The CP-Matrix Completion Problem

Zhou, Anwa; Fan, Junxuan

doi:10.1137/130919490

Cited by 20 publications

(27 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It might be possible that apλ˚, k q belongs to R A pKq while y˚, k | 2t is not flat for all t. In such cases, we can apply Algorithms given in [23,30] to check if apλ˚, k q P R A pKq or not. In computational experiments, the finite convergence always occurs.…”

Section: Remark 43 We Applymentioning

confidence: 99%

“…Since K as in (3.7) is nonempty compact and A as in (3.1) is finite, Rrxs A is K-full (cf. [30]). Note that (P 2 ) always has feasible points.…”

Section: Remark 43 We Applymentioning

confidence: 99%

“…Sponseldur & Dür [28] used polyhedral approximations to project a matrix to C n ; a CP-decomposition of a matrix can be obtained if it s an interior of C n . In [30], a semidefinite algorithm is proposed for solving the CP-matrix completion problem, which includes the CP-checking as a special case; a CP-decomposition for a general CP-matrix can be found by the algorithm.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Interiors of completely positive cones

Zhou

Fan

2015

J Glob Optim

Self Cite

View full text Add to dashboard Cite

ANWA ZHOU AND JINYAN FANÅ bstract. A symmetric matrix A is completely positive (CP) if there exists an entrywise nonnegative matrix B such that A " BB T . We characterize the interior of the CP cone. A semidefinite algorithm is proposed for checking interiors of the CP cone, and its properties are studied. A CP-decomposition of a matrix in Dickinson's form can be obtained if it is an interior of the CP cone. Some computational experiments are also presented. IntroductionA real nˆn symmetric matrix A is completely positive (CP) if there exist nonnegative vectors b 1 ,¨¨¨, b m P R n such that, where m is called the length of the decomposition (1.1). The smallest m in the above is called the CP-rank of A. If A is CP, we call (1.1) a CP-decomposition of A. Clearly, A is CP if and only if A " BB T for an entrywise nonnegative matrix B. Hence, a CP-matrix is not only positive semidefinite but also nonnegative entrywise.Let S n be the space of real nˆn symmetric matrices. For a cone C Ď S n , its dual cone is defined as C˚:" tG P S n : xA, Gy ě 0 for all A P Cu, where xA, Gy " tracepAGq is the trace inner product. Denote C n " tA P S n : A " BB T with B ě 0u, the completely positive cone, Cn " tG P S n : x T Gx ě 0 for all x ě 0u, the copositive cone.Both C n and Cn are proper cones (i.e. closed, pointed, convex and full-dimensional). Moreover, they are dual to each other [17]. The completely positive cone and copositive cone have wide applications in mixed binary quadratic programming [6], standard quadratic optimization problems and general quadratic programming [4], etc. Some NP-hard problems can also be formulated as linear optimization problems over the CP cone and the copositive cone (cf. [8]). We refer to [3,5,14] for the work in the field.The membership problems for the completely positive cone and the copositive cone are NP-hard (cf. [1,13]). To compute a CP-decomposition of a CP-matrix is also hard. Dickinson & Dür [9] studied the CP-checking and CP-decomposition of some sparse matrices. Sponseldur & Dür [28] used polyhedral approximations to project a matrix to C n ; a CP-decomposition of a matrix can be obtained if it 2000 Mathematics Subject Classification. Primary: 15A48, 65K05, 90C22, 90C26.

show abstract

Section: Remark 43 We Applymentioning

confidence: 99%

“…Since K as in (3.7) is nonempty compact and A as in (3.1) is finite, Rrxs A is K-full (cf. [30]). Note that (P 2 ) always has feasible points.…”

Section: Remark 43 We Applymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Interiors of completely positive cones

Zhou

Fan

2015

J Glob Optim

Self Cite

View full text Add to dashboard Cite

show abstract

“…Completely positive matrices (cp matrices) [6,28], as a special type of nonnegative matrices, have wide applications in combinatorial theory including the study of block designs [16], and in optimization especially in creating convex formulations of NP-hard problems, such as the quadratic assignment problem in combinatorial optimization and the polynomial optimization problems [1,2,3,17,31]. The verification of cp matrices is generally NP-hard unless for small scale matrices.…”

Section: Introductionmentioning

confidence: 99%

{0,1} completely positive tensors and multi-hypergraphs

Luo

et al. 2016

Linear Algebra and its Applications

View full text Add to dashboard Cite

Completely positive graphs have been employed to associate with completely positive matrices for characterizing the intrinsic zero patterns. As tensors have been widely recognized as a higher-order extension of matrices, the multi-hypergraph, regarded as a generalization of graphs, is then introduced to associate with tensors for the study of complete positivity. To describe the dependence of the corresponding zero pattern for a special type of completely positive tensorsthe {0, 1} completely positive tensors, the completely positive multihypergraph is defined. By characterizing properties of the associated multi-hypergraph, we provide necessary and sufficient conditions for any (0, 1) associated tensor to be {0, 1} completely positive. Furthermore, a necessary and sufficient condition for a uniform multihypergraph to be a completely positive multi-hypergraph is proposed as well.keywords: Completely positive tensor; {0, 1} completely positive tensor; multi-hypergraph ; (0, 1) tensor.

show abstract

“…They have wide applications in general quadratic programming [2], etc. Zhou and Fan [19] proposed a semidefinite algorithm for the CP-matrix completion problem, which includes the CP-checking as a special case. In this paper, we consider a more general problem: How do we check whether a matrix is partially positive for a given index set?…”

Section: Introductionmentioning

confidence: 99%

Partially positive matrices

Zhou

Fan

2014

Sci. China Math.

Self Cite

View full text Add to dashboard Cite

A real n × n symmetric matrix P is partially positive (PP) for a given index set I ⊆ {1, . . . , n} if there exists a matrix V such that V (I, :) 0 and P = V V T . We give a characterization of PP-matrices. A semidefinite algorithm is presented for checking whether a matrix is partially positive or not. Its properties are studied. A PP-decomposition of a matrix can also be obtained if it is partially positive.

show abstract

The CP-Matrix Completion Problem

Cited by 20 publications

References 37 publications

Interiors of completely positive cones

Interiors of completely positive cones

{0,1} completely positive tensors and multi-hypergraphs

Partially positive matrices

Contact Info

Product

Resources

About