Outer approximation methods for solving variational inequalities in Hilbert space

• Published in: Optimization Vol. 66; no. 3; pp. 417 - 437
• Main Authors: Gibali, Aviv, Reich, Simeon, Zalas, Rafał
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 04.03.2017; Taylor & Francis LLC
• Subjects: Approximation; Common fixed point; Convergence; Cutters; Hilbert space; Inequalities; iterative method; Iterative methods; Mathematical analysis; Operators; Optimization; Quantum theory; quasi-nonexpansive operator; subgradient projection; variational inequality
• ISSN: 0233-1934; 1029-4945
• DOI: 10.1080/02331934.2016.1271800

In this paper, we study variational inequalities in a real Hilbert space, which are governed by a strongly monotone and Lipschitz continuous operator F over a closed and convex set C. We assume that the set C can be outerly approximated by the fixed point sets of a sequence of certain quasi-nonexpansive operators called cutters. We propose an iterative method, the main idea of which is to project at each step onto a particular half-space constructed using the input data. Our approach is based on a method presented by Fukushima in 1986, which has recently been extended by several authors. In the present paper, we establish strong convergence in Hilbert space. We emphasize that to the best of our knowledge, Fukushima's method has so far been considered only in the Euclidean setting with different conditions on F. We provide several examples for the case where C is the common fixed point set of a finite number of cutters with numerical illustrations of our theoretical results.