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An interior-point method for approximate positive semidefinite completions

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Charles R. Johnson, Brenda Kroschel, Henry Wolkowicz

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
{\em unspecified} (free), while $H_{ij}$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.
University of Waterloo, CORR Report 95-11.

Contact:
hwolkowi@orion.math.uwaterloo.ca