HPR-QP: A dual Halpern Peaceman-Rachford method for solving large-scale convex composite quadratic programming

TL;DR

HPR-QP is a novel dual method for large-scale convex quadratic programming that uses restricted Wolfe duality, symmetric Gauss-Seidel updates, and adaptive strategies to achieve faster convergence and better scalability, especially on GPU.

Contribution

It introduces HPR-QP, a dual approach leveraging restricted Wolfe duality and symmetric Gauss-Seidel techniques, with adaptive strategies for improved performance over existing solvers.

Findings

Outperforms state-of-the-art solvers in speed.

Demonstrates high scalability on GPU.

Achieves faster convergence with adaptive updates.

Abstract

In this paper, we introduce HPR-QP, a dual Halpern Peaceman-Rachford (HPR) method designed for solving large-scale convex composite quadratic programming. One distinctive feature of HPR-QP is that, instead of working with the primal formulations, it builds on the novel restricted Wolfe dual introduced in recent years. It also leverages the symmetric Gauss-Seidel technique to simplify subproblem updates without introducing auxiliary slack variables that typically lead to slow convergence. By restricting updates to the range space of the Hessian of the quadratic objective function, HPR-QP employs proximal operators of smaller spectral norms to speed up the convergence. Shadow sequences are elaborately constructed to deal with the range space constraints. Additionally, HPR-QP incorporates adaptive restart and penalty parameter update strategies, derived from the HPR method's $O(1/k)$ convergence in terms of the Karush-Kuhn-Tucker residual, to further enhance its performance and robustness. Extensive numerical experiments on benchmark data sets using a GPU demonstrate that our Julia implementation of HPR-QP significantly outperforms state-of-the-art solvers in both speed and scalability.