An Algorithm for Solving the Variational Inequality Problem Over the Fixed Point Set of a Quasi-Nonexpansive Operator in Euclidean Space
Andrzej Cegielski, Aviv Gibali, Simeon Reich, Rafał Zalas · Numerical Functional Analysis and Optimization · 2013
This article is concerned with the variational inequality problem VIP(ℱ, Fix(T)): find such that for all z ∈ Fix(T), where T: ℝ n → ℝ n is quasi-nonexpansive, Fix(T) is its nonempty fixed point set, and ℱ: ℝ n → ℝ n 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.