Outer approximated projection and contraction method for solving variational inequalities

• Published in: Journal of inequalities and applications Vol. 2023; no. 1; pp. 141 - 28
• Main Authors: Uzor, V. A., Mewomo, O. T., Alakoya, T. O., Gibali, A.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 02.11.2023; Springer Nature B.V; SpringerOpen
• Subjects: Analysis; Applications of Mathematics; Approximation; Convex analysis; Inertial technique; Mathematical analysis; Mathematics; Mathematics and Statistics; Methods; Non-Lipschitz variational inequalities; Projection and contraction method; Relaxation technique; Self-adaptive step size; Sublevel sets; Self-adaptive step size; 47H10; 90C52; 47J40; 90C25; Non-Lipschitz variational inequalities; Sublevel sets; Inertial technique; 47J25; Relaxation technique; Projection and contraction method
• ISSN: 1029-242X; 1025-5834; 1029-242X
• DOI: 10.1186/s13660-023-03043-8

In this paper we focus on solving the classical variational inequality (VI) problem. Most common methods for solving VIs use some kind of projection onto the associated feasible set. Thus, when the involved set is not simple to project onto, then the applicability and computational effort of the proposed method could be arguable. One such scenario is when the given set is represented as a finite intersection of sublevel sets of convex functions. In this work we develop an outer approximation method that replaces the projection onto the VI’s feasible set by a simple, closed formula projection onto some “superset”. The proposed method also combines several known ideas such as the inertial technique and self-adaptive step size. Under standard assumptions, a strong minimum-norm convergence is proved and several numerical experiments validate and exhibit the performance of our scheme.