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

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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).

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Acknowledgements

The authors would like to thank the two anonymous referees for their constructive remarks that have helped us to improve the presentation of this paper.

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  1. School of Mathematical Sciences, University of Southampton, Southampton, SO17 1BJ, UK

    Alain B. Zemkoho & Shenglong Zhou

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  1. Alain B. Zemkoho
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  2. Shenglong Zhou
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Correspondence to Shenglong Zhou.

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This project was funded by the EPSRC Grant EP/P022553/1.

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Zemkoho, A.B., Zhou, S. Theoretical and numerical comparison of the Karush–Kuhn–Tucker and value function reformulations in bilevel optimization. Comput Optim Appl 78, 625–674 (2021). https://doi.org/10.1007/s10589-020-00250-7

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