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Erschienen in:
Metric and Topological Tools Kapitel lesen Erstes Kapitel lesen

2013 | OriginalPaper | Buchkapitel

4. Elementary and Viscosity Subdifferentials

verfasst von : Jean-Paul Penot

Erschienen in: Calculus Without Derivatives

Verlag: Springer New York

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Abstract

We devote the present chapter to some fundamental notions of nonsmooth analysis upon which some other constructions can be built. Their main features are easy consequences of the definitions. Normal cones have already been considered in connection with optimality conditions. Here we present their links with subdifferentials for nonconvex, nonsmooth functions. When possible, we mention the corresponding notions of tangent cones and directional derivatives; then one gets a full picture of four related objects that can be considered the four pillars of nonsmooth analysis, or even the six pillars if one considers graphical derivatives and coderivatives of multimaps. In the present framework, in contrast to the convex objects defined in Chap. 5, the passages from directional derivatives and tangent cones to subdifferentials and normal cones respectively are one-way routes, because the first notions are nonconvex, while a dual object exhibits convexity properties. On the other hand, the passages from analytical notions to geometrical notions and the reverse passages are multiple and useful. These connections are part of the attractiveness of nonsmooth analysis.

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Vorheriges Kapitel Elements of Convex Analysis
Nächstes Kapitel Circa-Subdifferentials, Clarke Subdifferentials
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Metadaten
Titel
Elementary and Viscosity Subdifferentials
verfasst von
Jean-Paul Penot
Copyright-Jahr
2013
Verlag
Springer New York
Buch
Calculus Without Derivatives

Print ISBN: 978-1-4614-4537-1

Electronic ISBN: 978-1-4614-4538-8

Copyright-Jahr: 2013

DOI
https://doi.org/10.1007/978-1-4614-4538-8_4

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