In inverse optimization problems, the goal is to modify the costs in an underlying optimization problem in such a way that a given solution becomes optimal, while the difference between the new and the original cost functions, called the deviation vector, is minimized with respect to some objective function. The $\ell_1$- and $\ell_\infty$-norms are standard objectives used to measure the size of the deviation. Minimizing the $\ell_1$-norm is a natural way of keeping the total change of the cost function low, while the $\ell_\infty$-norm achieves the same goal coordinate-wise. Nevertheless, none of these objectives is suitable to provide a balanced or fair change of the costs. In this paper, we initiate the study of a new objective that measures the difference between the largest and the smallest weighted coordinates of the deviation vector, called the weighted span. We give a min-max characterization for the minimum weighted span of a feasible deviation vector, and provide a Newton-type algorithm for finding one that runs in strongly polynomial time in the case of unit weights.

翻译：在逆优化问题中，目标是通过调整底层优化问题的成本函数，使得给定解成为最优解，同时最小化新成本函数与原成本函数之间的差异（称为偏差向量）相对于某个目标函数的度量。$\ell_1$范数和$\ell_\infty$范数是衡量偏差大小的标准目标函数。最小化$\ell_1$范数是一种保持成本函数总变化量较低的自然方式，而$\ell_\infty$范数则按坐标实现相同目标。然而，这些目标均不适用于实现成本变化的均衡或公平。本文首次研究一种新的目标函数，该函数衡量偏差向量中加权最大坐标与加权最小坐标之间的差异，称为加权跨度。我们给出了可行偏差向量最小加权跨度的极小-极大刻画，并提出了一个牛顿型算法，该算法在单位权重情形下具有强多项式时间复杂性。