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

TL;DR

This paper introduces new Newton's methods for efficiently computing Z-eigenpairs of symmetric tensors, demonstrating their global convergence and superior performance through numerical experiments.

Contribution

It proposes novel Newton's methods for symmetric tensor Z-eigenvalue problems with proven global and quadratic convergence properties.

Findings

Newton's methods are globally convergent and quadratically convergent.

Numerical experiments confirm high efficiency of the proposed methods.

Finitely many iterations ensure unit step acceptance.

Abstract

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.