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There's No Free Lunch: On the Hardness of Choosing a Correct Big-M in Bilevel Optimization

Thomas Kleinert 1 Martine Labbé 2 Fränk Plein 3 Martin Schmidt 4
2 INOCS - Integrated Optimization with Complex Structure
ULB - Université libre de Bruxelles, Inria Lille - Nord Europe, CRIStAL - Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189
Abstract : One of the most frequently used approaches to solve linear bilevel optimization problems consists in replacing the lower-level problem with its Karush-Kuhn-Tucker (KKT) conditions and by reformulating the KKT complementarity conditions using techniques from mixed-integer linear optimization. The latter step requires to determine some big-M constant in order to bound the lower level's dual feasible set such that no bilevel-optimal solution is cut off. In practice, heuristics are often used to find a big-M although it is known that these approaches may fail. In this paper, we consider the hardness of two proxies for the above mentioned concept of a bilevel-correct big-M. First, we prove that verifying that a given big-M does not cut off any feasible vertex of the lower level's dual polyhedron cannot be done in polynomial time unless P = NP. Second, we show that verifying that a given big-M does not cut off any optimal point of the lower level's dual problem (for any point in the projection of the high-point relaxation onto the leader's decision space) is as hard as solving the original bilevel problem.
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https://hal.inria.fr/hal-03137942
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Submitted on : Wednesday, February 10, 2021 - 6:02:18 PM
Last modification on : Friday, January 21, 2022 - 3:12:23 AM
Long-term archiving on: : Tuesday, May 11, 2021 - 9:06:16 PM

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Thomas Kleinert, Martine Labbé, Fränk Plein, Martin Schmidt. There's No Free Lunch: On the Hardness of Choosing a Correct Big-M in Bilevel Optimization. Operations Research, INFORMS, 2020, 68 (6), ⟨10.1287/opre.2019.1944⟩. ⟨hal-03137942⟩

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