Tame functions are semismooth

Jérôme Bolte; Aris Daniilidis; Adrian Lewis

doi:10.1007/s10107-007-0166-9

Tame functions are semismooth

Jérôme Bolte, Aris Daniilidis, Adrian Lewis

Department of Mathematics

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

26 Citations (Scopus)

Abstract

Superlinear convergence of the Newton method for nonsmooth equations requires a "semismoothness" assumption. In this work we prove that locally Lipschitz functions definable in an o-minimal structure (in particular semialgebraic or globally subanalytic functions) are semismooth. Semialgebraic, or more generally, globally subanalytic mappings present the special interest of being γ-order semismooth, where γ is a positive parameter. As an application of this new estimate, we prove that the error at the kth step of the Newton method behaves like O(2-(1+γ)k). © 2007 Springer-Verlag.

Original language	English
Pages (from-to)	5-19
Journal	Mathematical Programming
Volume	117
Issue number	1-2
DOIs	https://doi.org/10.1007/s10107-007-0166-9
Publication status	Published - 1 Mar 2009

Keywords

Nonsmooth Newton method
O-minimal structure
Semi-algebraic function
Semismoothness
Structured optimization problem
Superlinear convergence

Access to Document

10.1007/s10107-007-0166-9

Fingerprint Dive into the research topics of 'Tame functions are semismooth'. Together they form a unique fingerprint.

Cite this

@article{1b4ebfc7e1df45cd8ad3670bc24f890a,

title = "Tame functions are semismooth",

abstract = "Superlinear convergence of the Newton method for nonsmooth equations requires a {"}semismoothness{"} assumption. In this work we prove that locally Lipschitz functions definable in an o-minimal structure (in particular semialgebraic or globally subanalytic functions) are semismooth. Semialgebraic, or more generally, globally subanalytic mappings present the special interest of being γ-order semismooth, where γ is a positive parameter. As an application of this new estimate, we prove that the error at the kth step of the Newton method behaves like O(2-(1+γ)k). {\textcopyright} 2007 Springer-Verlag.",

keywords = "Nonsmooth Newton method, O-minimal structure, Semi-algebraic function, Semismoothness, Structured optimization problem, Superlinear convergence",

author = "J{\'e}r{\^o}me Bolte and Aris Daniilidis and Adrian Lewis",

year = "2009",

month = mar,

day = "1",

doi = "10.1007/s10107-007-0166-9",

language = "English",

volume = "117",

pages = "5--19",

journal = "Mathematical Programming",

issn = "0025-5610",

publisher = "Springer-Verlag GmbH and Co. KG",

number = "1-2",

}

TY - JOUR

T1 - Tame functions are semismooth

AU - Bolte, Jérôme

AU - Daniilidis, Aris

AU - Lewis, Adrian

PY - 2009/3/1

Y1 - 2009/3/1

N2 - Superlinear convergence of the Newton method for nonsmooth equations requires a "semismoothness" assumption. In this work we prove that locally Lipschitz functions definable in an o-minimal structure (in particular semialgebraic or globally subanalytic functions) are semismooth. Semialgebraic, or more generally, globally subanalytic mappings present the special interest of being γ-order semismooth, where γ is a positive parameter. As an application of this new estimate, we prove that the error at the kth step of the Newton method behaves like O(2-(1+γ)k). © 2007 Springer-Verlag.

AB - Superlinear convergence of the Newton method for nonsmooth equations requires a "semismoothness" assumption. In this work we prove that locally Lipschitz functions definable in an o-minimal structure (in particular semialgebraic or globally subanalytic functions) are semismooth. Semialgebraic, or more generally, globally subanalytic mappings present the special interest of being γ-order semismooth, where γ is a positive parameter. As an application of this new estimate, we prove that the error at the kth step of the Newton method behaves like O(2-(1+γ)k). © 2007 Springer-Verlag.

KW - Nonsmooth Newton method

KW - O-minimal structure

KW - Semi-algebraic function

KW - Semismoothness

KW - Structured optimization problem

KW - Superlinear convergence

U2 - 10.1007/s10107-007-0166-9

DO - 10.1007/s10107-007-0166-9

M3 - Article

VL - 117

SP - 5

EP - 19

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

IS - 1-2

ER -