Z-eigenvalue methods for a global polynomial optimization problem

  • Authors:

  • Liqun Qi;Fei Wang;Yiju Wang

  • Affiliations:

  • The Hong Kong Polytechnic University, Department of Applied Mathematics, Hung Hom, Kowloon, Hong Kong;Hunan City University, Department of Mathematics, Yiyang, Hunan, China;Qufu Normal University, School of Operations Research and Management Sciences, 276800, Rizhao Shandong, China

  • Venue:
  • Mathematical Programming: Series A and B
  • Year:
  • 2009

Quantified Score

Hi-index 0.00

Visualization

Abstract

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.