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

Abstract

We provide a complete classification of the extreme rays of the 6 × 6 copositive cone COP 6 . We proceed via a coarse intermediate classification of the possible minimal zero support set of an exceptional extremal matrix A ∈ COP 6 . To each such minimal zero support set we construct a stratified semi-algebraic manifold in the space of real symmetric 6 × 6 matrices S 6 , parameterized in a semitrigonometric way, which consists of all exceptional extremal matrices A ∈ 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.

The cone of $5\times 5$ completely positive matrices

“…This cone is Sym 5 which is self dual and its algebraic boundary is known to be given by matrices of low rank, i.e., solutions to the equation detpXq " detp1 ˝Xq " 0. The linear change of variables yields that K ˚has a boundary described by (1). The shared part of the boundary of K ˚and CP 5 satisfies then the desired equation.…”

“…In order to achieve this we define some relevant varieties. Let Z Horn Ă W be the hypersurface of all matrices of the form DB, where B is a matrix as in (1) and D is a diagonal matrix with positive entries.…”

“…Remark 3.1. Note that the factor matrix 1 2 B `1 2 B (or any other linear combination) does not yield a matrix close to A, even though they are in the same linear subspace described by (1). This is because the mixed terms B BT and BB T yield very unpredictable results.…”

“…It is shown in [1] that the nontrival extreme rays of the cone of 6 ˆ6 copositive matrices come in 42 types. In other words, instead of having to deal with two loci, an analogous analysis of that cone would involve 42 varieties, most of which are expected to yield varieties of much higher degree than the Hildebrand locus.…”

On equivalent representations and properties of faces of the cone of copositive matrices

“…It worth mentioning that several attempts have been done to study the facial structure of the cones of copositive and completely positive matrices. Thus in [3,9,11], the authors give explicit characterizations of extreme rays (faces of dimension one) of copositive cones of dimensions five and six. In [8,9], some properties of special types of faces (minimal and maximal ones) are studied.…”