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arxiv: 2601.07628 · v3 · submitted 2026-01-12 · 🧮 math.OC · cs.DC

D-PDLP: Scaling PDLP to Distributed Multi-GPU Systems

Hongpei Li , Yicheng Huang , Huikang Liu , Dongdong Ge , Yinyu Ye This is my paper

Pith reviewed 2026-05-16 15:13 UTC · model grok-4.3

classification 🧮 math.OC cs.DC
keywords distributed PDHGmulti-GPU linear programmingtwo-dimensional matrix partitioningPrimal-Dual Hybrid Gradientscalabilitylarge-scale LPcuPDLPx
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The pith

D-PDLP extends the Primal-Dual Hybrid Gradient method to multi-GPU systems through two-dimensional grid partitioning of the constraint matrix to solve large linear programs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces D-PDLP, a distributed framework that scales the PDHG algorithm for linear programming across multiple GPUs. It partitions the constraint matrix in two dimensions and applies block-wise random permutation together with nonzero-aware strategies to distribute the work. The goal is to overcome single-GPU memory and speed limits while keeping the same double-precision accuracy as the original method. A reader would care because many industrial linear programs are too large for one GPU, and this approach aims to make them solvable on ordinary clusters.

Core claim

D-PDLP extends the Primal-Dual Hybrid Gradient method to distributed multi-GPU systems via a practical two-dimensional grid partitioning of the constraint matrix, combined with block-wise random permutation and nonzero-aware matrix partitioning. This distributes the intensive computations of each PDHG iteration across GPUs with low communication overhead, yielding strong scalability and high performance while preserving full FP64 numerical accuracy on MIPLIB, Mittelmann, and real-world instances.

What carries the argument

Two-dimensional grid partitioning of the constraint matrix together with block-wise random permutation and nonzero-aware strategies, which spreads the matrix-vector products and updates required by PDHG iterations across GPUs.

If this is right

  • The framework achieves substantial speedups on massive LP instances by harnessing multiple GPUs.
  • Strong scalability occurs with relatively low communication overhead.
  • Full FP64 accuracy is retained, matching the numerical behavior of the single-node cuPDLPx solver.
  • Effective performance is observed on both standard benchmarks and huge-scale real-world datasets.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same partitioning ideas might extend to other first-order methods that rely on repeated matrix-vector multiplications.
  • Users could solve previously intractable industrial LPs on commodity multi-GPU servers in practical time.
  • Varying the grid dimensions could be tested to reach even larger problem sizes on bigger clusters.
  • The load-balancing approach might apply to heterogeneous GPU setups without major redesign.

Load-bearing premise

The two-dimensional grid partitioning with block-wise random permutation and nonzero-aware strategies will preserve the convergence behavior and numerical stability of the original PDHG method for arbitrary large-scale LP instances without introducing unacceptable load imbalance or communication bottlenecks.

What would settle it

A large LP instance where the multi-GPU version either diverges, requires many more iterations than the single-GPU baseline, or produces a solution whose objective or feasibility error exceeds machine epsilon would show the claim does not hold.

Figures

Figures reproduced from arXiv: 2601.07628 by Dongdong Ge, Hongpei Li, Huikang Liu, Yicheng Huang, Yinyu Ye.

Figure 1
Figure 1. Figure 1: Distributed memory layout under 2D grid partitioning. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of matrix permutation strategies for distributed sparse optimization. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
read the original abstract

We present a distributed framework of the Primal-Dual Hybrid Gradient (PDHG) algorithm for solving massive-scale linear programming (LP) problems. Although PDHG-based solvers demonstrate strong performance on single-node GPU architectures, their applicability to industrial-scale instances is often limited by single-GPU computational throughput. To overcome these challenges, we propose D-PDLP, the first Distributed PDLP framework, which extends PDHG to a multi-GPU setting via a practical two-dimensional grid partitioning of the constraint matrix. To improve load balance and computational efficiency, we introduce a block-wise random permutation strategy combined with nonzero-aware matrix partitioning. By distributing the intensive computation required in PDHG iterations, the proposed framework harnesses multi-GPU parallelism to achieve substantial speedups with relatively low communication overhead. Extensive experiments on standard LP benchmarks (including MIPLIB and Mittelmann instances) as well as huge-scale real-world datasets show that our distributed implementation, built upon cuPDLPx, achieves strong scalability and high performance while preserving full FP64 numerical accuracy.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper presents D-PDLP, a distributed multi-GPU extension of the Primal-Dual Hybrid Gradient (PDHG) algorithm for large-scale linear programming. It employs a two-dimensional grid partitioning of the constraint matrix together with block-wise random permutation and nonzero-aware strategies to improve load balance, built on top of cuPDLPx. The central claim is that this framework achieves strong scalability and high performance on MIPLIB, Mittelmann, and real-world instances while preserving full FP64 numerical accuracy and incurring only low communication overhead.

