The problem is to find the complex-valued solution $z$ of the real-valued cost function $f(z)$:
$$z^* = \mathop{argmin}_\limits{z\in \mathbb{C}} (f- |z|^2)^2 + \alpha|z-d|^2,$$ where $f, \alpha \in \mathbb{R+}$ and $d \in \mathbb{C}$ are some constants.
The natural idea is to find the stationary point of this formula. So I try to use Wirtinger calculus, the $f(z)$ can be expressed as: $f(z,\bar{z}) = (f^2 - 2fz\bar{z} + z^2\bar{z}^2) + \alpha \overline{(z-d)}(z-d)$.
Then, the derivative is: $$\frac{\partial f(z,\bar{z})}{\partial z} = (2z\bar{z}^2-2f\bar{z}) + \alpha (\bar{z} - \bar{d}) = 0.$$
Therefore, the derivative involves $z$ and $\bar{z}$. I dont know how to find the $z^*$ since $\bar{z}$ is also unknown for me. Is there any suggestion? Or is it a correct way to find the solution? Thanks in advance!
EDIT: change the objective function