Monge-Kantorovich interpolation with constraints and application to a parking problem

TL;DR

This paper introduces a constrained optimal transport framework involving an intermediate measure, applying it to a parking problem, with theoretical analysis and numerical simulations using entropic regularization.

Contribution

It develops a new Monge-Kantorovich interpolation model with constraints and applies it to urban parking location optimization.

Findings

Characterization of optimal pivot measures under constraints

Development of numerical methods using entropic regularization

Application to optimal parking region placement

Abstract

We consider optimal transport problems where the cost for transporting a given probability measure $\mu_0$ to another one $\mu_1$ consists of two parts: the first one measures the transportation from $\mu_0$ to an intermediate (pivot) measure $\mu$ to be determined (and subject to various constraints), and the second one measures the transportation from $\mu$ to $\mu_1$. This leads to Monge-Kantorovich interpolation problems under constraints for which we establish various properties of the optimal pivot measures $\mu$. Considering the more general situation where only some part of the mass uses the intermediate stop leads to a mathematical model for the optimal location of a parking region around a city. Numerical simulations, based on entropic regularization, are presented both for the optimal parking regions and for Monge-Kantorovich constrained interpolation problems.