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

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.

Strong convergence analysis for solving quasi-monotone variational inequalities and fixed point problems in reflexive Banach spaces

“…The concept of inertial technique introduced by Polyak [20] plays a vital role in speeding up the convergence rate of various iterative algorithms (see [3,9,18,19,31]). This technique originates from an implicit discretization method of the second-order dynamical systems in solving the smooth convex minimization problem.…”