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:在最低成本逆向优化问题中,我们得到一个可行的解决方案,解决一个潜在的优化问题,同时具有线性成本功能,目标是通过一个小偏移矢量来修改成本,使输入解决方案变得最佳。新功能和原始成本功能之间的差别可以用几种方式衡量。在本文中,我们侧重于两个目标:加权的瓶颈仓储距离和加权的美元/美元/印度法郎-诺尔姆。我们考虑了一个总模型,在这个模型中,偏移矢量的坐标必须位于给定的下限和上限之内。对于加权的瓶颈仓载距离目标,我们提出了一个简单、纯的组合式算法,在强烈的多元时段内决定最佳偏移矢量。对于加权的 $\ell<unk> infty$-norm目标,我们给出了一个微量的微量尺寸描述,为找到一个在单位重量情况下在高度聚度时间内运行的最佳偏移矢量矢量的矢量矢量矢量值。对于这两个目标,我们假设一种具有相同时间复杂性的算法,同时确定一个最优度的组合优化目标是如何得到的。</s>