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A finite, nonadjacent extreme-point search algorithm for optimization over the efficient set


The problem (P) of optimizing a linear function over the efficient set of a multiple-objective linear program serves many useful purposes in multiple-criteria decision making. Mathematically, problem (P) can be classified as a global optimization problem. Such problems are much more difficult to solve than convex programming problems. In this paper, a nonadjacent extreme-point search algorithm is presented for finding a globally optimal solution for problem (P). The algorithm finds an exact extreme-point optimal solution for the problem after a finite number of iterations. It can be implemented using only linear programming methods. Convergence of the algorithm is proven, and a discussion is included of its main advantages and disadvantages.

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  1. 1.

    Cohon, J. L.,Multiobjective Programming and Planning, Academic Press, New York, New York, 1978.

  2. 2.

    Evans, G. W.,An Overview of Techniques for Solving Multiobjective Mathematical Programs, Management Science, Vol. 30, pp. 1268–1282, 1984.

  3. 3.

    Goicoechea, A., Hansen, D. R., andDuckstein, L.,Multiobjective Decision Analysis with Engineering and Business Applications, John Wiley and Sons, New York, New York, 1982.

  4. 4.

    Hemming, T.,Multiobjective Decision Making under Certainty, Economic Research Institute, Stockholm School of Economics, Stockholm, Sweden, 1978.

  5. 5.

    Rosenthal, R. E.,Principles of Multiobjective Optimization, Decision Sciences, Vol. 16, pp. 133–152, 1985.

  6. 6.

    Roy, B.,Problems and Methods with Multiple Objective Functions, Mathematical Programming, Vol. 1, pp. 239–266, 1972.

  7. 7.

    Stadler, W.,A Survey of Multicriteria Optimization or the Vector Maximum Problem, Part 1: 1776–1960, Journal of Optimization Theory and Applications, Vol. 29, pp. 1–52, 1979.

  8. 8.

    Steuer, R. E.,Multiple Criteria Optimization: Theory, Computation, and Application, John Wiley and Sons, New York, New York, 1986.

  9. 9.

    Yu, P. L.,Multiple Criteria Decision Making, Plenum, New York, New York, 1985.

  10. 10.

    Zeleny, M.,Multiple Criteria Decision Making, McGraw-Hill, New York, New York, 1982.

  11. 11.

    Philip, J.,Algorithms for the Vector Maximization Problem, Mathematical Programming, Vol. 2, pp. 207–229, 1972.

  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.

  13. 13.

    Evans, J. P., andSteuer, R. E.,Generating Efficient Extreme Points in Linear Multiple Objective Programming: Two Algorithms and Computing Experience, Multiple-Criteria Decision Making, Edited by J. L. Cochrane and M. Zeleny, University of South Carolina Press, Columbia, South Carolina, pp. 349–365, 1973.

  14. 14.

    Marcotte, O., andSoland, R. M.,An Interactive Branch-and-Bound Algorithm for Multiple Criteria Optimization, Management Science, Vol. 32, pp. 61–75, 1986.

  15. 15.

    Steuer, R. E.,A Five-Phase Procedure for Implementing a Vector-Maximum Algorithm for Multiple Objective Linear Programming Problems, Multiple-Criteria Decision Making: Jouy-en-Josas, France, Edited by H. Thiriez and S. Zionts, Springer-Verlag, New York, New York, 1976.

  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.

  17. 17.

    Isermann, H., andSteuer, R. E.,Computational Experience Concerning Payoff Tables and Minimum Criteria Values over the Efficient Set, European Journal of Operational Research, Vol. 33, pp. 91–97, 1987.

  18. 18.

    Weistroffer, H. R.,Careful Use of Pessimistic Values Is Needed in Multiple Objectives Optimization, Operations Research Letters, Vol. 4, pp. 23–25, 1985.

  19. 19.

