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Tame functions are semismooth

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

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Authors and Affiliations

  1. Equipe Combinatoire et Optimisation, Case 189, Université Pierre et Marie Curie, 4 Place Jussieu, 75252, Paris Cedex 05, France

    Jérôme Bolte

  2. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra, Spain

    Aris Daniilidis

  3. School of ORIE, Cornell University, Ithaca, NY, 14853, USA

    Adrian Lewis

Authors
  1. Jérôme Bolte
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  2. Aris Daniilidis
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  3. Adrian Lewis
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Corresponding author

Correspondence to Jérôme Bolte.

Additional information

Dedicated to Stephen Robinson, who has so many of the best ideas first.

A. Daniilidis was supported by the MEC Grant MTM2005-08572-C03-03 (Spain) and A. Lewis was supported by the National Science Foundation Grant DMS-0504032 (USA).

Adrian Lewis: Research supported in part by National Science Foundation Grant DMS-0504032.

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Bolte, J., Daniilidis, A. & Lewis, A. Tame functions are semismooth. Math. Program. 117, 5–19 (2009). https://doi.org/10.1007/s10107-007-0166-9

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  • DOI: https://doi.org/10.1007/s10107-007-0166-9

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