A Complete Semidefinite Algorithm for Detecting Copositive Matrices and Tensors

Nie, Jiawang; Yang, Zi; Zhang, Xinzhen

doi:10.1137/17m115308x

Cited by 44 publications

(35 citation statements)

References 63 publications

(74 reference statements)

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“…To overcome this drawback, in [38], Li et al proposed an SDP relaxation algorithm to test the copositivity of higher-order tensors. Very recently, Nie et al gave a complete semidefinite relaxation algorithm for detecting the copositivity of a matrix or tensor [39]. If the potential tensor is copositive, the algorithm can get a certificate for the copositivity.…”

Section: Bymentioning

confidence: 99%

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An SDP Method for Copositivity of Partially Symmetric Tensors

Wang

Chen

Che

2020

Mathematical Problems in Engineering

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In this paper, we consider the problem of detecting the copositivity of partially symmetric rectangular tensors. We first propose a semidefinite relaxation algorithm for detecting the copositivity of partially symmetric rectangular tensors. Then, the convergence of the proposed algorithm is given, and it shows that we can always catch the copositivity of given partially symmetric tensors. Several preliminary numerical results confirm our theoretical findings.

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Section: Bymentioning

confidence: 99%

“…When the input tensor is copositive but not strictly copositive, the algorithm may not stop in general. To solve this, motivated by the algorithm of [38,39], we propose a new algorithm to check the copositivity of partially symmetric tensors in this paper. e remainder of this paper is organized as follows.…”

Section: Bymentioning

confidence: 99%

An SDP Method for Copositivity of Partially Symmetric Tensors

Wang

Chen

Che

2020

Mathematical Problems in Engineering

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“…The problem of deciding the copositivity of a tensor is therefore co-NP-hard [14,34]. When p = 1, discussions on copositive tensors can be found in [40,50] and references therein.…”

Section: Copositivitiy Of Tensors a Tensormentioning

confidence: 99%

“…), 10,10,10,10) :(40,854,376; 10,000) ((2,2,2,3),6,6,6,8) : ( 42,659,866; 9,720)Table 1. (d, n 1 , .…”

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confidence: 99%

Best Nonnegative Rank-One Approximations of Tensors

Sun

Toh

2019

SIAM J. Matrix Anal. Appl.

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In this paper, we study the polynomial optimization problem of multi-forms over the intersection of the multi-spheres and the nonnegative orthants. This class of problems is NP-hard in general, and includes the problem of finding the best nonnegative rank-one approximation of a given tensor. A Positivstellensatz is given for this class of polynomial optimization problems, based on which a globally convergent hierarchy of doubly nonnegative (DNN) relaxations is proposed. A (zero-th order) DNN relaxation method is applied to solve these problems, resulting in linear matrix optimization problems under both the positive semidefinite and nonnegative conic constraints. A worst case approximation bound is given for this relaxation method. Then, the recent solver SDPNAL+ is adopted to solve this class of matrix optimization problems. Typically, the DNN relaxations are tight, and hence the best nonnegative rank-one approximation of a tensor can be revealed frequently. Extensive numerical experiments show that this approach is quite promising.2010 Mathematics Subject Classification. 15A18; 15A42; 15A69; 90C22. Key words and phrases. Tensor, nonnegative rank-1 approximation, polynomial, multi-forms, doubly nonnegative semidefinite program, doubly nonnegative relaxation method.

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“…Recently, checking copositivity of tensors has attracted the attention of mathematical workers. For example, Chen-Huang-Qi [11] studied some basic theory of copositivity detection of symmetric tensors and gave corresponding numerical algorithms of testing copositivity based on the standard simplex and simplicial partitions; Chen-Huang-Qi [12] revised algorithm with a proper convex subcone of the copositive tensor cone; Nie-Yang-Zhang [35] proposed a complete semidefinite relaxation algorithm for detecting the copositivity of a symmetric tensor and showed such a detection can be done by solving a finite number of semidefinite relaxations for all tensors; Li-Zhang-Huang-Qi [28] presented an SDP relaxation algorithm to test the copositivity of higher order tensors. For more structured properties and numerical algorithms of copositive tensors, see [13,39,40].…”

Section: Introductionmentioning

confidence: 99%

Analytical expressions of copositivity for fourth-order symmetric tensors

Song

2020

Anal. Appl.

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In particle physics, scalar potentials have to be bounded from below in order for the physics to make sense. The precise expressions of checking lower bound of scalar potentials are essential, which is an analytical expression of checking copositivity and positive definiteness of tensors given by such scalar potentials. Because the tensors given by general scalar potential are 4th order and symmetric, our work mainly focuses on finding precise expressions to test copositivity and positive definiteness of 4th order tensors in this paper. First of all, an analytically sufficient and necessary condition of positive definiteness is provided for 4th order 2 dimensional symmetric tensors. For 4th order 3 dimensional symmetric tensors, we give two analytically sufficient conditions of (strictly) cpositivity by using proof technique of reducing orders or dimensions of such a tensor. Furthermore, an analytically sufficient and necessary condition of copositivity is 1 showed for 4th order 2 dimensional symmetric tensors. We also give several distinctly analytically sufficient conditions of (strict) copositivity for 4th order 2 dimensional symmetric tensors. Finally, we apply these results to check lower bound of scalar potentials, and to present analytical vacuum stability conditions for potentials of two real scalar fields and the Higgs boson.

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A Complete Semidefinite Algorithm for Detecting Copositive Matrices and Tensors

Cited by 44 publications

References 63 publications

An SDP Method for Copositivity of Partially Symmetric Tensors

An SDP Method for Copositivity of Partially Symmetric Tensors

Best Nonnegative Rank-One Approximations of Tensors

Analytical expressions of copositivity for fourth-order symmetric tensors

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