An Interior-Point Method for Approximate Positive Semidefinite Completions*


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Abstract

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 Aijbeing unspecified (free)), while Hijlarge in absolute value corresponds to the element Aijbeing 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

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