The Lipschitz Property of the Metric Projection in the Hilbert Space
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THE LIPSCHITZ PROPERTY OF THE METRIC PROJECTION IN THE HILBERT SPACE M. V. Balashov
UDC 517.982.22+517.982.252+517.982.256
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.
1. Introduction and Main Notation The metric projection operator on a subset A in a normed space PA x = a ∈ A | x − a = inf x − y y∈A
is very important both in theory and applications. It was well known from the times of H. Minkowski (or may be earlier) that metric projection (as a function of the point) on a closed convex subset in a finite-dimensional Euclidean space is singleton and Lipschitz continuous with Lipschitz constant 1. R. Phelps [20] was one of the first authors who paid attention to this fact in the Hilbert space. This property turned out to be very important in different branches of mathematics. First of all, J. Lindenstrauss proved [17, Sec. 3, Corollary 2] that if in a uniformly convex Banach space E ([15], the Hilbert space is uniformly convex) the metric projector operator on any closed subspace L is uniformly continuous, then there exists a bounded linear projector from E onto L (for any closed subspace L ⊂ E). Hence every closed linear subspace L ⊂ E in this case is complemented. J. Lindenstrauss proved (the complementary subspace problem [18]) that in this case the space E is isomorphic to the Hilbert space. Thus, the Lipschitz condition of the metric projector operator characterizes the Hilbert space. Another important question appeared in applications, in optimization problems. The property of the metric projection operator on the closed convex subset in the Hilbert space to be nonexpansive was useful in gradien
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