论文标题
分支运输和城市规划的二元性
Duality in branched transport and urban planning
论文作者
论文摘要
在最近的工作中，ARXIV：2109.07820我们显示了广泛使用的非凸（广义）分支运输问题的等效性，以及街道或铁路网络的形状优化问题，即（普遍）Urban Planning问题。该论点仅基于竞争对手的明确结构和表征。在当前文章中，我们相反分析了与这两个问题相关的双重视角。更详细的是，形状优化问题涉及两种度量之间的瓦斯汀距离，具体取决于街道网络。我们在轻度假设下显示了此类街道网络上的瓦斯坦斯坦距离的kantorovich $ \ unicode {x2013} $ rubinstein公式。此外，我们在假设下为这种瓦斯尔斯坦距离提供了贝克曼公式，该假设将我们先前的结果推广到arxiv：2109.07820。作为一种应用，我们提供了一个基于二元性的替代性，基于双重性的证明，证明了这两个问题在增长条件下对运输成本的等效性，这表明城市规划和分支运输都可以看作是两个双线性耦合凸优化问题。
In recent work arXiv:2109.07820 we have shown the equivalence of the widely used nonconvex (generalized) branched transport problem with a shape optimization problem of a street or railroad network, known as (generalized) urban planning problem. The argument was solely based on an explicit construction and characterization of competitors. In the current article we instead analyse the dual perspective associated with both problems. In more detail, the shape optimization problem involves the Wasserstein distance between two measures with respect to a metric depending on the street network. We show a Kantorovich$\unicode{x2013}$Rubinstein formula for Wasserstein distances on such street networks under mild assumptions. Further, we provide a Beckmann formulation for such Wasserstein distances under assumptions which generalize our previous result in arXiv:2109.07820. As an application we then give an alternative, duality-based proof of the equivalence of both problems under a growth condition on the transportation cost, which reveals that urban planning and branched transport can both be viewed as two bilinearly coupled convex optimization problems.
