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

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  • Shujun Bi
  • Le Han
  • Shaohua Pan

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. Copyright Springer Science+Business Media New York 2013

Suggested Citation

  • Shujun Bi & Le Han & Shaohua Pan, 2013. "Approximation of rank function and its application to the nearest low-rank correlation matrix," Journal of Global Optimization, Springer, vol. 57(4), pages 1113-1137, December.
    • Handle: RePEc:spr:jglopt:v:57:y:2013:i:4:p:1113-1137
    DOI: 10.1007/s10898-012-0007-0
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    References listed on IDEAS

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    1. Raoul Pietersz & Patrick Groenen, 2004. "Rank reduction of correlation matrices by majorization," Quantitative Finance, Taylor & Francis Journals, vol. 4(6), pages 649-662.
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    3. Igor Grubisic & Raoul Pietersz, 2005. "Efficient Rank Reduction of Correlation Matrices," Finance 0502007, University Library of Munich, Germany.
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    5. Defeng Sun, 2006. "The Strong Second-Order Sufficient Condition and Constraint Nondegeneracy in Nonlinear Semidefinite Programming and Their Implications," Mathematics of Operations Research, INFORMS, vol. 31(4), pages 761-776, November.
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