Skip to main content
SpringerLink
Log in

On a constructive approximation of the efficient outcomes in bicriterion vector optimization

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

For bicriterion quasiconvex optimization problems, we present a constructive procedure for an approximation of the efficient outcomes. Performing this procedure we can estimate the accuracy of the approximation. Conversely, if we prescribe an accuracy for the approximation, we can calculate the number of points which have to be computed by a certain scalarization method to remain under the given accuracy. Finally, we give a numerical example.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Y. Aksoy (1990), An Interactive Branch-and-Bound Algorithm for Bicriterion Nonconvex/Mixed Integer Programming,Naval Research Logistics 37, 403–417.

    Google Scholar 

  2. A. Bacopoulos and I. Singer (1977), On Convex Vectorial Optimization in Linear Spaces,Journal of Optimization Theory and Applications 21, 175–188.

    Google Scholar 

  3. H. P. Benson (1979), Vector Maximization with Two Objective Functions,Journal of Optimization Theory and Applications 28, 253–257.

    Google Scholar 

  4. H. Bernau (1987), Interactive Methods for Vector Optimization, in:Optimization in Mathematical Physics, edited by B. Brosowski and E. Martensen, Lang Verlag, Frankfurt, 21–36.

    Google Scholar 

  5. B. Brosowski (1989), A Recursive Procedure for the Solution of Linear and Nonlinear Vector Optimization Problems, in:Discretization Methods and Structural Optimization-Procedures and Applications, edited by H. A. Eschenauer and G. Thierauf, Springer-Verlag, Berlin, Heidelberg, New York, 95–101.

    Google Scholar 

  6. E. U. Choo and D. R. Atkins (1980), An Interactive Algorithm of Multicriteria Programming,Computers and Operations Research 7, 81–87.

    Google Scholar 

  7. E. U. Choo and D. R. Atkins (1982), Bicriteria Linear Fractional Programming,Journal of Optimization Theory and Applications 36, 203–220.

    Google Scholar 

  8. F. Gemibicki (1973), Vector Optimization for Control with Performance and Parameter Sensitivity Indices, Ph.D. doctoral thesis, Cleveland, Ohio, Case Western Reserve University.

    Google Scholar 

  9. S. Heibig (1990), On the Connectedness of the Set of Weakly Efficient Points of a Vector Optimization Problem in Locally Convex Spaces,Journal of Optimization Theory and Applications 65, 257–270.

    Google Scholar 

  10. S. Heibig (1991), Approximation of the Efficient Point Set by Perturbation of the Ordering Cone,Zeitschrift für Operations Research 35, 197–220.

    Google Scholar 

  11. S. Helbig (1992), On a Constructive Approximation of the Efficient Outcomes in Bicriterion Vector Optimization, Working Paper, University of Frankfurt.

  12. S. Helbig (1993), Quantitative and Qualitative Stability of a Scalarization Method, I, in:Parametric Optimization and Related Topics III, edited by J. Guddat, H. Th Jongen, B. Kummer and F. Nožička, Lang Verlag, Frankfurt, 255–270.

    Google Scholar 

  13. S. Helbig (1993), On Quantitative and Qualitative Stability of a Scalarization Method, II, in:Multicriteria Decision — Proceedings of the 14th Meeting of the German Working Group “Mehrkriterielle Entscheidungen”, edited by B. Brosowski, J. Ester, S. Helbig and R. Nehse, Lang Verlag, Frankfurt, 19–42.

    Google Scholar 

  14. D. T. Luc (1987), Connectedness of the Efficient Point Set in Quasiconcave Vector Maximization,Journal of Mathematical Analysis and Applications 122, 346–354.

    Google Scholar 

  15. J. Jahn (1988), A Method of Reference Point Approximation in Vector Optimization,Operations Research Proceedings 87, 576–587.

    Google Scholar 

  16. J. Jahn and A. Merkel (1990), An Interactive Method of Reference Point Approximation in Vector Optimization, to appear in:Journal of Optimization Theory and Applications.

  17. A. Pascoletti and P. Serafini (1984), Scalarizing Vector Optimization Problems,Journal of Optimization Theory and Applications 42, 499–524.

    Google Scholar 

  18. S. Schaible (1983), Bicriteria Quasiconcave Programs,Cahiers du Centre d'Etudes de Recherche Operationnelle 25, 93–101.

    Google Scholar 

  19. W. S. Shin, A. Ramachandran and S. F. Bullington (1992), Interactive Bicriterion Solution Method and Its Application to Critical Path Method Problems,Journal of Optimization Theory and Applications 72, 511–527.

    Google Scholar 

  20. R. M. Soland (1979), Multicriteria Optimization: A General Characterization of Efficient Solutions,Decision Sciences 10, 26–38.

    Google Scholar 

  21. R. E. Steuer and E.-U. Choo (1983), An Interactive Weighted Tschebycheff Procedure for Multiple Objective Programming,Mathematical Programming 26, 326–344.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich Mathematik, Johann Wolfgang Goethe-Universität, Robert-Mayer-Str. 6-10, 60325, Frankfurt 1, Germany

    Siegfried Helbig

Authors
  1. Siegfried Helbig
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Helbig, S. On a constructive approximation of the efficient outcomes in bicriterion vector optimization. J Glob Optim 5, 35–48 (1994). https://doi.org/10.1007/BF01097002

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01097002

Key words

Search

Navigation