Approximate convexity and submonotonicity

Aris Daniilidis, Pando Georgiev

Research output: Contribution to journalArticleResearchpeer-review

48 Citations (Scopus)

Abstract

It is shown that a locally Lipschitz function is approximately convex if, and only if, its Clarke subdifferential is a submonotone operator. Consequently, in finite dimensions, the class of locally Lipschitz approximately convex functions coincides with the class of lower-C1 functions. Directional approximate convexity is introduced and shown to be a natural extension of the class of lower-C1 functions in infinite dimensions. The following characterization is established: a multivalued operator is maximal cyclically submonotone if, and only if, it coincides with the Clarke subdifferential of a locally Lipschitz directionally approximately convex function, which is unique up to a constant. Furthermore, it is shown that in Asplund spaces, every regular function is generically approximately convex. © 2003 Elsevier Inc. All rights reserved.
Original languageEnglish
Pages (from-to)292-301
JournalJournal of Mathematical Analysis and Applications
Volume291
DOIs
Publication statusPublished - 1 Mar 2004

Keywords

  • Approximate convexity
  • Cyclicity
  • Lower-C function 1
  • Submonotone operator

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