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.
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This work is supported by the Research Grant Council of Hong Kong and the Natural Science Foundation of China (Grant No. 10771120).
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Qi, L., Wang, F. & Wang, Y. Z-eigenvalue methods for a global polynomial optimization problem. Math. Program. 118, 301–316 (2009). https://doi.org/10.1007/s10107-007-0193-6
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DOI: https://doi.org/10.1007/s10107-007-0193-6