A Newton algorithm for semi-discrete optimal transport with storage fees

TL;DR

This paper presents a convergent damped Newton algorithm for semi-discrete optimal transport with storage fees, addressing capacity constraints and applicable to data clustering, with proven stability and quantitative convergence rates.

Contribution

It introduces the first convergent numerical method for this variant of optimal transport, removing support connectedness assumptions and providing stability analysis.

Findings

Algorithm converges with proven rates

Applicable to classical semi-discrete optimal transport

Demonstrated stability of Laguerre cells

Abstract

We introduce and prove convergence of a damped Newton algorithm to approximate solutions of the semi-discrete optimal transport problem with storage fees, corresponding to a problem with hard capacity constraints. This is a variant of the optimal transport problem arising in queue penalization problems, and has applications to data clustering. Our result is novel as it is the first numerical method with proven convergence for this variant problem; additionally the algorithm applies to the classical semi-discrete optimal transport problem but does not require any connectedness assumptions on the support of the source measure, in contrast with existing results. Furthermore we find some stability results of the associated Laguerre cells. All of our results come with quantitative rates. We also present some numerical examples.