B tensors and tensor complementarity problems

TL;DR

This paper establishes bounds on the solution set of tensor complementarity problems involving B tensors, linking these bounds to the tensors' structural properties, and provides spectral radius estimates based on diagonal entries.

Contribution

It proves the boundedness of solutions for tensor complementarity problems with B tensors and derives spectral radius bounds depending only on diagonal entries.

Findings

Solution set boundedness depends on tensor structure

Upper bounds for spectral radius depend on diagonal entries

B tensors are strictly semi-positive

Abstract

In this paper, one of our main purposes is to prove the boundedness of solution set of tensor complementarity problem with B tensor such that the specific bounds only depend on the structural properties of tensor. To achieve this purpose, firstly, we present that each B tensor is strictly semi-positive and each B$_0$ tensor is semi-positive. Subsequencely, the strictly lower and upper bounds of different operator norms are given for two positively homogeneous operators defined by B tensor. Finally, with the help of the upper bounds of different operator norms, we show the strcitly lower bound of solution set of tensor complementarity problem with B tensor. Furthermore, the upper bounds of spectral radius and $E$-spectral radius of B (B$_0$) tensor are obtained, respectively, which achieves our another objective. In particular, such the upper bounds only depend on the principal diagonal entries of tensors.