  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.
