QPPAL: A Two-phase Proximal Augmented Lagrangian Method for High-dimensional Convex Quadratic Programming Problems

Author:

Liang Ling1,Li Xudong2,Sun Defeng3,Toh Kim-Chuan4

Affiliation:

1. Department of Mathematics, National University of Singapore, Singapore

2. School of Data Science, Fudan University, Shanghai, China

3. Department of Applied Mathematics, the Hong Kong Polytechnic University, Hung Hom, Hong Kong

4. Department of Mathematics and Institute of Operations Research and Analytics, National University of Singapore, Singapore

Abstract

In this article, we aim to solve high-dimensional convex quadratic programming (QP) problems with a large number of quadratic terms, linear equality, and inequality constraints. To solve the targeted QP problem to a desired accuracy efficiently, we consider the restricted-Wolfe dual problem and develop a two-phase Proximal Augmented Lagrangian 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 https://blog.nus.edu.sg/mattohkc/softwares/qppal/ .

Funder

National Key R&D Program of China

National Natural Science Foundation of China

Young Elite Scientists Sponsorship Program

NSFC/RGC Joint Research Scheme

Ministry of Education, Singapore

Publisher

Association for Computing Machinery (ACM)

Subject

Applied Mathematics,Software

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