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On Optimization over the Efficient Set in Linear Multicriteria Programming

Author

Listed:
  • R. Horst

    (University of Trier)

  • N. V. Thoai

    (University of Trier)

  • Y. Yamamoto

    (University of Tsukuba)

  • D. Zenke

    (University of Tsukuba)

Abstract

The efficient set of a linear multicriteria programming problem can be represented by a reverse convex constraint of the form g(z)≤0, where g is a concave function. Consequently, the problem of optimizing some real function over the efficient set belongs to an important problem class of global optimization called reverse convex programming. Since the concave function used in the literature is only defined on some set containing the feasible set of the underlying multicriteria programming problem, most global optimization techniques for handling this kind of reverse convex constraint cannot be applied. The main purpose of our article is to present a method for overcoming this disadvantage. We construct a concave function which is finitely defined on the whole space and can be considered as an extension of the existing function. Different forms of the linear multicriteria programming problem are discussed, including the minimum maximal flow problem as an example.

Suggested Citation

  • R. Horst & N. V. Thoai & Y. Yamamoto & D. Zenke, 2007. "On Optimization over the Efficient Set in Linear Multicriteria Programming," Journal of Optimization Theory and Applications, Springer, vol. 134(3), pages 433-443, September.
    • Handle: RePEc:spr:joptap:v:134:y:2007:i:3:d:10.1007_s10957-007-9219-8
    DOI: 10.1007/s10957-007-9219-8
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    References listed on IDEAS

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    1. R. Horst & N. V. Thoai, 1999. "DC Programming: Overview," Journal of Optimization Theory and Applications, Springer, vol. 103(1), pages 1-43, October.
    2. Horst, Reiner & Thoai, Nguyen V., 1999. "Maximizing a concave function over the efficient or weakly-efficient set," European Journal of Operational Research, Elsevier, vol. 117(2), pages 239-252, September.
    3. R. Horst & N. V. Thoai, 1997. "Utility Function Programs and Optimization over the Efficient Set in Multiple-Objective Decision Making," Journal of Optimization Theory and Applications, Springer, vol. 92(3), pages 605-631, March.
    4. Thoai, Nguyen V., 2000. "A class of optimization problems over the efficient set of a multiple criteria nonlinear programming problem," European Journal of Operational Research, Elsevier, vol. 122(1), pages 58-68, April.
    5. N.V. Thoai, 2002. "Convergence and Application of a Decomposition Method Using Duality Bounds for Nonconvex Global Optimization," Journal of Optimization Theory and Applications, Springer, vol. 113(1), pages 165-193, April.
    6. Le Thi, Hoai An & Pham, Dinh Tao & Thoai, Nguyen V., 2002. "Combination between global and local methods for solving an optimization problem over the efficient set," European Journal of Operational Research, Elsevier, vol. 142(2), pages 258-270, October.
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    Cited by:

    1. Henri Bonnel & Christopher Schneider, 2019. "Post-Pareto Analysis and a New Algorithm for the Optimal Parameter Tuning of the Elastic Net," Journal of Optimization Theory and Applications, Springer, vol. 183(3), pages 993-1027, December.
    2. Schöbel, Anita & Zhou-Kangas, Yue, 2021. "The price of multiobjective robustness: Analyzing solution sets to uncertain multiobjective problems," European Journal of Operational Research, Elsevier, vol. 291(2), pages 782-793.
    3. Henri Bonnel & Julien Collonge, 2015. "Optimization over the Pareto outcome set associated with a convex bi-objective optimization problem: theoretical results, deterministic algorithm and application to the stochastic case," Journal of Global Optimization, Springer, vol. 62(3), pages 481-505, July.
    4. N. V. Thoai, 2010. "Reverse Convex Programming Approach in the Space of Extreme Criteria for Optimization over Efficient Sets," Journal of Optimization Theory and Applications, Springer, vol. 147(2), pages 263-277, November.
    5. Nguyen Thoai, 2012. "Criteria and dimension reduction of linear multiple criteria optimization problems," Journal of Global Optimization, Springer, vol. 52(3), pages 499-508, March.
    6. Henri Bonnel & Julien Collonge, 2014. "Stochastic Optimization over a Pareto Set Associated with a Stochastic Multi-Objective Optimization Problem," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 405-427, August.
    7. Henri Bonnel & C. Yalçın Kaya, 2010. "Optimization Over the Efficient Set of Multi-objective Convex Optimal Control Problems," Journal of Optimization Theory and Applications, Springer, vol. 147(1), pages 93-112, October.
    8. Henri Bonnel & Léonard Todjihoundé & Constantin Udrişte, 2015. "Semivectorial Bilevel Optimization on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 167(2), pages 464-486, November.
    9. Alves, Maria João & Costa, João Paulo, 2009. "An exact method for computing the nadir values in multiple objective linear programming," European Journal of Operational Research, Elsevier, vol. 198(2), pages 637-646, October.

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