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In this paper, we present a majorized semismooth Newton-CG augmentedLagrangian method, called SDPNAL$+$, for semidefinite programming (SDP) withpartial or full nonnegative constraints on the matrix variable. SDPNAL$+$ is amuch enhanced version of SDPNAL introduced by Zhao, Sun and Toh [SIAM Journalon Optimization, 20 (2010), pp.~1737--1765] for solving generic SDPs. SDPNALworks very efficiently for nondegenerate SDPs but may encounter numericaldifficulty for degenerate ones. Here we tackle this numerical difficulty byemploying a majorized semismooth Newton-CG augmented Lagrangian method coupledwith a convergent 3-block alternating direction method of multipliersintroduced recently by Sun, Toh and Yang [arXiv preprint arXiv:1404.5378,(2014)]. Numerical results for various large scale SDPs with or withoutnonnegative constraints show that the proposed method is not only fast but alsorobust in obtaining accurate solutions. It outperforms, by a significantmargin, two other competitive publicly available first order methods basedcodes: (1) an alternating direction method of multipliers based solver calledSDPAD by Wen, Goldfarb and Yin [Mathematical Programming Computation, 2 (2010),pp.~203--230] and (2) a two-easy-block-decomposition hybrid proximalextragradient method called 2EBD-HPE by Monteiro, Ortiz and Svaiter[Mathematical Programming Computation, (2013), pp.~1--48]. In contrast to thesetwo codes, we are able to solve all the 95 difficult SDP problems arising fromthe relaxations of quadratic assignment problems tested in SDPNAL to anaccuracy of $10^{-6}$ efficiently, while SDPAD and 2EBD-HPE successfully solve30 and 16 problems, respectively.

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