Mathematics Colloquia and Seminars

Return to Colloquia & Seminar listing

Ramified optimal transportation models branching transport structures found in many living and non-living systems. In contrast to the well-known Monge-Kantorovich transport problems where the transportation cost is solely determined by a transport map/plan, the cost in ramified transport problems is determined by the actual transport path. In this talk, I will start with a brief introduction to the ramified/branched optimal transportation theory and its applications. Then, we study a variant of ramified/branched optimal transportation problems, using geometric measure theory. Given the distributions of production capacities and market sizes, a firm looks for an allocation of productions over factories, a distribution of sales across markets, and a transport path that delivers the product to maximize its profit. Mathematically, given any two measures $\mu$ and $\nu$ on $X$, and a payoff function $h$, the planner wants to minimize $\mathbf{M}_{\alpha }(T)-\int_{X}hd(\partial T)$ among all transport paths $T$ from $\tilde{\mu}$ to $\tilde{\nu}$ with $\tilde{\mu}\preceq \mu $ and $\tilde{\nu}\preceq \nu $, where $\mathbf{M}_{\alpha }$ is the standard cost functional used in ramified transportation. After proving the existence result, we provide a characterization of the boundary measures of the optimal solution. They turn out to be the original measures restricted on some Borel subsets up to a Delta mass on each connected component. Our analysis further finds that as the boundary payoff increases, the corresponding solution of the current problem converges to an optimal transport path, which is the solution of the standard ramified transportation.