求解大规模稀疏线性互补问题的移位分裂多参数模迭代法

中文摘要 AI 辅助

提出了一种移位分裂多参数模迭代法，用于求解大规模稀疏线性互补问题。该方法通过移位矩阵分裂引入两个额外参数，同时保留原始系数矩阵。分裂矩阵的适当选择可生成Jacobi、Gauss-Seidel、SOR和AOR型格式；数值实验中考虑了Gauss-Seidel型形式。利用相关迭代矩阵的谱半径，建立了$P$-矩阵和$H_+$-矩阵的充分收敛性结果。对于结构化稀疏矩阵，推导了确保所提方法收敛的充分参数区间。还开发了一种基于网格搜索的算法，以选择数值实现中高效的近最优参数对。通过敏感性分析检验了不同参数矩阵选择的影响。在标准大规模稀疏测试问题上的数值实验表明，与所测试的投影型方法、新型模矩阵分裂方法和模矩阵双分裂方法相比，所提方法在迭代次数、残差和CPU时间方面均具有高效性。

英文摘要

A relaxed shifted matrix-splitting modulus-based iteration method is proposed for solving large-scale sparse linear complementarity problems. The method extends the classical modulus-based matrix-splitting framework by introducing two shift parameters into the matrix splitting while preserving the original coefficient matrix. Different choices of the splitting lead to shifted Jacobi, Gauss-Seidel, SOR, and AOR-type schemes. Sufficient convergence conditions are established for \(P\)-matrices and \(H_+\)-matrices. For structured sparse matrices, practical AOR-type conditions and explicit admissible intervals for the shift parameters are also derived. These intervals are used together with a grid-search procedure to select efficient parameter pairs. Numerical experiments on four large-scale sparse test problems, including a quasi-variational inequality model, show that the proposed method is competitive with several existing modulus-based and projected methods. A direct comparison with the two-parameter MAOR method further illustrates the effect of the proposed shifted splitting.

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）
• Indian Institute of Technology, Kanpur（坎普尔印度理工学院）

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