Optimal transport through a toll

TL;DR

This paper studies optimal transport problems with flow constraints modeled as toll stations, providing detailed solutions in one dimension and discussing potential higher-dimensional generalizations.

Contribution

It introduces a formulation of optimal transport with flux constraints, proves existence and uniqueness in one dimension, and offers explicit solutions under regularity assumptions.

Findings

Existence and uniqueness of solutions in 1D

Explicit construction of transport plans

Framework extendable to higher dimensions

Abstract

We address the problem of optimal transport with a quadratic cost functional and a constraint on the flux through a constriction along the path. The constriction, conceptually represented by a toll station, limits the flow rate across. We provide a precise formulation which, in addition, is amenable to generalization in higher dimensions. We work out in detail the case of transport in one dimension by proving existence and uniqueness of solution. Under suitable regularity assumptions we give an explicit construction of the transport plan. Generalization of flux constraints to higher dimensions and possible extensions of the theory are discussed.