# Tensor absolute value equations

## TL;DR

This paper introduces tensor absolute value equations, explores their properties, establishes solution existence conditions, and proposes an algorithm with preliminary numerical validation.

## Contribution

It generalizes absolute value equations to tensors, links them to tensor complementarity problems, and develops a Levenberg-Marquardt-type algorithm for their solution.

## Key findings

- Established equivalence to tensor complementarity problems
- Provided sufficient conditions for solution existence
- Demonstrated algorithm efficiency through preliminary results

## Abstract

This paper is concerned with solving some structured multi-linear systems, which are called tensor absolute value equations. This kind of absolute value equations is closely related to tensor complementarity problems and is a generalization of the well-known absolute value equations in the matrix case. We prove that tensor absolute value equations are equivalent to some special structured tensor complementary problems. Some sufficient conditions are given to guarantee the existence of solutions for tensor absolute value equations. We also propose a Levenberg-Marquardt-type algorithm for solving some given tensor absolute value equations and preliminary numerical results are reported to indicate the efficiency of the proposed algorithm.