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The extreme rays of the $6\times6$ copositive cone

Andrei Afonin (afonin.ad***at***phystech.edu)
Roland Hildebrand (roland.hildebrand***at***univ-grenoble-alpes.fr)
Peter J.C. Dickinson (peter.jc.dickinson***at***googlemail.com)

Abstract: We provide a complete classification of the extreme rays of the $6 \times 6$ copositive cone ${\cal COP}^6$. We proceed via a coarse intermediate classification of the possible minimal zero support set of an exceptional extremal matrix $A \in {\cal COP}^6$. To each such minimal zero support set we construct a stratified semi-algebraic manifold in the space of real symmetric $6 \times 6$ matrices ${\cal S}^6$, parameterized in a semi-trigonometric way, which consists of all exceptional extremal matrices $A \in {\cal COP}^6$ having this minimal zero support set. Each semi-algebraic stratum is characterized by the supports of the minimal zeros $u$ as well as the supports of the corresponding matrix-vector products $Au$. The analysis uses recently and newly developed methods that are applicable also to copositive matrices of arbitrary order.

Keywords: copositive matrix, extreme ray, minimal zero, non-convex optimization

Category 1: Linear, Cone and Semidefinite Programming (Other )

Category 2: Global Optimization (Theory )


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Entry Submitted: 11/22/2019
Entry Accepted: 11/22/2019
Entry Last Modified: 04/19/2020

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