Mann-Type Inertial Projection and Contraction Method for Solving Split Pseudomonotone Variational Inequality Problem with Multiple Output Sets

Abstract

In this paper, we study the concept of split variational inequality problem with multiple output sets when the cost operators are pseudomonotone and non-Lipschitz. We introduce a new Mann-type inertial projection and contraction method with self-adaptive step sizes for approximating the solution of the problem in the framework of Hilbert spaces. Under some mild conditions on the control parameters and without prior knowledge of the operator norms, we prove a strong convergence theorem for the proposed algorithm. We point out that while the cost operators are non-Lipschitz, our proposed method does not require any linesearch method but uses a more efficient self-adaptive step size technique that generates a non-monotonic sequence of step sizes. Finally, we apply our result to study certain classes of optimization problems and we present several numerical experiments to illustrate the applicability of the proposed method. Several of the existing results in the literature could be viewed as special cases of our result in this study.