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：在反优化问题中，目标是通过修改底层优化问题中的成本，使得给定解成为最优解，同时最小化新成本函数与原始成本函数之间的差异（称为偏差向量）相对于某一目标函数。$\ell_1$范数和$\ell_\infty$范数是衡量偏差大小的标准目标。最小化$\ell_1$范数是保持成本函数总变化较低的自然方法，而$\ell_\infty$范数则在坐标方向上实现相同目标。然而，这些目标均不适用于实现成本的均衡或公平变化。本文首次研究一种衡量偏差向量最大与最小加权坐标之间差异的新目标，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，并提供了一种牛顿型算法，在单位权重情形下该算法可在强多项式时间内运行。