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）因其在求解大规模线性系统中的简洁性和高效性而日益受到关注。与此同时，绝对值方程由于与线性互补问题的关联也吸引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中的应用。与现有大多数工作不同，我们的方法处理了非方系数矩阵的广义绝对值方程。我们建立了更全面的充要条件来刻画广义绝对值方程的可解性，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随机迭代算法框架用于求解广义绝对值方程，该框架采用用户指定分布生成的采样矩阵，能够涵盖包括Picard迭代法和随机Kaczmarz法在内的多种经典方法。借助可解性与误差界的研究成果，我们不仅建立了该通用算法框架的几乎必然收敛性，还得到了线性收敛速率。最后，通过数值算例展示了新算法的优势。