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Block Coordinate Descent Network Simplex Methods for Optimal Transport

Lingrui Li, Nobuo Yamashita

TL;DR

The paper tackles large-scale discrete OT by marrying the Network Simplex method with a block coordinate descent strategy, producing the Block Coordinate Descent Network Simplex (BCDNS) that preserves feasibility via a basis-variable succession mechanism and achieves finite termination with an exact optimum. By introducing deterministic and grouped block-selection schemes, the method reduces per-iteration cost to $O(sN)$ while leveraging warm starts from the current basis; this yields substantial speedups over classical NS and competitive performance against high-precision Sinkhorn when exact solutions are needed. Theoretical guarantees of finite termination and global optimality are complemented by extensive experiments across synthetic and large-scale OT problems, showing dramatic reductions in reduced-cost evaluations and memory usage, as well as scalability up to $n=4000$. The results indicate that BCDNS is particularly well suited for large-scale OT where exact optimality is required, offering a favorable accuracy–efficiency balance and paving the way for further extensions to broader network flow problems.

Abstract

We propose the Block Coordinate Descent Network Simplex (BCDNS) method for solving large-scale discrete Optimal Transport (OT) problems. BCDNS integrates the Network Simplex (NS) algorithm with a block coordinate descent (BCD) strategy, decomposing the full problem into smaller subproblems per iteration and reusing basis variables to ensure feasibility. We prove that BCDNS terminates in a finite number of iterations with an exact optimal solution, and we characterize its per-iteration complexity as O(s N), where s is a user-defined parameter in (0,1) and N is the total number of variables. Numerical experiments demonstrate that BCDNS matches the classical NS method in solution accuracy, reduces memory footprint compared to the Sinkhorn algorithm, achieves speed-ups of up to tens of times over the classical NS method, and exhibits runtime comparable to a high-precision Sinkhorn implementation.

Block Coordinate Descent Network Simplex Methods for Optimal Transport

TL;DR

The paper tackles large-scale discrete OT by marrying the Network Simplex method with a block coordinate descent strategy, producing the Block Coordinate Descent Network Simplex (BCDNS) that preserves feasibility via a basis-variable succession mechanism and achieves finite termination with an exact optimum. By introducing deterministic and grouped block-selection schemes, the method reduces per-iteration cost to while leveraging warm starts from the current basis; this yields substantial speedups over classical NS and competitive performance against high-precision Sinkhorn when exact solutions are needed. Theoretical guarantees of finite termination and global optimality are complemented by extensive experiments across synthetic and large-scale OT problems, showing dramatic reductions in reduced-cost evaluations and memory usage, as well as scalability up to . The results indicate that BCDNS is particularly well suited for large-scale OT where exact optimality is required, offering a favorable accuracy–efficiency balance and paving the way for further extensions to broader network flow problems.

Abstract

We propose the Block Coordinate Descent Network Simplex (BCDNS) method for solving large-scale discrete Optimal Transport (OT) problems. BCDNS integrates the Network Simplex (NS) algorithm with a block coordinate descent (BCD) strategy, decomposing the full problem into smaller subproblems per iteration and reusing basis variables to ensure feasibility. We prove that BCDNS terminates in a finite number of iterations with an exact optimal solution, and we characterize its per-iteration complexity as O(s N), where s is a user-defined parameter in (0,1) and N is the total number of variables. Numerical experiments demonstrate that BCDNS matches the classical NS method in solution accuracy, reduces memory footprint compared to the Sinkhorn algorithm, achieves speed-ups of up to tens of times over the classical NS method, and exhibits runtime comparable to a high-precision Sinkhorn implementation.

Paper Structure

This paper contains 29 sections, 3 theorems, 45 equations, 10 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Discrete Optimal Transport
  3. Existing Methods for Optimal Transport
  4. Network Simplex Method for Optimal Transport
  5. Block Coordinate Descent Method for Optimal Transport
  6. Sinkhorn Algorithm for Optimal Transport
  7. Block Coordinate Network Simplex Methods with Basis Variable Succession
  8. Basis Variables Succession
  9. A Prototype of Block Coordinate Descent Network Simplex Methods and its Finite Termination
  10. Random Selection Block Coordinate Descent Network Simplex Method
  11. Grouped Selection Block Coordinate Descent Network Simplex method
  12. Numerical Experiments
  13. Experimental Setup
  14. Test instance generation and reproducibility.
  15. Algorithmic settings.
  16. ...and 14 more sections

Key Result

Theorem 1

The gap between the current objective function value and the optimal objective function value converges with rate $1 - v$, i.e., Here, $v$ satisfies where $\zeta$ is a predefined probability parameter in the range $(0,1)$, and $\tilde{p}$ is a parameter representing the properties of the matrix in the range $[3, n]$.

Figures (10)

  • Figure 1: Effect of $(s,t)$ on reduced-cost evaluations, pivot count, and runtime for the instance with $n=250$.
  • Figure 2: Effect of $(s,t)$ on reduced-cost evaluations, pivot count, and runtime for the instance with $n=500$.
  • Figure 3: Effect of $(s,t)$ on reduced-cost evaluations, pivot count, and runtime for the instance with $n=1000$.
  • Figure 4: Test Problem 2: runtime (left) and total reduced-cost evaluations (right) for NS, RS-BCDNS, and GS-BCDNS.
  • Figure 5: Test Problem 2: speedup relative to NS (NS time / method time).
  • ...and 5 more figures

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof