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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. [ABSTRACT FROM AUTHOR]

This article is concerned with the variational inequality problemVIP(ℱ, Fix(T)): findsuch thatfor allz ∈ Fix(T), whereT: ℝn → ℝnis quasi-nonexpansive, Fix(T) is its nonempty fixed point set, and ℱ: ℝn → ℝnis monotone. We propose, in particular, an algorithm which entails, at each step, projecting onto a suitably chosen half-space, and prove that the sequences it generates converge to the unique solution of the VIP. We also present an application of our result to a hierarchical optimization problem. [ABSTRACT FROM PUBLISHER]