HPR-QP: A GPU Solver for Convex Composite Quadratic Programming

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

Contact: {kaihuang.chen, guojun.zhang}@connect.polyu.hk, {yancheng.yuan, defeng.sun}@polyu.edu.hk, xyzhao@bjut.edu.cn

Find the open-source solver at https://github.com/PolyU-IOR/HPR-QP

Overview

HPR-QP is a Julia implementation of a dual Halpern Peaceman–Rachford (HPR) method for solving large-scale convex composite quadratic programming (CCQP) problems on the GPU. It efficiently handles problems of the form:

\[ \begin{array}{ll} \underset{x \in \mathbb{R}^n}{ min} \quad & \tfrac{1}{2}\langle x,Qx \rangle + \langle c, x \rangle + \phi(x) \\ \mathrm{s.t.} \quad & l \leq A x \leq u, \end{array} \]

\(Q\) is a positive semidefinite self-adjoint linear operator;
The matrix form of \(Q\) may be unavailable in large-scale instances (e.g., QAP relaxations, LASSO);
\(\phi\) is a proper closed convex function.

Numerical Results

HPR-QP is implemented in Julia and leverages CUDA for GPU acceleration
PDQP (GPU, downloaded in April 2025)
SCS (GPU, version 2.1.0) is written in C/C++ with a Julia interface. GPU acceleration is enabled via its indirect solver, which performs all matrix operations on the GPU
CuClarabel (GPU, version 0.10.0)
Gurobi (CPU, version 12.0.2, academic license) is executed on CPU using the barrier method
All benchmarks were conducted on a SuperServer SYS-420GP-TNR with an NVIDIA A100-SXM4-80GB GPU, Intel Xeon Platinum 8338C CPU @ 2.60 GHz, and 256 GB RAM

Maros–Mészáros Data Set (137 Instances; Tolerances \(10^{-6}\), \(10^{-8}\))

Solver	SGM10 \((10^{-6})\)	Solved \((10^{-6})\)	SGM10 \((10^{-8})\)	Solved \((10^{-8})\)
HPR-QP	10.5	129	12.6	128
PDQP	33.1	125	42.5	124
SCS	126.0	103	165.0	93
CuClarabel	3.7	130	7.8	124
Gurobi	0.4	137	1.2	135

QAP Relaxations (36 Instances; Tolerances \(10^{-6}\), \(10^{-8}\))

Solver	SGM10 \((10^{-6})\)	Solved \((10^{-6})\)	SGM10 \((10^{-8})\)	Solved \((10^{-8})\)
HPR-QP	1.8	36	4.7	36
PDQP	124.1	23	149.4	23
SCS	11.3	36	86.0	36
CuClarabel	13.6	33	114.9	22
Gurobi	24.8	36	26.8	36

LASSO Problems (11 Instances; Tolerance \(10^{-8}\))

Abbreviations: T = time-limit, F = failure (e.g., unbounded or infeasible)
Instance	HPR-QP	PDQP	SCS	CuClarabel	Gurobi
abalone7	10.5	372.5	T	24.4	127.3
bodyfat7	1.2	33.3	T	2.2	30.8
E2006.test	0.2	1.3	T	15.4	9.0
E2006.train	0.7	1.9	F	116.0	277.8
housing7	22.6	123.3	T	5.7	125.9
log1p.E2006.test	7.0	1416.9	T	196.0	137.0
log1p.E2006.train	17.3	2983.2	T	361.0	878.8
mpg7	0.6	18.1	2000.0	0.3	1.2
pyrim5	49.1	410.6	T	3.5	35.9
space_ga9	0.6	62.7	1210.0	6.7	38.1
triazines4	401.3	3533.3	T	26.0	843.1
SGM10 (Time)	13.2	161.8	3091.0	26.1	91.2

Citation

        @article{chen2025hpr,

        title={HPR-QP: A dual Halpern Peaceman-Rachford method for solving large-scale convex composite quadratic
        programming},

        author={Chen, Kaihuang and Sun, Defeng and Yuan, Yancheng and Zhang, Guojun and Zhao, Xinyuan},

        journal={arXiv preprint arXiv:2507.02470},

        year={2025}

        }