Proximal subgradient and a characterization of Lipschitz function on Riemannian manifolds

Abstract

A characterization of Lipschitz behavior of functions defined on Riemannian manifolds is given in this paper. First, it is extended the concept of proximal subgradient and some results of proximal analysis from Hilbert space to Riemannian manifold setting. A technique introduced by Clarke, Stern and Wolenski [F.H. Clarke, R.J. Stern, P.R. Wolenski, Subgradient criteria for monotonicity, the Lipschitz condition, and convexity, Canad. J. Math. 45 (1993) 1167-1183], for generating proximal subgradients of functions defined on a Hilbert spaces, is also extended to Riemannian manifolds in order to provide that characterization. A number of examples of Lipschitz functions are presented so as to show that the Lipschitz behavior of functions defined on Riemannian manifolds depends on the Riemannian metric.  2005 Elsevier Inc. All rights reserved.

Invex sets and preinvex functions on Riemannian manifolds

“…We recall the definition of a proximal subdifferential of a function defined on a Riemannian manifold and refer the reader to [4,7] for the discussion of proximal calculus on such a manifold. The set of all proximal subgradients of f at y ∈ M is denoted by ∂ p f (y) and is called the proximal subdifferential of f at y.…”

Generalized gradients and characterization of epi-Lipschitz sets in Riemannian manifolds

“…In the past few years, a number of results have been obtained on numerous aspects of nonsmooth analysis and their applications on Riemannian manifolds; see, e.g. [3][4][5][6][7].…”

Dini derivative and a characterization for Lipschitz and convex functions on Riemannian manifolds

“…Let M be a Riemannian manifold and S a closed subset of M . The distance function d S (p) = inf {d(p, s) : s ∈ S} is Lipschitz of rank 1, see [7].…”

“…In other words, if the metric on M is changed then the set of Lipschitz functions on M becomes different of previous one, see Example 4.4 in [7].…”

“…Works dealing with this subject include the ones by D. Azagra, J. Ferrera and F. López-Mesas [1], D. Azagra, J. Ferrera [2], O. P. Ferreira [7] and Yu. S. Ledyaev and Q. J. Zhu [11,12,13].…”