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On the Relationships among GPU-Accelerated First-Order Methods for Solving Linear Programming

Kaihuang Chen, Defeng Sun, Yancheng Yuan, Guojun Zhang, Xinyuan Zhao

TL;DR

This work analyzes GPU-accelerated first-order methods for large-scale linear programming, focusing on the relationships among HPR-LP, cuPDLPx, and EPR-LP. It proves that the base algorithm of cuPDLPx is a special case of HPR-LP, and that under strict complementarity the PR updates become affine on active faces, making HPR-LP and EPR-LP equivalent after active-set identification. The authors establish affine-face equivalence results and provide extensive GPU benchmarks showing that HPR-LP delivers the best overall performance among current GPU solvers. These findings position the HPR framework as a robust baseline for developing faster GPU-accelerated LP solvers and offer directions for extending HPR to broader convex problems and learning-based parameter strategies.

Abstract

This paper aims to understand the relationships among recently developed GPU-accelerated first-order methods (FOMs) for linear programming (LP), with particular emphasis on HPR-LP -- a Halpern Peaceman--Rachford (HPR) method for LP. Our findings can be summarized as follows: (i) the base algorithm of cuPDLPx, a recently released GPU solver, is a special case of the base algorithm of HPR-LP, thereby showing that cuPDLPx is another concrete implementation instance of HPR-LP; (ii) once the active sets have been identified, HPR-LP and EPR-LP -- an ergodic PR method for LP -- become equivalent under the same initialization; and (iii) extensive numerical experiments on benchmark datasets demonstrate that HPR-LP achieves the best overall performance among current GPU-accelerated LP solvers. These findings provide a strong motivation for using the HPR method as a baseline to further develop GPU-accelerated LP solvers and beyond.

On the Relationships among GPU-Accelerated First-Order Methods for Solving Linear Programming

TL;DR

This work analyzes GPU-accelerated first-order methods for large-scale linear programming, focusing on the relationships among HPR-LP, cuPDLPx, and EPR-LP. It proves that the base algorithm of cuPDLPx is a special case of HPR-LP, and that under strict complementarity the PR updates become affine on active faces, making HPR-LP and EPR-LP equivalent after active-set identification. The authors establish affine-face equivalence results and provide extensive GPU benchmarks showing that HPR-LP delivers the best overall performance among current GPU solvers. These findings position the HPR framework as a robust baseline for developing faster GPU-accelerated LP solvers and offer directions for extending HPR to broader convex problems and learning-based parameter strategies.

Abstract

This paper aims to understand the relationships among recently developed GPU-accelerated first-order methods (FOMs) for linear programming (LP), with particular emphasis on HPR-LP -- a Halpern Peaceman--Rachford (HPR) method for LP. Our findings can be summarized as follows: (i) the base algorithm of cuPDLPx, a recently released GPU solver, is a special case of the base algorithm of HPR-LP, thereby showing that cuPDLPx is another concrete implementation instance of HPR-LP; (ii) once the active sets have been identified, HPR-LP and EPR-LP -- an ergodic PR method for LP -- become equivalent under the same initialization; and (iii) extensive numerical experiments on benchmark datasets demonstrate that HPR-LP achieves the best overall performance among current GPU-accelerated LP solvers. These findings provide a strong motivation for using the HPR method as a baseline to further develop GPU-accelerated LP solvers and beyond.

Paper Structure

This paper contains 21 sections, 6 theorems, 39 equations, 4 figures, 6 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. HPR-LP: An HPR method for solving LP
  4. Base algorithm
  5. Algorithmic enhancements
  6. Restart strategy
  7. Update rules for Sigma
  8. Relationships among GPU-accelerated LP solvers
  9. The relationship between cuPDLPx and HPR-LP
  10. The relationship between HPR-LP and EPR-LP
  11. Numerical experiments
  12. Experimental setup
  13. Computing environment
  14. Datasets
  15. Shifted geometric mean
  16. ...and 6 more sections

Key Result

Proposition 2.3

Suppose that Assumption ass:CQ holds. Then the sequence $\{{\bar{w}}^k\} = \{({\bar{y}}^k, {\bar{z}}^k, {\bar{x}}^k)\}$ generated by the HPR method with semi-proximal terms in Algorithm alg:sp-HPR converges to a point $w^{*} = (y^*, z^*, x^*)$, where $(y^*, z^*)$ solves the dual problem model:dualLP

Figures (4)

  • Figure 1: Relationships among GPU-accelerated FOMs for solving LP.
  • Figure 1: Performance of HPR methods with different enhancements.
  • Figure 2: Performance of HPR methods under different restart strategies.
  • Figure 3: Performance of HPR methods under different penalty parameter strategies.

Theorems & Definitions (17)

  • Remark 2.1
  • Proposition 2.3: Corollary 3.5 in sun2025accelerating
  • Theorem 2.4: Proposition 2.9 and Theorem 3.7 in sun2025accelerating
  • Remark 2.5
  • Remark 2.6
  • Proposition 3.1
  • Proof 1
  • Remark 3.2
  • Remark 3.3
  • Proposition 3.4
  • ...and 7 more