Significance. If the empirical claims hold, the work would be significant for extending single-GPU PDHG solvers to distributed settings, enabling solution of industrial-scale LPs that exceed single-device memory limits. The practical heuristics for partitioning and the reported preservation of FP64 accuracy are strengths, but the absence of detailed quantitative speedup numbers, iteration counts, or convergence diagnostics in the abstract weakens the assessed impact.

major comments (2)
  1. [Abstract] Abstract: the claim of achieving 'strong scalability and high performance' rests on experiments whose quantitative results (speedup factors, iteration counts, wall-clock times, or error bars) are not supplied in the text, leaving the central performance assertion only partially supported.
  2. [Distributed framework description] The two-dimensional grid partitioning with block-wise random permutation and nonzero-aware splits: no analysis or proof is provided that the distributed matvec updates remain mathematically equivalent to the serial PDHG trajectory, so it is unclear whether the method preserves the original convergence rate or step-size selection for matrices with irregular sparsity; this directly affects the claim that numerical accuracy and iteration behavior are unchanged.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address each major comment below and outline targeted revisions to strengthen the manuscript's claims and clarity.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim of achieving 'strong scalability and high performance' rests on experiments whose quantitative results (speedup factors, iteration counts, wall-clock times, or error bars) are not supplied in the text, leaving the central performance assertion only partially supported.

    Authors: We agree that the abstract would benefit from explicit quantitative support. In the revised version we will incorporate representative metrics, including observed speedup factors (near-linear scaling to 8-16 GPUs), average iteration counts, wall-clock times, and a note on variance across MIPLIB, Mittelmann, and real-world instances. revision: yes

  2. Referee: [Distributed framework description] The two-dimensional grid partitioning with block-wise random permutation and nonzero-aware splits: no analysis or proof is provided that the distributed matvec updates remain mathematically equivalent to the serial PDHG trajectory, so it is unclear whether the method preserves the original convergence rate or step-size selection for matrices with irregular sparsity; this directly affects the claim that numerical accuracy and iteration behavior are unchanged.

    Authors: The 2D grid partitioning together with the communication pattern computes exactly the same floating-point matrix-vector products (A x and A^T y) as the serial implementation; results are assembled via all-reduce collectives in FP64 with no approximation or reordering of arithmetic. Consequently the PDHG iterates, step-size selection, and convergence behavior are identical regardless of sparsity pattern. We will add a dedicated subsection with pseudocode and a short equivalence argument to make this explicit. revision: yes

Circularity Check

0 steps flagged

No circularity; claims rest on empirical validation of distributed PDHG implementation

full rationale

The paper describes an engineering extension of the PDHG algorithm to multi-GPU via 2D grid partitioning, block-wise random permutation, and nonzero-aware splits. Central claims of strong scalability, low communication overhead, and preserved FP64 accuracy are supported by experiments on MIPLIB, Mittelmann, and real-world instances rather than any derivation. No equation, theorem, or prediction reduces to a fitted parameter, self-definition, or self-citation chain. Convergence preservation is asserted as a consequence of the partitioning design and verified empirically; it does not equate to its inputs by construction. The work is self-contained as a practical implementation contribution.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework inherits standard convergence assumptions of the PDHG algorithm for linear programs and standard models of GPU-to-GPU communication; no new free parameters or invented entities are introduced in the abstract description.

axioms (1)
  • domain assumption PDHG iterations converge to the optimal solution for feasible linear programs under standard step-size rules
    Invoked when claiming that distribution preserves solution quality and accuracy.

pith-pipeline@v0.9.0 · 5492 in / 1279 out tokens · 37788 ms · 2026-05-16T15:13:49.186442+00:00 · methodology

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