Critical and critical tangent cones in optimization problems

Zsolt Páles, Vera Zeidan

Research output: Contribution to journalArticle

Abstract

In this paper the notion of critical tangent cone CT (x|Q) to Q at x is introduced for the case when Q is a convex subset of a normed space X. If Q is closed with nonempty interior, and x ∈ Q, the nonemptiness of the Dubovitskii-Milyutin set of second-order admissible variations, V (x, d|Q), is then characterized by the condition d ∈ CT (x|Q). Furthermore, the support function of V (x, d|Q) is shown to be evaluated in terms of that support function of Q. For the cases when Q is the set of continuous or ℒ selections of a certain set-valued map, the corresponding characterization of the cone CT (x|Q) and the formula for the support function of V (x, d|Q) are obtained in terms of more verifiable conditions.

Original languageEnglish
Pages (from-to)241-258
Number of pages18
JournalSet-Valued Analysis
Volume12
Issue number1-2
DOIs
Publication statusPublished - Jan 1 2004

Keywords

  • Critical cone
  • Critical tangent cone
  • First- and second-order optimality conditions
  • Set-valued constraints

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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