GPU-Accelerated Preprocessing for Large-Scale Linear Programming

Abstract

GPU-accelerated first-order methods are increasingly effective for solving large-scale linear programs (LPs), yet LP preprocessing remains predominantly CPU-based and can become costly. A key challenge is that dependencies among presolve reductions and frequent modifications to sparse LP data make efficient GPU parallelization difficult. To address this challenge, we develop GPU-Preprocessor, a GPU-native LP preprocessing framework that integrates folding and presolve. To efficiently implement folding on GPUs, we adapt a randomized partition refinement algorithm to LP folding, using sparse matrix-vector products and parallel sorting to construct stable row and column partitions. For presolve, we introduce a new piecewise-linear reformulation rule that exploits underlying convex piecewise-linear structure to obtain sparser LP representations. We also develop a framework that combines phase scheduling with a Scan-Plan-Apply design to coordinate presolve rules and reduce sparse-matrix reconstruction overhead through batched updates. Numerical experiments on three LP benchmark sets show that GPU-Preprocessor achieves greater average problem-size reductions than PSLP and Gurobi Presolve, with SGM10 preprocessing-time speedups of up to $43.9\times$ and $677.7\times$, respectively. Here, SGM10 denotes the shifted geometric mean with a 10-second shift. On the largest tested quadratic assignment problem LP relaxation, these speedups reach $126.3\times$ and $2{,}207.9\times$. When integrated with the GPU-accelerated LP solver HPR-LP, GPU-Preprocessor achieves SGM10 speedups of $2.5\times$ to $5.0\times$ in combined preprocessing and solving time over HPR-LP without preprocessing. The code is available at https://github.com/PolyU-IOR/GPU-Preprocessor.