本项目将致力于研究源于协方差估计、信号和图像恢复等领域中若干条件数约束、Stiefel流形约束及Schattern-q范数下的矩阵最小二乘问题。结合矩阵理论和最优化方法，基于变量分裂和等价转化，以降低问题复杂度为目的，构造高效稳定的数值求解算法。具体研究内容为：1）设计最少计算量的网格搜索算法求条件数约束逼近问题，并寻求基于投影或分裂的算法及可行加速策略研究相关条件数约束矩阵最小二乘问题。2)构造基于变量分裂的部分非精确迭代算法求解Schattern-q范数下的约束矩阵最小二乘问题。3）结合Stiefel流形基本结论，设计可行或不可行方法求解Stiefel流形约束矩阵最小二乘问题，从搜索方向及搜索步长入手寻求基于Stiefel流形几何框架的可行迭代算法、或设计基于变量分裂的不可行迭代算法。

This project will study some matrix least squares problems under the condition number、Schattern-q norm and Stiefel manifold constraints, which come originally from covariance estimation, signal and image processing and linear systems and so on. Combining the matrix theory and the optimization theory together, meanwhile reducing both the computation quantity of algorithms and the complexity of the problems, the present research aims to design some efficient and stable algorithms to solve these relevant problems. The detailed research contents are listed as follows. 1) Establish the calculation of the projection onto the condition number constrained matrix set, and then design some effective algorithms and there acceleration schemes to solve the condition number constrained matrix approximation problems and associated matrix optimization problems. 2) Construct some partial inexact version of iteration methods to study the constrained matrix least squares problem with Schattern-q norm by using the combination of variable splitting and Bregman iteration. And tackle the subproblems generated by the variable splitting by linearizing or proposing some inexact inner iteration method with truly implementable inexactness criteria. 3) Present some feasible approach based on algorithms geometrical framework on the Stiefel manifolds, or design some splitting methods to study the matrix optimization problems with Stiefel manifold constraints. This is to be completed on the basis of the conclusion of Stiefel manifold, and starting from the search direction and the curvilinear search step.