Low rank methods for solving the nearest correlation matrix problem

The nearest correlation matrix problem is to find a positive semi-definite matrix with unit diagonal, that is, nearest in the Frobenius norm to a given symmetric matrix. Methods for solving this problem is considered. One approach is to formulate the problem as a quadratic convex programming problem in which the constraint is the intersection of two convex sets. This method is globally convergent but the rate of convergence is slow. Another method is an iterative method for finding solution of a fixed point problem which converge globally with the same rate of convergence. However, if the correct rank of the solution matrix is known, it is shown how to formulate the problem as a smooth nonlinear minimization problem, for which a rapid convergence can be obtained by SQP method. Another method presented is the quasi-Newton method. This paper studies various methods that attempt to combine the best features of all these method. Comparative numerical results are also reported.