Equivalent Conditions for Local Error Bounds

K. W. Meng, Xiaoqi Yang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

9 Citations (Scopus)

Abstract

In this paper we present two classes of equivalent conditions for local error bounds in finite dimensional spaces. We formulate conditions of the first class by using subderivatives, subdifferentials and strong slopes for nearby points outside the referenced set, and show that these conditions actually characterize a uniform version of the local error bound property. We demonstrate this uniformity for the max function of a finite collection of smooth functions, and as a consequence we show that quasinormality constraint qualifications guarantee the existence of local error bounds. We further present the second class of equivalent conditions for local error bounds by using the various limits defined on the boundary of the referenced set. In presenting these conditions, we exploit the variational geometry of the referenced set in a systematic way and unify some existing results in the literature.
Original languageEnglish
Pages (from-to)617-636
Number of pages20
JournalSet-Valued and Variational Analysis
Volume20
Issue number4
DOIs
Publication statusPublished - 1 Dec 2012

Keywords

  • Error bounds
  • Normal cones
  • Strong slope
  • Subderivative
  • Subdifferential

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Numerical Analysis
  • Geometry and Topology
  • Applied Mathematics

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