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Feasible Newton's methods for symmetric tensor Z-eigenvalue problems

Finding a Z-eigenpair of a symmetric tensor is equivalent to finding a KKT 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's method. We show that under appropriate conditions, the Newton's 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's method and establish its global and quadratic convergence. In the end, we do several numerical experiments to test the proposed Newton's methods. The results show that both Newton's methods are very efficient.