The error and perturbation bounds of the general absolute value equations

TL;DR

This paper investigates error and perturbation bounds for general absolute value equations, providing new theoretical bounds, computable estimates, and applications to linear complementarity problems with numerical validation.

Contribution

It introduces a class of absolute value functions and derives new error and perturbation bounds for AVEs, extending existing results and applying them to LCPs.

Findings

Derived new error bounds for AVEs

Provided computable estimates for bounds

Applied bounds to linear complementarity problems

Abstract

To our knowledge, the error and perturbation bounds of the general absolute value equations are not discussed. In order to fill in this study gap, in this paper, by introducing a class of absolute value functions, we study the error and perturbation bounds of two types of the general absolute value equations (AVEs): $Ax-B|x|=b$ and $Ax-|Bx|=b$. Some useful error bounds and perturbation bounds of the above two types of absolute value equations are provided. Without limiting the matrix type, some computable estimates for the above upper bounds are given. By applying the absolute value equations, a new approach for some existing perturbation bounds of the linear complementarity problem (LCP) in (SIAM J. Optim., 18 (2007), pp. 1250-1265) is provided. Some numerical examples for the AVEs from the LCP are given to show the feasibility of the perturbation bounds.