# How to solve the convex optimization problem with complex variables by ADMM?

$$\begin{split} &\arg\min_{C} \frac{1}{2}c^HAc+\lambda\|C\|_{*}\\ &s.t. C_{i,j}=M_{i,j}. \end{split}$$

where $A\in \mathbb{C}^{n\times n}$ is a Hermite matrix, $c$ is a complex variable which is the vectorization of $C\in \mathbb{C}^{n\times n}$ . How to solve this problem by ADMM? I see some references in real variable condition which is not the same as in complex condition, such as the derivatives of complex variables.

• Your objective function includes the variable $C$ as well as what looks like another variable vector $c$. Is $c=C$? Is $c$ a constant? Something else? – Brian Borchers Nov 7 '16 at 7:45
• I'm sorry, I have rewritten the problem above. – panmd Nov 7 '16 at 11:29
• You might want to take a look at the following article, which describes an elegant framework for employing ADMM to minimize a real function in complex variables: hindawi.com/journals/mpe/2015/104531/abs – thetouristbr Jul 18 '17 at 9:39