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

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 lowrank correlation matrix. Numerical comparisons with the convex relaxation method in [17] indicate that our method tends to yield a better local optimal solution.