CFPP of finitely many nonexpansive mappings and a strictly pseudocontractive mapping in a real Hilbert space. Some examples are presented to solve the VIP and CFPP

Preliminaries

Main results

An example