Recent Developments in HPR Methods for Large-Scale Linear Programming

Abstract

We present recent developments in Halpern Peaceman-Rachford (HPR) methods for solving large-scale linear programming problems. We first introduce the semi-proximal HPR framework and summarize its convergence guarantees, together with its connections to Halpern PDHG and ergodic Peaceman-Rachford iterations. To improve practical performance, we develop adaptive restart and penalty parameter update strategies. We then describe several computational enhancements, including GPU-based preprocessing, diagonal scaling, reusable execution plans for sparse matrix-vector multiplication, and sieving techniques. We further discuss multi-GPU parallelization and lossless model storage and loading for handling large-scale instances. These developments form the foundation of HPR-LP, a GPU-accelerated solver for large-scale linear programming.