Strong convergence of new iterative projection methods with regularization for solving monotone variational inequalities in Hilbert spaces

Abstract

In this paper, we introduce two new numerical methods for solving a variational inequality problem involving a monotone and Lipschitz continuous operator in a Hilbert space. We describe how to incorporate a regularization term depending on a parameter in the projection method and then establish the strong convergence of the resulting iterative regularization projection methods. Unlike known hybrid methods, the strong convergence of the new methods comes from the regularization technique. The first method is designed to work in the case where the Lipschitz constant of cost operator is known, whereas the second one is more easily implemented without this requirement. The reason is because the second method has used a simple computable stepsize rule. The variable stepsizes are generated by the second method at each iteration and based on the previous iterates. These stepsizes are found with only one cheap computation without line-search procedure. Several numerical experiments are implemented to show the computational effectiveness of the new methods over existing methods.

Multi-Step Inertial Regularized Methods for Hierarchical Variational Inequality Problems Involving Generalized Lipschitzian Mappings

“…In 2020, Hieu et al [3,4] proposed regularized subgradient extragradient method (Algorithm 1 RSEGM) and regularized Tseng's extragradient method (Algorithm 2 RTEGM) for solving HVIP (4). Both of these two methods have strong convergence results.…”

“…This algorithm has a weak convergence result under certain conditions. In this paper, motivated by the results of [3,4,7], we construct a multi-step inertial regularized subgradient extragradient method and a multi-step inertial regularized Tseng's extragradient method for solving HVIP (4) in a Hilbert space when F is a generalized Lipschitzian and hemicontinuous mapping (see in Section 2, Definitions 2 and 3). Then, we present two strong convergence theorems and give some numerical experiments to show the effectiveness and feasibility of our new iterative methods.…”