### Z-eigenvalue methods for a global polynomial optimization problem

SCI-EEI
WOS被引频次：77

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Qi, Liqun;Wang, Fei;Wang, Yiju

Qi, LQ

[Qi, Liqun] Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
[Wang, Fei] Department of Mathematics, Hunan City University, Yiyang, Hunan, China
[Wang, Yiju] School of Operations Research and Management Sciences, Qufu Normal University, Rizhao Shandong 276800, China

[Qi, Liqun] Hong Kong Polytech Univ, Dept Appl Math, Kowloon, Hong Kong, Peoples R China.

Applications. - Canonical forms - Eigenvalue methods - Eigenvalues - Higher dimensions - Higher orders - Local minimizers - Local minimums - Multidimensional cases - Numerical experiments - Optimization methods - Optimization solutions - Orthogonal transformation - Polynomial optimization - Polynomial optimization problems - Supersymmetric tensor - Z-eigenvalue

Mathematical Programming
ISSN：
0025-5610

2009

118

2

301-316

WOS:Article;EI:Journal article (JA)

ESI学科类别:计算机科学;WOS学科类别:Computer Science, Software Engineering;Mathematics, Applied;Operations Research & Management Science

WOS:000262316200005;EI:20090411867779

Natural Science Foundation of China 

As a global polynomial optimization problem, the best rank-one approximation to higher order tensors has extensive engineering and statistical applications. Different from traditional optimization solution methods, in this paper, we propose some Z-eigenvalue methods for solving this problem. We first propose a direct Z-eigenvalue method for this problem when the dimension is two. In multidimensional case, by a conventional descent optimization method, we may find a local minimizer of this problem. Then, by using orthogonal transformations, we convert the underlying supersymmetric tensor to a pseudo-canonical form, which has the same E-eigenvalues and some zero entries. Based upon these, we propose a direct orthogonal transformation Z-eigenvalue method for this problem in the case of order three and dimension three. In the case of order three and higher dimension, we propose a heuristic orthogonal transformation Z-eigenvalue method by improving the local minimum with the lower-dimensional Z-eigenvalue methods, and a heuristic cross-hill Z-eigenvalue method by using the two-dimensional Z-eigenvalue method to find more local minimizers. Numerical experiments show that our methods are efficient and promising.

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