Tensor eigenvalue complementarity problems

Fan, Junxuan; Nie, Jiawang; Zhou, Anwa

doi:10.1007/s10107-017-1167-y

Cited by 38 publications

(18 citation statements)

References 31 publications

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“…Copositive tensor has wide applications in vacuum stability [5,12], polynomial optimization [23,25], hypergraph theory [5], tensor complementarity problem [1,2,6,26] and tensor eigenvalue complementarity problem [8,18,19]. Specifically, Kannike [12] studied the vacuum stability of scalar potential with the help of copositive tensors.…”

Section: LI Xinzhen Zhang Zheng-hai Huang and Liqun Qimentioning

confidence: 99%

“…For such problem, the existence of solution was discussed by Che, Qi and Wei [2] with a strictly copositive tensor. Fan, Nie and Zhou [8] formulated a tensor eigenvalue complementarity problem as a polynomial optimization, and proposed a numerical algorithm with assumption that the related tensor is strictly copositive.…”

Section: LI Xinzhen Zhang Zheng-hai Huang and Liqun Qimentioning

confidence: 99%

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Test of copositive tensors

Zhang

Huang

et al. 2019

Journal of Industrial &Amp; Management Optimization

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In this paper, an SDP relaxation algorithm is proposed to test the copositivity of higher order tensors. By solving finitely many SDP relaxations, the proposed algorithm can determine the copositivity of higher order tensors. Furthermore, for any copositive but not strictly copositive tensor, the algorithm can also check it exactly. Some numerical results are reported to show the efficiency of the proposed algorithm.

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Section: LI Xinzhen Zhang Zheng-hai Huang and Liqun Qimentioning

confidence: 99%

Section: LI Xinzhen Zhang Zheng-hai Huang and Liqun Qimentioning

confidence: 99%

Test of copositive tensors

Zhang

Huang

et al. 2019

Journal of Industrial &Amp; Management Optimization

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show abstract

“…As an important special case of complementarity problems, the eigenvalue complementarity problem (EiCP) for matrices also has been studied extensively, see [1,3,12,19,20,21,31] for example. Most recently, the EiCP for matrices has been generalized to tensors in [23], where the authors called it tensor generalized eigenvalue complementarity problem (TGEiCP) which has been further studied from both theoretical and numerical perspective in [10,11,14,33]. It is well known that the second-order cone is an important class of cones in applied mathematics, whose high applicabil-…”

Section: Introductionmentioning

confidence: 99%

A Class of Second-Order Cone Eigenvalue Complementarity Problems for Higher-Order Tensors

Hou

Ling

2016

J. Oper. Res. Soc. China

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In this paper, we consider the second-order cone tensor eigenvalue complementarity problem (SOCTEiCP) and present three different reformulations to the model under consideration. Specifically, for the general SOCTEiCP, we first show its equivalence to a particular variational inequality under reasonable conditions. A notable benefit is that such a reformulation possibly provides an efficient way for the study of properties of the problem. Then, for the symmetric and sub-symmetric SOCTEiCPs, we reformulate them as appropriate nonlinear programming problems, which are extremely beneficial for designing reliable solvers to find solutions of the considered problem. Finally, we report some preliminary numerical results to verify our theoretical results.

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“…For instance, various tensors with special structures were given in [13,29,30,33,46], including copositive tensors, M tensors, P -tensors and positive-definite tensors. On the other hand, many kinds of tensor optimization problem have been proposed, such as tensor complementarity problems (TCP) in [3,4,14,15,17,18,31,35,36,38,39,47,50], tensor eigenvalue problems (TEiP) in [7,19,25,41,43] and tensor eigenvalue complementarity problems (TEiCP) in [9,10,16,21,22,44]. As an important special case of complementarity problems, tensor eigenvalue complementarity problems have been developing rapidly since the past decades.…”

mentioning

confidence: 99%

“…From above statement, we know that many theoretical results of tensor eigenvalue complementarity problems have been obtained. On the other hand, there are also many solution methods proposed for eigenvalue complementarity problems and tensor eigenvalue complementarity problems, such as Lattice projection method (LPM) [1], implementable splitting method [22], Lasserre's hierarchy of semidefinite relaxations [16], semi-smooth Newton method [10], spectral projected gradient method [44], shifted projected power method [23] and inexact Levenberg-Marquardt method [20]. As far as we know, the nonlinear conjugate gradient methods are not used to solve tensor eigenvalue complementarity problems.…”

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confidence: 99%

Modified spectral PRP conjugate gradient method for solving tensor eigenvalue complementarity problems

Li¹,

Du²,

Chen³

2022

JIMO

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Tensor eigenvalue complementary problems, as a special class of complementary problems, are the generalization of matrix eigenvalue complementary problems in higher-order. In recent years, tensor eigenvalue complementarity problems have been studied extensively. The research fields of tensor eigenvalue complementarity problems mainly focus on analysis of the theory and algorithms. In this paper, we investigate the solution method for four kinds of tensor eigenvalue complementarity problems with different structures. By utilizing an equivalence relation to unconstrained optimization problems, we propose a modified spectral PRP conjugate gradient method to solve the tensor eigenvalue complementarity problems. Under mild conditions, the global convergence of the given method is also established. Finally, we give related numerical experiments and numerical results compared with inexact Levenberg-Marquardt method, numerical results show the efficiency of the proposed method and also verify our theoretical results.

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Tensor eigenvalue complementarity problems

Cited by 38 publications

References 31 publications

Test of copositive tensors

Test of copositive tensors

A Class of Second-Order Cone Eigenvalue Complementarity Problems for Higher-Order Tensors

Modified spectral PRP conjugate gradient method for solving tensor eigenvalue complementarity problems

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