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

TL;DR

This paper proves that bilevel linear optimization problems are in NP and can be reformulated with polynomial-size KKT-based models, clarifying theoretical complexity and practical reformulation aspects.

Contribution

It provides the first rigorous proof that bilevel linear optimization belongs to NP and shows how to compute polynomial-size KKT reformulations efficiently.

Findings

Bilevel linear optimization is in NP.

Polynomial-size KKT reformulations are possible.

Efficient computation of large enough 'big M' is demonstrated.

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.