Mathematical Programming

, Volume 118, Issue 2, pp 301–316 | Cite as

Z-eigenvalue methods for a global polynomial optimization problem

  • Liqun QiEmail author
  • Fei Wang
  • Yiju Wang


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.


Polynomial optimization Supersymmetric tensor Orthogonal transformation Z-eigenvalue 

Mathematics Subject Classification (2000)



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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of Applied MathematicsThe Hong Kong Polytechnic UniversityKowloonHong Kong
  2. 2.Department of MathematicsHunan City UniversityHunanChina
  3. 3.School of Operations Research and Management SciencesQufu Normal UniversityRizhao ShandongChina

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