    Benayoun, R., de Montgolfier, J., Tergny, J., andLaritchev, O.,Linear Programming with Multiple Objective Functions: Step Method (STEM), Mathematical Programming, Vol. 1, pp. 366–375, 1971.

  20. 20.

    Belenson, S., andKapur, K. C.,An Algorithm for Solving Multicriterion Linear Programming Problems with Examples, Operational Research Quarterly, Vol. 24, pp. 65–77, 1973.

  21. 21.

    Kok, M., andLootsma, F. A.,Pairwise-Comparison Methods in Multiple Objective Programming, with Applications in a Long-Term Energy-Planning Model, European Journal of Operational Research, Vol. 22, pp. 44–55, 1985.

  22. 22.

    Benson, H. P.,On the Convergence of Two Branch-and-Bound Algorithms for Nonconvex Programming Problems, Journal of Optimization Theory and Applications, Vol. 36, pp. 129–134, 1982.

  23. 23.

    Horst, R.,An Algorithm for Nonconvex Programming Problems, Mathematical Programming, Vol. 10, pp. 312–321, 1976.

  24. 24.

    Horst, R.,Deterministic Global Optimization with Partition Sets Whose Feasibility Is Not Known: Application to Concave Minimization, Reverse Convex Constraints, DC Programming, and Lipschitzian Optimization, Journal of Optimization Theory and Applications, Vol. 58, pp. 11–37, 1988.

  25. 25.

    Horst, R.,Deterministic Global Optimization: Recent Advances and New Fields of Application, Naval Research Logistics, Vol. 37, pp. 433–471, 1990.

  26. 26.

    Horst, R.,A General Class of Branch-and-Bound Methods in Global Optimization with Some New Approaches for Concave Minimization, Journal of Optimization Theory and Applications, Vol. 51, pp. 271–291, 1986.

  27. 27.

    Horst, R., andTuy, H.,Global Optimization: Deterministic Approaches, Springer-Verlag, Berlin, Germany, 1990.

  28. 28.

    McCormick, G. P.,Computability of Global Solutions to Factorable Nonconvex Programs, Part 1: Convex Underestimating Problems, Mathematical Programming, Vol. 10, pp. 147–175, 1976.

  29. 29.

    Pardalos, P. M., andRosen, J. B.,Constrained Global Optimization: Algorithms and Applications, Springer-Verlag, Berlin, Germany, 1987.

  30. 30.

    Pardalos, P. M., andRosen, J. B.,Methods for Global Concave Minimization: A Bibliographic Survey, SIAM Review, Vol. 28, pp. 367–379, 1986.

  31. 31.

    Thoai, N. V., andTuy, H.,Convergent Algorithms for Minimizing a Concave Function, Mathematics of Operations Research, Vol. 5, pp. 556–566, 1980.

  32. 32.

    Benson, H. P.,Optimization over the Efficient Set, Journal of Mathematical Analysis and Applications, Vol. 98, pp. 562–580, 1984.

  33. 33.

    Ecker, J. G., andKouada, I. A.,Finding Efficient Points for Linear Multiple Objective Programs, Mathematical Programming, Vol. 8, pp. 375–377, 1975.

  34. 34.

    Benson, H. P.,Existence of Efficient Solutions for Vector Maximization Problems, Journal of Optimization Theory and Applications, Vol. 26, pp. 569–580, 1978.

  35. 35.

    Spivey, W. A., andThrall, R. M.,Linear Optimization, Holt, Rinehart, and Winston, New York, New York, 1970.

  36. 36.

    Murty, K. G.,Linear and Combinatorial Programming, John Wiley and Sons, New York, New York, 1976.

  37. 37.

    Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.

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The author owes thanks to two anonymous referees for their helpful comments.

Communicated by P. L. Yu

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Benson, H.P. A finite, nonadjacent extreme-point search algorithm for optimization over the efficient set. J Optim Theory Appl 73, 47–64 (1992).

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Key Words

  • Multiple-criteria decision making
  • extreme-point search
  • global optimization
  • efficient set
  • nonconvex programming