Abstract
In this paper we consider general rank minimization problems with rank appearing either in the objective function or as a constraint. We first establish that a class of special rank minimization problems has closed-form solutions. Using this result, we then propose penalty decomposition (PD) methods for general rank minimization problems in which each subproblem is solved by a block coordinate descent method. Under some suitable assumptions, we show that any accumulation point of the sequence generated by the PD methods satisfies the first-order optimality conditions of a nonlinear reformulation of the problems. Finally, we test the performance of our methods by applying them to the matrix completion and nearest low-rank correlation matrix problems. The computational results demonstrate that our methods are generally comparable or superior to the existing methods in terms of solution quality.
Acknowledgements
The authors would like to thank the two anonymous referees for their constructive comments which substantially improved the presentation of the paper.
Funding
This work was supported in part by NSERC Discovery Grant.
Notes
1. The code for Major used in our paper is the one modified by Defeng Sun, Department of Mathematics, National University of Singapore.