New inertial forward-backward type for variational inequalities with Quasi-monotonicity

• Published in: Journal of global optimization Vol. 84; no. 2; pp. 441 - 464
• Main Authors: Izuchukwu, Chinedu, Shehu, Yekini, Yao, Jen-Chih
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.10.2022; Springer; Springer Nature B.V
• Subjects: Computer Science; Convergence; Hilbert space; Inequalities; Iterative methods; Mathematical programming; Mathematics; Mathematics and Statistics; Methods; Operations Research/Decision Theory; Optimization; Real Functions; Weak convergence; Quasi-monotone; Linear convergence; 65K15; 47J20; Inertial projection method; 90C25; Hilbert spaces; 47H05; 47J25; Variational inequalities
• ISSN: 0925-5001; 1573-2916
• DOI: 10.1007/s10898-022-01152-0

In this paper, we present a modification of the forward-backward splitting method with inertial extrapolation step and self-adaptive step sizes to solve variational inequalities in a quasi-monotone setting. Our proposed method involves one computation of the projection onto the feasible set and one evaluation of the operator per iteration, which is simpler than most methods available in the literature to solve similar problems. We first establish weak convergence result when the set of solutions of the Minty formulation of the variational inequality is nonempty in infinite dimensional Hilbert spaces under appropriate conditions. Next, we give linear convergence result when the operator is strongly pseudo-monotone. We also give numerical implementations of our proposed method and some comparisons with some other methods available in the literature.