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

Thomas Kleinert (thomas.kleinert***at***fau.de)
Martine Labbé (martine.labbe***at***ulb.ac.be)
Fränk Plein (frank.plein***at***ulb.ac.be)
Martin Schmidt (martin.schmidt***at***uni-trier.de)

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.

Keywords: Bilevel optimization, Mathematical programs with complementarity constraints (MPCC), Bounding polyhedra, Big-$M$, Hardness

Category 1: Integer Programming ((Mixed) Integer Linear Programming )

Category 2: Complementarity and Variational Inequalities

Citation:

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Entry Submitted: 04/23/2019
Entry Accepted: 04/23/2019
Entry Last Modified: 08/07/2019

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