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

TL;DR

This paper investigates the geometric structure of $k$-PSD cone approximations, which are used in semidefinite programming to improve computational efficiency in applications like power flow optimization.

Contribution

It provides a characterization of certain generators of the $k$-PSD cone approximations, enhancing understanding of their geometric properties.

Findings

Characterization of some generators of $k$-PSD cones.

Insight into the geometric structure of $k$-PSD approximations.

Connections to applications like AC Optimal Power Flow.

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.