Outer approximation methods for solving variational inequalities in Hilbert space. (4th March 2017)

• Record Type: Journal Article
• Authors: Gibali, Aviv; Reich, Simeon; Zalas, Rafał
• Is Part Of: Optimization. Volume 66:Number 3(2017)
• Page Start: 417
• Page End: 437
• Publication Date: 2017-03-04
• Subjects: Common fixed point -- iterative method -- quasi-nonexpansive operator -- subgradient projection -- variational inequality; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2016.1271800
• Languages: English
• ISSNs: 0233-1934

Abstract

Abstract: 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.