- Title
- Rotund norms, Clarke subdifferentials and extensions of Lipschitz functions
- Creator
- Borwein, Jon; Giles, John; Vanderwerff, Jon
- Relation
- Nonlinear Analysis: Theory, Methods & Applications Vol. 48, Issue 2, p. 287-301
- Publisher Link
- http://dx.doi.org/10.1016/S0362-546X(00)00187-5
- Publisher
- Elsevier
- Resource Type
- journal article
- Date
- 2002
- Description
- The Clarke derivative of a locally Lipschitz function is defined by fo(x;v):=[formula cannot be replicated], and the Clarke subdifferential is defined by ∂cf(x) = {⏀∈X*:⏀(v) ≤ fo(x;v) for all v∈X}. This subdifferential has been widely used as a powerful tool in nonsmooth analysis with applications in diverse areas of optimization. Recently, substantial progress has been made on understanding the limitations of the Clarke derivative. Among other things, it is shown that on any Banach space X, the 1-Lipschitz functions for which ∂cf(x)=Bx* for all x∈X, is a residual set among all the 1-Lipschitz functions on X (where Bx* denotes the dual unit ball). That is, even though the Clarke derivative is an effective tool in a wide variety of both theoretical and applied optimization problems, just like the classical derivative, the class of pathological Lipschitz functions for which it provides no additional information is larger in the category sense. In this note, we begin by considering the following related question, which asks how profuse (from the point of view of extensions) the functions in the aforementioned result are.
- Subject
- Baire category; Clarke subdifferentials; Lipschitz functions; extensions; rotund norms
- Identifier
- http://hdl.handle.net/1959.13/940682
- Identifier
- uon:13076
- Identifier
- ISSN:0362-546X
- Language
- eng
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