An Efficient Augmented Lagrangian Method for Dynamic Optimal Transport on Surfaces Based on Second-Order Cone Programming

TL;DR

This paper introduces a fast, robust numerical method based on second-order cone programming for solving dynamic optimal transport problems on surfaces, with an open-source implementation demonstrating superior performance.

Contribution

It reformulates the dual dynamic optimal transport problem on surfaces into a linear second-order cone program and solves it efficiently with an augmented Lagrangian method.

Findings

Outperforms existing solvers in speed by several times

Successfully computes Wasserstein distances and transportation paths on surfaces

Demonstrates robustness and efficiency across diverse datasets

Abstract

This paper proposes an efficient numerical optimization approach for solving dynamic optimal transport (DOT) problems on general smooth surfaces, computing both the quadratic Wasserstein distance and the associated transportation path. Building on the convex DOT model of Benamou and Brenier, we first properly reformulate its dual problem, discretized on a triangular mesh for space together with a staggered grid for time, to a linear second-order cone programming. Then the resulting finite-dimensional convex optimization problem is solved via an inexact semi-proximal augmented Lagrangian method with a highly efficient numerical implementation, and the algorithm is guaranteed to converge to a Karush-Kuhn-Tucker solution without imposing any additional assumptions. Finally, we implement the proposed methodology as an open-source software package. The effectiveness, robustness, and computational efficiency of the software are demonstrated through extensive numerical experiments across diverse datasets, where it consistently outperforms state-of-the-art solvers by several times in speed.