, Volume 57, Issue 4, pp 1113-1137
Date: 18 Nov 2012

Approximation of rank function and its application to the nearest low-rank correlation matrix

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Abstract

The rank function rank(.) is neither continuous nor convex which brings much difficulty to the solution of rank minimization problems. In this paper, we provide a unified framework to construct the approximation functions of rank(.), and study their favorable properties. Particularly, with two families of approximation functions, we propose a convex relaxation method for the rank minimization problems with positive semidefinite cone constraints, and illustrate its application by computing the nearest low-rank correlation matrix. Numerical results indicate that this convex relaxation method is comparable with the sequential semismooth Newton method (Li and Qi in SIAM J Optim 21:1641–1666, 2011) and the majorized penalty approach (Gao and Sun, 2010) in terms of the quality of solutions.

This work was supported by National Young Natural Science Foundation (No. 10901058) and the Fundamental Research Funds for the Central Universities, and Project of Liaoning Innovative Research Team in University (WT2010004).