Solutions for Underdetermined Generalized Absolute Value Equations

TL;DR

This paper investigates conditions for the existence and multiplicity of solutions to underdetermined generalized absolute value equations, proposing iterative methods and extending results from square cases.

Contribution

It provides new sufficient conditions for solutions, especially infinitely many with specific sign patterns, and introduces iterative algorithms for solving underdetermined GAVEs.

Findings

Conditions guaranteeing solution existence

Characterization of solutions with specific sign patterns

Extension of results from square to underdetermined GAVEs

Abstract

An underdetermined generalized absolute value equation (GAVE) may have no solution, one solution, finitely many or infinitely many solutions. This paper is concerned with sufficient conditions that guarantee the existence of solutions to an underdetermined GAVE. Particularly, sufficient conditions are established for an underdetermined GAVE to have infinitely many solutions with no zero entry that possess a particular or any given sign pattern. Iterative methods are proposed for the case when the underdetermined GAVE does have a solution. Some existing results for square GAVE are also extended.