动态均衡流的通行费
Tolls for Dynamic Equilibrium Flows
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中文总结 AI 辅助
该研究针对异构用户模型，分别给出多源多目的地、多源单目的地场景下动态边流可实现为收费动态均衡流的刻画，还构建了无限维优化问题并证明相关假设成立。
中文摘要 AI 辅助
我们研究动态网络流，探讨以下问题：哪些动态边流可以被实现为收费动态均衡流？我们针对异构用户模型研究该问题，其中流粒子被划分为不同群体，每个群体具有各自的源-目的地对，以及与任意路径和出发时间相关联的成本函数。作为两项主要结果，我们首先针对多源多目的地情况，提供了动态边流可实现性的基于对偶性的刻画；其次，我们针对多源单目的地情况，推导出动态边流可实现性的组合式刻画与基于对偶性的刻画。两项结果均在相当通用的网络加载模型下得出。为完成证明，我们作出了多项技术贡献：我们构建了一个新颖的无限维优化问题，目标是针对给定边流诱导的固定网络加载，最小化粒子的总聚合成本。这需要用到最近提出的自主网络加载概念，我们针对该概念展示了若干新的结构见解，特别是通过推广M.A. Zareckiĭ关于绝对连续单调函数的Lusin N⁻¹性质的结果，为我们的场景提供了自主网络加载存在性的另一种（更严格的）刻画，该结果也可能具有独立研究价值。这些见解使我们能够在强对偶性假设下证明所述的刻画。最后，对于单目的地情况，我们能够给出一个非平凡的证明，即当成本表示加权旅行时间且边流具有有限支撑时，该假设总是成立。
英文摘要
We consider dynamic network flows and study the following question: Which dynamic edge flows can be implemented as tolled dynamic equilibrium flows? We study this question for the heterogeneous-user model, where the flow particles are partitioned into populations characterized by their own source,destination-pairs and a cost function associating with any walk and departure time some costs. As our two main results, we first provide a duality-based characterization of implementability of dynamic edge flows for the multi-source, multi-destination case. Secondly, we derive both, a combinatorial and duality-based characterization of implementability of dynamic edge flows for the multi-source, single-destination case. Both results are derived under a fairly general network loading model. For the proof, we make several technical contributions: We formulate a novel infinite dimensional optimization problem, where the goal is to minimize the aggregated costs of the particles with respect to the fixed network loading induced by the given edge flow. This requires the recently introduced concept of autonomous network loadings for which we show several new structural insights. In particular, we give an alternative (tighter) characterization of the existence of autonomous network loadings for our setting by deriving a generalization of a result of M.A. Zarecki\uı on the Lusin $N^{-1}$ property of absolutely continuous monotone functions which may also be of independent interest. These insights allow us to prove the stated characterizations under the assumption of strong duality. Finally, for the case of a single-destination, we are able to provide a non-trivial proof that this assumption is always fulfilled for finitely supported edge flows with costs representing weighted travel times.
