PDHCG¶

PDHCG is a high-performance, GPU-accelerated implementation of the Primal-Dual Hybrid Gradient (PDHG) algorithm for large-scale convex quadratic and quadratic conic programming.

Problem Formulation¶

PDHCG solves quadratic conic programs in the following form:

\[ \begin{aligned} \min_{x} \quad & \frac{1}{2}x^\top (Q + R^\top D R) x + c^\top x \\ \text{s.t.} \quad & \ell_c \le Ax \le u_c, \\ & Fx + g \in \mathcal{K}_a, \\ & \ell_v \le x \le u_v, \\ & x_J \in \mathcal{K}_v \quad \text{for variable-cone blocks } J. \end{aligned} \]

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
\(F\) and \(g\) define the native affine-cone map
\(c\) is the linear objective vector
\(\ell_c, u_c\) are constraint bounds
\(\ell_v, u_v\) are variable bounds
\(\mathcal{K}_a\) and \(\mathcal{K}_v\) are products of Standard SOC, Rotated SOC, Exponential, or Power cones

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
Conic constraints: fully GPU-accelerated SOC, Rotated SOC, Exponential, and Power cone projection on variable blocks or native affine maps \(Fx + g\)
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{li2026gpuacceleratedconicquadraticprogramming,
      title={GPU-Accelerated Conic Quadratic Programming with Local Linear Convergence under Strict Complementarity},
      author={Hongpei Li and Yicheng Huang and Huikang Liu and Dongdong Ge and Yinyu Ye},
      year={2026},
      eprint={2608.09159},
      archivePrefix={arXiv},
      primaryClass={math.OC},
      url={https://arxiv.org/abs/2608.09159},
}

License¶

Licensed under the Apache License, Version 2.0.