Advertisement

About the Duality Gap in Vector Optimization

  • G. Bigi
  • M. Pappalardo
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 79)

Abstract

Since a vector program has not just an optimal value but a set of optimal ones, the analysis of duality gap requires at least the comparison between two sets of vector optimal values. Relying only on a weak duality property, the situations that can occur are analysed in detail and some concepts of duality gap are proposed. Some numerical examples are also provided.

Key words

vector optimization duality gap 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Aubin, J.P. and Ekeland, I. (1976), “Estimates of the Duality Gap in Nonconvex Optimization”, Mathematics of Operations Research, Vol. 1, pp. 225–245.zbMATHMathSciNetGoogle Scholar
  2. [2]
    Di Guglielmo, F. (1977), “Nonconvex Duality in Multiobjective Optimization”, Mathematics of Operations Research, Vol. 2, pp. 285–291.zbMATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    Egudo, R.R. (1989), “Efficiency and Generalized Convex Duality for Multiobjective Programs”, Journal of Mathematical Analysis and Applications, Vol. 138, pp. 84–94.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Isermann, H. (1978), “On Some Relations Between a Dual Pair of Multiple Objective Linear Programs”, Zeitschrift für Operations Research, Vol. 22, pp. 33–41.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Jahn, J. (1983), “Duality in Vector Optimization”, Mathematical Programming, Vol. 25, pp. 343–353.zbMATHMathSciNetGoogle Scholar
  6. [6]
    Luc, D.T. and Jahn J. (1991), “Axiomatic Approach to Duality in Optimization”, Numerical Functional Analysis and Optimization, Vol. 13, pp. 305–326.MathSciNetGoogle Scholar
  7. [7]
    Sawaragi, Y., Nakayama, H. and Tanino, T. (1985), Theory of Multiobjective Optimization, Academic Press.Google Scholar
  8. [8]
    Tanino, T. Sawaragi, Y. (1979), “Duality Theory in Multiobjective Programming”, Journal of Optimization Theory and Applications, Vol. 27, pp. 509–529.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Tanino, T. and Sawaragi, Y. (1980), “Conjugate Maps and Duality in Multiobjective Programming”, Journal of Optimization Theory and Applications, Vol. 31, pp. 473–499.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Weir, T. and Mond, B. (1988), “Pre-invex Functions in Multiple Objective Optimization”, Journal of Mathematical Analysis and Applications, Vol. 136, pp. 29–38.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Weir, T., Mond, B. and Craven, B.D. (1987), “Weak Minimization and Duality”, Numerical Functional Analysis and Optimization, Vol. 9, pp. 181–192.zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • G. Bigi
    • 1
  • M. Pappalardo
    • 2
  1. 1.Dept. of Computer SciencesUniversity of PisaPisaItaly
  2. 2.Dept. of Applied MathematicsUniversity of PisaPisaItaly

Personalised recommendations