发表机构
Peking University; MIT, Sloan School of Management（北京大学; 麻省理工学院斯隆管理学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出无分解一阶求解器PDNQP，结合近端增广拉格朗日与重启加速原始-对偶混合梯度，在GPU上高效求解大规模非凸二次规划，在CUTEst和百万变量合成问题上表现优越。
AI 中文摘要
大规模非凸二次规划仍然具有挑战性，因为许多现有求解器使用的稀疏矩阵分解会带来大量的计算和内存成本，并且难以在GPU上高效并行化。我们提出了PDNQP，一种无分解的一阶求解器，用于寻找大规模非凸二次规划的稳定点。该方法将近端增广拉格朗日框架与重启的加速原始-对偶混合梯度相结合，以求解由此产生的强凸QP子问题。其关键要素包括：一种残差形式的重新表述，避免了依赖于惩罚参数的正常矩阵，并在外部迭代中保持固定的稀疏约束算子；以及一种自适应策略，协调内部求解精度与外部进展。由此产生的计算依赖于稀疏矩阵-向量乘积、投影和适用于GPU执行的向量运算。在107个非凸CUTEst实例上，PDNQP在两种精度水平下均取得了测试求解器中最高的成功率。当绝对和相对容差设置为$10^{-4}$时，其在移位几何平均运行时间上实现了$2.6\times$--$6.0\times$的加速；在$10^{-6}$时，其总运行时间与测试的最快求解器相当。在30个每个约有一百万个变量的大规模合成非凸QP问题上，PDNQP在27个实例上返回满足常见外部终止准则的点，而其他测试求解器在相同实验协议下均未成功。
英文摘要
Large-scale nonconvex quadratic programming remains challenging because the sparse matrix factorizations used by many existing solvers can incur substantial computational and memory costs and are difficult to parallelize efficiently on GPUs. We present PDNQP, a factorization-free first-order solver for finding stationary points of large-scale nonconvex quadratic programs. The method combines a proximal augmented-Lagrangian framework with restarted accelerated primal--dual hybrid gradient to solve the resulting strongly convex QP subproblems. Its key ingredients are a residual-form reformulation that avoids penalty-dependent normal matrices and preserves a fixed sparse constraint operator across outer iterations, and an adaptive strategy that coordinates inner-solve accuracy with outer progress. The resulting computations rely on sparse matrix--vector products, projections, and vector operations suited to GPU execution. On $107$ nonconvex CUTEst instances, PDNQP achieves the highest success rate among the tested solvers at both accuracy levels. With absolute and relative tolerances set to $10^{-4}$, it achieves speedups of $2.6\times$--$6.0\times$ in shifted geometric mean runtime; at $10^{-6}$, its aggregate runtime remains comparable to that of the fastest tested solvers. On $30$ large-scale synthetic nonconvex QPs with approximately one million variables each, PDNQP returns points satisfying the common external termination criteria on $27$ instances, while none of the other tested solvers succeeds under the same experimental protocol.
