广义绝对值方程的松弛极大范数S-迭代法

中文摘要 AI 辅助

提出了一种用于求解广义绝对值方程（GAVEs）的松弛极大范数S-迭代法（RMNSI）。该方法将基于极大值的定点公式与恒定松弛参数相结合，避免了辅助矩阵的选择。假设A+B非奇异，两个阶段均需解具有相同系数矩阵A+B的线性系统，从而在整个迭代过程中可重复使用单次分解。建立了一个谱条件，该条件保证了GAVE的唯一可解性以及从任意初始向量出发的全局收敛性。刻画了松弛参数的允许范围，并推导了R-线性收敛性和误差估计。由于全局条件可能较为保守，还提出了基于解符号模式的局部收敛性分析。给出了互补衍生、稠密混合符号和非对称岭回归问题的数值实验，并与几种近期方法进行了比较，结果表明RMNSI具有竞争性的效率和精度。结果进一步表明，优选的松弛参数主要取决于矩阵结构和对角位移，而其对问题维度的依赖性通常较弱。

英文摘要

A relaxed maximum-based normal $S$-iteration method (RMNSI) is proposed for solving generalized absolute value equations (GAVEs). The method combines a maximum-based fixed-point formulation with a constant relaxation parameter and avoids the selection of an auxiliary matrix. Assuming that $A+B$ is non-singular, both stages require linear systems with the same coefficient matrix $A+B$, allowing a single factorization to be reused throughout the iteration. A single spectral condition is established that guarantees unique solvability of the GAVE and global convergence from an arbitrary initial vector. An admissible range of the relaxation parameter is characterized, and $R$-linear convergence and error estimates are derived. Since the global condition can be conservative, a local convergence analysis based on the solution sign pattern is also presented. Numerical experiments on complementarity-derived, dense mixed-sign, and asymmetric ridge-regression problems are presented, and comparisons with several recent methods demonstrate the competitive efficiency and accuracy of RMNSI. The results further indicate that the preferred relaxation parameter depends mainly on the matrix structure and diagonal shift, while its dependence on the problem dimension is generally weak.

发表机构

• Indian Institute of Information Technology Design and Manufacturing（印度信息技术设计与制造学院）

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