网络上具有出发-到达约束和节点容量限制的灵活时间运输调度
Temporally Flexible Transport Scheduling on Networks with Departure-Arrival Constriction and Nodal Capacity Limits
AI总结：
本文提出了一种灵活时间运输调度方法，通过引入出发-到达约束和节点容量限制，扩展了传统最优运输框架，解决了网络中运输时间和资源分配问题。
摘要：
我们研究了在网络上的最优运输（OT）问题，其中供应和需求被概念化为时间边缘，控制粒子从源节点的出发率和到达sink节点的到达率。这种设定扩展了经典的OT框架，其中所有质量通常假设在t=0时出发并在t=tf时到达。我们的推广允许在指定时间出发和到达，称为出发-到达（DA）约束。特别地，我们在源节点和sink节点处施加节点-时间流量约束，刻画了两种不同的场景：（i）独立的DA约束，其中出发和到达率被独立规定；（ii）耦合的DA约束，其中每个粒子的运输时间跨度被明确指定。我们证明，具有独立DA约束的OT问题可以采用多边缘最优运输公式，而耦合DA情况则与非均等维度OT框架一致。对于线图，我们分析了解的存在性和唯一性。对于一般图，我们使用构造性的路径还原方法，并在给定的路径集上进行优化。从计算角度来看，我们考虑对原始问题进行熵正则化，以高效地提供解决方案，基于多边缘Sinkhorn方法，利用成本的图结构进一步提高可扩展性。我们的数值模拟进一步展示了在边际违反方面线性收敛率。
英文摘要：
We investigate the optimal transport (OT) problem over networks, wherein supply and demand are conceptualized as temporal marginals governing departure rates of particles from source nodes and arrival rates at sink nodes. This setting extends the classical OT framework, where all mass is conventionally assumed to depart at $t = 0$ and arrive at $t = t_f$. Our generalization accommodates departures and arrivals at specified times, referred as departure--arrival(DA) constraints. In particular, we impose nodal-temporal flux constraints at source and sink nodes, characterizing two distinct scenarios: (i) Independent DA constraints, where departure and arrival rates are prescribed independently, and (ii) Coupled DA constraints, where each particle's transportation time span is explicitly specified. We establish that OT with independent DA constraints admits a multi-marginal optimal transport formulation, while the coupled DA case aligns with the unequal-dimensional OT framework. For line graphs, we analyze the existence and uniqueness of the solution path. For general graphs, we use a constructive path-based reduction and optimize over a prescribed set of paths. From a computational perspective, we consider entropic regularization of the original problem to efficiently provide solutions based on multi-marginal Sinkhorn method, making use of the graphical structure of the cost to further improve scalability. Our numerical simulation further illustrates the linear convergence rate in terms of marginal violation.
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