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A quadratically convergent Newton method for computing the nearest correlation matrix. (English) Zbl 1120.65049
Given a symmetric matrix $G$, one has to find a correlation matrix $X$ (i.e., a positive semidefinite and symmetric matrix with diagonal elements equal to 1) which is closest to $G$ in Frobenius norm. The dual of this constrained optimization problem is an unconstrained problem with an objective function that is not continuously differentiable. Hence a direct approach would result in at most linear convergence. However, its solution is obtained by solving a system of nonlinear equations involving the Frobenius norm of a projection onto the cone of positive semidefinite symmetric matrices. Since this projection is strongly semismooth, the authors succeed in designing a (nonsmooth) Newton method which they prove to be quadratically convergent which is confirmed by numerical experiments. Extensions are discussed which include minimizing a weighted Frobenius norm, imposing a lower bound on $X$, and allowing $X$ to be nonsymmetric.

##### MSC:
 65F20 Overdetermined systems, pseudoinverses (numerical linear algebra) 65C60 Computational problems in statistics 90C25 Convex programming 62H20 Statistical measures of associations 65K05 Mathematical programming (numerical methods)
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