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.
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Communicated by X.-Q. Yang.
D. Li’s research is supported by the major project of the Ministry of Education of China Grant 309023 and the NSF of China Grant 10771057.
H. Qi’s research was partially supported by EPSRC Grant EP/D502535/1.
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Li, Q., Li, D. & Qi, H. Newton’s Method for Computing the Nearest Correlation Matrix with a Simple Upper Bound. J Optim Theory Appl 147, 546–568 (2010). https://doi.org/10.1007/s10957-010-9738-6
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DOI: https://doi.org/10.1007/s10957-010-9738-6