An adaptive cubic regularization algorithm for computing H- and Z-eigenvalues of real even-order supersymmetric tensors

Abstract

In this paper, we compute the H- and Z-eigenvalues of real even-order supersymmetric tensors by using the adaptive cubic regularization algorithm. First, the equation of eigenvalues of the tensor is represented by a spherically constrained optimization problem. Owing to the nice geometry of the spherical constraint, we minimize the objective function and preserve the constraint in an alternating way. The objective function of the optimization model is approximated by a cubic function with a tunable parameter, which is solved inexactly to obtain a trial step. Then the Cayley transform is applied to the trial step. Based on the ratio of actual and predicted reductions, a parameter is regulated to make sure that the cubic function is a good estimation of the original objective function. Finally we obtain an adaptive cubic regularization algorithm for computing an eigenvalue of a tensor (ACRCET). Furthermore, we prove that the sequence of iterations generated by ACRCET converges to an eigenvalue of a given tensor globally. In order to improve the computational efficiency, we propose a fast computing skill for Txr−2 which is the matrix-valued product of a hypergraph related tensor T and a vector x. Numerical experiments illustrate that the fast computing skill for Txr−2 is efficient and ACRCET is effective when computing H- and Z-eigenvalues of real even-order supersymmetric tensors.