Subdifferential Determination of Essentially Directionally Smooth Functions in Banach Space

It is known that the subdifferential of a semismooth or essentially smooth locally Lipschitz continuous function f over a Banach space determines this function up to an additive constant in the sense that any other function of the same type g whose subdifferential coincides with that of f at every p...

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Bibliographic Details
Published in:SIAM journal on optimization Vol. 20; no. 5; pp. 2300 - 2326
Main Authors: Thibault, Lionel, Zagrodny, Dariusz
Format: Journal Article
Language:English
Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2010
Subjects:
Additives
Banach space
Banach spaces
Convex analysis
Expansion
Inclusions
Optimization
Studies
Symbols
ISSN:1052-6234, 1095-7189
Online Access:Get full text
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Summary:It is known that the subdifferential of a semismooth or essentially smooth locally Lipschitz continuous function f over a Banach space determines this function up to an additive constant in the sense that any other function of the same type g whose subdifferential coincides with that of f at every point is equal to f plus a constant, i.e, ... . Unfortunately, those classes of locally Lipschitz continuous functions do not include proper lower semicontinuous convex functions taking the value ... at some points. In this paper a new concept of essentially directionally smooth functions is introduced, and it is also shown, by a detailed analysis of enlarged inclusions of their subdifferentials, that these functions are subdifferentially determined up to an additive constant. It is also proved that the class of such functions contains proper lower semicontinuous convex functions and locally Lipschitz continuous functions which are arcwise essentially smooth. (ProQuest: ... denotes formulae/symbols omitted.)
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ISSN:1052-6234
1095-7189
DOI:10.1137/090754571