Advertisement

Quasiconcavity of Sets and Connectedness of the Efficient Frontier in Ordered Vector Spaces

  • Elena Molho
  • Alberto Zaffaroni
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 27)

Abstract

We introduce new notions of quasiconcavity of sets in ordered vector spaces, extending the properties of sets which are images of convex sets by quasiconcave functions. This allows us to generalize known results and obtain new ones on the connectedness of the sets of various types of efficient solutions.

Keywords

Convex Cone Efficient Solution Vector Optimization Image Space Efficient Frontier 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Bitran, G.R. and Magnanti, T.L, “The structure of admissible points with respect to cone dominance,” J. Optim. Theory Appl, 29, pp. 573–614, 1979.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Borwein, J.M. and Zhuang, D, “Super efficiency in vector optimization,” Trans. Amer. Math. Soc, 338, pp. 105–122, 1993.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    Danilidis, A, Hadjisavvas, N. and Schaible, S, “Connectedness of the efficient set for three-objective quasiconcave maximization problems,” to appear on J. Optim. Th. Appl, 93, 1997.Google Scholar
  4. [4]
    Gong, X.H, “Connectedness of efficient solution sets for set-valued maps in normed spaces,” J. Optim. Theory Appl, 83, pp. 83–96, 1994.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Hiriart-Urruty, J.B, “Images of connected sets by semicontinuous multi-functions,” J. Math. Anal. Appl, 111, pp. 407–422, 1985.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    Hu, Y.D. and Sun, E.J, “Connectedness of the Efficient Set in Strictly Quasiconcave Vector Maximization,” J. Optim. Theory Appl, 78, pp. 613–622, 1993.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Jahn, J, “Mathematical Vector Optimization in Partially Ordered Linear Spaces,” Verlag Peter Lang, Frankfurt, 1986.Google Scholar
  8. [8]
    Jameson, G, “Ordered Linear Spaces,” LNM 141, Springer Verlag, Berlin, 1970.Google Scholar
  9. [9]
    Luc, D. T, “Connectedness of the Efficient Point Sets in Quasiconcave Vector Maximization,” Journal of Mathematical Analysis and Applications, 122, pp. 346–354, 1987.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    Luc, D.T, “Theory of Vector Optimization,” LNMES 319, Springer Verlag, Berlin and New York, 1989.Google Scholar
  11. [11]
    Makarov, E.K. and Rachkowski, N.N., “Density theorems for generalized proper efficiency,” J. Optim. Theory Appl., to appear 1996.Google Scholar
  12. [12]
    Molho, E., “On a notion of quasiconcave set and some applications to vector optimization,” in Scalar and Vector optimization in economic and financial problems, E. Castagnoli and G. Giorgi eds., pp. 113–118, 1995.Google Scholar
  13. [13]
    Naccache, P. H., “Connectedness of the Set of Nondominated Outcomes in Multicriteria Optimization,” Journal of Optimization Theory and Applications, 25, pp. 549–467, 1978.MathSciNetCrossRefGoogle Scholar
  14. [14]
    Peressini, A. L., “Ordered Topological Vector Spaces,” Harper and Row, London, 1967.Google Scholar
  15. [15]
    Sawaragi, Y., Nakayama, H. and Tanino, T., “Theory of Multiobjective Optimization,” Academic Press, New York, 1985.MATHGoogle Scholar
  16. [16]
    Schaible, S., “Bicriteria Quasiconcave Programs,” Cahiers du Centre d’Études de Recherche Operationelle, 25, pp. 93–101, 1983.MathSciNetMATHGoogle Scholar
  17. [17]
    Sun, E.J., “On the connectedness of the efficient set for strictly quasiconvex vector minimization problem,” J. Optim. Theory Appl., 89, pp. 475–481, 1996.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    Warbourton, A. R., “Quasiconcave Vector Maximization: Connectedness of the Sets of Pareto-Optimal and Weak Pareto-Optimal Alternatives,” Journal of Optimization Theory and Applications, 40, pp. 537–557, 1983.MathSciNetCrossRefGoogle Scholar
  19. [19]
    Wen, S., “A note on connectedness of efficient solutions sets,” Institute of Mathematics, Polish Academy of Sciences, Preprint 533, 1995.Google Scholar

Copyright information

© Kluwer Academic Publishers. Printed in the Netherlands 1998

Authors and Affiliations

  • Elena Molho
    • 1
  • Alberto Zaffaroni
    • 2
  1. 1.Dipartimento di Ricerche AziendaliUniversita di PaviaPaviaItaly
  2. 2.Istituto di Metodi QuantitativeUniversitá BocconiMilanoItaly

Personalised recommendations