A Two-Step Inertial Subgradient Extragradient Method for Solving Non-Monotone Variational Inequality Problems in Reflexive Banach Spaces

Abstract

In this paper, we introduce a novel two-step iterative algorithm based on a double inertial subgradient extragradient method for solving non-monotone variational inequality problems in reflexive Banach spaces. Without imposing any monotonicity conditions on the cost operator, we establish strong convergence of the proposed algorithm by employing a tailored Armijo-type line search rule and without imposing a global Lipschitz continuity assumption on the cost operator. Our analysis relies on mild assumptions concerning the algorithm's control parameters and the assumption that the solution set of the corresponding dual variational inequality is non-empty. Furthermore, we present numerical test problems and an optimal-control application framework illustrating the proposed method with the revised line-search implementation. The findings of this work extend and generalize several important results in the current literature on variational inequalities.