黎曼流形上若干最优化方法及理论的研究

本项目将黎曼几何、数值分析、数值优化等有机的统一起来，充分利用黎曼几何的内在性质及线性空间中已有的关于最优化方法及理论的研究结果，对黎曼流形上的若干最优化方法及理论进行分析和研究。建立求解全能量最小值问题及多元特征值问题的Ulm法的收敛判据及收敛半径的估计；建立一般黎曼流形上的逼近点方法及次梯度算法的收敛性分析及收敛阶的估计；建立一般黎曼流形上的弱锐最优解等价刻画，并应用到流形上的各种数值方法的稳定性研究中；建立流形上的Clarke次微分的相关性质及等价刻画，并应用到流形上的优化控制问题的求解中。本项目是属于黎曼几何、变分分析、数值分析、非光滑分析、优化理论等多个分支的交叉学科，无论在理论上还是在应用前景上都有重要的研究价值和学术意义。

By Riemannian geometry, numerical analysis and optimization techniques, we will exploit several optimization problems on Riemannian manifolds. We will investigate convergence analysis of Ulm method for total energy minimization problems and multivariate eigenvalue problems. On the other hand, we will investigate the convergence of proximal point algorithm and subgradient algorithm for convex feasibility on Riemannian manifolds. We will exploit some equivalent conditions of weak sharp minima on Riemannian manifolds and apply our theoretical results to some important problems arising from applied areas. Finally, we will investigate nonsmooth analysis on Riemannian manifolds. We will establish the corresponding results of Clarke subdifferentials on Riemannian manifolds and apply our results to optimization and control problems on Riemannian manifolds. Our project is about Riemannian manifold, variational analysis, numerical analysis, nonsmooth analysis, optimization theory and so on, which is important for both theoretical and pratical areas.