Outer Approximation Methods for Solving Variational Inequalities Defined over the Solution Set of a Split Convex Feasibility Problem
Andrzej Cegielski, Aviv Gibali, Simeon Reich, Rafał Zalas · Numerical Functional Analysis and Optimization · 2020
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 H1 and H2, 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.