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.

翻译：受Zamani和Hladík先前发表工作[Math. Program., 198 (2023), pp. 85-113]的启发，本文聚焦于一般绝对值方程的误差界与摄动界研究。据我们所知，目前尚未有文献讨论一般绝对值方程的这类界值。为填补这一研究空白，本文通过引入一类绝对值函数，研究了两类绝对值方程(AVEs)：$Ax-B|x|=b$ 与 $Ax-|Bx|=b$ 的误差界与摄动界。我们给出了上述两类绝对值方程若干有用的误差界与摄动界，并借助绝对值方程得到了水平线性互补问题(HLCP)的误差界与摄动界。此外，本文还给出了无约束条件下线性互补问题(LCP)的新摄动界，该结果在某种意义上弱于现有工作[SIAM J. Optim., 2007, 18: 1250-1265]。同时，在不限制矩阵类型的前提下，我们给出了上述上界的可计算估计值，在某些条件下比现有结果更优。通过LCP导出的AVEs数值算例验证了摄动界的可行性。