A nonsmooth Morse-Sard theorem for subanalytic functions

Jérôme Bolte, Aris Daniilidis, Adrian Lewis

Research output: Contribution to journalArticleResearchpeer-review

24 Citations (Scopus)

Abstract

According to the Morse-Sard theorem, any sufficiently smooth function on a Euclidean space remains constant along any arc of critical points. We prove here a theorem of Morse-Sard type suitable as a tool in variational analysis: we broaden the definition of a critical point to the standard notion in nonsmooth optimization, while we restrict the functions under consideration to be semialgebraic or subanalytic. We make no assumption of subdifferential regularity. Łojasiewicz-type inequalities for nonsmooth functions follow quickly from tools of the kind we develop, leading to convergence theory for subgradient dynamical systems. © 2005 Elsevier Inc. All rights reserved.
Original languageEnglish
Pages (from-to)729-740
JournalJournal of Mathematical Analysis and Applications
Volume321
DOIs
Publication statusPublished - 15 Sep 2006

Keywords

  • Critical point
  • Morse-Sard theorem
  • Nonregular function
  • Nonsmooth analysis
  • Semialgebraic function
  • Subanalytic function

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