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.

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技术说明—没有免费的午餐：关于在双层优化中选择正确的big-M的难度

解决线性双层优化问题的最常用方法之一是用其Karush-Kuhn-Tucker（KKT）条件替换较低层的问题，并使用混合整数线性优化技术重新构造KKT互补条件。后面的步骤需要确定一些big-M常数，以便绑定较低级别的对偶可行集，这样就不会切断任何双层最优解。在实践中，尽管已知这些方法可能会失败，但常常使用启发式方法来找到big-M。在本文中，我们考虑了上述双水平校正big-M概念的两个代理的硬度。首先，我们证明，除非P = NP，否则无法在多项式时间内验证给定的big-M不会切断下层对偶多面体的任何可行顶点。