A subgradient extragradient algorithm for solving monotone variational inequalities in Banach spaces

Fei Ma

doi:10.1186/s13660-020-2295-0

Abstract

In this paper, we introduce an algorithm for solving classical variational inequalities problem with Lipschitz continuous and monotone mapping in Banach space. We modify the subgradient extragradient methods with a new and simple iterative step size, the strong convergence of algorithm is established without the knowledge of the Lipschitz constant of the mapping. Finally, a numerical experiment is presented to show the efficiency and advantage of the proposed algorithm. Our results generalize some of the work in Hilbert spaces to Banach spaces.

Highlights

The variational inequality problem (VIP) which was first introduced by Hartman and Stampacchia [1] in 1966, is a very important tool in studying engineering mechanics, physics, economics, optimization theory and applied sciences in a unified and general framework
The subgradient extragradient-type algorithm was introduced by Censor et al in [5] for solving variational inequalities in real Hilbert space
Inspired by the work mentioned, in this work, we extend subgradient extragradient algorithm proposed by [8] for solving variational inequalities from Hilbert spaces to Banach spaces

Summary

Introduction

The variational inequality problem (VIP) which was first introduced by Hartman and Stampacchia [1] in 1966, is a very important tool in studying engineering mechanics, physics, economics, optimization theory and applied sciences in a unified and general framework (see [2, 3]). Many projection-type algorithms for solving the variational inequalities problem have been proposed and analyzed by many authors [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]. The subgradient extragradient-type algorithm was introduced by Censor et al in [5] for solving variational inequalities in real Hilbert space. Inspired by the work mentioned, in this work, we extend subgradient extragradient algorithm proposed by [8] for solving variational inequalities from Hilbert spaces to Banach spaces.

Mathematical preliminaries

Conclusions