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Subject: irreducibility × WGL Institute: WIAS ×

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    Considering copositivity locally
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Dickinson, Peter J.C.; Hildebrand, Roland
    Let $A$ be an element of the copositive cone $coposn$. A zero $vu$ of $A$ is a nonnegative vector whose elements sum up to one and such that $vu^TAvu = 0$. The support of $vu$ is the index set $Suppvu subset 1,dots,n$ 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 $Suppvv$ is a strict subset of $Suppvu$. Our main result is a characterization of the cone of feasible directions at $A$, i.e., the convex cone $VarKA$ of real symmetric $n times n$ matrices $B$ such that there exists $delta > 0$ satisfying $A + delta B in coposn$. 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 $coposn$, and of the minimal exposed face of $A$ in $coposn$, 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 $coposn$ 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 $A - delta C notin coposn$.
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    Minimal zeros of copositive matrices
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Hildebrand, Roland
    Let A be an element of the copositive cone Cn. A zero u of A is a nonzero nonnegative vector such that uT Au = 0. The support of u is the index set supp u c {1,..., 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 supp v is a strict subset of 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 Nn of entry-wise nonnegative matrices. For n = 5 matrices which are irreducible respect to both S+(5) and N5 are extremal. For n = 6 a list of candidate combinations of supports of minimal zeros which an exceptional extremal matrix can have is provided.