An efficient second-order cone programming approach for dynamic optimal transport on staggered grid discretization
Liang Chen, Youyicun Lin, Yuxuan Zhou
TL;DR
The paper tackles efficient numerical solution of dynamic optimal transport with quadratic cost on staggered grids. It develops an exact SOCP reformulation of the discretized DOT problem and solves it with an inexact proximal ALM that leverages diagonalized linear systems and SOC projections, achieving substantial speedups and robustness. The approach scales to large 3D grids, handles Dirac measures and irregular domains, and is released as open-source software. Together, these advances enable reliable, high-precision DOT computations for complex applications in Wasserstein geometry and related areas.
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.