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Global Optimization Approach to Optimizing Over the Efficient Set

  • Le Tu Luc
  • Le Dung Muu
Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 452)

Abstract

We consider optimization problem over the efficient set of a linear vector problem where the objective function is a composite convex function of the criteria. We show that in this case the problem can be reduced to a single linear constrained convex maximization or a single convex-concave programming problem. The number of the “nonconvexity variables” in the both reduced forms is just equal to the number of the criteria. Some algoritmic aspects are discussed.

Key Words

Optimization over the Pareto set convex-concave programming convex maximization inner approximation decomposition 

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References

  1. [1]
    An, Le T., Tao Pham D., Muu Le D. “ D. C. Optimization Approach for Optimizing over the Efficient Set”. Operations Research Letters 19 (1996) 117–128.CrossRefGoogle Scholar
  2. [2]
    Benson H.P. “ An All-Linear Programming Relaxation Algorithm for Optimizing over the Efficient Set”. J. of Global Optimization 1 (1991) 83–104.CrossRefGoogle Scholar
  3. [3]
    Benson H.P. “ A Finite, Nonadjacent Extreme Point Search Algorithm for Optimization over the Efficient Set”. J. of Optimization Theory and Application 73 (1992) 47–63.CrossRefGoogle Scholar
  4. [4]
    Benson H.P. “ A Bisection-Extreme Point Search Algorithm for Optimizing over the Efficient Set in the Linear Dependence Case”. J. of Global Optimization, 3 (1993) 95–111.CrossRefGoogle Scholar
  5. [5]
    Bolintineanu, S., “ Minimization of a Quasi-concave Function over an Efficient Set”, Mathematical Programming 61 (1993) 89–110.CrossRefGoogle Scholar
  6. [6]
    Dan, Ng. D. and Muu Le D., “A parametric Simplex Method for Optimizing a Linear Function over the Efficient Set of A Bicriteria Linear Problem”. Acta Mathematica Vietnamica 21 (1996) 59–67.Google Scholar
  7. [7]
    Dauer, J. P., and Fosnaugh, T. A., “ Optimization over the Efficient Set”. Journal of Global Optimization 7 (1995) 261–277.CrossRefGoogle Scholar
  8. [8]
    Ecker, J. G., and Song, J. H., “ Optimizing a linear function over an Efficient Set”. Journal of Optimization Theory and Applications, 83 (1994) 541–563.CrossRefGoogle Scholar
  9. [9]
    Fulop, J., “ A Cutting Plane Algorithm for Linear Optimization over the Efficient Set”. In S. Komlosi, T. Rapcsak and S. Schaible eds., Generalized Convexity (Springer, Berlin 1994) 374–385.CrossRefGoogle Scholar
  10. [10]
    Grotschel, M., Lovasz, L. and Schrijver, G. Geo metric Algorithms and Combinatorical Optimization (Springer-Verlag, Berlin 1988).CrossRefGoogle Scholar
  11. [11]
    Horst, R., and Tuy, H., Glob al Optimization? (Deterministic Approaches). (Springer-Verlag, Berlin, 1996).Google Scholar
  12. [12]
    Horst, R., and Thoai, N. V., “Utility Function Programs and Optimization over the Efficient Set in Multiple Objective Decision Making”. Universitat Trier, Forschungsbericht Nr. 1996–03.Google Scholar
  13. [13]
    Isermann H. and Steuer R.E. “ Computational Experience Concerning Pazoff Tables and Minimum Criterion Values over the Efficient Set”. European J. of Operations Research 33 (1987) 91–97.CrossRefGoogle Scholar
  14. [14]
    Muu, Le D., and Oettli, W., “ Method for Minimizing a Convex-Concave Function Over a Convex Set”. J. of Optimization Theory and Applications 70 (1991) 377–384.CrossRefGoogle Scholar
  15. [15]
    Muu, Le D., “ An Algorithm for Solving Convex Programs with and Additional Convex-Concave Constraints”. Mathematical Programming 61 (1993) 75–87.CrossRefGoogle Scholar
  16. [16]
    Muu Le D. and Luc Le T., “On Convex maximization Formulations and Decomposable Property of Optimization Problem over the Efficient Set”. Preprint, Institute of Mathematics, Hanoi 17 (1995).Google Scholar
  17. [17]
    Muu, Le D., “ Computational Aspects of Optimization Problems over the Efficient Set”. Vietnam Journal of Mathematics 23 (1995) 85–106.Google Scholar
  18. [18]
    Philip J. “ Algorithms for the Vector Maximization Problem”. Mathematical Programming 2 (1972), 207–229.CrossRefGoogle Scholar
  19. [19]
    Rockafellar R.T. Convex Analysis. (Princeton University Press 1970).Google Scholar
  20. [20]
    Thach, P. T., Konno, H., and Yokota, D., “ Dual Approach to Optimization on the Set of Pareto-Optimal Solutions”. Journal of Optimization Theory and Applications 88 (1976) 689–707.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Le Tu Luc
    • 1
  • Le Dung Muu
    • 2
  1. 1.RMITMelbourneAustralia
  2. 2.Institute of MathematicsHanoiVietnam

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