Further study on tensor absolute value equations

TL;DR

This paper introduces tensor absolute value equations (TAVEs), explores their solution existence using degree and fixed point theories, and proposes a generalized Newton method for solving them, extending matrix AVEs theory to multilinear systems.

Contribution

It is the first to analyze solution existence and bounds for TAVEs, and to apply a generalized Newton method for their solutions.

Findings

Existence of solutions under certain conditions

Bound estimates for solutions in special cases

Preliminary success of the generalized Newton method

Abstract

In this paper, we consider the {\it tensor absolute value equations} (TAVEs), which is a newly introduced problem in the context of multilinear systems. Although the system of TAVEs is an interesting generalization of matrix {\it absolute value equations} (AVEs), the well-developed theory and algorithms for AVEs are not directly applicable to TAVEs due to the nonlinearity (or multilinearity) of the problem under consideration. Therefore, we first study the solutions existence of some classes of TAVEs with the help of degree theory, in addition to showing, by fixed point theory, that the system of TAVEs has at least one solution under some checkable conditions. Then, we give a bound of solutions of TAVEs for some special cases. To find a solution to TAVEs, we employ the generalized Newton method and report some preliminary results.