A Relaxed Inertial Tseng’s Extragradient Method for Solving Split Variational Inequalities with Multiple Output Sets

Abstract

Recently, the split inverse problem has received great research attention due to its several applications in diverse fields. In this paper, we study a new class of split inverse problems called the split variational inequality problem with multiple output sets. We propose a new Tseng extragradient method, which uses self-adaptive step sizes for approximating the solution to the problem when the cost operators are pseudomonotone and non-Lipschitz in the framework of Hilbert spaces. We point out that while the cost operators are non-Lipschitz, our proposed method does not involve any linesearch procedure for its implementation. Instead, we employ a more efficient self-adaptive step size technique with known parameters. In addition, we employ the relaxation method and the inertial technique to improve the convergence properties of the algorithm. Moreover, under some mild conditions on the control parameters and without the knowledge of the operators’ norm, we prove that the sequence generated by our proposed method converges strongly to a minimum-norm solution to the problem. Finally, we apply our result to study certain classes of optimization problems, and we present several numerical experiments to demonstrate the applicability of our proposed method. Several of the existing results in the literature in this direction could be viewed as special cases of our results in this study.

Keywords

• split inverse problems
• non-Lipschitz operators
• pseudomonotone operators
• Tseng’s extragradient method
• relaxation and inertial techniques