Completely Positive Matrices

Berman, Abraham; Shaked-Monderer, Naomi

doi:10.1142/9789812795212

Cited by 186 publications

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“…It can also be seen that Y a is a set of n − 1 linearly independent vectors and so from the previous lemma we see that U gives us a set of 1 2 n(n − 1) linearly independent matrices contained in the face. 2…”

Section: Maximal Faces Of the Completely Positive Conementioning

confidence: 82%

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Geometry of the copositive and completely positive cones

Dickinson

2011

Journal of Mathematical Analysis and Applications

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The copositive cone, and its dual the completely positive cone, have useful applications in optimisation, however telling if a general matrix is in the copositive cone is a co-NPcomplete problem. In this paper we analyse some of the geometry of these cones. We discuss a way of representing all the maximal faces of the copositive cone along with a simple equation for the dimension of each one. In doing this we show that the copositive cone has faces which are isomorphic to positive semidefinite cones. We also look at some maximal faces of the completely positive cone and find their dimensions. Additionally we consider extreme rays of the copositive and completely positive cones and show that every extreme ray of the completely positive cone is also an exposed ray, but the copositive cone has extreme rays which are not exposed rays.

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Section: Maximal Faces Of the Completely Positive Conementioning

confidence: 82%

“…Theorem 2.14. (See [1,Chapter 1].) If K is a proper cone then so is its dual K * and we have that K * * = K.…”

Section: Geometry Of General Proper Conesmentioning

confidence: 99%

Geometry of the copositive and completely positive cones

Dickinson

2011

Journal of Mathematical Analysis and Applications

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“…From the definitions, there exists v ∈ R n ++ such that M − vv T ∈ CP n \ {O} and from Corollary 2.11 we have cp + (M) ≤ cp + (vv T ) + cp(M − vv T ) ≤ 1 + p n . While (6) and (7) are well known since long, see for example [4], the bounds in (8) and (9) were established quite recently, namely in [8] and in [19], respectively. For n = 5 we have p n = n 2 /4 [20].…”

Section: Perron-frobenius Perturbationsmentioning

confidence: 99%

The structure of completely positive matrices according to their CP-rank and CP-plus-rank

Bomze

Dickinson

Still

2015

Linear Algebra and its Applications

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We study the topological properties of the cp-rank operator cp(A) and the related cp-plus-rank operator cp + (A) (which is introduced in this paper) in the set S n of symmetric n × n-matrices. For the set of completely positive matrices, CP n , we show that for any fixed p the set of matrices A satisfying cp(A) = cp + (A) = p is open in S n \ bd (CP n ). We also prove that the set A n of matrices with cp(A) = cp + (A) is dense in S n . By applying the theory of semi-algebraic sets we are able to show that membership in A n is even a generic property. We furthermore answer several questions on the existence of matrices satisfying cp(A) = cp + (A) or cp(A) = cp + (A), and establish genericity of having infinitely many minimal cp-decompositions.

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“…This justifies the notation C * (and the title of the paper). The comprehensive monograph [1] introduces basic concepts such as the cp-rank of a completely positive matrix. In particular, in Theorem 3.5 in [1], it is shown, that the number n + 1 2 in the above definition of C * can be reduced to n + 1 2 − 1 without changing C * .…”

Section: Notationmentioning

confidence: 99%

“…The comprehensive monograph [1] introduces basic concepts such as the cp-rank of a completely positive matrix. In particular, in Theorem 3.5 in [1], it is shown, that the number n + 1 2 in the above definition of C * can be reduced to n + 1 2 − 1 without changing C * . A recent characterization of the interior of the completely positive cone is given in [12], see also [10].…”

Section: Notationmentioning

confidence: 99%

Quadratic factorization heuristics for copositive programming

2011

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Copositive optimization problems are particular conic programs: optimize linear forms over the copositive cone subject to linear constraints. Every quadratic program with linear constraints can be formulated as a copositive program, even if some of the variables are binary. So this is an NP-hard problem class. While most methods try to approximate the copositive cone from within, we propose a method which approximates this cone from outside. This is achieved by passing to the dual problem, where the feasible set is an affine subspace intersected with the cone of completely positive matrices, and this cone is approximated from within. We consider feasible descent directions in the completely positive cone, and regularized strictly convex subproblems. In essence, we replace the intractable completely positive cone with a nonnegative cone, at the cost of a series of nonconvex quadratic subproblems. Proper adjustment of the regularization parameter results in short steps for the nonconvex quadratic programs. This suggests to approximate their solution by standard linearization techniques. Preliminary numerical results on three different classes of test problems are quite promising.

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Completely Positive Matrices

Cited by 186 publications

References 0 publications

Geometry of the copositive and completely positive cones

Geometry of the copositive and completely positive cones

The structure of completely positive matrices according to their CP-rank and CP-plus-rank

Quadratic factorization heuristics for copositive programming

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