凸约束下多变量线性矩阵方程问题的交替投影算法研究

The multivariable linear matrix equation problem appears frequently in many fields, such as mathematics, mechanics, linear system and control theory and so on, and it is an important branch of the numerical algebra. Alternating projecting method can often be a very effective means of finding a point in the intersection of two or more convex sets, which has been widely used in inverse eigenvalue problem, covariance control design and image restoration and so on. In this project we will for the first time apply this algorithm and its variants to solve the convex constrained multivariable linear matrix equation problems. The detailed research contents are: 1) combining with optimal approximation theory, we characterize numerically the projection onto every one of the closed convex sets and affine subspaces involved in the algorithms; 2) we establish alternating projection method to solve the problems proposed in this project, and also study various accelerated versions of the alternating projection method, and then observe how the relaxation parameter and the step size influence both the convergence and the convergence rate of these accerated algorithms; 3) in addition, we analyze the convergence of these accelerate algorithms by operator theory. In conclusion, apart from enriching and developing the research of matrix equation, this project can be also capable of promoting the development and application of matrix theory.

多变量线性矩阵方程问题出现在数学、力学、线性系统和控制理论等众多学科领域，是数值代数的重要分支。交替投影类算法是数值求解相交集合交点的最有效方法之一，该算法目前已广泛地应用于逆特征值问题、协方差控制和图像复原等诸多领域。本项目将首次系统地应用该算法理论研究凸约束下多变量线性矩阵方程求解问题。具体研究内容为：1）结合最佳逼近理论，研究任意矩阵对（束）在所涉及的闭凸集和仿射子空间上投影矩阵的高效计算；2）建立求解问题的交替投影算法及其加速算法，并分析加速算法中各参数对算法收敛性和收敛速度的影响；3）利用算子理论分析加速算法的收敛性。本项目的研究将丰富和发展矩阵方程问题的研究，促进矩阵理论的应用和发展。

多变量线性矩阵方程问题出现在数学、力学、线性系统和控制理论等众多学科领域，是数值代数的重要分支。本项目利用交替投影算法理论研究了闭凸约束下的多变量线性矩阵方程求解问题。.第一、 提出了线性子空间约束、非负约束和半正定约束下线性矩阵方程求解的交替投影算法。利用矩阵QR分解、奇异值分解、广义逆和矩阵形式的Krylov子空间等多种方法研究矩阵在仿射子空间内的投影矩阵；研究了交替投影算法的加速形式—定向交替投影算法；提出了求解最佳逼近问题的Dykstra交替投影算法；通过大量数值算例说明算法的可行性，并通过数值比较说明交替投影算法及其加速形式在迭代效率上比传统的算法有明显的优势。.第二、提出了有界约束、Q-正定约束和矩阵不等式（正定意义下的不等式）约束下矩阵方程求解的松弛交替投影算法，结合松弛交替投影算子的拟非扩张性给出了松弛交替投影算法的收敛性分析,通过大量数值算例说明算法的可行性和高效性。.第三、研究了矩阵不等式（非负意义下的不等式）约束下矩阵方求解问题。通过将问题等价转化为矩阵不等式非负偏差最小二乘问题，给出了基于投影的不动点形式的迭代求解算法，进而利用极分解理论证明了算法的收敛性，并给出了数值算例验证了算法的可行性。.第四、研究了对称半正定矩阵低秩逼近问题和广义Karhunen-Loeve变换中的低秩逼近问题；研究了混合Lyapunov 矩阵方程的Hermitian正定解和一类矩阵方程的扰动分析。