  We develop a decomposition method based on the augmented Lagrangian framework
to solve a broad family of semidefinite programming problems, possibly with
nonlinear objective functions, nonsmooth regularization, and general linear
equality/inequality constraints. In particular, the positive semidefinite
variable along with a group of linear constraints can be transformed into a
variable on a smooth manifold via matrix factorization. The nonsmooth
regularization and other general linear constraints are handled by the
augmented Lagrangian method. Therefore, each subproblem can be solved by a
semismooth Newton method on a manifold. Theoretically, we show that the first
and second-order necessary optimality conditions for the factorized subproblem
are also sufficient for the original subproblem under certain conditions.
Convergence analysis is established for the Riemannian subproblem and the
augmented Lagrangian method. Extensive numerical experiments on large-scale
semidefinite programming problems such as max-cut, nearest correlation
estimation, clustering, and sparse principal component analysis demonstrate the
strength of our proposed method compared to other state-of-the-art methods.
