本项目主要研究正交约束优化问题的非光滑算法和有约束的正交约束优化问题。正交约束优化问题在数据科学、统计学、分子模拟、电子结构计算等领域上有着重要应用。目前，信赖域算法在正交约束问题的无约束光滑问题计算效果非常良好。但是因为很多实际应用问题都是非光滑的并且带有约束，研究非光滑算法和约束流形优化问题成为必需。本项目的一个研究目标是非光滑算法，本项目将以欧式空间的(1)信赖域算法(2)束方法(3)次梯度算法为基础，把这些欧式空间的理论和算法推广到正交流约束优化问题的非光滑算法。因为稀疏主成分分析和稀疏线性判别分析等统计模型都是非光滑的，本项目还要将这些算法应用到这些模型中。另一个研究目标是研究有约束的正交约束优化问题。鉴于信赖域方法目前是无约束光滑流形优化问题中计算效果最好的，同时欧式空间带约束的信赖域方法已经存在，本项目致力于把欧式空间带约束的信赖域方法推广到正交约束优化问题。

This project is devoted to the nonsmooth algorithms for optimal problems with orthogonality constraints and constrained optimal problems with orthogonality constraints. Optimal problems with orthogonality constraints has many important applications in fields such as data science, statisticas, molecular simulation and electronic structures computations. Now some results on unconstrained smooth problems over orthogonality constraints has been established well. Since in real applictions the objective function is always nonsmooth and constrained, it is necessary to study nonsmooth constrained problems on manifolds. One goal of this project is to generalize the theory and algorithm of (1)trust region method (2) bundle method (3) subgradient method to optimal problems with orthogonality constraints. Because real world models such as sparse principal components analysis and sparse linear discriminant analysis always have nonsmooth objective functions, we will apply these methods to such models. Another goal of this project is to study the constrained optimal problems with orthogonality constraints. Since trust region methods is the best algorithms for smooth problems on manifolds and constrained trust region methods on Euclidean space also exists, we will generalize these methods to optimal problems to problems with orthogonality constraints.