Advertisement

Dual approach to minimization on the set of pareto-optimal solutions

  • P. T. Thach
  • H. Konno
  • D. Yokota
Contributed Papers

Abstract

LetX* be the set of Pareto-optimal solutions of a multicriteria programming problem. We are interested in finding a vectorxεX* which minimizes another criterion. SinceX* is a nonconvex set, our problem is that of minimization over a nonconvex set. By exploiting the fact that the number of criteria is often very small compared with the number of variables, we use a dual approach to obtain a practical algorithm. We report preliminary numerical results on problems with up to 100 variables and 5 criteria.

Key Words

Pareto-optimal solutions duality 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Geoffrion, A. M.,Proper Efficiency and the Theory of Vector Maximization, Journal of Mathematical Analysis and Applications, Vol. 22, pp. 618–630, 1968.Google Scholar
  2. 2.
    Cohon, I. L.,Multiobjective Programming and Planning, Academic Press, New York, New York, 1978.Google Scholar
  3. 3.
    Yu, P. L.,Multiple-Criteria Decision Making, Plenum Press, New York, New York, 1985.Google Scholar
  4. 4.
    Hansen, P., Editor,Essays and Surveys on Multiple-Criteria Decision Making, Springer Verlag, Berlin, Germany, 1983.Google Scholar
  5. 5.
    Sawaragi, Y., Nakayama, H., andTanino, T.,Theory of Multiobjective Optimization, Academic Press, Orlando, Florida, 1985.Google Scholar
  6. 6.
    Steuer, R. E.,Multiple-Criteria Optimization, Wiley, New York, New York, 1985.Google Scholar
  7. 7.
    Lootsma, F. A., Optimization with Multiple Objectives, Mathematical Programming: Recent Developments and Applications, Edited by M. Iri and K. Tanabe, Kluwer Academic Publishers, Dordrecht, Holland, pp. 333–364, 1989.Google Scholar
  8. 8.
    Benson, H. P.,Optimization over the Efficient Set, Journal of Mathematical Analysis and Applications, Vol. 98, pp. 562–580, 1984.Google Scholar
  9. 9.
    Ishizuka, Y.,Optimality Conditions for Directionally Differentiable Multiobjective Programming Problems, Journal of Optimization Theory and Applications, Vol. 72, pp. 91–112, 1992.Google Scholar
  10. 10.
    Bolintineau, S.,Minimization of a Quasiconcave Function over an Efficient Set, Mathematical Programming, Vol. 61, pp. 83–120, 1993.Google Scholar
  11. 11.
    Philip, J.,Algorithms for the Vector Maximization Problem, Mathematical Programming, Vol. 2, pp. 207–229, 1972.Google Scholar
  12. 12.
    Benson, H. P.,An All-Linear Programming Relaxation Algorithm for Optimizing over the Efficient Set, Journal of Global Optimization, Vol. 1, pp. 83–104, 1991.Google Scholar
  13. 13.
    Benson, H. P.,A Finite, Nonadjacent Extreme-Point Search Algorithm for Optimization over the Efficient Set, Journal of Optimization Theory and Applications, Vol. 73, pp. 47–64, 1992.Google Scholar
  14. 14.
    Benson, H. P.,A Bisection Extreme-Point Search Algorithm for Optimizing over the Efficient Set in the Linear Dependence Case, Journal of Global Optimization, Vol. 3, pp. 95–111, 1993.Google Scholar
  15. 15.
    Benson, H. P., andSayin, S.,A Face Search Heuristic Algorithm for Optimizing over the Efficient Set, Naval Research Logistics, Vol. 40, pp. 103–116, 1993.Google Scholar
  16. 16.
    Dessouky, M. I., Ghiassi, M., andDavis, W. J.,Estimates of the Minimum Nondominated Criterion Values in Multiple-Criteria Decision Making, Engineering Costs and Production Economics, Vol. 10, pp. 95–104, 1986.Google Scholar
  17. 17.
    Isermann, H., andSteuer, R. E.,Computational Experience Concerning Payoff Tables and Minimum Criterion Values over the Efficient Set, European Journal of Operational Research, Vol. 33, pp. 91–97, 1987.Google Scholar
  18. 18.
    Dauer, J. P.,Optimization over the Efficient Set Using an Active Constraint Approach, Zeitschrift für Operations Research, Vol. 35, pp. 185–195, 1991.Google Scholar
  19. 19.
    Dyer, J. S.,Interactive Goal Programming, Management Science, Vol. 19, pp. 62–70, 1972.Google Scholar
  20. 20.
    Konno, H., andInori, M.,Bond Portfolio Optimization by Bilinear Fractional Programming, Journal of the Operations Research Society of Japan, Vol. 32, pp. 143–158, 1989.Google Scholar
  21. 21.
    Thach, P. T.,Quasiconjugates of Functions, Duality Relationship between Quasiconvex Minimization under a Reverse Convex Constraint and Quasiconvex Maximization under a Convex Constraint, and Applications, Journal of Mathematical Analysis and Applications, Vol. 159, pp. 299–322, 1991.Google Scholar
  22. 22.
    Tuy, H.,On Outer-Approximation Methods for Solving Concave Minimization Problems, Acta Mathematica Vietnamica, Vol. 8, pp. 3–34, 1983.Google Scholar
  23. 23.
    Thieu, T. V., Tam, B. T., andBan, V. T.,An Outer-Approximation Method for Globally Minimizing a Concave Function over a Compact Convex Set, Acta Mathematica Vietnamica, Vol. 8, pp. 21–40, 1983.Google Scholar
  24. 24.
    Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.Google Scholar
  25. 25.
    Eaves, B. C., andZangwill, W. I.,Generalized Cutting Plane Algorithms, SIAM Journal on Control, Vol. 9, pp. 529–542, 1971.Google Scholar
  26. 26.
    Horst, R., andTuy, H.,Global Optimization, Springer Verlag, Berlin, Germany, 1990.Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • P. T. Thach
    • 1
  • H. Konno
    • 1
  • D. Yokota
    • 1
  1. 1.Department of Industrial Engineering and ManagementTokyo Institute of TechnologyTokyoJapan

Personalised recommendations