On generators of $k$-PSD closures of the positive semidefinite cone

Abstract

Positive semidefinite (PSD) cone is the cone of positive semidefinite matrices, and is the object of interest in semidefinite programming (SDP). A computational efficient approximation of the PSD cone is the $k$-PSD closure, $1 \leq k < n$, cone of $n\times n$ real symmetric matrices such that all of their $k\times k$ principal submatrices are positive semidefinite. For $k=1$, one obtains a polyhedral approximation, while $k=2$ yields a second order conic (SOC) approximation of the PSD cone. These approximations of the PSD cone have been used extensively in real-world applications such as AC Optimal Power Flow (ACOPF) to address computational inefficiencies where SDP relaxations are utilized for convexification the non-convexities. However a theoretical discussion about the geometry of these conic approximations of the PSD cone is rather sparse. In this short communication, we attempt to provide a characterization of some family of generators of the aforementioned conic approximations.