Modified subgradient extragradient method to solve variational inequalities

Abstract

In this paper, we introduce a new method to solve pseudomonotone variational inequalities with the Lipschitz condition in a real Hilbert space. This problem is a general mathematical problem in the sense that it unifies a number of the mathematical problems as a particular case, such as the optimization problems, the equilibrium problems, the fixed point problems, the saddle point problems and Nash equilibrium point problems. The new method is constructed around two methods: the extragradient method and the inertial method. The proposed method uses a new stepsize rule based on local operator information rather than its Lipschitz constant or any other line search method. The proposed method does not require any knowledge of the Lipschitz constant of an operator. The strong convergence of the proposed method is well-established. Finally, we conduct a number of numerical experiments to determine the performance and superiority of the proposed method.