Technical Note—There’s No Free Lunch: On the Hardness of Choosing a Correct Big-M in Bilevel Optimization

There's No Free Lunch in Bilevel Optimization Linear bilevel problems are often reformulated as single-level problems by using the KKT optimality conditions of the lower level. The resulting KKT complementarity conditions are usually linearized by the classical big-M approach. However, this approach requires to bound the lower-level dual variables to obtain a correct single-level reformulation of the original bilevel problem. If the big-M value is not large enough, this yields to a reformulation admitting wrong solutions w.r.t. the original problem. In this technical note we show that verifying that a big-M does not lead to cutting off any feasible vertex of the lower-level dual polyhedron cannot be done in polynomial time unless P = NP. Moreover, we prove that verifying that a big-M does not lead to cutting off any optimal point of the lower-level dual problem is as hard as solving the original bilevel problem. 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.