Set intersection theorems and existence of optimal solutions

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16 Scopus citations

Abstract

The question of nonemptiness of the intersection of a nested sequence of closed sets is fundamental in a number of important optimization topics, including the existence of optimal solutions, the validity of the minimax inequality in zero sum games, and the absence of a duality gap in constrained optimization. We consider asymptotic directions of a sequence of closed sets, and introduce associated notions of retractive, horizon, and critical directions, based on which we provide new conditions that guarantee the nonemptiness of the corresponding intersection. We show how these conditions can be used to obtain simple and unified proofs of some known results on existence of optimal solutions, and to derive some new results, including a new extension of the Frank-Wolfe Theorem for (nonconvex) quadratic programming.

Original languageEnglish (US)
Pages (from-to)287-314
Number of pages28
JournalMathematical Programming
Volume110
Issue number2
DOIs
StatePublished - Jul 2007
Externally publishedYes

Keywords

  • Asymptotic direction
  • Frank-Wolfe theorem
  • Global minimum
  • Quasiconvex function
  • Recession direction
  • Set intersection

ASJC Scopus subject areas

  • Software
  • Mathematics(all)

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