Pareto eigenvalue inclusion intervals for tensors

A Pareto eigenvalue of a tensor $ {\mathcal A} $ of order $ m $ and dimension $ n $ is a real number $ \lambda $ for which the complementarity problem \begin{document}$ \mathbf{0}\leq {\mathbf x} \bot (\lambda{\mathcal E}{\mathbf x}^{m-1}- {\mathcal A}{\mathbf x}^{m-1}) \geq \mathbf{0} $\end{document} admits a nonzero solution $ {\mathbf x}\in \mathbb{R}^n $, where $ {\mathcal E} $ is an identity tensor. In this paper, we investigate some basic properties of Pareto eigenvalues, including an equivalent condition for the existence of strict Pareto eigenvalues and the nonnegative conditions of Pareto eigenvalues. Then we focus on the estimation of the bounds of Pareto eigenvalues. Specifically, we propose several Pareto eigenvalue inclusion intervals, and discuss the relationships among them and the known result, which demonstrate that the inclusion intervals obtained are tighter than the known one. Finally, as an application of an achieved inclusion intervals, we propose a sufficient condition for judging that a tensor is strictly copositive.