Hyperbolic Relaxation of $k$-Locally Positive Semidefinite Matrices

Abstract

In order to solve positive semidefinite (PSD) programs efficiently, a successful computational trick is to consider a relaxation, where PSD-ness is enforced only on a collection of submatrices. In order to study this formally, we consider the class of n × n symmetric matrices where we enforce PSD-ness on all k × k principal submatrices. We call a matrix in this class k-locally PSD. In order to compare the set of k-locally PSD matrices (denoted as S n,k ) to the set of PSD matrices, we study eigenvalues of k-locally PSD matrices. The key insight in this paper is that the eigenvalues of a matrix in S n,k are contained in a convex set, H(e n k ) (this can be defined as the hyperbolicity cone of the elementary symmetric polynomial e k n (where e n k (x) = S⊆[n]:|S|=k i∈S x i ) with respect to the all ones vector). Using this insight, we are able to improve previously known upper bounds on the Frobenius distance between matrices in S n,k from PSD matrices. We also study the quality of the convex relaxation H(e n k ). We first show that this relaxation is tight for the case of k = n − 1, that is, for every vector in H(e n n−1 ) there exists a matrix in S n,n−1 whose eigenvalues are equal to the components of the vector. We then prove a structure theorem that precisely characterizes the non-singular matrices in S n,k whose eigenvalues belong to the boundary of H(e n k ). This result indicates that "large parts" of the boundary of H(e n k ) do not intersect with the eigenvalues of matrices in S n,k when k < n − 1.

On eigenvalues of symmetric matrices with PSD principal submatrices

“…In this work we are interested in the set λ( n,k ) = λ(X) σ : X ∈ n,k , σ ∈ S n of all possible (permuted) eigenvalue vectors of k-locally PSD matrices. As observed in [2], the set λ( n,k ) is contained in the closed hyperbolicity cone…”

“…2.2]). The authors of [2] pointed out that it is unknown whether the set λ( n,k ) of eigenvalue vectors of matrices in n,k is convex for any k and n. We give a negative answer to this question, showing that λ( 4,2 ) H(e 4 2 ) is not convex. Theorem 1.1.…”

“…, n} let n,k denote the set of those matrices in Sym n , whose all k × k principal submatrices are positive semidefinite. Elements of n,k are known as k-locally positive semidefinite matrices, they have been systematically studied in [1] and [2]. Note that n,n consists of all positive semidefinite matrices in Sym n .…”

“…Cor. 3] and, surprisingly, one has equality λ( n,n−1 ) = H(e n n−1 ) for k = n − 1 (see [2,Thm. 2.2]).…”

“…We now discuss one application of this result. It is shown in [2] that every point on the boundary of H(e 4 2 ) with exactly one negative entry is an eigenvalue vector of some matrix in 4,2 . We observe that the same holds for all points in H(e 4 2 ) with this property.…”