SIAM Journal on Applied Mathematics, Volume 80, Issue 1, Page 599-619, January 2020.
Discrete optimal transportation problems arise in various contexts in engineering, the sciences, and the social sciences. Often the underlying cost criterion is unknown, or only partly known, and the observed optimal solutions are corrupted by noise. In this paper we propose a systematic approach to infer unknown costs from noisy observations of optimal transportation plans. The algorithm requires only the ability to solve the forward optimal transport problem, which is a linear program, and to generate random numbers. It has a Bayesian interpretation and may also be viewed as a form of stochastic optimization. We illustrate the developed methodologies using the example of international migration flows. Reported migration flow data captures (noisily) the number of individuals moving from one country to another in a given period of time. It can be interpreted as a noisy observation of an optimal transportation map, with costs related to the geographical position of countries. We use a graph-based formulation of the problem, with countries at the nodes of graphs and nonzero weighted adjacencies only on edges between countries which share a border. We use the proposed algorithm to estimate the weights, which represent cost of transition, and to quantify uncertainty in these weights.

中文翻译：

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逆最优运输

SIAM应用数学杂志，第80卷，第1期，第599-619页，2020年1月。
离散的最优运输问题出现在工程，科学和社会科学的各种环境中。通常，潜在的成本标准是未知的，或者只是部分已知的，并且观察到的最佳解决方案会被噪声破坏。在本文中，我们提出了一种从最优运输计划的嘈杂观测中推断未知成本的系统方法。该算法仅需要解决线性规划的前向最优运输问题并生成随机数的能力。它具有贝叶斯解释，也可以看作是随机优化的一种形式。我们以国际移民流为例说明开发的方法。报告的移民流数据捕获（嘈杂）在给定时间段内从一个国家迁移到另一个国家的人数。可以将其解释为最佳交通图的嘈杂观测，其成本与国家/地区的地理位置有关。我们使用基于图的问题表示法，将国家置于图的节点处，并且非零加权邻接仅在共享边界的国家之间的边缘上。我们使用提出的算法来估计代表过渡成本的权重，并量化这些权重中的不确定性。