In quantitative finance, one often needs to find a nearest correlation matrix to a given symmetric matrix, measured by the componentwise weighted Frobenius norm, with a prescribed rank and bound constraints on its correlations. This is in general a non convex and difficult problem due to the combined nature of the rank and bound constraints. To deal with this difficulty, we first consider a penalized version of this problem and then apply the essential ideas of the majorization method to the penalized problem by solving iteratively a sequence of least squares correlation matrix problems without the rank constraint. The latter problems can be solved by a recently developed quadratically convergent smoothing Newton-BiCGStab method. Interestingly, in the case of equal weights, our approach can be regarded as adding a weighted trace term iteratively to the objective function with the weighted positive semidefinite matrix to be of low rank except possibly for the first iteration starting with the zero matrix. Numerical examples demonstrate that our approach is very efficient for obtaining a nearest correlation matrix with both rank and bound constraints. [This is a joint work with Yan Gao]
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