两类特征值互补问题及反问题的黎曼优化方法及其应用

In this project, we mainly consider two special kinds of eigenvalue complementarity problem and inverse eigenvalue problem: generalized symmetric eigenvalue complementarity problem and inverse Pareto eigenvalue problem. These two kinds of problems belong to special types of complementarity problems, most existing researches for these two kinds of problems devise numerical methods from constrained Euclidean optimization viewpoint, while we wish to construct some stable and efficient Riemannian optimization algorithms from Riemannian optimization viewpoint. Based on the analysis of the characteristics of the original problems, we can transformed the first kind of problem into equivalent Riemannian convex constrained optimization problem, and reformulate the second kind of problem into equivalent nonlinear matrix equation with Riemannian constraint. For the first problem, we will try to construct the efficient Riemannian projected descent direction method and the Riemannian feasible direction method for effective solution. For the second problem, we will try to construct the efficient Riemannian nonlinear conjugate gradient method and the Riemannian quasi-Newton method for effective solution. At last, we will investigate the convergence and stability properties of these algorithms, and extent these methods to solve some important realistic application problems.

在本项目中，我们主要考虑两类特殊的特征值互补问题及反问题：广义对称特征值互补问题和Pareto特征值反问题。这两类问题属于特殊类型的互补问题，现有的关于这两类问题的研究大多是从欧式空间约束优化的角度设计算法，现在我们希望从黎曼优化的角度设计稳定而有效的黎曼优化算法来求解这两类特殊问题。通过分析问题本身的特性，我们将第一类问题等价转化为黎曼凸约束优化问题，将第二类问题等价转化为具有黎曼约束的非线性矩阵方程。对于第一类问题，我们将设计有效的黎曼投影型算法和黎曼可行方向法进行求解；对于第二类问题，我们将设计黎曼非线性共轭梯度法和黎曼拟牛顿型算法进行求解。最后，我们分析算法的收敛性和稳定性，并将算法推广应用于求解一些实际问题。

在本项目中，我们主要考虑两类特殊的特征值互补问题和反问题以及延伸出的相关黎曼优化问题。研究内容主要包括针对给定部分特征对的随机矩阵特征值反问题、参数化最小二乘特征值反问题、切向量场的零点求解问题和自伴随切向量场的零点求解问题，我们设计了相应的黎曼修正Fletcher-Reeves非线性共轭梯度型算法、预处理黎曼高斯-牛顿型算法、非单调黎曼Derivative-Free PRP型算法以及黎曼非单调谱梯度法等有效的算法并分析了算法的收敛性。所有算法均使用Matlab进行了数值模拟试验，数值实验结果表明这些算法可以稳定且有效地应用于求解大规模问题，其数值效果在计算迭代次数和计算时间等方面相对于已有的算法都有一定程度的改进。同时项目组成员还针对非线性薛定谔方程高阶奇点解的解析性问题做了相关研究。