In this paper, we focus on using optimization methods to solve matrix equations by transforming the problem of solving the Sylvester matrix equation or continuous algebraic Riccati equation into an optimization problem. Initially, we use a constrained convex optimization method (CCOM) to solve the Sylvester matrix equation with $\ell_{2,1}$-norm, where we provide a convergence analysis and numerical examples of CCOM; however, the results show that the algorithm is not efficient. To address this issue, we employ classical quasi-Newton methods such as DFP and BFGS algorithms to solve the Sylvester matrix equation and present the convergence and numerical results of the algorithm. Additionally, we compare these algorithms with the CG algorithm and AR algorithm, and our results demonstrate that the presented algorithms are effective. Furthermore, we propose a unified framework of the alternating direction multiplier method (ADMM) for directly solving the continuous algebraic Riccati equation (CARE), and we provide the convergence and numerical results of ADMM. Our experimental results indicate that ADMM is an effective optimization algorithm for solving CARE. Finally, to improve the effectiveness of the optimization method for solving Riccati equation, we propose the Newton-ADMM algorithm framework, where the outer iteration of this method is the classical Newton method, and the inner iteration involves using ADMM to solve Lyapunov matrix equations inexactly. We also provide the convergence and numerical results of this algorithm, which our results demonstrate are more efficient than ADMM for solving CARE.

翻译：本文聚焦于利用优化方法求解矩阵方程，通过将求解Sylvester矩阵方程或连续代数Riccati方程的问题转化为优化问题。首先，采用约束凸优化方法（CCOM）求解带有$\ell_{2,1}$范数的Sylvester矩阵方程，并给出了CCOM的收敛性分析与数值算例；然而结果表明该算法效率不高。为解决此问题，我们运用经典拟牛顿方法（如DFP和BFGS算法）求解Sylvester矩阵方程，并给出了算法的收敛性及数值结果。此外，我们将这些算法与CG算法及AR算法进行比较，结果表明所提算法是有效的。进一步地，我们提出了一种交替方向乘子法（ADMM）的统一框架，用于直接求解连续代数Riccati方程（CARE），并给出了ADMM的收敛性与数值结果。实验结果表明，ADMM是求解CARE的有效优化算法。最后，为提升求解Riccati方程优化方法的有效性，我们提出了Newton-ADMM算法框架，其中该方法的迭代外层为经典的牛顿法，内层则利用ADMM不精确求解Lyapunov矩阵方程，同时给出了该算法的收敛性与数值结果。结果表明，在求解CARE时，该算法的效率优于ADMM。