Feasible Newton methods for symmetric tensor Z-eigenvalue problems

Finding a Z-eigenpair of a symmetric tensor is equivalent to finding a Karush-Kuhn-Tucker point of a sphere constrained minimization problem. Based on this equivalency, in this paper, we first propose a class of iterative methods to get a Z-eigenpair of a symmetric tensor. Each method can generate a...

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Bibliographic Details
Published in:Optimization methods & software 2023-05, Vol.38 (3), p.510-528
Main Authors: Xu, Jiefeng, Li, Dong-Hui, Bai, Xueli
Format: Article
Language:English
Subjects:
Convergence
descent method
Eigenvalues
Equivalence
global convergence
Iterative methods
Mathematical analysis
Newton method
Newton methods
Nonlinear equations
Optimization
quadratic convergence
Tensors
Z-eigenvalue problem
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Summary:Finding a Z-eigenpair of a symmetric tensor is equivalent to finding a Karush-Kuhn-Tucker point of a sphere constrained minimization problem. Based on this equivalency, in this paper, we first propose a class of iterative methods to get a Z-eigenpair of a symmetric tensor. Each method can generate a sequence of feasible points such that the sequence of function evaluations is decreasing. These methods can be regarded as extensions of the descent methods for unconstrained optimization problems. We pay particular attention to the Newton method. We show that under appropriate conditions, the Newton method is globally and quadratically convergent. Moreover, after finitely many iterations, the unit steplength will always be accepted. We also propose a nonlinear equations-based Newton method and establish its global and quadratic convergence. In the end, we do several numerical experiments to test the proposed Newton methods. The results show that both Newton methods are very efficient.
ISSN:1055-6788
1029-4937
DOI:10.1080/10556788.2022.2142586