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

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 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 NewtonCG 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