Clarke critical values of subanalytic Lipschitz continuous functions

Clarke critical values of subanalytic Lipschitz continuous functions

We prove that any subanalytic locally Lipschitz function has the Sard property. Such functions are typically nonsmooth and their lack of regularity necessitates the choice of some generalized notion of gradient and of critical point. In our framework these notions are defined in terms of the Clarke and of the convex-stable subdifferentials. The main result of this note asserts that for any suba...

Lipschitz Functions with Minimal Clarke Subdiierential Mappings

In this paper we characterise, in terms of the upper Dini derivative, when the Clarke subdiierential mapping of a real-valued locally Lipschitz function is a minimal weak cusco. We then use this characterisation to deduce some new results concerning Lips-chitz functions with minimal subdiierential mappings. In the papers 7] and 1], the authors investigate a class of locally Lipschitz functions ...

Rotund Norms, Clarke Subdifferentials and Extensions of Lipschitz Functions

Lipschitz functions with maximal Clarke subdifferentials are staunch

In a recent paper we have shown that most non-expansive Lipschitz functions (in the sense of Baire’s category) have a maximal Clarke subdifferential. In the present paper, we show that in a separable Banach space the set of non-expansive Lipschitz functions with a maximal Clarke subdifferential is not only of generic, but also staunch. 1991 Mathematics Subject Classification: Primary 49J52.

Lipschitz Functions with Maximal Clarke Subdi erentials Are Generic

We show that on a separable Banach space most Lipschitz functions have maximal Clarke subdiierential mappings. In particular, the generic nonexpansive function has the dual unit ball as its Clarke subdiierential at every point. Diverse corollaries are given.