Nonsmooth and nonconvex optimal transport problems

近年来，数学领域对所谓的“分支运输”模型产生了浓厚的兴趣，这种模型可以描述发生在道路系统、河流流域、通信网络、脉管系统以及许多其他自然和人工环境中的运输网络。在经典的最优输送中，一定数量的物质需要从给定的初始质量分布有效地输送到最终质量分布。然而，在分支运输中，运输成本在运输质量中不是成比例的，而是次相加的，如果质量是散装运输，则运输效率会增加。这自然有利于质量通量集中在一个复杂的、分散的一维线路网络上的运输方案。分支输运问题是描述质量通量的Radon测度（实际上是法向电流）上的一个复杂的非凸、非光滑变分问题。各种不同的公式被开发和分析（包括pi的工作），然而，他们都采取几何测量理论的观点，使用平链，利普希茨曲线空间上的概率测量，或类似的。完全缺乏的是优化和最优控制的视角（尽管在现有的变分论证中，如通过必要的最优性条件进行规律性分析或与对偶优化变量相关的校准概念）。这种情况也反映在分支输运的数值领域相当不发达，主要由特殊情况下的特设图优化方法和二维相场近似组成。我们将在Radon测度优化和最优控制的框架下重新表述分支运输，在分支运输网络的变分分析中提出这种优化观点，并将结果应用于新的数值方法中。新的视角将有助于变分分析，促进对测度的非光滑、非凸优化问题的理解，并为获得高效的运输网络提供数值方法。

In recent years a strong interest has developed within mathematics in so-called "branched Transport" models, which allow to describe transportation networks as they occur in road systems, river basins, communication networks, vasculature, and many other natural and artificial contexts. As in classical optimal transport, an amount of material needs to be transported efficiently from a given initial to a final mass distribution. In branched transport, however, the transportation cost is not proportional, but subadditive in the transported mass, modelling an increased transport efficiency if mass is transported in bulk. This automatically favours transportation schemes in which the mass flux concentrates on a complicated, ramified network of one-dimensional lines. The branched transport problem is an intricate nonconvex, nonsmooth variational problem on Radon measures (in fact on normal currents) that describe the mass flux. Various different formulations were developed and analysed (including work by the PIs), however, they all all take the viewpoint of geometric measure theory, working with flat chains, probability measures on the space of Lipschitz curves, or the like. What is completely lacking is an optimization and optimal control perspective (even though some ideas of optimization shimmer through in the existing variational arguments such as regularity analysis via necessary optimality conditions or the concept of calibrations which are related to dual optimization variables). This situation is also reflected in the fact that the field of numerics for branched transport is rather underdeveloped and consists of ad hoc graph optimization methods for special cases and two-dimensional phase field approximations. We will reformulate branched transport in the framework of optimization and optimal control for Radon measures, work out this optimization viewpoint in the variational analysis of branched transport networks, and exploit the results in novel numerical approaches. The new perspective will at the same time help variational analysts, advance the understanding of nonsmooth, nonconvex optimization problems on measures, and provide numerical methods to obtain efficient transport networks.