Hyperbolic Relaxation of $k$-Locally Positive Semidefinite Matrices

A successful computational approach for solving large-scale positive semidefinite (PSD) programs is to enforce PSD-ness on only a collection of submatrices. For our study, we let $\mathcal{S}^{n,k}$ be the convex cone $n\times n$ symmetric matrices where all $k\times k$ principal submatrices are PSD. We call matrix in this $k$-locally In order compare PSD matrices, study eigenvalues matrices. The key insight paper that there $H(e_k^n)$ so if $X \in \mathcal{S}^{n,k}$, then vector $X$ contained $H(e_k^n)$. hyperbolicity elementary polynomial $e_k^n$ (where $e_k^n(x) = \sum_{S \subseteq [n] : |S| k} \prod_{i S} x_i$) with respect ones vector. Using insight, able improve previously known upper bounds Frobenius distance between and also quality relaxation $H(e^n_k)$. first show tight case $k n -1$, is, every $H(e^n_{n -1})$ exists $\mathcal{S}^{n, -1}$ whose equal components prove structure theorem nonsingular minors zero, which believe independent interest. This result shows $1< k < -1$ “large parts” boundary do not intersect $\mathcal{S}^{n,k}$.