A chain rule for essentially smooth Lipschitz functions

Title: A chain rule for essentially smooth Lipschitz functions
Creator: Borwein, Jonathan M.; Moors, Warren B.
Relation: SIAM Journal on Optimization Vol. 8, Issue 2, p. 300-308
Publisher Link: http://dx.doi.org/10.1137/S1052623496297838
Publisher: Society for Industrial and Applied Mathematics (SIAM)
Resource Type: journal article
Date: 1998
Description: 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 : R^m → R is arcwise essentially smooth on R^m and each function f_j : R^n → R, 1 ≤ j ≤ m, is strictly differentiable almost everywhere in Rⁿ, then g ○ f is strictly differentiable almost everywhere in Rⁿ, where f ≡ (f₁,f₂,...,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.
Subject: Lipschitz functions; chain rule; Haar-null sets; differentiability; essentially smooth
Identifier: http://hdl.handle.net/1959.13/940494
Identifier: uon:13023
Identifier: ISSN:1052-6234
Language: eng
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