New results on the cp-rank and related properties of co(mpletely )positive matrices

Shaked-Monderer, Naomi; Berman, Abraham; Bomze, Immanuel M.; Jarre, Florian; Schachinger, Werner

doi:10.1080/03081087.2013.869591

Cited by 33 publications

(23 citation statements)

References 17 publications

(30 reference statements)

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“…(note that [44,Theorem 2.1(b)] says that the jth row of Z + (y * , w * )X * vanishes if either j = 0 or if x * j > 0, which, by (5.3), amounts exactly to the same). Hence (x * ; u * , w * ) ∈ (F ∩ P ) × R m + × R p form a generalized KKT pair for (2.2).…”

Section: Semi-lagrangian Tightness and Second-order Optimality Conditmentioning

confidence: 93%

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Copositive Relaxation Beats Lagrangian Dual Bounds in Quadratically and Linearly Constrained Quadratic Optimization Problems

Bomze¹

2015

SIAM J. Optim.

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We study nonconvex quadratic minimization problems under (possibly nonconvex) quadratic and linear constraints, characterizing both Lagrangian and semi-Lagrangian dual bounds in terms of conic optimization. While the Lagrangian dual is equivalent to the SDP relaxation (which has been known for quite a while, although the presented form, incorporating explicitly linear constraints, seems to be novel), we show that the semi-Lagrangian dual is equivalent to a natural copositive relaxation (and this has apparently not been observed before). This way, we arrive at conic bounds tighter than the usual Lagrangian dual (and thus than the SDP) bounds. Any of the known tractable inner approximations of the copositive cone can be used for this tightening, but in particular, the above-mentioned characterization with explicit linear constraints is a natural, much cheaper, relaxation than the usual zero-order approximation by doubly nonnegative (DNN) matrices and still improves upon the Lagrangian dual bounds. These approximations are based on LMIs on matrices of basically the original order plus additional linear constraints (in contrast to more familiar sum-of-squares or moment approximation hierarchies) and thus may have merits in particular for large instances where it is important to employ only a few inequality constraints (e.g., n instead of n(n−1) 2 for the DNN relaxation). Further, we specify sufficient conditions for tightness of the semi-Lagrangian relaxation and show that copositivity of the slack matrix guarantees global optimality for KKT points of this problem, thus significantly improving upon a well-known secondorder global optimality condition.

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Section: Semi-lagrangian Tightness and Second-order Optimality Conditmentioning

confidence: 93%

“…Note that the factor matrix F has many more columns than rows. The upper bound s n on the necessary number of columns was recently established in [44] and is asymptotically tight as n → ∞ [14]. Nevertheless, a perhaps more amenable representation is…”

Section: Notation and Terminologymentioning

confidence: 98%

Copositive Relaxation Beats Lagrangian Dual Bounds in Quadratically and Linearly Constrained Quadratic Optimization Problems

Bomze¹

2015

SIAM J. Optim.

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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: 98%

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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“…Since the extreme rays of a cone determine the facets of its dual cone, they are also important tools for the study of this dual cone. The extreme rays of the copositive cone have been used in a number of papers on its dual, the completely positive cone [8,19,5,6,18,17].…”

Section: Introductionmentioning

confidence: 99%

Extremal copositive matrices with minimal zero supports of cardinality two

Hildebrand

2017

Linear and Multilinear Algebra

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Let A ∈ C n be an extremal copositive matrix with unit diagonal. Then the minimal zeros of A all have supports of cardinality two if and only if the elements of A are all from the set {−1, 0, 1}. Thus the extremal copositive matrices with minimal zero supports of cardinality two are exactly those matrices which can be obtained by diagonal scaling from the extremal {−1, 0, 1} unit diagonal matrices characterized by Hoffman and Pereira in 1973.

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New results on the cp-rank and related properties of co(mpletely )positive matrices

Cited by 33 publications

References 17 publications

Copositive Relaxation Beats Lagrangian Dual Bounds in Quadratically and Linearly Constrained Quadratic Optimization Problems

Copositive Relaxation Beats Lagrangian Dual Bounds in Quadratically and Linearly Constrained Quadratic Optimization Problems

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

Extremal copositive matrices with minimal zero supports of cardinality two

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