Z-eigenvalue methods for a global polynomial optimization problem

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-eigenvaluemethod for this problemwhen the dimension is two. Inmultidimensional 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. This work is supported by the Research Grant Council of Hong Kong and the Natural Science Foundation of China (Grant No. 10771120). L. Qi (B) Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong e-mail: [email protected] F. Wang Department of Mathematics, Hunan City University, Yiyang, Hunan, China e-mail: [email protected] Y. Wang School of Operations Research and Management Sciences, Qufu Normal University, Rizhao Shandong 276800, China e-mail: [email protected]