An Interior Point Method for Approximate Positive Semidefinite Completions

Given a nonnegative, symmetric matrix of weights, H , we study the problem of finding an Hermitian, positive semidefinite matrix which is closest to a given Hermitian matrix, A, with respect to the weighting H. This extends the notion of exact matrix completion problems in that, Hij = 0 corresponds to the element Aij being unspecified (free), while Hij large in absolute value corresponds to the element Aij being approximately specified (fixed). We present optimality conditions, duality theory, and two primal-dual interior-point algorithms. Because of sparsity considerations, the dual-step-first algorithm is more efficient for a large number of free elements, while the primal-step-first algorithm is more efficient for a large number of fixed elements. Included are numerical tests that illustrate the efficiency and robustness of the algorithms