PDNQP: A GPU-based Factorization-free Method for Large-scale Nonconvex Quadratic Programming

Abstract

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.