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Newton’s method for computing the nearest correlation matrix with a simple upper bound

Newton’s method for computing the nearest correlation matrix with a simple upper bound
Newton’s method for computing the nearest correlation matrix with a simple upper bound
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.
semismooth newton method, constraint nondegeneracy, quadratic convergence, correlation matrix
0022-3239
546-568
Li, Qingna
a189d836-f8f0-407b-9983-0a73bf8a214a
Li, Donghui
0ff94eb7-4ae8-4b81-bae1-a6ee962cbdbf
Qi, Houduo
e9789eb9-c2bc-4b63-9acb-c7e753cc9a85
Li, Qingna
a189d836-f8f0-407b-9983-0a73bf8a214a
Li, Donghui
0ff94eb7-4ae8-4b81-bae1-a6ee962cbdbf
Qi, Houduo
e9789eb9-c2bc-4b63-9acb-c7e753cc9a85

Li, Qingna, Li, Donghui and Qi, Houduo (2010) Newton’s method for computing the nearest correlation matrix with a simple upper bound. Journal of Optimization Theory and Applications, 147 (3), 546-568. (doi:10.1007/s10957-010-9738-6).

Record type: Article

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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Published date: 2010
Keywords: semismooth newton method, constraint nondegeneracy, quadratic convergence, correlation matrix

Identifiers

Local EPrints ID: 181535
URI: https://eprints.soton.ac.uk/id/eprint/181535
ISSN: 0022-3239
PURE UUID: 570772b3-4493-4ac9-a86b-b3c32d183745
ORCID for Houduo Qi: ORCID iD orcid.org/0000-0003-3481-4814

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Date deposited: 19 Apr 2011 08:48
Last modified: 17 Sep 2019 00:51

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