Outer Approximation Methods for Solving Variational Inequalities Defined over the Solution Set of a Split Convex Feasibility Problem. (3rd July 2020)

• Record Type: Journal Article
• Authors: Cegielski, Andrzej; Gibali, Aviv; Reich, Simeon; Zalas, Rafał
• Is Part Of: Numerical functional analysis and optimization. Volume 41:Number 9(2020)
• Page Start: 1089
• Page End: 1108
• Publication Date: 2020-07-03
• Subjects: CQ-method -- split convex feasibility problem -- variational inequality; 47H09 -- 47H10 -- 47J20 -- 47J25 -- 65K15; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2020.1737938
• Languages: English
• ISSNs: 0163-0563

Abstract

Abstract: We study variational inequalities which are governed by a strongly monotone and Lipschitz continuous operator F over a closed and convex set S . We assume that S = C ∩ A − 1 ( Q ) is the nonempty solution set of a (multiple-set) split convex feasibility problem, where C and Q are both closed and convex subsets of two real Hilbert spaces H 1 and H 2, respectively, and the operator A acting between them is linear. We consider a modification of the gradient projection method the main idea of which is to replace at each step the metric projection onto S by another metric projection onto a half-space which contains S . We propose three variants of a method for constructing the above-mentioned half-spaces by employing the multiple-set and the split structure of the set S . For the split part we make use of the Landweber transform.