A Chain Rule for Essentially Smooth Lipschitz Functions

In this paper we introduce a new class of real-valued locally Lipschitz functions (that are similar in nature and definition to Valadier's saine functions), which we call arcwise essentially smooth, and we show that if g : Rm \rightarrow R$ is arcwise essentially smooth on Rm and each function...

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Bibliographic Details
Published inSIAM journal on optimization Vol. 8; no. 2; pp. 300 - 308
Main Authors Borwein, Jonathan M., Moors, Warren B.
Format Journal Article
LanguageEnglish
Published Philadelphia Society for Industrial and Applied Mathematics 01.05.1998
Subjects
Applied mathematics
Banach spaces
Mathematical functions
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Summary:In this paper we introduce a new class of real-valued locally Lipschitz functions (that are similar in nature and definition to Valadier's saine functions), which we call arcwise essentially smooth, and we show that if g : Rm \rightarrow R$ is arcwise essentially smooth on Rm and each function $f_j : R^n \rightarrow R,\ 1 \leq j \leq m$, is strictly differentiable almost everywhere in Rn, then $g \circ f$ is strictly differentiable almost everywhere in Rn, where $f \equiv (f_1,f_2,\ldots,f_m)$. We also show that all the semismooth and all the pseudoregular functions are arcwise essentially smooth. Thus, we provide a large and robust lattice algebra of Lipschitz functions whose generalized derivatives are well behaved.
ISSN:1052-6234
1095-7189
DOI:10.1137/S1052623496297838