Z-eigenvalue methods for a global polynomial optimization problem

Liqun Qi, Fei Wang, Yiju Wang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

131 Citations (Scopus)


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.
Original languageEnglish
Pages (from-to)301-316
Number of pages16
JournalMathematical Programming
Issue number2
Publication statusPublished - 1 May 2009


  • Orthogonal transformation
  • Polynomial optimization
  • Supersymmetric tensor
  • Z-eigenvalue

ASJC Scopus subject areas

  • Software
  • Mathematics(all)


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