Journal of Global Optimization

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

Reduction of finite exhausters

  • Jerzy Grzybowski
  • Diethard Pallaschke
  • Ryszard Urbański


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).


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

Mathematics Subject Classification (2000)

46B20 52A05 54B20 


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  1. 1.
    Birkhoff, G.: Lattice Theory. Corrected reprint of the 1967 third edition. Amer. Math. Soc. Colloquium Publications, Vol. 25, American Mathematical Society, Providence, RI (1979)Google Scholar
  2. 2.
    Clarke F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)Google Scholar
  3. 3.
    Cristescu R.: Topological Vector Spaces. Noordhoff International Publishing Leyden, The Netherlands (1977)Google Scholar
  4. 4.
    Debreu, G.: Integration of correspondences. In: Proceedings of the Fifth Berkeley Symposium Mathematics, Statistics and Probability, Berkeley and Los Angeles, University of California Press, vol. 2, part 1, pp. 351–372 (1967)Google Scholar
  5. 5.
    Demyanov V.F.: Exhausters of a positively homogeneous function. Optimization 45, 13–29 (1999)CrossRefGoogle Scholar
  6. 6.
    Demyanov V.F.: Exhausters ond convexificators new tools in nonsmooth analysis. In: Demyanov, V.F., Rubinov, A.M. (eds) Quasidifferentiability and Related Topics, pp. 85–137. Kluwer Academic Publishers, Dortrecht (2000)Google Scholar
  7. 7.
    Demyanov V.F., Roshchina V.A.: Constrained optimality conditions in terms of upper and lower exhausters. Appl. Comput. Math. 4, 25–35 (2005)Google Scholar
  8. 8.
    Demyanov V.F., Roshchina V.: Optimality conditions in terms of upper and lower exhausters. Optimization 55, 525–540 (2006)CrossRefGoogle Scholar
  9. 9.
    Demyanov V.F., Roshchina V.: Exhausters and subdifferentials in non-smooth analysis. Optimization 57, 41–56 (2008)CrossRefGoogle Scholar
  10. 10.
    Demyanov V.F., Roshchina V.: Exhausters, optimality conditions and related problems. J. Global Optim. 40, 71–85 (2008)CrossRefGoogle Scholar
  11. 11.
    Demyanov V.F., Roshchina V.A.: Optimality conditions in term of upper and lower exhausters. Optimization 55(5–6), 525–540 (2006)CrossRefGoogle Scholar
  12. 12.
    Demyanov V.F., Rubinov A.M.: Quasidifferential Calculus. Optimization Software Inc., Publication Division, New York (1986)Google Scholar
  13. 13.
    Demyanov V.F., Rubinov A.M.: Quasidifferentiability and Related Topics. Nonconvex Optimization and its Applications, vol. 43. Kluwer Academic Publishers, Dortrecht–Boston–London (2000)Google Scholar
  14. 14.
    Drewnowski L.: Additive and countably additive correspondences. Comment. Math. 19, 25–54 (1976)Google Scholar
  15. 15.
    Goossens P.: Completeness of spaces of closed bounded convex sets. J. Math. Anal. Appl. 115(1), 192–201 (1986)CrossRefGoogle Scholar
  16. 16.
    Grzybowski J.: Minimal pairs of compact sets. Arch. Math 63, 173–181 (1994)CrossRefGoogle Scholar
  17. 17.
    Grzybowski J., Urbański R.: Minimal pairs of bounded closed convex sets. Studia Math. 126, 95–99 (1997)Google Scholar
  18. 18.
    Grzybowski J., Urbański R.: On inclusion and summands of bounded closed convex sets. Acta. Math. Hungar 106(4), 293–300 (2005)CrossRefGoogle Scholar
  19. 19.
    Hörmander L.: Sur la fonction d’appui des ensembles convexes dans un espace localement convexe. Arkiv för Mathematik 3, 181–186 (1954)CrossRefGoogle Scholar
  20. 20.
    Kurosz A.G.: Algebra Ogólna. PWN - Polish Scientific Publishers, Warszawa (1965)Google Scholar
  21. 21.
    Pallaschke D., Urbański R.: Pairs of Compact Convex Sets, Fractional Arithmetic with Convex Sets, Mathematics and Its Applications. Kluwer Academic Publisher, Dortrecht–Boston–London (2002)Google Scholar
  22. 22.
    Pallaschke D., Przybycień H., Urbański R.: On partialy ordered semigroups. J. Set-Valued Anal. 16, 257–265 (2007)CrossRefGoogle Scholar
  23. 23.
    Pallaschke D., Scholtes S., Urbański R.: On minimal pairs of convex compact sets. Bull. Polish Acad. Sci. Math. 39, 1–5 (1991)Google Scholar
  24. 24.
    Praksash P., Sertel M.R.: Hyperspaces of topological vector spaces: their embedding in topological vector spaces. Proc. Amer. Math. Soc. 61, 163–168 (1976)CrossRefGoogle Scholar
  25. 25.
    Rådström H.: An embedding theorem for spaces of convex sets. Proc. Amer. Math. Soc. 3, 165–169 (1952)CrossRefGoogle Scholar
  26. 26.
    Ratschek H., Schröder G.: Representation of semigroups as systems of compact convex sets. Proc. American Math. Soc. 65, 24–28 (1977)CrossRefGoogle Scholar
  27. 27.
    Roshchina V.: Reducing exhausters. J. Optim. Theory Appl. 136, 261–273 (2008)CrossRefGoogle Scholar
  28. 28.
    Roshchina V.: Relationships between upper exhausters and the basic subdifferential in variational analysis. J. Math. Anal. Appl 334, 261–272 (2007)CrossRefGoogle Scholar
  29. 29.
    Roshchina V.: On conditions for the minimality of exhausters. J. Convex Anal. 15(4), 859–868 (2008)Google Scholar
  30. 30.
    Scholtes S.: Minimal pairs of convex bodies in two dimensions. Mathematika 39, 267–273 (1992)CrossRefGoogle Scholar
  31. 31.
    Urbański R.: A generalization of the Minkowski–Rådström–Hörmander theorem. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys 24, 709–715 (1976)Google Scholar
  32. 32.
    Urbański R.: On minimal convex pairs of convex compact sets. Arch. Math. 67, 226–238 (1996)CrossRefGoogle Scholar

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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