Faces of homogeneous cones and applications to homogeneous chordality

TL;DR

This paper characterizes the facial structure of homogeneous cones, showing they are projectionally exposed, and applies these findings to cones of positive semidefinite matrices with homogeneous chordal sparsity, impacting PSD completion problems.

Contribution

It provides a detailed analysis of the facial structure of homogeneous cones, including an algorithmic method for automorphisms, and extends results to cones of PSD matrices with homogeneous chordal structure.

Findings

Faces of homogeneous cones are mapped to principal faces via automorphisms.

Homogeneous cones are proven to be projectionally exposed.

Analysis of PSD cones with homogeneous chordality impacts PSD completion problems.

Abstract

A convex cone $\mathcal{K}$ is said to be homogeneous if its group of automorphisms acts transitively on its relative interior. Important examples of homogeneous cones include symmetric cones and cones of positive semidefinite (PSD) matrices that follow a sparsity pattern given by a homogeneous chordal graph. Our goal in this paper is to elucidate the facial structure of homogeneous cones and make it as transparent as the faces of the PSD matrices. We prove that each face of a homogeneous cone $\mathcal{K}$ is mapped by an automorphism of $\mathcal{K}$ to one of its finitely many so-called principal faces. Furthermore, constructing such an automorphism can be done algorithmically by making use of a generalized Cholesky decomposition. Among other consequences, we give a proof that homogeneous cones are projectionally exposed, which strengthens the previous best result that they are amenable. Using our results, we will carefully analyze the facial structure of cones of PSD matrices satisfying homogeneous chordality and discuss consequences for the corresponding family of PSD completion problems.