Self-adaptive subgradient extragradient-type methods for solving variational inequalities

Abstract

In this paper, we introduce two subgradient extragradient-type algorithms for solving variational inequality problems in the real Hilbert space. The first one can be applied when the mapping f is strongly pseudomonotone (not monotone) and Lipschitz continuous. The first algorithm only needs two projections, where the first projection onto closed convex set C and the second projection onto a half-space C_{k}. The strong convergence theorem is also established. The second algorithm is relaxed and self-adaptive; that is, at each iteration, calculating two projections onto some half-spaces and the step size can be selected in some adaptive ways. Under the assumption that f is monotone and Lipschitz continuous, a weak convergence theorem is provided. Finally, we provide numerical experiments to show the efficiency and advantage of the proposed algorithms.