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Multicriteria Approach to Bilevel Optimization

Author

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  • J. Fliege

    (University of Birmingham)

  • L. N. Vicente

    (Universidade de Coimbra)

Abstract

In this paper, we study the relationship between bilevel optimization and multicriteria optimization. Given a bilevel optimization problem, we introduce an order relation such that the optimal solutions of the bilevel problem are the nondominated points with respect to the order relation. In the case where the lower-level problem of the bilevel optimization problem is convex and continuously differentiable in the lower-level variables, this order relation is equivalent to a second, more tractable order relation. Then, we show how to construct a (nonconvex) cone for which we can prove that the nondominated points with respect to the order relation induced by the cone are also nondominated points with respect to any of the two order relations mentioned before. We comment also on the practical and computational implications of our approach.

Suggested Citation

  • J. Fliege & L. N. Vicente, 2006. "Multicriteria Approach to Bilevel Optimization," Journal of Optimization Theory and Applications, Springer, vol. 131(2), pages 209-225, November.
    • Handle: RePEc:spr:joptap:v:131:y:2006:i:2:d:10.1007_s10957-006-9136-2
    DOI: 10.1007/s10957-006-9136-2
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    References listed on IDEAS

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    1. A. Haurie & G. Savard & D. J. White, 1990. "A Note on: An Efficient Point Algorithm for a Linear Two-Stage Optimization Problem," Operations Research, INFORMS, vol. 38(3), pages 553-555, June.
    2. Jörg Fliege, 2004. "Gap-free computation of Pareto-points by quadratic scalarizations," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 59(1), pages 69-89, February.
    3. Marcotte, Patrice, 1988. "A note on a bilevel programming algorithm by Leblanc and Boyce," Transportation Research Part B: Methodological, Elsevier, vol. 22(3), pages 233-236, June.
    4. P. A. Clark & A. W. Westerberg, 1988. "A note on the optimality conditions for the bilevel programming problem," Naval Research Logistics (NRL), John Wiley & Sons, vol. 35(5), pages 413-418, October.
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    Citations

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    Cited by:

    1. Fliege, Jörg & Werner, Ralf, 2014. "Robust multiobjective optimization & applications in portfolio optimization," European Journal of Operational Research, Elsevier, vol. 234(2), pages 422-433.
    2. Jörg Fliege & Huifu Xu, 2011. "Stochastic Multiobjective Optimization: Sample Average Approximation and Applications," Journal of Optimization Theory and Applications, Springer, vol. 151(1), pages 135-162, October.
    3. Shahabeddin Najafi & Masoud Hajarian, 2023. "Multiobjective Conjugate Gradient Methods on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 197(3), pages 1229-1248, June.
    4. J. Glackin & J. G. Ecker & M. Kupferschmid, 2009. "Solving Bilevel Linear Programs Using Multiple Objective Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 140(2), pages 197-212, February.
    5. Sauli Ruuska & Kaisa Miettinen & Margaret M. Wiecek, 2012. "Connections Between Single-Level and Bilevel Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 153(1), pages 60-74, April.
    6. A. Georgantas, 2020. "Robust Optimization Approaches for Portfolio Selection: A Computational and Comparative Analysis," Papers 2010.13397, arXiv.org.
    7. Chi-Bin Cheng & Hsu-Shih Shih & Boris Chen, 2017. "Subsidy rate decisions for the printer recycling industry by bi-level optimization techniques," Operational Research, Springer, vol. 17(3), pages 901-919, October.
    8. Rihab Said & Maha Elarbi & Slim Bechikh & Lamjed Ben Said, 2022. "Solving combinatorial bi-level optimization problems using multiple populations and migration schemes," Operational Research, Springer, vol. 22(3), pages 1697-1735, July.

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