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Israel Journal of Mathematics

, Volume 212, Issue 2, pp 757–790 | Cite as

Sard theorems for Lipschitz functions and applications in optimization

  • Luc BarbetEmail author
  • Marc Dambrine
  • Aris DaniilidisEmail author
  • Ludovic RiffordEmail author
Article

Abstract

We establish a “preparatory Sard theorem” for smooth functions with a partially affine structure. By means of this result, we improve a previous result of Rifford [17, 19] concerning the generalized (Clarke) critical values of Lipschitz functions defined as minima of smooth functions. We also establish a nonsmooth Sard theorem for the class of Lipschitz functions from R d to R p that can be expressed as finite selections of C k functions (more generally, continuous selections over a compact countable set). This recovers readily the classical Sard theorem and extends a previous result of Barbet–Daniilidis–Dambrine [1] to the case p > 1. Applications in semi-infinite and Pareto optimization are given.

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Copyright information

© Hebrew University of Jerusalem 2016

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques et de leurs ApplicationsUMR CNRS 5142 Université de Pau et des Pays de l’AdourPauFrance
  2. 2.DIM–CMM, UMI CNRS 2807Universidad de Chile Beauchef 581Santiago de ChileChile
  3. 3.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterraSpain
  4. 4.Laboratoire J. A. DieudonnéUniversité Nice Sophia Antipolis Parc ValroseNice Cedex 2France

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