A trust region algorithm for computing extreme eigenvalues of tensors

Abstract

<p style='text-indent:20px;'>Eigenvalues and eigenvectors of high order tensors have crucial applications in sciences and engineering. For computing H-eigenvalues and Z-eigenvalues of even order tensors, we transform the tensor eigenvalue problem to a nonlinear optimization with a spherical constraint. Then, a trust region algorithm for the spherically constrained optimization is proposed in this paper. At each iteration, an unconstrained quadratic model function is solved inexactly to produce a trial step. The Cayley transform maps the trial step onto the unit sphere. If the trial step generates a satisfactory actual decrease of the objective function, we accept the trial step as a new iterate. Otherwise, a second order line search process is performed to exploit valuable information contained in the trial step. Global convergence of the proposed trust region algorithm is analyzed. Preliminary numerical experiments illustrate that the novel trust region algorithm is efficient and promising.