Strictly semi-positive tensors and the boundedness of tensor complementarity problems

TL;DR

This paper investigates the properties of strictly semi-positive tensors, establishing eigenvalue positivity, defining new constants for bounds, and analyzing the boundedness and error bounds of tensor complementarity problems.

Contribution

It introduces new constants related to eigenvalues, proves positivity of eigenvalues for principal sub-tensors, and provides global error bounds for tensor complementarity problems.

Findings

All H^+(Z^+)-eigenvalues of principal sub-tensors are positive.

Established upper bounds for key quantities related to semi-positivity.

Proved boundedness and monotonicity of these quantities and provided error bounds for related problems.

Abstract

In this paper, we prove that all H$^+$(Z$^+$)-eigenvalues of each principal sub-tensor of a strictly semi-positive tensor are positive. We define two new constants associated with H$^+$(Z$^+$)eigenvalues of a strictly semi-positive tensor. With the help of these two constants, we establish upper bounds of an important quantity whose positivity is a necessary and sufficient condition for a general tensor to be a strictly semi-positive tensor. The monotonicity and boundedness of such a quantity are established too. Furthermore, we present global error bound analysis for a class of the nonlinear complementarity problem defined by a strictly semi-positive tensor.