Complex numbers are commonly used in power systems optimization problems, in particular, AC optimal power flow (OPF) problems. We have constraints of the form
$$p_{ij} + iq_{ij} = f(v_i,v_j),$$
where $p_{ij}$ and $q_{ij}$ are real-valued decision variables, $i=\sqrt{-1}$, $v_i$ and $v_j$ are complex-valued decision variables, and $f(\cdot)$ is a complex-valued function. Writing each $v_i$ as
$$v_i = |v_i|(\cos \theta_i + i\sin \theta_i),$$
where $|v_i|,\theta_i\in \mathbb{R}$,
and doing a lot of simplifying, we get a constraint of the form
$$p_{ij} + iq_{ij} = f_1(|v_i|,|v_j|,\theta_i,\theta_j) + if_2(|v_i|,|v_j|,\theta_i,\theta_j),$$
where $f_1(\cdot)$ and $f_2(\cdot)$ are real-valued functions.
The typical approach is then to decompose the complex constraints into two real-valued constraints:
$$\begin{align}
p_{ij} & = f_1(|v_i|,|v_j|,\theta_i,\theta_j) \\
q_{ij} & = f_2(|v_i|,|v_j|,\theta_i,\theta_j).
\end{align}$$
The functions $f_1$ and $f_2$ are nonconvex, so a lot of the research in OPF problems is on convexifying or approximating these constraints.
But the main point here is just that each complex constraint gets rewritten as a pair of constraints that have only real-valued parameters and decision variables. I'm curious as to whether the same approach is used in other optimization problems that involve complex numbers.
Also, the objective function is real-valued, although it can be a function of complex decision variables.
