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We study variational inequalities which are governed by a strongly monotoneand Lipschitz continuous operator $F$ over a closed and convex set $S$. Weassume that $S=C\cap A^{-1}(Q)$ is the nonempty solution set of a(multiple-set) split convex feasibility problem, where $C$ and $Q$ are bothclosed and convex subsets of two real Hilbert spaces $\mathcal H_1$ and$\mathcal H_2$, respectively, and the operator $A$ acting between them islinear. We consider a modification of the gradient projection method the mainidea of which is to replace at each step the metric projection onto $S$ byanother metric projection onto a half-space which contains $S$. We proposethree variants of a method for constructing the above-mentioned half-spaces byemploying the multiple-set and the split structure of the set $S$. For thesplit part we make use of the Landweber transform.

Outer Approximation Methods for Solving Variational Inequalities Defined over the Solution Set of a Split Convex Feasibility Problem