两类自对偶锥上的最大最小互补特征值问题研究

The Eigenvalue Complementarity Problem (EiCP) is the generalization of the classical Matrix Eigenvalue Problem and plays an important role in some science and engineering fields such as the stability analysis of frictional elastic systems and the dynamic analysis of structural mechanical systems. At present, the numerical algorithms for solving the Eigenvalue Complementarity Problems can only be designed from the view of nonlinear optimization because of insufficient theoretical basis and it would be a NP hard problem for computing the whole.eigenvalues due to the fact that the number of the eigenvalues may grow exponentially with the dimension of the problems. This project focuses on the research on the Eigenvalue Complementarity Problems induced by two kinds of the self-dual cones (Nonnegative cone and Second-Order cone), especially the variational theories about the maximum or minimum eigenvalue and numerical algorithms which be suitable for computing the maximum and minimum eigenvalues or the part of special eigenvalues as well as the corresponding eigenvectors of the large scale Eigenvalue Complementarity Problems. On this basis of that, the Generalized Eigenvalue Complementarity Problem which comes from the unilateral frictional elastic systems will be further studied in this project, and the theories and efficient numerical methods will be provided for the related practical problems.

互补特征值问题是经典矩阵特征值问题的推广，它在摩擦弹性系统的稳定性分析和结构机械系统的动力学分析等科学工程领域均有重要应用。由于缺少必要的理论基础，目前只能从非线性最优化的角度设计求解的数值算法。由于特征值个数可能会随着问题的规模呈指数增长，因此求解中等或大规模矩阵的全部互补特征值是一个NP难问题。本项目主要研究两类自对偶锥（非负锥和二阶锥）上的互补特征值问题，特别是最大或最小互补特征值的变分理论和适合求解大规模互补特征值问题的最大最小特征值或部分特定特征值及其特征向量的数值算法，并进一步研究来源于单边摩擦弹性系统的广义互补特征值问题，设计适应于此类实际问题的高效数值求解方法。

本项目研究的互补特征值问题具有广泛的应用性，由于其理论和高效数值方法具有重要的实际意义，近年来它迅速成为了科学计算领域的热点关注问题之一。经过项目组成员四年的团结合作与努力，在对称与非对称互补特征值问题、广义对称互补特征值问题以及两类特殊最小二乘问题的理论与高效数值算法研究方面取得了较大进展。本项目的研究成果不仅丰富了自对偶锥上互补特征值问题和相关问题的理论及数值计算方法，并能为单边接触摩擦系统的稳定性分析、结构与流体机械系统的动力学分析与模拟等实际应用问题提供理论与方法支撑。同时，本项目研究过程中产生的模型、理论、方法与思想，也为进一步深入研究自对偶和非自对偶锥上互补特征值问题的理论与数值方法奠定坚实的基础。