A partial parallel splitting augmented Lagrangian method for solving constrained matrix optimization problems

Alternating directions methods (ADMs) are very effective for solving convex optimization problems with separable structure. However, when these methods are applied to solve convex optimization problems with three separable operators, their convergence results have not been established as yet. In this paper, we consider a class of constrained matrix optimization problems. The problem is first reformulated into a convex optimization problem with three separable operators, then it is solved by a proposed partial parallel splitting method. The proposed method combines the parallel splitting (augmented Lagrangian) method (PSALM) and the alternating directions method (ADM), and it is referred to as PADALM in short. The main difference between PADALM and PSALM is that in PADALM, two operators are handled first by a parallel method, then the third operator and the former two are dealt with by an alternating method. Finally, the convergence result for PADALM is established and numerical results are provided to show the efficacy of PADALM and its superiority over PSALM.