Proximal Calculus on Riemannian Manifolds

Abstract

We introduce a proximal subdifferential and develop a calculus for nonsmooth functions defined on any Riemannian manifold M . We give some applications of this theory, concerning, for instance, a Borwein-Preiss type variational principle on a Riemannian manifold M , as well as differentiability and geometrical properties of the distance function to a closed subset C of M . (2000). 49J52, 58E30, 58C30, 47H10. Mathematics Subject Classification

Location of Nash equilibria: A Riemannian geometrical approach

“…It seems that the class of locally Lipschitz functions (in the appropriate variable) is the optimal one among not necessarily convex functions for which the existence of Nash critical points can be guaranteed. The proximal calculus for lower semicontinuous functions (see Azagra and Ferrera [1], Ledyaev and Zhu [8]) could be another good candidate for developing a similar concept as Nash critical points. Unfortunately, the lack of a suitable regularity of the proximal subdifferential leaves this question open.…”

Proximal subgradient and a characterization of Lipschitz function on Riemannian manifolds

“…In this paper, we assume that all manifolds are complete and finite dimensional. The exponential map exp p : (1), where γ v is the geodesic defined by its position p and velocity v at p. We can prove that exp p tv = γ v (t) for any values of t. Now, for p ∈ M, let…”

“…In Section 4 we will give a characterization and some examples of Lipschitz functions defined on Riemannian manifolds. We conclude this paper by making some general comments about extensions of our results from finite to infinite dimensional Riemannian manifolds as applications of the recent results due to Azagra and Ferrera [1].…”

Fixed Points and Zeros for Set Valued Mappings on Riemannian Manifolds: A Subdifferential Approach

“…We will also use the definition of the proximal subdifferential ∂ P f (x 0 ) of a function f defined on a Riemannian manifold, which was introduced in [2] and in [3], as well as the following characterization: ζ ∈ ∂ P f (x 0 ) if and only if there exists a positive number σ and a neighborhood of x 0 on which…”