Optimal transport is the problem of designing a joint distribution for two random variables with fixed marginals. In virtually the entire literature on this topic, the objective is to minimize expected cost. This paper is the first to study a variant in which the goal is to minimize a quantile of the cost, rather than the mean. For the semidiscrete setting, where one distribution is continuous and the other is discrete, we derive a complete characterization of the optimal transport plan and develop simulation-based methods to efficiently compute it. One particularly novel aspect of our approach is the efficient computation of a tie-breaking rule that preserves marginal distributions. In the context of geographical partitioning problems, the optimal plan is shown to produce a novel geometric structure.

翻译：最优传输问题旨在为两个具有固定边缘分布的随机变量设计联合分布。在关于该主题的几乎所有文献中，目标都是最小化期望成本。本文首次研究了一种变体，其目标是最小化成本的分位数而非均值。针对半离散情形（即一个分布为连续分布，另一个为离散分布），我们推导了最优传输方案的完整刻画，并开发了基于模拟的方法以高效计算该方案。我们方法中一个特别新颖的方面在于高效计算能够保持边缘分布的平局决胜规则。在地理分区问题的背景下，最优方案被证明会产生一种新颖的几何结构。