Interval B-Tensors and Interval Double B-Tensors
Li Ye, Yisheng Song
TL;DR
This work extends interval matrix theory to tensors by introducing interval B-tensors and interval double B-tensors and by deriving verifiable criteria that depend only on the interval endpoints $\underline{\mathcal{A}}$ and $\overline{\mathcal{A}}$. It establishes a concrete, extreme-point based characterization for interval B-tensors and interval double B-tensors, explores their relationships with interval Z-tensors and interval P-tensors, and provides simplified checks for circulant structures. Under even order $m$ and symmetry, these interval tensor classes imply interval $P$-tensors, linking uncertainty in tensor entries to positivity properties relevant to optimization and complementarity. The results supply a rigorous foundation and practical tools for applications in polynomial optimization and uncertain multilinear systems, with explicit reductions in special cases like circulant tensors.
Abstract
This paper systematically investigates the properties and characterization of interval B-tensors and interval double B-tensors. We propose verifiable necessary and sufficient conditions that allow for determining whether an entire interval tensor family belongs to these classes based solely on its extreme point tensors. The study elucidates profound connections between these interval tensors and other structured ones such as interval Z-tensors and P-tensors, while also providing simplified criteria for special cases like circulant structures. Furthermore, under the condition of even order and symmetry, we prove that interval B-tensors (double B-tensors) ensure the property of being an interval P-tensor. This work extends interval matrix theory to tensors, offering new analytical tools for fields such as polynomial optimization and complementarity problems involving uncertainty.