Equivalence among various derivatives and subdifferentials of the distance function

Zili Wu; Jane J. Ye

doi:10.1016/S0022-247X(03)00213-0

Equivalence among various derivatives and subdifferentials of the distance function

Zili Wu^*
, Jane J. Ye

^*Corresponding author for this work

University of Victoria BC

Research output: Contribution to journal › Article › peer-review

23 Citations (Scopus)

Abstract

For a nonempty closed set C in a normed linear space X with uniformly Gâteaux differentiable norm, it is shown that the distance function d_C is strictly differentiable at x ∈ X\C iff it is regular at x iff its modified upper or lower Dini subdifferential at x is a singleton iff its upper or lower Dini subdifferential at x is nonempty iff its upper or lower Dini derivative at x is subadditive. Moreover if X is a Hilbert space, then d_C is Fréchet differentiable at x ∈ X \ C iff its Fréchet subdifferential at x is nonempty. Many characteristics of proximally smooth sets and convex closed sets in a Hilbert space are also given.

Original language	English
Pages (from-to)	629-647
Number of pages	19
Journal	Journal of Mathematical Analysis and Applications
Volume	282
Issue number	2
DOIs	https://doi.org/10.1016/S0022-247X(03)00213-0
Publication status	Published - 15 Jun 2003
Externally published	Yes

Keywords

Distance function
Proximal smoothness
Proximal, Fréchet, Dini, and modified Dini subdifferentials
Strict Gâteaux, and Fréchet derivatives
Uniformly Gâteaux differentiable norm

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10.1016/S0022-247X(03)00213-0

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title = "Equivalence among various derivatives and subdifferentials of the distance function",

abstract = "For a nonempty closed set C in a normed linear space X with uniformly G{\^a}teaux differentiable norm, it is shown that the distance function dC is strictly differentiable at x ∈ X\textbackslash{}C iff it is regular at x iff its modified upper or lower Dini subdifferential at x is a singleton iff its upper or lower Dini subdifferential at x is nonempty iff its upper or lower Dini derivative at x is subadditive. Moreover if X is a Hilbert space, then dC is Fr{\'e}chet differentiable at x ∈ X \textbackslash{} C iff its Fr{\'e}chet subdifferential at x is nonempty. Many characteristics of proximally smooth sets and convex closed sets in a Hilbert space are also given.",

keywords = "Distance function, Proximal smoothness, Proximal, Fr{\'e}chet, Dini, and modified Dini subdifferentials, Strict G{\^a}teaux, and Fr{\'e}chet derivatives, Uniformly G{\^a}teaux differentiable norm",

author = "Zili Wu and Ye, \{Jane J.\}",

note = "Funding Information: * Corresponding author. E-mail addresses: ziliwu@email.com (Z. Wu), janeye@math.uvic.ca (J.J. Ye). 1 The research of J.J. Ye was supported by NSERC and University of Victoria Internal Research Grant.",

year = "2003",

month = jun,

day = "15",

doi = "10.1016/S0022-247X(03)00213-0",

language = "English",

volume = "282",

pages = "629--647",

journal = "Journal of Mathematical Analysis and Applications",

issn = "0022-247X",

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TY - JOUR

T1 - Equivalence among various derivatives and subdifferentials of the distance function

AU - Wu, Zili

AU - Ye, Jane J.

N1 - Funding Information: * Corresponding author. E-mail addresses: ziliwu@email.com (Z. Wu), janeye@math.uvic.ca (J.J. Ye). 1 The research of J.J. Ye was supported by NSERC and University of Victoria Internal Research Grant.

PY - 2003/6/15

Y1 - 2003/6/15

N2 - For a nonempty closed set C in a normed linear space X with uniformly Gâteaux differentiable norm, it is shown that the distance function dC is strictly differentiable at x ∈ X\C iff it is regular at x iff its modified upper or lower Dini subdifferential at x is a singleton iff its upper or lower Dini subdifferential at x is nonempty iff its upper or lower Dini derivative at x is subadditive. Moreover if X is a Hilbert space, then dC is Fréchet differentiable at x ∈ X \ C iff its Fréchet subdifferential at x is nonempty. Many characteristics of proximally smooth sets and convex closed sets in a Hilbert space are also given.

AB - For a nonempty closed set C in a normed linear space X with uniformly Gâteaux differentiable norm, it is shown that the distance function dC is strictly differentiable at x ∈ X\C iff it is regular at x iff its modified upper or lower Dini subdifferential at x is a singleton iff its upper or lower Dini subdifferential at x is nonempty iff its upper or lower Dini derivative at x is subadditive. Moreover if X is a Hilbert space, then dC is Fréchet differentiable at x ∈ X \ C iff its Fréchet subdifferential at x is nonempty. Many characteristics of proximally smooth sets and convex closed sets in a Hilbert space are also given.

KW - Distance function

KW - Proximal smoothness

KW - Proximal, Fréchet, Dini, and modified Dini subdifferentials

KW - Strict Gâteaux, and Fréchet derivatives

KW - Uniformly Gâteaux differentiable norm

UR - https://www.scopus.com/pages/publications/0037778834

U2 - 10.1016/S0022-247X(03)00213-0

DO - 10.1016/S0022-247X(03)00213-0

M3 - Article

AN - SCOPUS:0037778834

SN - 0022-247X

VL - 282

SP - 629

EP - 647

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 2

ER -

Equivalence among various derivatives and subdifferentials of the distance function

Abstract

Keywords

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