SDPNAL$+$: A Majorized Semismooth Newton-CG Augmented Lagrangian Method for Semidefinite Programming with Nonnegative Constraints

TL;DR

This paper introduces SDPNAL$+$, an advanced augmented Lagrangian method utilizing majorized semismooth Newton-CG techniques, significantly improving the efficiency and robustness of solving large-scale semidefinite programs with nonnegative constraints.

Contribution

The paper presents SDPNAL$+$, a novel algorithm that enhances previous methods by effectively handling degenerate SDPs and outperforming existing solvers in accuracy and speed.

Findings

Successfully solved all 95 challenging SDP problems with high accuracy.

Outperformed existing methods SDPAD and 2EBD-HPE in speed and robustness.

Demonstrated efficiency on large-scale SDPs from quadratic assignment relaxations.

Abstract

In this paper, we present a majorized semismooth Newton-CG augmented Lagrangian method, called SDPNAL$+$, for semidefinite programming (SDP) with partial or full nonnegative constraints on the matrix variable. SDPNAL$+$ is a much enhanced version of SDPNAL introduced by Zhao, Sun and Toh [SIAM Journal on Optimization, 20 (2010), pp.~1737--1765] for solving generic SDPs. SDPNAL works very efficiently for nondegenerate SDPs but may encounter numerical difficulty for degenerate ones. Here we tackle this numerical difficulty by employing a majorized semismooth Newton-CG augmented Lagrangian method coupled with a convergent 3-block alternating direction method of multipliers introduced recently by Sun, Toh and Yang [arXiv preprint arXiv:1404.5378, (2014)]. Numerical results for various large scale SDPs with or without nonnegative constraints show that the proposed method is not only fast but also robust in obtaining accurate solutions. It outperforms, by a significant margin, two other competitive publicly available first order methods based codes: (1) an alternating direction method of multipliers based solver called SDPAD by Wen, Goldfarb and Yin [Mathematical Programming Computation, 2 (2010), pp.~203--230] and (2) a two-easy-block-decomposition hybrid proximal extragradient method called 2EBD-HPE by Monteiro, Ortiz and Svaiter [Mathematical Programming Computation, (2013), pp.~1--48]. In contrast to these two codes, we are able to solve all the 95 difficult SDP problems arising from the relaxations of quadratic assignment problems tested in SDPNAL to an accuracy of $10^{-6}$ efficiently, while SDPAD and 2EBD-HPE successfully solve 30 and 16 problems, respectively.