A sequential subspace projection method for extreme Z-eigenvalues of supersymmetric tensors

Summary Z-eigenvalues of tensors, especially extreme ones, are quite useful and are related to many problems, such as automatic control, quantum physics, and independent component analysis. For supersymmetric tensors, calculating the smallest/largest Z-eigenvalue is equivalent to solving a global minimization/maximization problem of a homogenous polynomial over the unit sphere. In this paper, we utilize the sequential subspace projection method (SSPM) to find extreme Z-eigenvalues and the corresponding Z-eigenvectors. The main idea of SSPM is to form a 2-dimensional subspace at the current point and then solve the original optimization problem in the subspace. SSPM benefits from the fact that the 2-dimensional subproblem can be solved by a direct method. Global convergence and linear convergence are established for supersymmetric tensors under certain assumptions. Preliminary numerical results over several testing problems show that SSPM is very promising. Besides, the globalization strategy of random phase can be easily incorporated into SSPM, which promotes the ability to find extreme Z-eigenvalues. Copyright © 2014 John Wiley & Sons, Ltd.

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