An Algorithm for Solving the Variational Inequality Problem Over the Fixed Point Set of a Quasi-Nonexpansive Operator in Euclidean Space

This article is concerned with the variational inequality problem VIP(F, Fix(T)): find such that for all zFix(T), where T: (nn) is quasi-nonexpansive, Fix(T) is its nonempty fixed point set, and F: (nn) is 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.