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Theoretical and numerical comparison of the Karush–Kuhn–Tucker and value function reformulations in bilevel optimization

Author

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  • Alain B. Zemkoho

    (University of Southampton)

  • Shenglong Zhou

    (University of Southampton)

Abstract

The Karush–Kuhn–Tucker and value function (lower-level value function, to be precise) reformulations are the most common single-level transformations of the bilevel optimization problem. So far, these reformulations have either been studied independently or as a joint optimization problem in an attempt to take advantage of the best properties from each model. To the best of our knowledge, these reformulations have not yet been compared in the existing literature. This paper is a first attempt towards establishing whether one of these reformulations is best at solving a given class of the optimistic bilevel optimization problem. We design a comparison framework, which seems fair, considering the theoretical properties of these reformulations. This work reveals that although none of the models seems to particularly dominate the other from the theoretical point of view, the value function reformulation seems to numerically outperform the Karush–Kuhn–Tucker reformulation on a Newton-type algorithm. The computational experiments here are mostly based on test problems from the Bilevel Optimization LIBrary (BOLIB).

Suggested Citation

  • Alain B. Zemkoho & Shenglong Zhou, 2021. "Theoretical and numerical comparison of the Karush–Kuhn–Tucker and value function reformulations in bilevel optimization," Computational Optimization and Applications, Springer, vol. 78(2), pages 625-674, March.
    • Handle: RePEc:spr:coopap:v:78:y:2021:i:2:d:10.1007_s10589-020-00250-7
    DOI: 10.1007/s10589-020-00250-7
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    References listed on IDEAS

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    1. Stephan Dempe & Alain B. Zemkoho, 2011. "The Generalized Mangasarian-Fromowitz Constraint Qualification and Optimality Conditions for Bilevel Programs," Journal of Optimization Theory and Applications, Springer, vol. 148(1), pages 46-68, January.
    2. Lorenzo Lampariello & Simone Sagratella, 2020. "Numerically tractable optimistic bilevel problems," Computational Optimization and Applications, Springer, vol. 76(2), pages 277-303, June.
    3. Lorenzo Lampariello & Simone Sagratella, 2017. "A Bridge Between Bilevel Programs and Nash Games," Journal of Optimization Theory and Applications, Springer, vol. 174(2), pages 613-635, August.
    4. S. Dempe & S. Franke, 2016. "On the solution of convex bilevel optimization problems," Computational Optimization and Applications, Springer, vol. 63(3), pages 685-703, April.
    5. Shenglong Zhou & Alain B. Zemkoho & Andrey Tin, 2020. "BOLIB: Bilevel Optimization LIBrary of Test Problems," Springer Optimization and Its Applications, in: Stephan Dempe & Alain Zemkoho (ed.), Bilevel Optimization, chapter 0, pages 563-580, Springer.
    6. Liqun Qi & Houyuan Jiang, 1997. "Semismooth Karush-Kuhn-Tucker Equations and Convergence Analysis of Newton and Quasi-Newton Methods for Solving these Equations," Mathematics of Operations Research, INFORMS, vol. 22(2), pages 301-325, May.
    7. Mengwei Xu & Jane Ye, 2014. "A smoothing augmented Lagrangian method for solving simple bilevel programs," Computational Optimization and Applications, Springer, vol. 59(1), pages 353-377, October.
    8. Liqun Qi, 1993. "Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations," Mathematics of Operations Research, INFORMS, vol. 18(1), pages 227-244, February.
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    Cited by:

    1. Qingna Li & Zhen Li & Alain Zemkoho, 2022. "Bilevel hyperparameter optimization for support vector classification: theoretical analysis and a solution method," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 96(3), pages 315-350, December.

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