广义特征值极小扰动问题的一类黎曼共轭梯度法

研究含参数$l$非方矩阵对广义特征值极小扰动问题所导出的一类复乘积流形约束矩阵最小二乘问题.与已有工作不同,本文直接针对复问题模型,结合复乘积流形的几何性质和欧式空间上的改进Fletcher-Reeves共轭梯度法,设计一类适用于问题模型的黎曼非线性共轭梯度求解算法,并给出全局收敛性分析.数值实验和数值比较表明该算法比参数$l=1$的已有算法收敛速度更快,与参数$l=n$的已有算法能得到相同精度的解.与部分其它流形优化相比与已有的黎曼Dai非线性共轭梯度法具有相当的迭代效率,与黎曼二阶算法相比单步迭代成本较低、总体迭代时间较少,与部分非流形优化算法相比在迭代效率上有明显优势.

A RIEMANNIAN CONJUGATE GRADIENT APPROACH FOR SOLVING THE GENERALIZED EIGENVALUE PROBLEM WITH MINIMAL PERTURBATION

This paper presents an efficient approach for solving a kind of complex product manifold constrained matrix least squares problem, which derived from the $l$ parameterized generalized eigenvalue problem for nonsquare matrix pencils with minimal perturbation. Different from the existing work, the paper directly focuses on the complex problem model, combining the geometric properties of the considered complex product manifold and basing on the modified Fletcher-Reeves nonlinear conjugate gradient method on Euclidean space, a class of Riemannian nonlinear conjugate gradient algorithm is designed for solving the underlying problem, and the global convergence analysis is given. Numerical experiments and numerical comparisons are given to show that the proposed algorithm converges faster than the existing algorithm with parameter $l=1$, and can get the same accuracy as the existing algorithm with parameter $l=n$. Detailed comparisons with some latest methods, including some other gradient-based methods, some Riemannian second-order algorithms, and two non-manifold optimization algorithms are also provided to show the merits of the proposed approach.