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Newton’s Method for Computing the Nearest Correlation Matrix with a Simple Upper Bound

Author

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  • Qingna Li

    (Hunan University)

  • Donghui Li

    (South China Normal University)

  • Houduo Qi

    (The University of Southampton)

Abstract

The standard nearest correlation matrix can be efficiently computed by exploiting a recent development of Newton’s method (Qi and Sun in SIAM J. Matrix Anal. Appl. 28:360–385, 2006). Two key mathematical properties, that ensure the efficiency of the method, are the strong semismoothness of the projection operator onto the positive semidefinite cone and constraint nondegeneracy at every feasible point. In the case where a simple upper bound is enforced in the nearest correlation matrix in order to improve its condition number, it is shown, among other things, that constraint nondegeneracy does not always hold, meaning Newton’s method may lose its quadratic convergence. Despite this, the numerical results show that Newton’s method is still extremely efficient even for large scale problems. Through regularization, the developed method is applied to semidefinite programming problems with simple bounds.

Suggested Citation

  • Qingna Li & Donghui Li & Houduo Qi, 2010. "Newton’s Method for Computing the Nearest Correlation Matrix with a Simple Upper Bound," Journal of Optimization Theory and Applications, Springer, vol. 147(3), pages 546-568, December.
    • Handle: RePEc:spr:joptap:v:147:y:2010:i:3:d:10.1007_s10957-010-9738-6
    DOI: 10.1007/s10957-010-9738-6
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    References listed on IDEAS

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    1. Defeng Sun & Jie Sun, 2002. "Semismooth Matrix-Valued Functions," Mathematics of Operations Research, INFORMS, vol. 27(1), pages 150-169, February.
    2. 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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    Cited by:

    1. Kawee Numpacharoen & Amporn Atsawarungruangkit, 2012. "Generating Correlation Matrices Based on the Boundaries of Their Coefficients," PLOS ONE, Public Library of Science, vol. 7(11), pages 1-7, November.

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