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Generalized Perfect Matrices

Alexander Oertel, Achill Schürmann

TL;DR

This work generalizes Voronoi’s theory of perfect quadratic forms to the setting of generalized copositive matrices over a closed convex cone $K$, introducing the $K$-copositive minimum and perfect $K$-copositive matrices and the Interior Ryshkov (IR) property. It establishes that rationally generated cones are IR, analyzes a two-dimensional non-IR example to reveal deviations from the classical theory, and connects these ideas to Diophantine approximation and Pell equations. The paper then develops inner and outer polyhedral approximations for the generalized completely positive cone ${\mathcal{CP}}_{K}$ and describes how to obtain rational certificates for (non-)membership, including adaptations of Voronoi-type tessellations to the generalized setting. Overall, it provides a unified framework linking perfectness, Ryshkov structures, and certificate-based approaches for generalized positivity cones, with algorithmic implications for complete positivity over arbitrary cones. The results pave the way for practical computations and further geometric understandings of the Ryshkov set in generalized contexts.

Abstract

We generalize Voronoi's theory of perfect quadratic forms to generalized copositive matrices over a closed convex and full-dimensional cone K. We introduce a notion of a K-copositive minimum and of perfect K-copositive matrices. We consider a key feature of a given cone, which we call Interior Ryshkov (IR) property. Under this property the classical theory and its applications generalize nicely and we prove that rationally generated cones possess this IR property. For contrast, we give a detailed example of a simple cone without the IR property, showing various differences to the classical case. Moreover, this example yields connections to questions of number theory, in particular to Diophantine approximation and the Pell Equation. Finally, as an application, we give inner and outer polyhedral approximations for the generalized completely positive cone and a method to find rational certificates for (non-)membership in this cone.

Generalized Perfect Matrices

TL;DR

This work generalizes Voronoi’s theory of perfect quadratic forms to the setting of generalized copositive matrices over a closed convex cone , introducing the -copositive minimum and perfect -copositive matrices and the Interior Ryshkov (IR) property. It establishes that rationally generated cones are IR, analyzes a two-dimensional non-IR example to reveal deviations from the classical theory, and connects these ideas to Diophantine approximation and Pell equations. The paper then develops inner and outer polyhedral approximations for the generalized completely positive cone and describes how to obtain rational certificates for (non-)membership, including adaptations of Voronoi-type tessellations to the generalized setting. Overall, it provides a unified framework linking perfectness, Ryshkov structures, and certificate-based approaches for generalized positivity cones, with algorithmic implications for complete positivity over arbitrary cones. The results pave the way for practical computations and further geometric understandings of the Ryshkov set in generalized contexts.

Abstract

We generalize Voronoi's theory of perfect quadratic forms to generalized copositive matrices over a closed convex and full-dimensional cone K. We introduce a notion of a K-copositive minimum and of perfect K-copositive matrices. We consider a key feature of a given cone, which we call Interior Ryshkov (IR) property. Under this property the classical theory and its applications generalize nicely and we prove that rationally generated cones possess this IR property. For contrast, we give a detailed example of a simple cone without the IR property, showing various differences to the classical case. Moreover, this example yields connections to questions of number theory, in particular to Diophantine approximation and the Pell Equation. Finally, as an application, we give inner and outer polyhedral approximations for the generalized completely positive cone and a method to find rational certificates for (non-)membership in this cone.
Paper Structure (26 sections, 11 theorems, 72 equations, 3 figures)

This paper contains 26 sections, 11 theorems, 72 equations, 3 figures.

Table of Contents

  1. Introduction and Overview
  2. A Generalized Framework for Perfectness
  3. Basic Definitions and Notation
  4. Generalized Completely Positive and Copositive Matrices
  5. Generalized Minima and Perfectness
  6. Symmetry
  7. Interior Ryshkov Property
  8. Embedding the Classical Theory
  9. Preliminaries from Diophantine Approximation
  10. Rationally Generated Cones are IR
  11. Squareroot Two Example
  12. Characterizing a 2-Face of the K-Copositive Cone
  13. The Relative Boundary of the 2-Face
  14. The Relative Interior of F
  15. Vertices of the Ryshkov Polyhedral Set
  16. ...and 11 more sections

Key Result

Lemma 2.2

Let $K \in \{ \mathbb{R}^n, \mathbb{R}_{\geq 0}^n \}$ and $Q \in {\mathcal{S}}^n$. Then

Figures (3)

  • Figure 1: Sketch of ${\mathcal{CP}}_{K}$ with the dark gray part being the matrices of ${\mathcal{CP}}_{K}$ with trace $3$.
  • Figure 2: Graph of $f$.
  • Figure 3: Sketch of $\mathcal{R}_{K}\cap F$.

Theorems & Definitions (25)

  • Example 2.1
  • Lemma 2.2
  • Example 2.3
  • Definition 2.4: Interior Ryshkov Property
  • Remark 2.5
  • Lemma 2.6
  • Theorem 2.7
  • Lemma 2.8
  • proof
  • Theorem 2.9
  • ...and 15 more