Tame functions are semismooth

  • Authors:

  • Jérôme Bolte;Aris Daniilidis;Adrian Lewis

  • Affiliations:

  • Université Pierre et Marie Curie, Equipe Combinatoire et Optimisation, Case 189, 4 Place Jussieu, 75252, Paris Cedex 05, France;Universitat Autònoma de Barcelona, Departament de Matemàtiques, 4 Place Jussieu, 08193, Bellaterra, Spain;Cornell University, School of ORIE, 4 Place Jussieu, 14853, Ithaca, NY, USA

  • Venue:
  • Mathematical Programming: Series A and B
  • Year:
  • 2008

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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 $$O(2^{-{(1+\gamma)}^k})$$.