AI 中文总结
本文提出GPU加速的一阶求解器PDHCG-CQP，其在严格互补性下具备局部线性收敛性，可高效扩展至8个GPU，在多类大规模规划基准上展现出先进鲁棒性。
AI 中文摘要
本文提出了PDHCG-CQP，一款面向大规模锥凸二次规划的GPU加速一阶求解器，其支持仿射约束以及非负锥、二阶锥、旋转二阶锥、指数锥和三维幂锥的笛卡尔积。该求解器的核心是重启平均原始对偶混合梯度（PDHG）方法，其原始更新通过带投影梯度迭代求解锥二次邻近子问题的方式不精确计算。我们在平滑原始对偶间隙的一致局部二次增长条件下，证明了该重启平均方案在原始邻近评估为精确或不精确时均具备局部线性收敛性；进一步通过结合旋转二阶锥提升以及局部原始和对偶正则性条件，证明了该条件在严格互补性下成立。本文的C/CUDA实现结合了无矩阵线性代数、批处理锥投影、自适应内部求解、反射哈尔彭加速以及完全驻留在设备上的KKT残差计算，还可通过对问题数据的二维划分支持多GPU执行。在标准及大规模二次规划（QP）、凸二次约束二次规划（QCQP）、二阶锥规划（SOCP）以及拟线性费舍尔均衡基准上开展的大量实验表明，PDHCG-CQP在一阶求解器中达到了先进的鲁棒性，同时可高效扩展至8个GPU以及存储了多达4.4×10⁸个原始坐标的实例。PDHCG-CQP为开源软件，可在该httpsURL获取。
英文摘要
We present PDHCG-CQP, a GPU-accelerated first-order solver for large-scale conic convex quadratic programming. PDHCG-CQP supports affine constraints and Cartesian products of nonnegative, second-order, rotated second-order, exponential, and three-dimensional power cones. At its core is a restarted averaged primal-dual hybrid gradient (PDHG) method, whose primal update is computed inexactly by solving a conic quadratic proximal subproblem with projected gradient iterations. We establish local linear convergence of the restarted averaged scheme with both exact and inexact primal proximal evaluations under a uniform local quadratic-growth condition on the smoothed primal-dual gap. We further show that this condition holds under strict complementarity by exploiting a rotated second-order-cone lifting together with local primal and dual regularity conditions. Our C/CUDA implementation combines matrix-free linear algebra, batched cone projections, adaptive inner solves, reflected-Halpern acceleration, and fully device-resident KKT residual computations. It also supports multi-GPU execution through a two-dimensional partitioning of the problem data. Extensive experiments on standard and large-scale quadratic programming (QP), convex quadratically constrained quadratic programming (QCQP), second-order cone programming (SOCP), and quasilinear Fisher equilibrium benchmarks demonstrate that PDHCG-CQP achieves state-of-the-art robustness among first-order solvers while scaling efficiently to 8 GPUs and instances with up to $4.4\times10^8$ stored primal coordinates. PDHCG-CQP is open source and available at https://github.com/Lhongpei/PDHCG.
CommentsSource code available at https://github.com/Lhongpei/PDHCG
