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arXiv · 2307.06639

Bilevel linear optimization belongs to NP and admits polynomial-size KKT-based reformulations

Christoph Buchheim
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Abstract

It is a well-known result that bilevel linear optimization is NP-hard. In many publications, reformulations as mixed-integer linear optimization problems are proposed, which suggests that the decision version of the problem belongs to NP. However, to the best of our knowledge, a rigorous proof of membership in NP has never been published, so we close this gap by reporting a simple but not entirely trivial proof. A related question is whether a large enough "big M" for the classical KKT-based reformulation can be computed efficiently, which we answer in the affirmative. In particular, our big M has polynomial encoding length in the original problem data.

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Christoph Buchheim. 2023-07-13. Bilevel linear optimization belongs to NP and admits polynomial-size KKT-based reformulations. https://doi.org/10.1016/j.orl.2023.10.006

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