In minimum-cost inverse optimization problems, we are given a feasible solution to an underlying optimization problem together with a linear cost function, and the goal is to modify the costs by a small deviation vector so that the input solution becomes optimal. The difference between the new and the original cost functions can be measured in several ways. In this paper, we focus on two objectives: the weighted bottleneck Hamming distance and the weighted $\ell_\infty$-norm. We consider a general model in which the coordinates of the deviation vector are required to fall within given lower and upper bounds. For the weighted bottleneck Hamming distance objective, we present a simple, purely combinatorial algorithm that determines an optimal deviation vector in strongly polynomial time. For the weighted $\ell_\infty$-norm objective, we give a min-max characterization for the optimal solution, and provide a pseudo-polynomial algorithm for finding an optimal deviation vector that runs in strongly polynomial time in the case of unit weights. For both objectives, we assume that an algorithm with the same time complexity for solving the underlying combinatorial optimization problem is available. For both objectives, we also show how to extend the results to inverse optimization problems with multiple cost functions.

翻译：在最小成本逆优化问题中，给定一个底层优化问题的可行解及其相应的线性成本函数，目标是通过一个微小偏差向量调整成本，使该输入解成为最优解。新旧成本函数之间的差异可通过多种方式度量。本文重点研究两个目标：加权瓶颈汉明距离和加权$\ell_\infty$-范数。我们考虑一个通用模型，其中偏差向量的各坐标需位于给定的上下界内。针对加权瓶颈汉明距离目标，我们提出了一种简单的纯组合算法，可在强多项式时间内确定最优偏差向量。针对加权$\ell_\infty$-范数目标，我们给出了最优解的极小-极大刻画，并提供了一种伪多项式算法来寻找最优偏差向量；在单位权重情况下，该算法可在强多项式时间内运行。对于这两个目标，我们均假设存在一个求解底层组合优化问题的算法，且其时间复杂度相同。此外，我们还展示了如何将结果推广至具有多个成本函数的逆优化问题。