Strong convergence of extragradient methods with a new step size for solving variational inequality problems

Abstract

In this paper, we propose two different kinds of extragradient methods with a new step size for finding an element of the set of solutions of the variational inequality problem for a monotone and Lipschitz continuous operator in real Hilbert spaces. We only use one projection to design the algorithms and the strong convergence theorems proved without the prior knowledge of the Lipschitz constant of cost operator. Numerical experiments illustrate the performances of our new algorithms and provide a comparison with related algorithms.