The error and perturbation bounds for the absolute value equations with some applications

In this paper, inspired by in the pervious published work in [Math. Program., 198 (2023), pp. 85-113] by Zamani and Hlad\'{\i}k, we focus on the error and perturbation bounds for the general absolute value equations because so far, to our knowledge, the error and perturbation bounds for 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 bounds and perturbation bounds for two types of absolute value equations (AVEs): $Ax-B|x|=b$ and $Ax-|Bx|=b$. Some useful error bounds and perturbation bounds for the above two types of absolute value equations are presented. By applying the absolute value equations, we also obtain the error and perturbation bounds for the horizontal linear complementarity problem (HLCP). In addition, a new perturbation bound for the LCP without constraint conditions is given as well, which are weaker than the presented work in [SIAM J. Optim., 2007, 18: 1250-1265] in a way. Besides, without limiting the matrix type, some computable estimates for the above upper bounds are given, which are sharper than some existing results under certain conditions. Some numerical examples for the AVEs from the LCP are given to show the feasibility of the perturbation bounds.