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

Theoretical and numerical comparison of the Karush-Kuhn-Tucker and value function reformulations in bilevel optimization
Theoretical and numerical comparison of the Karush-Kuhn-Tucker and value function reformulations in bilevel optimization
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).
Bilevel optimization, Karush–Kuhn–Tucker reformulation, Newton method, Value function reformulation
0926-6003
625–674
Zemkoho, Alain
30c79e30-9879-48bd-8d0b-e2fbbc01269e
Zhou, Shenglong
d183edc9-a9f6-4b07-a140-a82213dbd8c3
Zemkoho, Alain
30c79e30-9879-48bd-8d0b-e2fbbc01269e
Zhou, Shenglong
d183edc9-a9f6-4b07-a140-a82213dbd8c3

Zemkoho, Alain and Zhou, Shenglong (2021) Theoretical and numerical comparison of the Karush-Kuhn-Tucker and value function reformulations in bilevel optimization. Computational Optimization and Applications, 78 (2), 625–674. (doi:10.1007/s10589-020-00250-7).

Record type: Article

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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Zemkoho-Zhou2021_Article_TheoreticalAndNumericalCompari - Author's Original
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More information

Accepted/In Press date: 1 December 2020
Published date: 2 June 2021
Keywords: Bilevel optimization, Karush–Kuhn–Tucker reformulation, Newton method, Value function reformulation

Identifiers

Local EPrints ID: 444486
URI: http://eprints.soton.ac.uk/id/eprint/444486
ISSN: 0926-6003
PURE UUID: 183bb31a-193f-4c20-9111-087ef9443804
ORCID for Alain Zemkoho: ORCID iD orcid.org/0000-0003-1265-4178
ORCID for Shenglong Zhou: ORCID iD orcid.org/0000-0003-2843-1614

Catalogue record

Date deposited: 21 Oct 2020 16:31
Last modified: 26 Nov 2021 03:03

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Contributors

Author: Alain Zemkoho ORCID iD
Author: Shenglong Zhou ORCID iD

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