HALPERN TSENG’S EXTRAGRADIENT METHODS FOR SOLVING VARIATIONAL INEQUALITIES INVOLVING SEMISTRICTLY QUASIMONOTONE OPERATOR

Abstract

In this paper, we study the strong convergence of new methods for solving classical variational inequalities problems involving semistrictly quasimonotone and Lipschitz- continuous operators in a real Hilbert space. Three proposed methods are based on Tseng’s extragradient method and use a simple self-adaptive step size rule that is independent of the Lipschitz constant. The step size rule is built around two techniques: the monotone and the non-monotone step size rule. We establish strong convergence theorems for the proposed methods that do not require any additional projections or knowledge of an involved operator’s Lipschitz constant. Finally, we present some numerical experiments that demonstrate the efficiency and advantages of the proposed methods.