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 rank 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 t...

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Zhaosong Lu, Yong Zhang
• Keywords: CORR 2010, Education, General Rank Minimization, Penalty Decomposition Methods, Rank Minimization Problems