New approximations for the cone of copositive matrices and its dual

We provide convergent hierarchies for the cone C of copositive matrices and its dual, the cone of completely positive matrices. In both cases the corresponding hierarchy consists of nested spectrahedra and provide outer (resp. inner) approximations for C (resp. for its dual), thus complementing previous inner (resp. outer) approximations for C (for the dual). In particular, both inner and outer approximations have a very simple interpretation. Finally, extension to K-copositivity and K-complete positivity for a closed convex cone K, is straightforward.Comment: 8

Sur la structure algébrique du cône copositif

We decompose the copositive cone $COP^n$ into a disjoint union of a finite number of open subsets $S_E$ of algebraic sets $Z_E$. Each set $S_E$ consists of interiors of faces of $COP^n$. On each irreducible component of $Z_E$ these faces generically have the same dimension. Each algebraic set $Z_E$ is characterized by a finite collection $E = {(I_\alpha, J_\alpha)}, \alpha=1,...,|E|$ of pairs of index sets. Namely, $Z_E$ is the set of symmetric matrices $A$ such that the submatrices $A_{I_\alpha\times J_\alpha}$ are rank-deficient for all $\alpha$. For every copositive matrix $A \in S_E$, the index sets $I_\alpha$ are the minimal zero supports of $A$. If $u^\alpha$ is a corresponding minimal zero of $A$, then $J_\alpha$ is the set of indices $j$ such that $(Au^\alpha)j = 0$. We call the pair $(I_\alpha, J_\alpha)$ the extended support of the zero $u^\alpha$ , and $E$ the extended minimal zero support set of $A$. We provide some necessary conditions on $E$ for $S_E$ to be non-empty, and for a subset $S_E$ to intersect the boundary of another subset $S_E$

Minimal zeros of copositive matrices

Let $A$ be an element of the copositive cone ${\cal C}_n$. A zero $u$ of $A$ is a nonzero nonnegative vector such that $u^TAu = 0$. The support of $u$ is the index set \mbox{supp}u \subset \{1,\dots,n\} corresponding to the positive entries of $u$. A zero $u$ of $A$ is called minimal if there does not exist another zero $v$ of $A$ such that its support \mbox{supp}v is a strict subset of \mbox{supp}u. We investigate the properties of minimal zeros of copositive matrices and their supports. Special attention is devoted to copositive matrices which are irreducible with respect to the cone $S_+(n)$ of positive semi-definite matrices, i.e., matrices which cannot be written as a sum of a copositive and a nonzero positive semi-definite matrix. We give a necessary and sufficient condition for irreducibility of a matrix $A$ with respect to $S_+(n)$ in terms of its minimal zeros. A similar condition is given for the irreducibility with respect to the cone ${\cal N}_n$ of entry-wise nonnegative matrices. For $n = 5$ matrices which are irreducible with respect to both $S_+(5)$ and ${\cal N}_5$ are extremal. For $n = 6$ a list of candidate combinations of supports of minimal zeros which an exceptional extremal matrix can have is provided.Comment: Some conditions and proofs simplifie

On the structure of the 6×66 \times 6 copositive cone

In this work we complement the description of the extreme rays of the $6 \times 6$ copositive cone with some topological structure. In a previous paper we decomposed the set of extreme elements of this cone into a disjoint union of pieces of algebraic varieties of different dimension. In this paper we link this classification to the recently introduced combinatorial characteristic called extended minimal zero support set. We determine those components which are essential, i.e., which are not embedded in the boundary of other components. This allows to drastically decrease the number of cases one has to consider when investigating different properties of the $6 \times 6$ copositive cone. As an application, we construct an example of a copositive $6 \times 6$ matrix with unit diagonal which does not belong to the Parrilo inner sum of squares relaxation ${\cal K}^{(1)}_6$

Considering copositivity locally

Let $A$ be an element of the copositive cone \copos{n}. A zero \vu of $A$ is a nonnegative vector whose elements sum up to one and such that \vu^TA\vu = 0. The support of \vu is the index set \Supp{\vu} \subset \{1,\dots,n\} corresponding to the nonzero entries of \vu. A zero \vu of $A$ is called minimal if there does not exist another zero \vv of $A$ such that its support \Supp{\vv} is a strict subset of \Supp{\vu}. Our main result is a characterization of the cone of feasible directions at $A$, i.e., the convex cone \VarK{A} of real symmetric $n \times n$ matrices $B$ such that there exists $\delta > 0$ satisfying A + \delta B \in \copos{n}. This cone is described by a set of linear inequalities on the elements of $B$ constructed from the set of zeros of $A$ and their supports. This characterization furnishes descriptions of the minimal face of $A$ in \copos{n}, and of the minimal exposed face of $A$ in \copos{n}, by sets of linear equalities and inequalities constructed from the set of minimal zeros of $A$ and their supports. In particular, we can check whether $A$ lies on an extreme ray of \copos{n} by examining the solution set of a system of linear equations. In addition, we deduce a simple necessary and sufficient condition on the irreducibility of $A$ with respect to a copositive matrix $C$. Here $A$ is called irreducible with respect to $C$ if for all $\delta > 0$ we have A - \delta C \not\in \copos{n}