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

This paper proposes an efficient numerical optimization framework for solving dynamic optimal transport (DOT) problems on surfaces, computing both the quadratic Wasserstein distance and the associated interpolation.

Building on the convex DOT model of Benamou-Brenier-Lisini, we first properly reformulate its dual problem, discretized on a triangular mesh in space and a staggered grid in time, into a linear second-order cone programming (SOCP) problem.

Then the resulting SOCP is solved via an inexact proximal augmented Lagrangian method with a highly efficient numerical implementation, and the algorithm is guaranteed to converge to a Karush-Kuhn-Tucker point without imposing any additional assumptions.

Finally, we implement the proposed framework as an open-source software package.

The effectiveness, robustness, and computational efficiency of the software are validated through extensive numerical experiments across diverse datasets, demonstrating that it consistently outperforms state-of-the-art surface DOT solvers by several times in speed, while the commercial solvers Gurobi and MOSEK either fail to solve the same SOCP reformulation due to out-of-memory or require substantially prolonged computation times.