An inertial‐like Tseng's extragradient method for solving pseudomonotone variational inequalities in reflexive Banach spaces

Abstract

A survey of the existing results in the literature shows that several of the results on variational inequality problem were established under some stringent conditions and employed some form of linesearch technique even in the framework of Hilbert spaces. However, due to the loop nature of the linesearch technique, the implementation of such algorithms might be economically nonviable. In this article, we study the pseudomonotone variational inequality problem. We propose a new inertial‐like Tseng's extragradient method with non‐monotonic self‐adaptive step size that does not involve any linesearch technique for finding the solution of the problem in reflexive Banach spaces. Under some mild conditions, we prove a strong convergence result for the proposed method without the sequentially weakly continuity condition often assumed by authors to guarantee convergence when studying the pseudomonotone variational inequality problems. Moreover, we apply our result to study a constrained convex minimization problem. Finally, we present several theoretical and real applications to optimal control numerical experiments to illustrate and show the performance of our method in comparison with related methods in the literature.