Computational Semi-Discrete Optimal Transport with General Storage Fees

TL;DR

This paper introduces a modified damped Newton algorithm for semi-discrete optimal transport problems with storage fees, proving global linear convergence under independence conditions and demonstrating stability and quantitative convergence rates.

Contribution

The paper develops a new algorithm for semi-discrete optimal transport with storage fees and proves its convergence and stability under broad conditions.

Findings

Global linear convergence for a wide range of storage fee functions

Stability of optimizers under perturbations of storage fees

Quantitative convergence rates provided

Abstract

We propose and analyze a modified damped Newton algorithm to solve the semi-discrete optimal transport with storage fees. We prove global linear convergence for a wide range of storage fee functions, the main assumption being that each warehouse's storage costs are independent. We show that if $F$ is an arbitrary storage fee function that satisfies this independence condition then $F$ can be perturbed into a new storage fee function so that our algorithm converges. We also show that the optimizers are stable under these perturbations. Furthermore, our results come with quantitative rates.