Randomized iterative methods, such as the Kaczmarz method and its variants, have gained growing attention due to their simplicity and efficiency in solving large-scale linear systems. Meanwhile, absolute value equations (AVE) have attracted increasing interest due to their connection with the linear complementarity problem. In this paper, we investigate the application of randomized iterative methods to generalized AVE (GAVE). Our approach differs from most existing works in that we tackle GAVE with non-square coefficient matrices. We establish more comprehensive sufficient and necessary conditions for characterizing the solvability of GAVE and propose precise error bound conditions. Furthermore, we introduce a flexible and efficient randomized iterative algorithmic framework for solving GAVE, which employs sampling matrices drawn from user-specified distributions. This framework is capable of encompassing many well-known methods, including the Picard iteration method and the randomized Kaczmarz method. Leveraging our findings on solvability and error bounds, we establish both almost sure convergence and linear convergence rates for this versatile algorithmic framework. Finally, we present numerical examples to illustrate the advantages of the new algorithms.

翻译：随机迭代方法（如Kaczmarz方法及其变体）因其在大规模线性系统求解中的简洁性与高效性而备受关注。与此同时，绝对值方程(AVE)由于其与线性互补问题的关联，也引起了越来越多的研究兴趣。本文研究了随机迭代方法在广义绝对值方程(GAVE)中的应用。与现有大多数研究不同，我们的方法针对的是非方系数矩阵的GAVE问题。我们建立了更全面的GAVE可解性充要条件，并提出了精确的误差界条件。此外，我们引入了一种灵活高效的GAVE求解随机迭代算法框架，该框架采用根据用户指定分布抽取的采样矩阵，能够涵盖包括Picard迭代法和随机Kaczmarz法在内的多种经典方法。基于可解性与误差界的分析结果，我们证明了该通用算法框架的几乎必然收敛性与线性收敛速率。最后通过数值算例展示了新算法的优势。