The Lipschitz Property of the Metric Projection in the Hilbert Space

Abstract

In this survey, we consider the metric projection operator from the real Hilbert space onto a closed subset. We discuss the following question: When is this operator Lipschitz continuous? First, we consider the class of strongly convex sets of radius R, i.e., each set from this class is a nonempty intersection of closed balls of radius R. We prove that the restriction of the metric projection operator on the complement of the neighborhood of radius r of a strongly convex set of radius R is Lipschitz continuous with Lipschitz constant C = R/(r + R) ∈ (0, 1). Vice versa, if for a closed convex set from the real Hilbert space the metric projection operator is Lipschitz continuous with Lipschitz constant C ∈ (0, 1) on the complement of the neighborhood of radius r of the set, then the set is strongly convex of radius R = Cr/(1 − C). It is known that if a closed subset of a real Hilbert space has Lipschitz continuous metric projection in some neighborhood, then this set is proximally smooth. We show that if a closed subset of the real Hilbert space has Lipschitz continuous metric projection on the neighborhood of radius r with Lipschitz constant C > 1, then this set is proximally smooth with constant of proximal smoothness R = Cr/(C − 1), and, if the constant C is the smallest possible, then the constant R is the largest possible. We apply the obtained results to the question concerning the rate of convergence for the gradient projection algorithm.