Table of Content

30 March 2017, Volume 5 Issue 1

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Preface

Li-Qun Qi · Zong-Ben Xu · Qing-Zhi Yang

2017, 5(1):  1.  doi:10.1007/s40305-017-0152-8

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A Method with Parameter for Solving the Spectral Radius of Nonnegative Tensor

Yi-Yong Li· Qing-Zhi Yang · Xi He

2017, 5(1):  3.  doi:10.1007/s40305-016-0132-4

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In this paper, a method with parameter is proposed for finding the spectral radius of weakly irreducible nonnegative tensors. What is more, we prove this method has an explicit linear convergence rate for indirectly positive tensors. Interestingly, the algorithm is exactly the NQZ method (proposed by Ng, Qi and Zhou in Finding the largest eigenvalue of a non-negative tensor SIAM J Matrix Anal Appl 31:1090–1099, 2009) by taking a specific parameter. Furthermore, we give a modified NQZ method, which has an explicit linear convergence rate for nonnegative tensors and has an error bound for nonnegative tensors with a positive Perron vector. Besides, we promote an inexact power-type algorithm. Finally, some numerical results are reported.

The Adjacency and Signless Laplacian Spectra of Cored Hypergraphs and Power Hypergraphs

Jun-Jie Yue · Li-Ping Zhang· Mei Lu· Li-Qun Qi

2017, 5(1):  27.  doi:10.1007/s40305-016-0141-3

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In this paper, we study the adjacency and signless Laplacian tensors of cored hypergraphs and power hypergraphs. We investigate the properties of their adjacency and signlessLaplacian H-eigenvalues.Especially,wefind out the largest H-eigenvalues of adjacency and signless Laplacian tensors for uniform squids. We also compute the H-spectra of sunflowers and some numerical results are reported for the H-spectra.

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

Jiao-Jiao Hou · Chen Ling · Hong-Jin He

2017, 5(1):  45.  doi:10.1007/s40305-016-0137-z

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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.

An Inequality for the Perron Pair of an Irreducible and Symmetric Nonnegative Tensor with Application

Mao-Lin Che · Yi-Min Wei

2017, 5(1):  65.  doi:10.1007/s40305-016-0138-y

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The main purpose of this paper is to consider the Perron pair of an irreducible and symmetric nonnegative tensor and the smallest eigenvalue of an irreducible and symmetric nonsingularM-tensor. We analyze the analytical property of an algebraic simple eigenvalue of symmetric tensors.We also derive an inequality about the Perron pair of nonnegative tensors based on plane stochastic tensors. We finally consider the perturbation of the smallest eigenvalue of nonsingular M-tensors and design a strategy to compute its smallest eigenvalue.We verify our results via random numerical examples.

Upper Bounds for the Spectral Radii of Nonnegative Tensors

Jing-Jing Jia · Qing-Zhi Yang

2017, 5(1):  83.  doi:10.1007/s40305-016-0150-2

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In this paper, we present several sharper upper bounds for the M-spectral radius and Z-spectral radius based on the eigenvalues of some unfolding matrices of nonnegative tensors. Meanwhile, we show that these bounds could be tight for some special tensors. For a general nonnegative tensor which can be transformed into a matrix, we prove the maximal singular value of this matrix is an upper bound of its Z-eigenvalues. Some examples are provided to show these proposed bounds greatly improve some existing ones.

A Hybrid Second-Order Method for Homogenous Polynomial Optimization over Unit Sphere

Yi-Ju Wang· Guang-Lu Zhou

2017, 5(1):  99.  doi:10.1007/s40305-016-0148-9

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In this paper, we propose a hybrid second-order method for homogenous polynomial optimization over the unit sphere in which the new iterate is generated by employing the second-order information of the objective function. To guarantee the convergence, we recall the shifted power method when the second-order method does not make an improvement to the objective function. As the Hessian of the objective function can easily be computed and no line search is involved in the second-order iterative step, the method is not time-consuming. Further, the new iterate is generated in a relatively larger region and thus the global maximum can be likely obtained. The given numerical experiments show the efficiency of the proposed method.

Computing Geometric Measure of Entanglement for Symmetric Pure States via the Jacobian SDP Relaxation Technique

Bing Hua · Gu-Yan Ni · Meng-Shi Zhang

2017, 5(1):  111.  doi:10.1007/s40305-016-0135-1

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The problem of computing geometric measure of quantum entanglement for symmetric pure states can be regarded as the problem of finding the largest unitary symmetric eigenvalue (US-eigenvalue) for symmetric complex tensors, which can be taken as a multilinear optimization problem in complex number field. In this paper, we convert the problem of computing the geometric measure of entanglement for symmetric pure states to a real polynomial optimization problem. Then we use Jacobian semidefinite relaxation method to solve it. Some numerical examples are presented.

On the Bound of the Eigenvalue in Module for a Positive Tensor

Wen Li · Wei-Hui Liu · Seak-Weng Vong

2017, 5(1):  123.  doi:10.1007/s40305-016-0142-2

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In this paper, we propose a bound for ratio of the largest eigenvalue and second largest eigenvalue in module for a higher-order tensor. From this bound, one may deduce the bound of the second largest eigenvalue in module for a positive tensor, and the bound can reduce to the matrix cases.