Solutions for Underdetermined Generalized Absolute Value Equations
Cairong Chen, Xuehua Li, Ren-Cang Li
TL;DR
The paper addresses the underdetermined generalized absolute value equation $A x - B|x| = b$ with $m<n$, establishing sufficient conditions for existence and, in particular, for infinitely many solutions with no zero entry under prescribed sign patterns. It introduces contraction-based fixed-point arguments that guarantee a unique solution for all $b$ in a $b$-independent setting, and complementary rank-based submatrix conditions that yield infinitely many solutions for any $b$ in a $b$-dependent setting. In the dependent-on-$b$ case, the analysis leverages a sign-pattern transformation $y= ext{diag}(s)x$ to derive infinite solution families for certain $s$, along with criteria for nonnegative solutions and a variety of illustrative examples. Practical iterative methods are derived to compute solutions when they exist, and several results extend known square-GAVE theory to the underdetermined regime, with implications for related optimization problems and interval analyses.
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.