Optimal transport with constraints: from mirror descent to classical mechanics

TL;DR

This paper introduces a physics-inspired method to incorporate realistic constraints into optimal transport problems, using classical mechanics principles to improve flexibility and computational efficiency.

Contribution

It develops a novel approach that integrates constraints into mirror descent dynamics via the D'Alembert-Lagrange principle, enabling closed-form updates.

Findings

The method allows flexible constraint incorporation in optimal transport.

It results in sparse, local, and linear feasible set approximations.

Closed-form updates are achieved in many cases.

Abstract

Finding optimal trajectories for multiple traffic demands in a congested network is a challenging task. Optimal transport theory is a principled approach that has been used successfully to study various transportation problems. Its usage is limited by the lack of principled and flexible ways to incorporate realistic constraints. We propose a principled physics-based approach to impose constraints flexibly in such optimal transport problems. Constraints are included in mirror descent dynamics using the principle of D'Alembert-Lagrange from classical mechanics. This leads to a sparse, local and linear approximation of the feasible set leading in many cases to closed-form updates.