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

shujun Bi(beamilan***at***163.com)
shaohua Pan(shhpan***at***scut.edu.cn)

Abstract: The rank function $\rank(\cdot)$ 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 $\rank(\cdot)$, 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 comparisons with the convex relaxation method in \cite{LQ09} indicate that our method tends to yield a better local optimal solution.

Keywords: rank optimization problem; approximation; convex relaxation; nearest low-rank correlation matrix; semismooth Newton method

Category 1: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Category 2: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Category 3: Applications -- Science and Engineering

Citation: Department of Mathematics, South China University of Technology, Guangzhou City, China, July 10, 2011

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Entry Submitted: 07/10/2011
Entry Accepted: 07/11/2011
Entry Last Modified: 07/10/2011

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