Positive definite completions and determinant maximization

AbstractA method is described for determining whether a positive definite completion of a given partial Hermitian matrix exists and, if so, for finding the determinant maximizing positive definite completion. No assumption is made about the arrangement of the specified entries. The method employs iterative application to a modified problem of an explicit formula for the maximum determinant in case there is only one symmetrically placed pair of unspecified entries. A robust and fast algorithm based upon this approach is shown to have global convergence to the necessarily unique solution