Z-eigenvalue methods for a global polynomial optimization problem
Z-eigenvalue methods for a global polynomial optimization problem
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全局多项式优化问题的 Z 特征值方法
DOI：
10.1007/s10107-007-0193-6
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发表时间：
2009-05-01
影响因子：
2.7
通讯作者：
Wang, Yiju
中科院分区：
文献类型：
--
作者：
Qi, Liqun;Wang, Fei;Wang, Yiju
高阶张量的最佳秩一逼近作为一个全局多项式优化问题，在工程和统计领域有着广泛的应用。与传统的优化求解方法不同，本文提出了求解该问题的Z-特征值方法。首先，我们提出了一个直接的Z-特征值方法时，该问题的维数为2。在多维情形下，通过传统的下降优化方法，我们可以找到该问题的局部极小点。然后，通过正交变换，我们将潜在的超对称张量转换为一个伪标准型，它具有相同的E-特征值和一些零元。在此基础上，我们提出了一种直接的正交变换Z-特征值法，用于求解三阶和三维情况下的这类问题。在3阶及更高维的情形下，利用低维Z-特征值方法改进局部极小点，提出了一种启发式正交变换Z-特征值方法，利用二维Z-特征值方法寻找更多的局部极小点，提出了一种启发式跨山Z-特征值方法.数值实验表明，我们的方法是有效的和有前途的。
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.