cuPDQPx: A GPU-Accelerated Restarted Halpern PDHG for Large-Scale Convex Quadratic Programming

Abstract

We present cuPDQPx, a GPU-accelerated, factorization-free first-order solver for large-scale convex quadratic programming (QP). Building on the restarted reflected Halpern primal-dual framework of cuPDLPx, we introduce an auxiliary-variable splitting that yields explicit updates without linearizing the quadratic objective or solving inner quadratic subproblems. A preconditioned proximal point interpretation yields both sublinear and linear fixed-point residual convergence guarantees. For practical efficiency, cuPDQPx integrates diagonal preconditioning, step-size selection, adaptive restart, and adaptive primal and quadratic weight updates. cuPDQPx is implemented in C and CUDA with optimized sparse matrix-vector products, kernel fusion, and CUDA Graphs. Numerical experiments on the Maros-Mészáros collection, a convex subset of QPLIB, and synthetic large-scale QPs demonstrate that cuPDQPx has strong numerical performance at both $10^{-4}$ and $10^{-8}$ tolerances.