Self-adaptive subgradient extragradient method with inertial modification for solving monotone variational inequality problems and quasi-nonexpansive fixed point problems

Ming Tian,Mengying Tong

doi:10.1186/s13660-019-1958-1

Abstract

In this paper, we introduce a new algorithm with self-adaptive method for finding a solution of the variational inequality problem involving monotone operator and the fixed point problem of a quasi-nonexpansive mapping with a demiclosedness property in a real Hilbert space. The algorithm is based on the subgradient extragradient method and inertial method. At the same time, it can be considered as an improvement of the inertial extragradient method over each computational step which was previously known. The weak convergence of the algorithm is studied under standard assumptions. It is worth emphasizing that the algorithm that we propose does not require one to know the Lipschitz constant of the operator. Finally, we provide some numerical experiments to verify the effectiveness and advantage of the proposed algorithm.

Highlights

Throughout this paper, let H be a real Hilbert space with the inner product ·, · and norm·
In this paper, motivated and inspired by the above results, we introduce a new algorithm with self-adaptive subgradient extragradient method and inertial modification for finding a solution of the variational inequality problem involving monotone operator and the fixed point problem of a quasi-nonexpansive mapping with a demiclosedness property in a real Hilbert space
3 Main results we propose a new iterative algorithm with self-adaptive method for solving monotone variational inequality problems and quasi-nonexpansive fixed point problems in a Hilbert space

Summary

Introduction

Throughout this paper, let H be a real Hilbert space with the inner product ·, · and norm. In this paper, motivated and inspired by the above results, we introduce a new algorithm with self-adaptive subgradient extragradient method and inertial modification for finding a solution of the variational inequality problem involving monotone operator and the fixed point problem of a quasi-nonexpansive mapping with a demiclosedness property in a real Hilbert space. Lemma 2.10 ([27]) Let C be a nonempty closed and convex subset of a real Hilbert space H and {xn} be a sequence in H. By Lemma 2.10, we get the conclusion that the sequence {xn} converges weakly to an element of Fix(T) ∩ VI(C, A).

Numerical experiments

Conclusion