A self-adaptive inertial extragradient method for a class of split pseudomonotone variational inequality problems

Abstract

Abstract In this article, we study a class of pseudomonotone split variational inequality problems (VIPs) with non-Lipschitz operator. We propose a new inertial extragradient method with self-adaptive step sizes for finding the solution to the aforementioned problem in the framework of Hilbert spaces. Moreover, we prove a strong convergence result for the proposed algorithm without prior knowledge of the operator norm and under mild conditions on the control parameters. The main advantages of our algorithm are: the strong convergence result obtained without prior knowledge of the operator norm and without the Lipschitz continuity condition often assumed by authors; the minimized number of projections per iteration compared to related results in the literature; the inertial technique employed, which speeds up the rate of convergence; and unlike several of the existing results in the literature on VIPs with non-Lipschitz operators, our method does not require any linesearch technique for its implementation. Finally, we present several numerical examples to illustrate the usefulness and applicability of our algorithm.