On the quality of the $k-$PSD closure approximation

TL;DR

This paper analyzes the approximation quality of the $k$-PSD closure of the PSD cone, providing new bounds and characterizing extreme rays, with implications for optimization problems like ACOPF.

Contribution

It offers a new dominant bound on the approximation quality of the $k$-PSD closure and characterizes the extreme rays of the 2-PSD closure.

Findings

Introduces a new dominant bound on the $k$-PSD approximation quality.

Characterizes the extreme rays of the 2-PSD closure.

Revisits and extends previous bounds on approximation quality.

Abstract

Postive 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. In a recent series of articles Blekharman et al. provided bounds on the quality of these approximations. In this work, we revisit some of their results and also propose a new dominant bound on quality of the $k$-PSD closure approximation of the PSD cone. In addition, we characterize the extreme rays of the $2$-PSD closure.