PDHCG-II¶

PDHCG-II is a high-performance, GPU-accelerated implementation of the Primal-Dual Hybrid Gradient (PDHG) algorithm designed for solving large-scale Convex Quadratic Programming (QP) problems.

Problem Formulation¶

PDHCG solves quadratic programs in the following form:

\[ \begin{aligned} \min_{x} \quad & \frac{1}{2}x^\top (Q + R^\top D R) x + c^\top x \\ \text{s.t.} \quad & \ell_c \le Ax \le u_c, \\ & \ell_v \le x \le u_v. \end{aligned} \]

Where:

\(Q\) is a sparse symmetric matrix (optional)
\(R \in \mathbb{R}^{k\times n}\) is a low-rank factor of rank \(k\) (optional)
\(D \in \mathbb{R}^{k\times k}\) is an optional middle matrix; defaults to the identity, recovering the standard \(Q + R^\top R\) form. May be diagonal, sparse, dense, or indefinite — the backend auto-detects the cheapest representation
\(A\) is the constraint matrix
\(c\) is the linear objective vector
\(\ell_c, u_c\) are constraint bounds
\(\ell_v, u_v\) are variable bounds

Key Features¶

GPU Acceleration: Fully leverages NVIDIA CUDA for extreme-scale QP problems
Flexible Problem Structure: Supports sparse, low-rank, and middle-weighted low-rank (\(R^\top D R\)) quadratic terms — alone or combined
High Performance: Competitive with commercial solvers on large-scale problems
SpMVOp Auto-Detection: Automatically uses cuSPARSE SpMVOp on CUDA 13+ while falling back to standard SpMV on CUDA 12.x
Multi-GPU Distributed Solving: Supports parallel solving across multiple GPUs via MPI and NCCL (optional, enabled at compile time)

Quick Links¶

Citation¶

If you use this software in your research, please cite:

@misc{li2026pdhcgiienhancedversionpdhcg,
      title={PDHCG-II: An Enhanced Version of PDHCG for Large-Scale Convex QP},
      author={Hongpei Li and Yicheng Huang and Huikang Liu and Dongdong Ge and Yinyu Ye},
      year={2026},
      eprint={2602.23967},
      archivePrefix={arXiv},
      primaryClass={math.OC},
      url={https://arxiv.org/abs/2602.23967},
}

License¶

Licensed under the Apache License, Version 2.0.