闭凸集约束下线性矩阵方程求解的松弛交替投影算法

Relaxed Alternating Projection Method for Solving Linear Matrix Equation Problem under Closed Convex Constraint

研究线性矩阵方程AXB=C在闭凸集合R约束下的数值迭代解法.所考虑的闭凸集合R为(1)有界矩阵集合,(2)Q-正定矩阵集合和(3)矩阵不等式解集合.构造松弛交替投影算法求解上述问题,并用算子理论证明了由该算法生成的序列具有弱收敛性.给出了矩阵方程AXB=C求对称非负解和对称半正定解的数值算例,大量数值实验验证了该算法的可行性和高效性,并说明该算法与交替投影算法和谱投影梯度算法比较在迭代效率上的明显优势.

We discuss the existing relaxed alternating projection method for solving the linear matrix equation AXB = C under some closed convex constraints to X.The considered closed convex constrained set, denoted by ~, is （1） the set of bounded matrices, （2） the set of Q-positive definite matrices, （3） the solution set of a linear matrix inequality. We prove the weak convergence of the matrix sequence generated by the proposed algorithm, and present some numerical examples for solving AXB = C under symmetric nonnegative and symmetric positive semidefinite matrices constraint to illustrate the feasibility and efficiency of the proposed algorithm, and to show its clear superiority comparing with alternating projection method and spectral projected gradient method.