Clarke critical values of subanalytic Lipschitz continuous functions

LIPSCHITZ CONTINUITY PROPERTIES FOR p-ADIC SEMI-ALGEBRAIC AND SUBANALYTIC FUNCTIONS

We prove that a (globally) subanalytic function f : X ⊂ Qp → Qp which is locally Lipschitz continuous with some constant C is piecewise (globally on each piece) Lipschitz continuous with possibly some other constant, where the pieces can be taken to be subanalytic. We also prove the analogous result for a subanalytic family of functions fy : Xy ⊂ Qp → Qp depending on p-adic parameters. The stat...

An effective optimization algorithm for locally nonconvex Lipschitz functions based on mollifier subgradients

Regular Covers of Open Relatively Compact Subanalytic Sets

Let U be an open relatively compact subanalytic subset of a real analytic manifold. We show that there exists a finite linear covering (in the sense of Guillermou and Schapira) of U by subanalytic open subsets of U homeomorphic to a unit ball. We also show that the algebra of open relatively compact subanalytic subsets of a real analytic manifold is generated by subsets subanalytically and bi-l...

A differential operator and weak topology for Lipschitz maps

We show that the Scott topology induces a topology for real-valued Lipschitz maps on Banach spaces which we call the L-topology. It is the weakest topology with respect to which the L-derivative operator, as a second order functional which maps the space of Lipschitz functions into the function space of non-empty weak* compact and convex valued maps equipped with the Scott topology, is continuo...

Tame Mappings Are Semismooth

Superlinear convergence of the Newton method for nonsmooth equations requires a “semismoothness” assumption. In this work we prove that locally Lipschitz functions definable in an o-minimal structure (in particular semialgebraic or globally subanalytic functions) are semismooth. Semialgebraic, or more generally, globally subanalytic mappings present the special interest of being γ-order semismo...