Simultaneous Optimal Transport

TL;DR

This paper introduces a novel framework called simultaneous optimal transport (SOT) for transporting multiple types of resources simultaneously, addressing complex real-world scenarios like economic matching and resource allocation.

Contribution

It develops the mathematical foundation of SOT, contrasts it with classical optimal transport, and establishes existence, duality, and explicit solutions through a connection with martingale optimal transport.

Findings

Established existence conditions and duality formulas for SOT.

Connected SOT to martingale optimal transport, enabling explicit solutions.

Demonstrated the applicability of SOT in various scenarios.

Abstract

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.