Positive definite completions of partial Hermitian matrices

Abstract

The question of which partial Hermitian matrices (some entries specified, some free) may be completed to positive definite matrices is addressed. It is shown that if the diagonal entries are specified and principal minors, composed of specified entries, are positive, then, if the undirected graph of the specified entries is chordal, a positive definite completion necessarily exists. Furthermore, if this graph is not chordal, then examples exist without positive definite completions. In case a positive definite completion exists, there is a unique matrix, in the class of all positive definite completions, whose determinant is maximal, and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original

Sums of Squares, Moment Matrices and Optimization Over Polynomials

“…. , k. Using a result of Grone et al [44] (which claims that any partial positive semidefinite matrix whose specified entries form a chordal graph can be completed to a fully specified positive semidefinite matrix), we can complete y to a vectorỹ ∈ R N n 2 satisfying M 1 (ỹ) 0. ; that is, [152] verified that 0 = p sos 2 < p sos 2 = p min ∼ 0.84986. Waki et al [153] have implemented the above sparse SDP relaxations.…”

Fast and Accurate PSD Matrix Estimation by Row Reduction

“…PSD Completion completes a matrix representing a chordal graph when it is interpreted to be an adjacency matrix of the graph having non-zeros at known elements and zeros at missing elements [5]. A graph is chordal if any minimal cycles in the graph have at most three vertices.…”

“…"Positive Semi-Definite (PSD) Completion" is known to be a subclass of Semidefinite programming, and derives a matrixà in which missing elements of an original PSD matrix A are completed by maximizing the determinant det(Ã) under PSD constraints [5], [6]. Its program named dualcomp is opened to the public [7].…”

“…(5). Note that all data needed to compute q,k+1 for all q ∈ OB n−k−1 is the elements a q,p max in A corresponding to the relations between the moved object p max ∈ OA k+1 with all objects q ∈ OB n−k−1 , and thus L(k + 1) is progressively computed only from A k+1 and B n−k−1 by Eq.…”

Faster, but weaker, relaxations for quadratically constrained quadratic programs

“…The following proposition allows us to eliminate all entries of X except for the blocks X C k C k : Proposition 3 (Grone et. al [7]). Given a vector x and symmetric blocks X C 1 C 1 , .…”