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, H ij =0 corresponds to the element A ij being unspecified (free), while H ij large in absolute value corresponds to the element A ij 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
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Johnson, C.R., Kroschel, B. & Wolkowicz, H. An Interior-Point Method for Approximate Positive Semidefinite Completions. Computational Optimization and Applications 9, 175–190 (1998). https://doi.org/10.1023/A:1018363021404
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DOI: https://doi.org/10.1023/A:1018363021404