﻿ Ph.D. Candidate Research Innovation Program ## “The research of Tensors and Related Polynomial Optimization” and “Low Rank Decomposition for Tensors”

First, I would like to express my heartfelt thanks to the Ph.D. Candidate Research Innovation Fund of Nankai University. We have the opportunity to participate in academic exchanges and academic conferences under the support of the Ph.D. Candidate Research Innovation Fund of Nankai University. So that we improve our works greatly. Under the support of the Ph.D. Candidate Research Innovation Fund of Nankai University, we have published three papers and have completed four papers for now. Those seven papers as following:

1. A new definition of geometric multiplicity of tensor eigenvalues and some results based on it.( Yiyong Li, Qingzhi Yang, Yuning Yang)

We give a new definition of geometric multiplicity for nonnegative tensors and based on this, we study the geometric and algebraic multiplicity of irreducible tensors' eigenvalues. We get the result that the eigenvalues with modulus spectral radius have the same geometric multiplicity. We alsoprove that two dimensional nonnegative tensors' geometric multiplicity of eigenvalues is equal to algebraic.

The following is the classification of tensors. 2. The solution methods for the largest eigenvalue (singular value) of nonnegative tensors and convergence analysis.( Zhongming Chen, Liqun Qi, Qingzhi Yang, Yuning Yang)

In this paper we study two solution methods for finding the largest eigenvalue (singular value) of general square (rectangular) nonnegative tensors. For a positive tensor, one can find the largest eigenvalue (singular value) based on the properties of the positive tensor and the power-type method. While for a general nonnegative tensor, we use a series of decreasing positive perturbations of the original tensor and repeatedly recall power-type method for finding the largest eigenvalue (singular value) of a positive tensor with an inexact strategy. We prove the convergence of the method for the general nonnegative tensor. Under a certain assumption, the computing complexity of the method is established. Motivated by the interior-point method for the convex optimization, we put
forward a one-step inner iteration power-type method, whose convergence is also established under certain assumption. Additionally, by using embedding technique, we show the relationship between the singular values of the rectangular tensor and the eigenvalues of related square tensor, which suggests another way for finding
the largest singular value of nonnegative rectangular tensor besides direct power-type method for this problem. Finally, numerical examples of our algorithms are reported, which demonstrate the convergence behaviors of our methods and show that the algorithms presented are promising.

One numerical result as following: 1. An inexact alternating Direction Method for structured variational Inequalities.( Zhongming Chen , Li Wan , Qingzhi Yang)

Recently, the alternating direction method of multipliers has attracted great attention. For a class of variational inequalities, this method is efficient when the subproblems can be solved exactly. However, the subproblems could be too difficult or impossible to be solved exactly in many practical applications. In this paper, we propose an inexact method for structured variational inequality problem based on the projection and contraction method. Instead of solving the subproblems exactly, we use the simple projection to get a predictor and correct it to approximate the subproblems’ real solutions. The convergence of the proposed method is proved under mild assumptions and its efficiency is also verified by some numerical experiments.

One numerical result as following: 1. A method with parameter for solving the spectral radius of nonnegative tensor. (Yiyong Li, Qingzhi Yang, Xi He)

In this paper, a method with parameter is proposed for solving the spectral radius of weakly irreducible nonnegative tensors. What’s more, we prove this method has an explicit linear convergence rate for indirectly positive tensors. Interestingly, the algorithm is exactly the NQZ method 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 explicit error for nonnegative tensors with a positive Perron vector. Besides, we promote an inexact power-type algorithm. Finally some numerical results are reported.

One numerical result as following: 1. A new model for solving the Maximum-Cut problem.(Yiyong Li, Qingzhi Yang)

In this paper, a new model is proposed for solving the Maximum-Cut problem. What's more, we get a bigger approximation rate and a approximation solution directly in our model. Furthermore, this model can be used to solve other quadratic \$0-1\$ programming. Besides, a new algorithm is proposed for solving the huge scale Maximum-Cut problem, which cost less time. Numerical results show that the model and the algorithm are efficient.

One numerical result as following: 1. Standard tensor and applications in studying the problem of singular values of tensors. (Yiyong Li, Qingzhi Yang)

In this paper, first we give the definition of standard tensors. Then we clarify the relationship between weakly irreducible tensors and weakly irreducible polynomial maps by the definition of standard tensors. And we prove that the singular values of rectangular tensors are the special cases of the eigenvalues of square tensors by the definition of standard tensors. In these views we present a generalized version of the weak Perron-Frobenius Theorem from nonnegative square tensors to nonnegative rectangular tensors, which get the weak Perron-Frobenius Theorem for nonnegative rectangular tensors under weaker conditions. Furthermore, by studying standard tensors we get some new results for rectangular tensors. Besides, considering the special structure of standard tensors corresponding to nonnegative rectangular tensors, we reach that the largest singular value is really geometrically simple under some weaker conditions.

One example as following: 7. A necessary and sufficient condition on the existence of positive perron eigenvectors of a square tensor.( Xi He, Yiyong Li, Qingzhi Yang)

In this paper, we give a necessary and sufficient condition concerning the existence of the positive perron eigenvector of a tensor. The existence of positive perron eigenvector is an important assumption in several theorems. What's more, in this paper, we show some other results that depend on the existence of positive perron eigenvector. Besides, we develop the algorithms which aim at computing the spectral radius of a general nonnegative tensor. Our algorithm is more suitable to computing the spectral radius in different situations. Then we give a more adoptable algorithm to finding the spectral radius and state the convergence analysis under some certain assumption. At last, we list some numerical experiment to show our suggestion algorithm is more efficient and effective and offer some application on our work.

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