Results 1 
6 of
6
Lipschitz Functions with Prescribed Derivatives and Subderivatives
, 1995
"... . In general it is difficult to construct Lipschitz functions which are not directly built up from either convex or distance functions. One impediment to such constructions is that outside of the real line it is difficult to find antiderivatives. The main result of this paper provides, under suitab ..."
Abstract

Cited by 11 (4 self)
 Add to MetaCart
. In general it is difficult to construct Lipschitz functions which are not directly built up from either convex or distance functions. One impediment to such constructions is that outside of the real line it is difficult to find antiderivatives. The main result of this paper provides, under suitable circumstances, a technique for constructing such antiderivatives. More precisely, we show that if f 1 , f 2 ,: : :, f n are continuously differentiable realvalued locally Lipschitz functions defined on a nonempty open subset A of a separable Banach space X, then there exists a realvalued locally Lipschitz function g defined on A such that at each point x 2 A the Clarke subgradient of g at x equals cofrf 1 (x); rf 2 (x); : : : ; rf n (x)g. This same construction also shows that for any finite family fT 1 ; T 2 ; : : : ; T n g of maximal cyclically monotone mappings from A into nonempty subsets of X , there exists a realvalued locally Lipschitz function g defined on A such that at e...
Subgradient Representation of Multifunctions
 CECM RESEARCH REPORT 97090, J. AUST. MATH. SOC., SERIES B
, 1997
"... We provide necessary and sufficient conditions for a minimal upper semicontinuous multifunction defined on a separable Banach space to be the subdifferential mapping of a Lipschitz function. ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
We provide necessary and sufficient conditions for a minimal upper semicontinuous multifunction defined on a separable Banach space to be the subdifferential mapping of a Lipschitz function.
Lipschitz functions with minimal Clarke subdifferential mappings
, 1996
"... In this paper we characterise, in terms of the upper Dini derivative, when the Clarke subdifferential mapping of a realvalued locally Lipschitz function is a minimal weak cusco. We then use this characterisation to deduce some new results concerning Lipschitz functions with minimal subdifferenti ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
In this paper we characterise, in terms of the upper Dini derivative, when the Clarke subdifferential mapping of a realvalued locally Lipschitz function is a minimal weak cusco. We then use this characterisation to deduce some new results concerning Lipschitz functions with minimal subdifferential mappings.
A chain rule for essentially strictly differentiable Lipschitz functions
, 1996
"... In this paper we introduce a new class of realvalued 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 n ! R is arcwise essentially smooth on R n and each function f ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
In this paper we introduce a new class of realvalued 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 n ! R is arcwise essentially smooth on R n and each function f j : R m ! R; 1 j n is strictly differentiable almost everywhere in R m , then g ffi f is strictly differentiable almost everywhere in R m , where f j (f 1 ; f 2 ; :::f n ). We also show that all the semismooth and pseudoregular functions are arcwise essentially smooth. Thus, we provide a large and robust lattice algebra of Lipschitz functions whose generalized derivatives are wellbehaved.
Abstract subdifferential calculus and semiconvex functions
 Serdica Math. J
, 1997
"... Abstract. We develop an abstract subdifferential calculus for lower semicontinuous functions and investigate functions similar to convex functions. As application we give sufficient conditions for the integrability of a lower semicontinuous function. 1. Introduction. Throughout this paper ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We develop an abstract subdifferential calculus for lower semicontinuous functions and investigate functions similar to convex functions. As application we give sufficient conditions for the integrability of a lower semicontinuous function. 1. Introduction. Throughout this paper
JeanMichel Morel President du jury
, 2007
"... on the variational models and dictionary learning ..."
(Show Context)