黎曼流形上向量优化问题的稳定性与算法研究

Vector optimization has become a significant research direction in optimization theory and application, it has achieved very successful applications in many fields such as industrial design、 national defense military、transportation、production management、chemical engineering、financial investment and ecological protection and so on. Recently, based on the idea of Riemannian geometry, how to extend the optimization theory and method from linear spaces to Riemannian manifolds has become a hot research spot of many scholars. The main purpose of this project will concentrate on the stability and algorithm for vector optimization problems on Riemannian manifolds, several aspects are studied, including: Firstly, we will study the stability of vector optimization problems on Riemannian manifolds, including the stability of Lipschitz type, ill-posedness and the regularity of measure; Secondly, resorting to the nonmonotone technique in linear spaces and the Quasi-Newton algorithm, the global convergence and local convergence rate for the nonmonotone Quasi-Newton algorithm of vector optimization problems is studied on Riemannian manifolds; Finally, on the applications, we will convert the joint diagonalization method from blind source separation to the unconstrained optimization problem on Stiefel manifolds, the stability of this problem is studied, and the convergence result is obtained by using the nonmonotone Quasi-Newton algorithm on Riemannian manifolds.

向量优化是优化理论及应用研究的一个重要领域，并在工业设计、国防军事、交通运输、生产管理、化学工程、金融投资以及生态保护等领域有着很成功的应用。近年来，基于黎曼几何的思想，如何将线性空间中的优化理论与方法推广到黎曼流形上已成为众多学者关注和研究的热点。本项目主要研究黎曼流形上向量优化问题的稳定性与算法，包括以下几个方面：首先，研究黎曼流形上向量优化问题的定量稳定性，包括解映射的Lipschitz稳定性、不适定性以及度量正则性；其次，将线性空间中的非单调搜索技术和黎曼流形上的拟牛顿算法相结合，探讨黎曼流形上向量优化问题非单调拟牛顿算法的全局收敛性和局部收敛速度；最后在应用上，将盲源分离中的联合对角化方法转化为求解Stiefel流形上的无约束优化问题，研究该问题解的稳定性，并将非单调拟牛顿算法应用于此问题，研究算法的收敛性。

黎曼流形优化理论与算法已成功应用于相关学科，如航空航天、工业设计、国防军事、交通运输、 生产管理、化学工程、金融投资、机器学习等领域。基于黎曼几何的思想，本项目主要研究黎曼流形上向量优化问题的稳定性与算法，已取得如下研究成果：首先，本项目将线性空间中向量优化问题的稳定性结论推广到黎曼流形上，得到了黎曼流形上向量优化问题解映射的Lipschitz稳定性、不适定性与度量正则性等；其次，本项目研究了黎曼流形上的非单调拟牛顿算法，并将黎曼流形上的非单调拟牛顿算法应用到向量优化问题中，得到算法的全局收敛性与局部收敛速度；最后，本项目将盲源分离中的问题转化为Stiefel流形上的无约束优化问题，并将黎曼流形上的非单调拟牛顿算法应用于该问题，得到了算法的收敛性质。本项目共完成6篇论文，其中3篇已发表，3篇已投稿。这些研究成果在一定程度上有助于该学科的发展。