Tame functions are semismooth

Jérôme Bolte, Aris Daniilidis, Adrian Lewis

Research output: Contribution to journalArticleResearchpeer-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 languageEnglish
Pages (from-to)5-19
JournalMathematical Programming
Volume117
Issue number1-2
DOIs
Publication statusPublished - 1 Mar 2009

Keywords

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

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