Generalized tensor equations with leading structured tensors

TL;DR

This paper introduces generalized tensor equations (GTEs), explores their solution properties using Z+ tensors, and applies a Levenberg-Marquardt algorithm, expanding the theoretical and computational understanding of tensor systems.

Contribution

It defines Z+ tensors, proves existence of solutions for GTEs with such tensors, and develops an algorithm for solving these equations.

Findings

GTEs with Z+ tensors have at least one solution.

Local error bounds are established under certain conditions.

Preliminary numerical results demonstrate the algorithm's effectiveness.

Abstract

The system of tensor equations (TEs) has received much considerable attention in the recent literature. In this paper, we consider a class of generalized tensor equations (GTEs). An important difference between GTEs and TEs is that GTEs can be regarded as a system of non-homogenous polynomial equations, whereas TEs is a homogenous one. Such a difference usually makes the theoretical and algorithmic results tailored for TEs not necessarily applicable to GTEs. To study properties of the solution set of GTEs, we first introduce a new class of so-named ${\rm Z}^+$-tensor, which includes the set of all P-tensors as its proper subset. With the help of degree theory, we prove that the system of GTEs with a leading coefficient ${\rm Z}^+$-tensor has at least one solution for any right-hand side vector. Moreover, we study the local error bounds under some appropriate conditions. Finally, we employ a Levenberg-Marquardt algorithm to find a solution to GTEs and report some preliminary numerical results.