关于逆混合整数线性优化的复杂性
On the Complexity of Inverse Mixed Integer Linear Optimization
AI总结：
本文研究逆 MILP 的计算复杂性，证明其原始界验证为 coNP 完全、最优值验证为 D^P 完全，并通过割平面算法揭示逆问题、正向问题与分离问题的等价性。
摘要：
逆优化是这样一个问题：为相关的正向问题确定缺失输入参数的值，这些值既要最接近给定估计值，又要使给定目标向量成为最优。本研究关注一个特定的逆混合整数线性优化问题（MILP）与其正向问题以及与其可行域相关的分离问题之间的关系。我们证明，验证一个原始界的逆 MILP 判定版本是 coNP 完全的，而相关正向问题的原始界验证是 NP 完全的；并且逆问题与相关正向问题的最优值验证问题对于复杂性类 D^P 都是完全的。我们还描述了一种用于求解逆 MILP 的割平面算法，该算法阐明了给定 MILP 解的凸包的分离问题与相关逆问题之间的密切关系。逆问题被证明等价于径向锥的分离问题，该径向锥由所有既对正向问题解的凸包有效、又在目标向量处紧的不等式定义。因此，可以说逆问题、正向问题和分离问题是等价的。
英文摘要：
Inverse optimization is the problem of determining the values of missing input parameters for an associated forward problem that are closest to given estimates and that will make a given target vector optimal. This study is concerned with the relationship of a particular inverse mixed integer linear optimization problem (MILP) to both the forward problem and the separation problem associated with its feasible region. We show that a decision version of the inverse MILP in which a primal bound is verified is coNP-complete, whereas primal bound verification for the associated forward problem is NP-complete, and that the optimal value verification problems for both the inverse problem and the associated forward problem are complete for the complexity class D^P. We also describe a cutting-plane algorithm for solving inverse MILPs that illustrates the close relationship between the separation problem for the convex hull of solutions to a given MILP and the associated inverse problem. The inverse problem is shown to be equivalent to the separation problem for the radial cone defined by all inequalities that are both valid for the convex hull of solutions to the forward problem and binding at the target vector. Thus, the inverse, forward, and separation problems can be said to be equivalent.
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