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Refined Estimates on the Dimensions of Maximal Faces of Completely Positive Cones

O. I. Kostyukova, T. V. Tchemisova

Abstract

The structure of maximal faces of the cone of completely positive matrices is still not well understood in higher dimensions, mainly due to the lack of a general characterization of extreme exposed rays of the copositive cone beyond small matrix orders. This paper contributes to the study of maximal faces of the cone of completely positive matrices by establishing sharper bounds on their dimensions than those currently available. For every odd dimension $n$, we prove that the exact lower bound on the dimensions of maximal faces of the cone of $n \times n$ completely positive matrices equals $n$. For even dimensions $n \geq 8$, we derive a new upper estimate for this lower bound and show that it lies between $n$ and $n+3$. These results substantially refine the previously known bounds.

Refined Estimates on the Dimensions of Maximal Faces of Completely Positive Cones

Abstract

The structure of maximal faces of the cone of completely positive matrices is still not well understood in higher dimensions, mainly due to the lack of a general characterization of extreme exposed rays of the copositive cone beyond small matrix orders. This paper contributes to the study of maximal faces of the cone of completely positive matrices by establishing sharper bounds on their dimensions than those currently available. For every odd dimension , we prove that the exact lower bound on the dimensions of maximal faces of the cone of completely positive matrices equals . For even dimensions , we derive a new upper estimate for this lower bound and show that it lies between and . These results substantially refine the previously known bounds.
Paper Structure (9 sections, 13 theorems, 95 equations)

This paper contains 9 sections, 13 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Basic notations and definitions
  3. Convex cones, their faces and some properties
  4. The cones of copositive and completely positive matrices
  5. The Tight Lower Bound on the Dimensions of Maximal Faces of the Cone ${\mathbb {CP}}(n)$ of Odd-Order Matrices
  6. Estimation of the Maximal Face Dimensions for the Cone ${\mathbb {COP}}(n)$ of Even-Order Matrices
  7. Preliminary Results
  8. Estimation of $low(n)$ for even $n$
  9. Conclusion

Key Result

Theorem 1

[Dickinson, Theorem 2.20] If ${\mathcal{K}}$ is a proper cone and $x\in R^+v$, where ${R^+v}$ is an exposed ray of ${\mathcal{K}}^*$ generated by some $v \in \mathcal{K}^*\setminus \{\textbf{0}\}$, then ${\mathcal{F}}:={\mathcal{K}}\cap x^\bot$ is a maximal face of ${\mathcal{K}}$.

Theorems & Definitions (16)

  • Theorem 1
  • Definition 1
  • Lemma 1
  • Corollary 1
  • Proposition 1
  • Lemma 2
  • Theorem 2
  • Theorem 3
  • Proposition 2
  • Proposition 3
  • ...and 6 more