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This paper is concerned with the variational inequality problem (VIP) overthe fixed point set of a quasi-nonexpansive operator. We propose, inparticular, an algorithm which entails, at each step, projecting onto asuitably chosen half-space, and prove that the sequences it generates convergeto the unique solution of the VIP. We also present an application of our resultto a hierarchical optimization problem.

An algorithm for solving the variational inequality problem over the fixed point set of a quasi-nonexpansive operator in Euclidean space