Journal of Global Optimization

, Volume 46, Issue 4, pp 589–601 | Cite as

Reduction of finite exhausters

  • Jerzy Grzybowski
  • Diethard Pallaschke
  • Ryszard Urbański
Article

Abstract

In this paper we introduce the notation of shadowing sets which is a generalization of the notion of separating sets to the family of more than two sets. We prove that \({\bigcap_{i\in I}A_{i}}\) is a shadowing set of the family \({\{A_{i}\}_{i\in I}}\) if and only if \({\sum_{i\in I}A_{i}=\bigvee_{i\in I}\sum_{k\in I\setminus \{i\}}A_{i} + \bigcap_{i\in I}A_{i}}\). It generalizes the theorem stating that \({A\cap B}\) is separating set for A and B if and only if \({A+B=A\cap B+A\vee B}\). In terms of shadowing sets, we give a criterion for an arbitrary upper exhauster to be an exhauster of sublinear function and a criterion for the minimality of finite upper exhausters. Finally we give an example of two different minimal upper exhausters of the same function, which answers a question posed by Vera Roshchina (J Convex Anal, to appear).

Keywords

Minkowski–Rådström–Hörmander spaces Exhausters Pairs of closed bounded convex sets 

Mathematics Subject Classification (2000)

46B20 52A05 54B20 

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Copyright information

© Springer Science+Business Media, LLC. 2009

Authors and Affiliations

  • Jerzy Grzybowski
    • 1
  • Diethard Pallaschke
    • 2
  • Ryszard Urbański
    • 1
  1. 1.Faculty of Mathematics and Computer ScienceAdam Mickiewicz UniversityPoznańPoland
  2. 2.Institute of Operations ResearchUniversity of KarlsruheKarlsruheGermany

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