I have a least square optimization problem stated as below

[tex]\xi(z_1, z_2) = \sum_{i=1}^{M} ||r(z_1, z_2)||^2[/tex]

where [tex]\xi[/tex] denotes the cost function and [tex]r[/tex] denotes the residual and is a complex function of [tex]z_1, z_2[/tex].

My question is around [tex]||\cdot||[/tex]. Many textbooks only deal with real functions and say that this is the Euclidean norm, which is defined as the conjugated inner product of the residual, i.e. [tex]||r||^2 = conj(r)*r[/tex].

My question is that when I apply the gradient descent method to solve this problem, how to calculate [tex]\nabla \xi[/tex]? In particular, as [tex]\xi[/tex] includes [tex]conj(r)[/tex], we cannot take the derivative with respect to [tex]z_1, z_2[/tex] as [tex]conj(r)[/tex] is not an analytic function.

Should I use the un-conjugated inner product for the definition of the norm for this LS optimization with a complex residual function?

Any feedback is welcome. Thank you.

elgen