how to solve a minimization problem when a variable is complex, and an objective function is real?

Question

Thank you for your help. I'd like to solve an optimization problem, with real-valued objective function and complex-valued control variable

for example, my minimization problem is minimize conj(f(x))*f(x) with respect to x (x is a complex variable)

how can I solve this problem?? Is it possible to use First order necessary condition just like as for real optimization problem??

Answer 1

Replace $x = a + bi$ and work with $a$ and $b$ as your decision variables. Nothing changes