A Newton-type algorithm for the tensor eigenvalue complementarity problem and some applications

Abstract

We focus on establishing an algorithm to solve the tensor eigenvalue complementarity problem (TEiCP), and we have two contributions in this paper. First, a smoothing Newton-type algorithm is proposed for the TEiCP based on the CHKS smoothing function. Its global convergence is established under some mild conditions. Numerical experiments are reported to show that the proposed algorithm is efficient and could detect more solutions than some existing methods. Second, we apply the proposed algorithm to solve the eigenvalue problem of nonnegative tensors. We analyze the relationship between the TEiCP and the H-eigenpair and Z-eigenpair problems of an irreducible nonnegative tensor. We show that the TEiCP with an irreducible nonnegative tensor and unit tensor has a unique solution, which is just the unique positive H-eigenpair of the irreducible nonnegative tensor. We also show that the solution set of the TEiCP with an irreducible nonnegative tensor and identity tensor is nonempty and its solutions are positive. Moreover, we can obtain positive Z-eigenpairs of the irreducible nonnegative tensor from these solutions. Finally, we also apply the proposed algorithm to find the unique positive H-eigenpair and a positive Z-eigenpair of an irreducible nonnegative tensor; the numerical results indicate its efficiency and promising performance.