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The Double Regularization Method for Capacity Constrained Optimal Transport

Tianhao Wu, Qihao Cheng, Zihao Wang, Chaorui Zhang, Bo Bai, Zhongyi Huang, Hao Wu

TL;DR

The paper tackles capacity-constrained OT by introducing the Double Regularization Method (DRM), which uses two entropic-like terms to enforce both upper and (via an auxiliary reformulation) effective lower bounds while preserving convexity. It derives a dual formulation with $O(N)$ dual variables and reduces updates to solving $2N$ monotone scalar equations in each iteration through Newton's method, achieving $O(N^2)$ time and $O(N)$ memory. The method converges to the original discretized COT solution as the regularization parameter $\varepsilon$ tends to zero, with proven uniqueness and stability. Empirical results on 1D and 2D grids with uniform and marginal-capacity constraints show DRM substantially outperforms Iterative Bregman Projection in accuracy and speed and uses dramatically less memory, making large-scale capacity-constrained OT practical for applications in finance and network flow.

Abstract

Capacity constrained optimal transport is a variant of optimal transport, which adds extra constraints on the set of feasible couplings in the original optimal transport problem to limit the mass transported between each pair of source and sink. Based on this setting, constrained optimal transport has numerous applications, e.g., finance, network flow. However, due to the large number of constraints in this problem, existing algorithms are both time-consuming and space-consuming. In this paper, inspired by entropic regularization for the classical optimal transport problem, we introduce a novel regularization term for capacity constrained optimal transport. The regularized problem naturally satisfies the capacity constraints and consequently makes it possible to analyze the duality. Unlike the matrix-vector multiplication in the alternate iteration scheme for solving classical optimal transport, in our algorithm, each alternate iteration step is to solve several single-variable equations. Fortunately, we find that each of these equations corresponds to a single-variable monotonic function, and we convert solving these equations into finding the unique zero point of each single-variable monotonic function with Newton's method. Extensive numerical experiments demonstrate that our proposed method has a significant advantage in terms of accuracy, efficiency, and memory consumption compared with existing methods.

The Double Regularization Method for Capacity Constrained Optimal Transport

TL;DR

The paper tackles capacity-constrained OT by introducing the Double Regularization Method (DRM), which uses two entropic-like terms to enforce both upper and (via an auxiliary reformulation) effective lower bounds while preserving convexity. It derives a dual formulation with dual variables and reduces updates to solving monotone scalar equations in each iteration through Newton's method, achieving time and memory. The method converges to the original discretized COT solution as the regularization parameter tends to zero, with proven uniqueness and stability. Empirical results on 1D and 2D grids with uniform and marginal-capacity constraints show DRM substantially outperforms Iterative Bregman Projection in accuracy and speed and uses dramatically less memory, making large-scale capacity-constrained OT practical for applications in finance and network flow.

Abstract

Capacity constrained optimal transport is a variant of optimal transport, which adds extra constraints on the set of feasible couplings in the original optimal transport problem to limit the mass transported between each pair of source and sink. Based on this setting, constrained optimal transport has numerous applications, e.g., finance, network flow. However, due to the large number of constraints in this problem, existing algorithms are both time-consuming and space-consuming. In this paper, inspired by entropic regularization for the classical optimal transport problem, we introduce a novel regularization term for capacity constrained optimal transport. The regularized problem naturally satisfies the capacity constraints and consequently makes it possible to analyze the duality. Unlike the matrix-vector multiplication in the alternate iteration scheme for solving classical optimal transport, in our algorithm, each alternate iteration step is to solve several single-variable equations. Fortunately, we find that each of these equations corresponds to a single-variable monotonic function, and we convert solving these equations into finding the unique zero point of each single-variable monotonic function with Newton's method. Extensive numerical experiments demonstrate that our proposed method has a significant advantage in terms of accuracy, efficiency, and memory consumption compared with existing methods.
Paper Structure (13 sections, 2 theorems, 24 equations, 3 figures, 4 tables, 1 algorithm)

This paper contains 13 sections, 2 theorems, 24 equations, 3 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Double Regularization Method
  3. Numerical Algorithm
  4. Numerical Experiments
  5. 1D grid with uniform capacity constraints
  6. 1D grid with capacity constraints regarding marginals
  7. 2D grid with uniform capacity constraints
  8. 2D grid with capacity constraints regarding marginals
  9. Comparison of memory usage
  10. Conclusion
  11. PROOFS
  12. Proof of Lemma \ref{['lemma:equivalence']}
  13. Proof of Theorem \ref{['thm:convergence']}

Key Result

Lemma 2.1

The discretized COT problem eq:COT can be equivalently converted to the problem without extra lower bound constraint, which is defined as Correspondingly, the double regularized problem eq:DRM-double can be equivalently converted to the following regularized problem which corresponds to the discretized COT problem without lower bound constraint eq:COT-upper.

Figures (3)

  • Figure 1: The 1D random distribution problem with capacity constraints regarding marginals. Left: The comparison of computational time between Gurobi, IBP, and DRM with different numbers of grid points $N$. Right: The relative error between numerical results generated by IBP or DRM and the ground truth w.r.t. computational time.
  • Figure 2: The 2D random distribution problem with capacity constraints regarding marginals. Left: The comparison of computational time between IBP and DRM with different number of grid points $N \times N$. Right: The relative error between numerical results generated by IBP or DRM and the ground truth w.r.t. computational time.
  • Figure 3: The memory consumption of Gurobi, IBP and our proposed DRM. Left: The comparison of memory consumption among these three methods with different number of grid points $N$ and upper bound matrix $\bm{\eta} = 2 \bm{u} \bm{v}^T$ in the 1D case. Right: The comparison of memory consumption among these three methods with different number of grid points $N \times N$ and $\bm{\eta} = 2 \bm{u} \bm{v}^T$ in the 2D case.

Theorems & Definitions (3)

  • Lemma 2.1
  • Theorem 2.1
  • Remark 2.1