Sard theorems for Lipschitz functions and applications in optimization

Luc Barbet, Marc Dambrine, Aris Daniilidis, Ludovic Rifford

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3 Citations (Scopus)

Abstract

© 2016, Hebrew University of Jerusalem. 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 Rd to Rp that can be expressed as finite selections of Ck 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.
Original languageEnglish
Pages (from-to)757-790
JournalIsrael Journal of Mathematics
Volume212
Issue number2
DOIs
Publication statusPublished - 1 May 2016

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