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

A Decomposition Augmented Lagrangian Method for Low-rank Semidefinite Programming