广义绝对值方程的随机迭代方法：可解性与误差界
Randomized iterative methods for generalized absolute value equations: Solvability and error bounds
AI总结：
本文研究非方系数矩阵广义绝对值方程的随机迭代求解，建立可解性充要条件与误差界，并提出涵盖 Picard 迭代和随机 Kaczmarz 方法的统一算法框架，证明其几乎必然收敛和线性收敛速率。
摘要：
随机迭代方法，如 Kaczmarz 方法及其变体，由于其在求解大规模线性系统时的简单性和高效性而受到越来越多的关注。与此同时，绝对值方程（AVE）因其与线性互补问题的联系而引起了越来越多的兴趣。本文研究了随机迭代方法在广义 AVE（GAVE）中的应用。我们的方法与大多数现有工作的不同之处在于，我们处理的是具有非方系数矩阵的 GAVE。我们建立了更全面的充分必要条件来刻画 GAVE 的可解性，并提出了精确的误差界条件。此外，我们引入了一个灵活且高效的随机迭代算法框架来求解 GAVE，该框架采用从用户指定分布中抽取的随机草图矩阵。该框架能够涵盖许多众所周知的方法，包括 Picard 迭代方法和随机 Kaczmarz 方法。利用我们在可解性和误差界方面的发现，我们为这一通用算法框架建立了几乎必然收敛性和线性收敛速率。最后，我们给出了数值例子来说明新算法的优势。
英文摘要：
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 randomized sketching 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.
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