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Faces of homogeneous cones and applications to homogeneous chordality

João Gouveia, Masaru Ito, Bruno F. Lourenço

TL;DR

This work provides a comprehensive, algorithmic description of the facial structure of homogeneous cones via Vinberg’s T-algebra framework. It proves that every face of a homogeneous cone is transformable by a triangular automorphism to a principal face, and that a generalized Cholesky factorization yields the required automorphisms, establishing projectional exposedness for these cones. The theory is then specialized to cones of PSD matrices with homogeneous chordal sparsity, showing faces correspond to induced subgraphs and enabling PSD completion results that preserve sparsity. The findings offer practical tools for facial reduction in conic optimization and illuminate the interplay between rank, faces, and duality within this broad class of cones. The paper also situates these results among Siegel-cone and matrix-realization perspectives and connects to Ishi’s orbit analysis, contributing to a unified view of homogeneous cones and their applications to chordal PSD 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.

Faces of homogeneous cones and applications to homogeneous chordality

TL;DR

This work provides a comprehensive, algorithmic description of the facial structure of homogeneous cones via Vinberg’s T-algebra framework. It proves that every face of a homogeneous cone is transformable by a triangular automorphism to a principal face, and that a generalized Cholesky factorization yields the required automorphisms, establishing projectional exposedness for these cones. The theory is then specialized to cones of PSD matrices with homogeneous chordal sparsity, showing faces correspond to induced subgraphs and enabling PSD completion results that preserve sparsity. The findings offer practical tools for facial reduction in conic optimization and illuminate the interplay between rank, faces, and duality within this broad class of cones. The paper also situates these results among Siegel-cone and matrix-realization perspectives and connects to Ishi’s orbit analysis, contributing to a unified view of homogeneous cones and their applications to chordal PSD problems.

Abstract

A convex cone 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 is mapped by an automorphism of 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.
Paper Structure (19 sections, 20 theorems, 80 equations, 2 algorithms)

This paper contains 19 sections, 20 theorems, 80 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Homogeneous cones and T-algebras
  4. Dual cone
  5. Automorphisms
  6. Principal subalgebras
  7. Facial structure of homogeneous cones
  8. Unique Cholesky factorization in homogeneous cones
  9. Faces and their orbits under the action of triangular matrices
  10. Some consequences and discussion
  11. The dual cone
  12. Other approaches and related results
  13. Siegel cones
  14. Matrix realizations
  15. Connection to Ishi's results
  16. ...and 4 more sections

Key Result

Lemma 2.1

If ${\mathcal{F}} \mathrel{\unlhd} {\mathcal{K}}$ is an exposed extreme ray, then ${\mathcal{F}}^\Delta$ is a maximal proper face of ${\mathcal{K}}^*$.

Theorems & Definitions (41)

  • Lemma 2.1
  • proof
  • Remark 2.2: Analogy with the usual matrices
  • Theorem 2.3: V63
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • Lemma 3.1
  • proof
  • Proposition 3.2: Unique Cholesky factorization
  • ...and 31 more