Skip to main content
SpringerLink
Log in

Generic Uniqueness of the Solution of “Max Min” Problems

  • Conference paper
Optimization, Parallel Processing and Applications

Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 304))

  • 454 Accesses

Abstract

Let X and Y be compact spaces and f be from C(X × Y) ( the space of all continuous real-valued functions on X × Y). Consider the “max min” problem ( Pf ) of maximizing ( over X ) the function min { f(x,y) : y ∈ Y }. A solution to this problem is every pair (x0,y0) ∈ X × Y for which

$$f\left( {{x}_{0}},{{y}_{0}} \right)\,=\,\min \,\left\{ \,f\left( {{x}_{0}},y \right)\,:\,y\,\in \,Y \right\}\,=\,\max \,\left\{ \min \,\left\{ f\left( x,y \right)\,:\,y\,\in \,Y\, \right\}\,:\,x\,\in \,X\, \right\}$$

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. H.P.R.Christensen, Theorems of Namioka and R.E.Johnson type for upper semicontinuous and compact valued set-valued mappings, PAMS, Vol. 86, No. 4 (1982), 649–655.

    Article  Google Scholar 

  2. H.P.R. Christensen, P.S. Kenderov, Dense strong H.P.R.Christensen, P.S.Kenderov, continuity of mappings and the Radon-Nikodym property, Univ. Copenhagen Preprint Ser.#17, 1982;

    Google Scholar 

  3. H.P.R.Christensen, P.S.Kenderov, published in Math. Scand. 54 (1984), 70–78.

    Google Scholar 

  4. M.M.Čoban, P.S.Kenderov, Generic Gateaux differentiability of convex functionals in C(T) and the topological properties of T, C.R.Acad. Bulgare Sci. 38 (1985), 1603–1604.

    Google Scholar 

  5. M.M.Čoban, P.S.Kenderov, Generic Gateaux differentiability of convex functionals in C(T) and the topological properties of T, Math. and Education in Math. 1986 Proc. 15-th Conf. Union of Bulg. Mathematicians, Sunny beach, April 1986, 141–149.

    Google Scholar 

  6. G.Debs, Espaces K-analytiques et espaces de Baire de fonctions continues, Mathematika 32 (1985), 218–228.

    Article  Google Scholar 

  7. F.Deutsch, P.S.Kenderov, Continuous selections and approximate selections for set-valued mappings and applications to metric projections, SIAM J. Math. Anal., Vol. 14, No. 1, 1983.

    Google Scholar 

  8. P.S.Kenderov, Most of the optimization problems have unique solution, C.R.Acad. Bulgare Sci. 37,#3(1984).

    Google Scholar 

  9. P.S.Kenderov, Most of the optimization problems have unique solution, International Series of Numerical Mathematics, Vol.72 (1984), Birkhäuser Verlag Basel, 203–216.

    Google Scholar 

  10. P.S.Kenderov, C(T) is weak Asplund for every Gul’ko compact T, C.R.Acad. Bulgare Sci. 40,#2(1987).

    Google Scholar 

  11. I.Namioka, Radon-Nikodym Compact Spaces and Fragmentability, preprint.

    Google Scholar 

  12. N.K.Ribarska, Internal characterization of fragmented spaces, to appear.

    Google Scholar 

  13. C.Stegall, A class of topological spaces and differentiation of functions on Banach spaces, Vorlesungen aus dem Fachbereich Mathematik der Universitat Essen, 1983.

    Google Scholar 

  14. C.Stegall, More Gateaux differentiability spaces, in Banach spaces, Proceedings, Missouri 1984, Springer Lecture Notes no. 1166, Berlin, 1985.

    Google Scholar 

  15. C.Stegall, Topological spaces with dense subspaces that are homeomrphic to complete metric spaces and the classification of C(K) Banach spaces, preprint.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Bulgarian Academy of Sciences, “Acad. G.Bonchev”-Street, Block 8, 1113, Sofia, Bulgaria

    P. S. Kenderov & N. K. Ribarska

Authors
  1. P. S. Kenderov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. K. Ribarska
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. International Institute for Applied Systems Analysis (IIASA), Schloßplatz 1, A-2361, Laxenburg, Austria

    Alexander Kurzhanski

  2. Institute of Mathematics and Mechanics of the Ural Scientific Center of the Academy of Sciences of the USSR, Sverdlovsk, USSR

    Alexander Kurzhanski

  3. Institut für Wirtschaftstheorie und Operations Research, Universität Karlsruhe, Kaiserstr. 12, D-7500, Karlsruhe, Germany

    Klaus Neumann

  4. Institut für Statistik und Mathematische Wirtschaftstheorie, Universität Karlsruhe, Postfach 6980, D-7500, Karlsruhe 1, Germany

    Diethard Pallaschke

Rights and permissions

Reprints and permissions

Copyright information

© 1988 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Kenderov, P.S., Ribarska, N.K. (1988). Generic Uniqueness of the Solution of “Max Min” Problems. In: Kurzhanski, A., Neumann, K., Pallaschke, D. (eds) Optimization, Parallel Processing and Applications. Lecture Notes in Economics and Mathematical Systems, vol 304. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46631-1_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-46631-1_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-19053-0

  • Online ISBN: 978-3-642-46631-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation