An efficient second-order cone programming approach for dynamic optimal transport on staggered grid discretization

TL;DR

This paper introduces an efficient SOCP-based numerical method for solving dynamic optimal transport problems on staggered grids, significantly improving computational efficiency and robustness.

Contribution

The paper presents a novel SOCP reformulation for DOT problems that eliminates interpolation matrices, enabling faster and more robust solutions with an open-source implementation.

Findings

The proposed method outperforms existing software in efficiency.

It demonstrates high robustness to non-negative measure problems.

Numerical experiments confirm significant computational savings.

Abstract

This paper proposes an efficient numerical method based on second-order cone programming (SOCP) to solve dynamic optimal transport (DOT) problems with quadratic cost on staggered grid discretization. By properly reformulating discretized DOT problems into a linear SOCP, the proposed method eliminates the interpolation matrices and thus avoids solving a series of cubic equations and linear systems induced by interpolation. Then, by taking advantage of the SOCP reformulation, we can solve them efficiently by a computationally highly economical implementation of an inexact decomposition-based proximal augmented Lagrangian method. Moreover, we have made the proposed approach an open-source software package. Numerical experiments on various DOT problems suggest that the proposed approach performs significantly more efficiently than state-of-the-art software packages. In addition, it exhibits prominent robustness to problems with non-negative measures.