Least-squares approximation of an improper by a proper correlation matrix using a semi-infinite convex program

Abstract

An algorithm is presented for the best least-squares fitting correlation matrix approximating a given missing value or improper correlation matrix. The proposed algorithm is based on a solution for C. I. Mosier's oblique Procrustes rotation problem offered by J. M. F. ten Berge and K. Nevels (1977). It is shown that the minimization problem belongs to a certain class of convex programs in optimization theory. A necessary and sufficient condition for a solution to yield the unique global minimum of the least-squares function is derived from a theorem by A. Shapiro (1985). A computer program was implemented to yield the solution of the minimization problem with the proposed algorithm. This empirical verification of the condition indicates that the occurrence of non-optimal solutions with the proposed algorithm is very unlikely