Explicit Formula for Preimages of Relaxed One-Sided Lipschitz Mappings with Negative Lipschitz Constants: A Geometric Approach

Andrew S. Eberhard, Boris S. Mordukhovich, Janosch Rieger

Research output: Contribution to journalArticleResearchpeer-review


This paper addresses Lipschitzian stability issues, that play an important role in both theoretical and numerical aspects of variational analysis, optimization, and their applications. We particularly concentrate on the so-called relaxed one-sided Lipschitz property of set-valued mappings with negative Lipschitz constants. This property has been much less investigated than more conventional Lipschitzian behavior, while being well recognized in a variety of applications. Recent work has revealed that set-valued mappings satisfying the relaxed one-sided Lipschitz condition with negative Lipschitz constant possess a localization property, that is stronger than uniform metric regularity, but does not imply strong metric regularity. The present paper complements this fact by providing a characterization, not only of one specific single point of a preimage, but of entire preimages of such mappings. Developing a geometric approach, we derive an explicit formula to calculate preimages of relaxed one-sided Lipschitz mappings between finite-dimensional spaces and obtain a further specification of this formula via extreme points of image sets.

Original languageEnglish
Pages (from-to)34-43
Number of pages10
JournalJournal of Optimization Theory and Applications
Publication statusPublished - 28 Feb 2020


  • Explicit formula
  • Multivalued mapping
  • Preimages
  • Relaxed one-sided Lipschitz property
  • Well-posedness in variational analysis

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