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Copositive and completely positive cones over symmetric cones of rank at least 5

Mitsuhiro Nishijima

Abstract

We focus on copositive and completely positive cones over symmetric cones of rank at least $5$, and in particular investigate whether these cones are spectrahedral shadows. We extend known results for nonnegative orthants of dimension at least $5$ to general symmetric cones of rank at least $5$. Specifically, we prove that when the rank of a symmetric cone is at least $5$, neither the copositive nor the completely positive cone over it is a spectrahedral shadow. We then generalize the Horn matrix to the setting of symmetric cones of rank at least $5$ by introducing Horn transformations and analyzing their geometric and algebraic properties. We show that Horn transformations generate exposed rays of copositive cones over symmetric cones. We also show that Horn transformations evade the zeroth level of a sum-of-squares inner-approximation hierarchy for copositive cones over symmetric cones.

Copositive and completely positive cones over symmetric cones of rank at least 5

Abstract

We focus on copositive and completely positive cones over symmetric cones of rank at least , and in particular investigate whether these cones are spectrahedral shadows. We extend known results for nonnegative orthants of dimension at least to general symmetric cones of rank at least . Specifically, we prove that when the rank of a symmetric cone is at least , neither the copositive nor the completely positive cone over it is a spectrahedral shadow. We then generalize the Horn matrix to the setting of symmetric cones of rank at least by introducing Horn transformations and analyzing their geometric and algebraic properties. We show that Horn transformations generate exposed rays of copositive cones over symmetric cones. We also show that Horn transformations evade the zeroth level of a sum-of-squares inner-approximation hierarchy for copositive cones over symmetric cones.
Paper Structure (15 sections, 15 theorems, 104 equations, 2 figures)

This paper contains 15 sections, 15 theorems, 104 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic Notation and Terminology
  4. Symmetric Cones and Euclidean Jordan Algebras
  5. Linear Mappings
  6. Copositive and Completely Positive Cones
  7. Spectrahedra and Their Shadows
  8. Nonexistence of Spectrahedral Shadow Representations for Copositive and Completely Positive Cones over Symmetric Cones
  9. Horn Transformations
  10. Exposedness of the Rays Generated by the Horn Transformations
  11. Non-Exactness of a Sum-of-Squares Inner-Approximation Hierarchy for Copositive Cones over Symmetric Cones
  12. Notation for Polynomials
  13. Main Result
  14. Proof of Theorem \ref{['thm:not_sos']}
  15. Final Remarks

Key Result

Lemma 2.1

Let $\phi\colon \mathbb{E} \to \mathbb{R}^n$ be an isometry, and define $\mathbin{\lozenge}$ and $\mathbin{\blacklozenge}$ as for $\bm{x},\bm{y}\in \mathbb{R}^n$, respectively. Then the mapping $\phi$ satisfies $\phi(x\circ y) = \phi(x) \mathbin{\lozenge} \phi(y)$ and $x\bullet y = \phi(x) \mathbin{\blacklozenge} \phi(y)$ for all $x,y\in \mathbb{E}$. In particular, $(\mathbb{R}^n,\mathbin{\lozeng

Figures (2)

  • Figure 1: Illustration of the seven cases from Case \ref{['enum:case_a']} to Case \ref{['enum:case_g']}. The cells marked with $*$ correspond to the indices $(ij,kl)$ in $\mathcal{J}$, for which $A_{ij,kl} = aH_{ij,kl}$ holds as shown in \ref{['eq:A_eq_aH_on_size3block']}.
  • Figure 2: Illustration of the six cases from Case \ref{['enum:case_a_rkg']} to Case \ref{['enum:case_f_rkg']} for $r = 5$. The cells marked with $*$ correspond to the indices $(ij,kl) \in \llbracket r\rrbracket$, for which $A_{ij,kl} = aH_{ij,kl}$ holds as shown in \ref{['eq:A_eq_aH_on_Ec1']}. The cell marked with $\heartsuit$ indicates that the corresponding element is $0$ because of \ref{['eq:Ar+r+r+r+_eq_0']}, those marked with $\clubsuit$ indicate that the corresponding elements are $0$ because of \ref{['eq:Aijr+r+_eq_0']}, and those marked with $\spadesuit$ indicate that the corresponding elements are $0$ because of \ref{['eq:Air+ir+_eq_0']}.

Theorems & Definitions (24)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Lemma 4.1
  • ...and 14 more