We introduce a new class of inverse optimization problems in which an input solution is given together with $k$ linear weight functions, and the goal is to modify the weights by the same deviation vector $p$ so that the input solution becomes optimal with respect to each of them, while minimizing $\|p\|_1$. In particular, we concentrate on three problems with multiple weight functions: the inverse shortest $s$-$t$ path, the inverse bipartite perfect matching, and the inverse arborescence problems. Using LP duality, we give min-max characterizations for the $\ell_1$-norm of an optimal deviation vector. Furthermore, we show that the optimal $p$ is not necessarily integral even when the weight functions are so, therefore computing an optimal solution is significantly more difficult than for the single-weighted case. We also give a necessary and sufficient condition for the existence of an optimal deviation vector that changes the values only on the elements of the input solution, thus giving a unified understanding of previous results on arborescences and matchings.
翻译:我们引入了一个新的反优化问题类别, 输入解决方案与 $k$ 线性重量函数一同给出, 目标是用相同的偏向矢量 $p$来修改重量, 以使输入解决方案对每种矢量都达到最佳, 同时尽量减少$p ⁇ 1$。 特别是, 我们集中关注三个具有多重重量功能的问题: 逆向最短的美元- 美元路径, 反向的双方完美匹配, 反向的损耗问题。 使用 LP 双重性, 我们给最佳偏向矢量的 $_ 1 美元- 诺尔 进行最小最大量的描述。 此外, 我们显示即使重量功能如此, 最优的 $p$ 也不一定是不可或缺的, 因此, 计算最佳的解决方案比单一加权的要困难得多。 我们还为最佳偏差矢量的存在提供了必要和充分的条件, 最佳偏差矢量只改变输入解决方案的值, 从而统一理解 perboresercerence 和匹配的先前结果 。