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，我们给定一个底层优化问题的可行解及其线性成本函数，目标是通过一个微小偏差向量修改成本，使得输入解成为最优解。新旧成本函数之间的差异可通过多种方式度量。本文聚焦于两种目标：加权瓶颈汉明距离与加权$\ell_\infty$范数。我们考虑一个通用模型，要求偏差向量的各分量落在给定的上下界范围内。针对加权瓶颈汉明距离目标，我们提出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，我们给出最优解的最小-最大刻画，并提供一个伪多项式算法，该算法在单位权重情形下可在强多项式时间内找到最优偏差向量。对于这两种目标，我们假设存在具有相同时间复杂度的算法可求解底层组合优化问题。此外，我们还展示了如何将结果扩展至含多个成本函数的逆优化问题。