Lipschitz-like property relative to a set and the generalized Mordukhovich criterion

K. W. Meng, M. H. Li, W. F. Yao, X. Q. Yang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

1 Citation (Scopus)

Abstract

In this paper we will establish some necessary condition and sufficient condition respectively for a set-valued mapping to have the Lipschitz-like property relative to a closed set by employing regular normal cone and limiting normal cone of a restricted graph of the set-valued mapping. We will obtain a complete characterization for a set-valued mapping to have the Lipschitz-property relative to a closed and convex set by virtue of the projection of the coderivative onto a tangent cone. Furthermore, by introducing a projectional coderivative of set-valued mappings, we establish a verifiable generalized Mordukhovich criterion for the Lipschitz-like property relative to a closed and convex set. We will study the representation of the graphical modulus of a set-valued mapping relative to a closed and convex set by using the outer norm of the corresponding projectional coderivative value. For an extended real-valued function, we will apply the obtained results to investigate its Lipschitz continuity relative to a closed and convex set and the Lipschitz-like property of a level-set mapping relative to a half line.

Original languageEnglish
Pages (from-to)455 - 489
Number of pages35
JournalMathematical Programming
Volume189
Issue number1-2
DOIs
Publication statusE-pub ahead of print - 25 Sep 2020

Keywords

  • Graphical modulus
  • Level-set mapping
  • Lipschitz-like property relative to a set
  • Mordukhovich criterion
  • Projectional coderivative

ASJC Scopus subject areas

  • Software
  • Mathematics(all)

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