Tensor Eigenvalue Complementarity Problems

TL;DR

This paper explores tensor eigenvalue complementarity problems, proposing methods to compute eigenvalues via polynomial optimization and semidefinite relaxations, with proven finite convergence and demonstrated efficiency through experiments.

Contribution

It introduces a novel formulation of tensor eigenvalue complementarity problems as polynomial optimization problems and provides an efficient solution approach with theoretical guarantees.

Findings

Finite convergence of the polynomial optimization approach.

Efficient computation of complementarity eigenvalues.

Numerical experiments confirm method effectiveness.

Abstract

This paper studies tensor eigenvalue complementarity problems. Basic properties of standard and complementarity tensor eigenvalues are discussed. We formulate tensor eigenvalue complementarity problems as constrained polynomial optimization. When one tensor is strictly copositive, the complementarity eigenvalues can be computed by solving polynomial optimization with normalization by strict copositivity. When no tensor is strictly copositive, we formulate the tensor eigenvalue complementarity problem equivalently as polynomial optimization by a randomization process. The complementarity eigenvalues can be computed sequentially. The formulated polynomial optimization can be solved by Lasserre's hierarchy of semidefinite relaxations. We show that it has finite convergence for general tensors. Numerical experiments are presented to show the efficiency of proposed methods.