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On the number of criteria needed to decide Pareto optimality

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  • Matthias Ehrgott
  • Stefan Nickel

Abstract

In this paper we address the question of how many objective functions are needed to decide whether a given point is a Pareto optimal solution for a multicriteria optimization problem. We extend earlier results showing that the set of weakly Pareto optimal points is the union of Pareto optimal sets of subproblems and show their limitations. We prove that for strictly quasi-convex problems in two variables Pareto optimality can be decided by consideration of at most three objectives at a time. Our results are based on a geometric characterization of Pareto, strict Pareto, and weak Pareto solutions and Helly's Theorem. We also show that a generalization to quasi-convex objectives is not possible and state a weaker result for this case. Furthermore, we show that an analogous result for deciding strict Pareto optimality is impossible, even in the convex case. Copyright Springer-Verlag Berlin Heidelberg 2002

Suggested Citation

  • Matthias Ehrgott & Stefan Nickel, 2002. "On the number of criteria needed to decide Pareto optimality," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 55(3), pages 329-345, June.
    • Handle: RePEc:spr:mathme:v:55:y:2002:i:3:p:329-345
    DOI: 10.1007/s001860200207
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    Citations

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    Cited by:

    1. Frank Plastria, 2020. "On the Structure of the Weakly Efficient Set for Quasiconvex Vector Minimization," Journal of Optimization Theory and Applications, Springer, vol. 184(2), pages 547-564, February.
    2. Alexander Engau & Margaret M. Wiecek, 2008. "Interactive Coordination of Objective Decompositions in Multiobjective Programming," Management Science, INFORMS, vol. 54(7), pages 1350-1363, July.
    3. Melissa Gardenghi & Trinidad Gómez & Francisca Miguel & Margaret M. Wiecek, 2011. "Algebra of Efficient Sets for Multiobjective Complex Systems," Journal of Optimization Theory and Applications, Springer, vol. 149(2), pages 385-410, May.
    4. Alzorba, Shaghaf & Günther, Christian & Popovici, Nicolae & Tammer, Christiane, 2017. "A new algorithm for solving planar multiobjective location problems involving the Manhattan norm," European Journal of Operational Research, Elsevier, vol. 258(1), pages 35-46.
    5. Naoki Hamada & Shunsuke Ichiki, 2022. "Free Disposal Hull Condition to Verify When Efficiency Coincides with Weak Efficiency," Journal of Optimization Theory and Applications, Springer, vol. 192(1), pages 248-270, January.
    6. Engau, Alexander, 2009. "Tradeoff-based decomposition and decision-making in multiobjective programming," European Journal of Operational Research, Elsevier, vol. 199(3), pages 883-891, December.
    7. Lindroth, Peter & Patriksson, Michael & Strömberg, Ann-Brith, 2010. "Approximating the Pareto optimal set using a reduced set of objective functions," European Journal of Operational Research, Elsevier, vol. 207(3), pages 1519-1534, December.

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