A Chain Rule for Essentially Smooth Lipschitz Functions

Borwein, Jonathan M.; Moors, Warren B.

doi:10.1137/s1052623496297838

Cited by 18 publications

(28 citation statements)

References 8 publications

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“…Haar-null in E in the sense that there exists a Radon probability measure P on E such that P (N + y) = 0 for all y ∈ E. The essential smoothness is not stable by composition; that is, there are locally Lipschitz continuous functions f and mappings F with values in R n which are essentially smooth (i.e., each component of F is essentially smooth) such that f • F fails to be essentially smooth (see [5]). …”

Section: Essentially Directionally Smooth Functions and The Main Resumentioning

confidence: 99%

“…That nonclosedness property under composition led Borwein and Moors [5] to introduce for E = R m , as a large subclass, the concept of arcwise essentially smooth functions, which is in line with Valadier's sound functions (fonctions saines in French; see [36]) and which is preserved under composition. Observing that the equality…”

Section: Essentially Directionally Smooth Functions and The Main Resumentioning

confidence: 99%

“…It is a direct consequence of Propositions 4.3 and 4.4. Our next step is to prove that the class of locally Lipschitz continuous and segmentwise essentially smooth functions of Borwein and Moors [5] is contained in the class of essentially ∂-directionally smooth functions g (with D = dom g) given in Definition 3.1. Before that, let us introduce another class.…”

Section: By Lemmas 41 and 42 For Everymentioning

confidence: 99%

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Subdifferential Determination of Essentially Directionally Smooth Functions in Banach Space

Thibault¹,

Zagrodny²

2010

SIAM J. Optim.

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It is known (see [J. M. 157-174]) 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., g = f + c. 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. Moreover, it is established that the essentially directional smoothness property is preserved under addition. It is also shown that the class of essentially directionally smooth functions includes that of directionally stable functions (studied in [L. Thibault and D. Zagrodny, Canad. Math. Bull., 48 (2005), pp. 283-301]) as well as several other classes of functions.

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Section: Essentially Directionally Smooth Functions and The Main Resumentioning

confidence: 99%

Section: Essentially Directionally Smooth Functions and The Main Resumentioning

confidence: 99%

Section: By Lemmas 41 and 42 For Everymentioning

confidence: 99%

See 1 more Smart Citation

Subdifferential Determination of Essentially Directionally Smooth Functions in Banach Space

Thibault¹,

Zagrodny²

2010

SIAM J. Optim.

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“…Chain rules are of vital importance from the point of view of applications and have been the focus of intensive research, see, e.g., the papers by Borwein and Moors [3], Jourani and Thibault [18], Mordukhovich [24], Mordukhovich and Shao [28], Ralph [36], Thibault [44]. A chain rule for the composition of a smooth (finite dimensional) and a nonsmooth function has been obtained by the authors in [30] in terms of the generalized Jacobians for functions with finite dimensional range.…”

Section: Chain Rules and Their Consequencesmentioning

confidence: 99%

Generalized Jacobian for Functions with Infinite Dimensional Range and Domain

Zeidan

2007

Set-Valued Anal

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In this paper, locally Lipschitz functions acting between infinite dimensional normed spaces are considered. When the range is a dual space and satisfies the Radon-Nikodým property, Clarke's generalized Jacobian will be extended to this setting. Characterization and fundamental properties of the extended generalized Jacobian are established including the nonemptiness, the β-compactness, the β-upper semicontinuity, and a mean-value theorem. A connection with known notions is provided and chain rules are proved using key results developed. This included the vectorization and restriction theorem, and the extension theorem. Therefore, the generalized Jacobian introduced in this paper is proved to enjoy all the properties required of a derivative like-set.

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“…To see that CSC(0) is the smallest cusco containing 0 it suffices to observe that for any cusco 9 containing 0, Gr(CSC(0)) Gr(CSC (9 ))=Gr (9 ). K Note.…”

Section: Introductionmentioning

confidence: 99%

Essentially Smooth Lipschitz Functions

Borwein

Moors

1997

Journal of Functional Analysis

Self Cite

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In this paper we address some of the most fundamental questions regarding the differentiability structure of locally Lipschitz functions defined on separable Banach spaces. For example, we examine the relationship between integrability, D-representability, and strict differentiability. In addition to this, we show that on any separable Banach space there is a significant family of locally Lipschitz functions that are integrable, D-representable and possess desirable differentiability properties. We also present some striking applications of our results to distance functions. Academic Press

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A Chain Rule for Essentially Smooth Lipschitz Functions

Cited by 18 publications

References 8 publications

Subdifferential Determination of Essentially Directionally Smooth Functions in Banach Space

Subdifferential Determination of Essentially Directionally Smooth Functions in Banach Space

Generalized Jacobian for Functions with Infinite Dimensional Range and Domain

Essentially Smooth Lipschitz Functions

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