SDPNAL+: a majorized semismooth Newton-CG augmented Lagrangian method for semidefinite programming with nonnegative constraints

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 et al. (SIAM J Optim 20:1...

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Veröffentlicht in:Mathematical programming computation Jg. 7; H. 3; S. 331 - 366
Hauptverfasser: Yang, Liuqin, Sun, Defeng, Toh, Kim-Chuan
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2015
Schlagworte:
Full Length Paper
Mathematics
Mathematics and Statistics
Mathematics of Computing
Operations Research/Decision Theory
Optimization
Theory of Computation
Semidefinite programming
Degeneracy
65F10
90C25
90C22
Augmented Lagrangian
90C06
Semismooth Newton-CG method
ISSN:1867-2949, 1867-2957
Online-Zugang:Volltext
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Zusammenfassung: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 et al. (SIAM J Optim 20:1737–1765, 2010 ) 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 et al. (SIAM J Optim, to appear). 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 et al. (Math Program Comput 2:203–230, 2010 ) and (2) a two-easy-block-decomposition hybrid proximal extragradient method called 2EBD-HPE by Monteiro et al. (Math Program Comput 1–48, 2014 ). 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. In addition, SDPNAL + appears to be the only viable method currently available to solve large scale SDPs arising from rank-1 tensor approximation problems constructed by Nie and Wang (SIAM J Matrix Anal Appl 35:1155–1179, 2014 ). The largest rank-1 tensor approximation problem we solved (in about 14.5 h) is nonsym(21,4), in which its resulting SDP problem has matrix dimension n = 9261 and the number of equality constraints m = 12 , 326 , 390 .
ISSN:1867-2949
1867-2957
DOI:10.1007/s12532-015-0082-6