Simultaneous Optimal Transport

We propose a general framework of mass transport between vector-valued measures, which will be called simultaneous optimal transport (SOT). The new framework is motivated by the need to transport resources of different types simultaneously, i.e., in single trips, from specified origins to destinations; similarly, in economic matching, one needs to couple two groups, e.g., buyers and sellers, by equating supplies and demands of different goods at the same time. The mathematical structure of simultaneous transport is very different from the classic setting of optimal transport, leading to many new challenges. The Monge and Kantorovich formulations are contrasted and connected. Existence conditions and duality formulas are established. More interestingly, by connecting SOT to a natural relaxation of martingale optimal transport (MOT), we introduce the MOT-SOT parity, which allows for explicit solutions of SOT in many interesting cases.