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Facial structure of copositive and completely positive cones over a second-order cone

Mitsuhiro Nishijima, Bruno F. Lourenço

TL;DR

The paper provides a complete facial analysis of COP(\mathbb{L}^n) and CP(\mathbb{L}^n), delivering a four-type classification for COP faces, a detailed description of extreme rays, and precise measurements of longest face chains and polyhedrality distances. It leverages semidefinite representations and subspace projections to compute dimensions of CP faces arising from intersections with subspaces, and it derives analogous chain-length results for CP(\mathbb{L}^n). The results illuminate when faces are exposed and how these properties scale with dimension, offering both explicit results for second-order cones and a pathway to understanding faces in more general symmetric cones. The extensions discuss limitations and counterexamples for general COP/CP cones, clarifying which aspects carry over and which do not, with implications for convex reformulations of nonconvex problems.

Abstract

We classify the faces of copositive and completely positive cones over a second-order cone and investigate their dimension and exposedness properties. Then we compute two parameters related to chains of faces of both cones. At the end, we discuss some possible extensions of the results with a view toward analyzing the facial structure of general copositive and completely positive cones.

Facial structure of copositive and completely positive cones over a second-order cone

TL;DR

The paper provides a complete facial analysis of COP(\mathbb{L}^n) and CP(\mathbb{L}^n), delivering a four-type classification for COP faces, a detailed description of extreme rays, and precise measurements of longest face chains and polyhedrality distances. It leverages semidefinite representations and subspace projections to compute dimensions of CP faces arising from intersections with subspaces, and it derives analogous chain-length results for CP(\mathbb{L}^n). The results illuminate when faces are exposed and how these properties scale with dimension, offering both explicit results for second-order cones and a pathway to understanding faces in more general symmetric cones. The extensions discuss limitations and counterexamples for general COP/CP cones, clarifying which aspects carry over and which do not, with implications for convex reformulations of nonconvex problems.

Abstract

We classify the faces of copositive and completely positive cones over a second-order cone and investigate their dimension and exposedness properties. Then we compute two parameters related to chains of faces of both cones. At the end, we discuss some possible extensions of the results with a view toward analyzing the facial structure of general copositive and completely positive cones.

Paper Structure

This paper contains 17 sections, 45 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Cones and their faces
  5. Copositive and completely positive cones
  6. Facial structure of $\mathcal{COP}(\mathbb{L}^n)$
  7. Faces of $\mathcal{COP}(\mathbb{L}^n)$
  8. Case 1: TEXT
  9. Case 2: TEXT
  10. Extreme rays of $\mathcal{COP}(\mathbb{L}^n)$
  11. The length of a longest chain of faces and the distance to polyhedrality of $\mathcal{COP}(\mathbb{L}^n)$
  12. Facial structure of $\mathcal{CP}(\mathbb{L}^n)$
  13. Dimensions of faces of the form $\mathcal{CP}(\mathbb{X} \cap \mathbb{L}^n)$
  14. Dimensions of faces of the form $\mathcal{CP}(\mathbb{X} \cap \partial\mathbb{L}^n)$
  15. The length of a longest chain of faces and the distance to polyhedrality of $\mathcal{CP}(\mathbb{L}^n)$
  16. ...and 2 more sections

Key Result

Lemma 2.1

Let $\bm{v}\in\mathbb{R}^{n-1}$. Then the Euclidean space $\mathbb{R}^n$ has the following orthogonal direct sum decomposition:

Theorems & Definitions (86)

  • Lemma 2.1
  • Lemma 2.2: Pataki2000
  • Lemma 2.3
  • Example 2.4
  • Lemma 2.5: LMT2018
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • ...and 76 more