The Lipschitz property of the metric projection in the Hilbert space

M. V. Balashov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 22 (2018) no. 1, pp. 13-29

Résumé

In the survey, we consider the metric projection operator from the real Hilbert space onto a closed subset. We discuss the question: when this operator is Lipschitz continuous? Firstly, we consider the class of strongly convex sets of radius $R$, i.e., each set from this class is 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 the Lipschitz constant $C=R/(r+R)\in (0,1)$. Vice versa, if for a closed convex set from the real Hilbert space the metric projection operator is Lipschitz continuous with the Lipschitz constant $C\in (0,1)$ 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 the 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 the Lipschitz continuous metric projection on the neighborhood of radius $r$ with the Lipschitz constant $C>1$, then this set is proximally smooth with constant of proximal smoothness $R=Cr/(C-1)$, and, if constant $C$ is the smallest possible, then constant $R$ is the largest possible. We apply obtained results to the question concerning the rate of convergence for the gradient projection algorithm.