线性互补问题的数值分析

Linear complementarity problem has very wide applications. Hence, the numerical analysis on the problems has been a hot research topic in scientific computing. The project focuses on the error bounds, perturbation bounds and fast iterative algorithms for the linear complementarity problem containing: improve the current error bounds with theoretical analysis by presenting new parameter matrices; improve the current perturbation bounds by considering the equivalent linear complementarity problem; generalize the modulus-based matrix splitting iteration method by presenting new preconditioners, in particular, laying stress on the class of non-diagonal preconditioners; construct new algorithms for solving the special linear complementarity problems raised from application and analyze the stability and sensitivity of the new methods. The aims of this research are to promote the study on the numerical analysis of the linear complementarity problem, provide more methods and theories for solve the linear complementarity problem and perfect the theory of the nonlinear complementarity problem and other relative mathematical problems. Therefore, the project has profoundly theoretical values and wide applications in not only scientific and engineering computing but also computational mathematics itself.

线性互补问题具有极其广泛的应用背景，这类问题的数值分析是计算数学界的研究热点.本项目主要研究线性互补问题的误差界、扰动界以及快速迭代算法，具体内容包括：设计新的参数矩阵，改进已有的误差界，作相应的理论分析；从线性互补问题的等价问题入手，改进已有的扰动界；设计新的预条件子，改进推广模基矩阵分裂迭代法，其重点在于研究非对角的预条件矩阵类；针对应用问题中出现的特殊线性互补问题，构建新型算法，并分析新算法的稳定性和敏感性。本项目旨在促进线性互补问题的数值分析研究，为更有效地求解线性互补问题提供更多的方法和理论.本项目的开展也能促进非线性互补问题以及其他相关数学问题的研究.因此本项目的立项无论对工程和科学计算还是对计算数学本身的发展都有非常重要的理论和实际意义.

项目主要研究了互补问题的扰动界以及快速迭代算法，具体内容包括：①从线性互补问题的符号分析入手，改进已有的扰动界；②通过松弛、二步预处理技术等技术改进了求解线性互补问题的模基矩阵分裂迭代法;③把线性互补问题的数值分析推广应用到了非线性互补问题、隐互补问题、拟互补问题和水平线性互补问题的研究中。本研究旨在促进线性互补问题的数值分析研究，为更有效地求解互补问题提供更多的方法和理论.本研究也能促进非线性互补问题以及其他相关数学问题的研究.因此本研究无论对工程和科学计算还是对计算数学本身的发展都有非常重要的理论和实际意义。