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It is a well-known result that bilevel linear optimization is NP-hard. Inmany publications, reformulations as mixed-integer linear optimization problemsare proposed, which suggests that the decision version of the problem belongsto NP. However, to the best of our knowledge, a rigorous proof of membership inNP has never been published, so we close this gap by reporting a simple but notentirely trivial proof. A related question is whether a large enough "big M"for the classical KKT-based reformulation can be computed efficiently, which weanswer in the affirmative. In particular, our big M has polynomial encodinglength in the original problem data.

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