QPPAL: A two-phase proximal augmented Lagrangian method for high dimensional convex quadratic programming problems

TL;DR

This paper introduces QPPAL, a two-phase proximal augmented Lagrangian method designed to efficiently solve high-dimensional convex quadratic programming problems with many quadratic terms and constraints, outperforming existing solvers.

Contribution

The paper develops a novel two-phase algorithm combining sGS-based semi-proximal and proximal augmented Lagrangian methods for large-scale convex QP problems, with demonstrated superior performance.

Findings

QPPAL outperforms Gurobi, OSQP, and QPALM in numerical tests.

The method is robust across various large-scale convex QP problems.

MATLAB implementation of QPPAL is publicly available.

Abstract

In this paper, we aim to solve high dimensional convex quadratic programming (QP) problems with a large number of quadratic terms, linear equality and inequality constraints. In order to solve the targeted {\bf QP} problems to a desired accuracy efficiently, we develop a two-phase {\bf P}roximal {\bf A}ugmented {\bf L}agrangian method {(QPPAL)}, with Phase I to generate a reasonably good initial point to warm start Phase II to obtain an accurate solution efficiently. More specifically, in Phase I, based on the recently developed symmetric Gauss-Seidel (sGS) decomposition technique, we design a novel sGS based semi-proximal augmented Lagrangian method for the purpose of finding a solution of low to medium accuracy. Then, in Phase II, a proximal augmented Lagrangian algorithm is proposed to obtain a more accurate solution efficiently. Extensive numerical results evaluating the performance of {QPPAL} against {existing state-of-the-art solvers Gurobi, OSQP and QPALM} are presented to demonstrate the high efficiency and robustness of our proposed algorithm for solving various classes of large-scale convex QP problems. {The MATLAB implementation of the software package QPPAL is available at: \url{https://blog.nus.edu.sg/mattohkc/softwares/qppal/}.