Penalty Decomposition Methods for Rank Minimization

In this paper we consider general rank minimization problems with rank appearing in either objective function or constraint. We first show that a class of matrix optimization problems can be solved as lower dimensional vector optimization problems. As a consequence, we establish that a class of r… ank minimization problems have closed form solutions. Using this result, we then propose penalty decomposition methods for general rank minimization problems in which each subproblem is solved by a block coordinate descend method. Under some suitable assumptions, we show that any accumulation point of the sequence generated by our method when applied to the rank constrained minimization problem is a stationary point of a nonlinear reformulation of the problem. Finally, we test the performance of our methods by applying them to matrix completion and nearest low-rank correlation matrix problems. The computational results demonstrate that our methods generally outperform the existing methods in terms of solution quality and/or speed.