Lagrange Multipliers Optimization - Complex Functions
I found the following in Charles Therrien's "Discrete Random Signals and Statistical Signal Processing" in one of the Appendicies.

Say you have the function $Q(a)$ you wish to minimize such that $C(a)=0$, where $C(a)$ may be complex valued and $a$ may be a complex vector. The constraint really represents two real-valued constraints.
$$C_r(a)=0,\qquad 
C_i(a)=0,
$$
where $C_r(a)$ and $C_i(a)$ represent the real and imaginary components. The Lagrangian can be written as:
$$\mathcal{L}=Q(a) +\mu_1C_r(a)+\mu_2C_i(a),
$$
where $\mu_1$ and $\mu_2$ are real valued.

This can also be written more compactly using a single complex parameter $\lambda=\lambda_r+j\lambda_i$ as
$$\mathcal{L}=Q(a) +\lambda C(a)+\lambda ^*C^*(a)
$$
where $\lambda^*$  and $C^*(a)$ denote the complex conjugate of the quantities. The reason for this is that
$$
\lambda C(a)+\lambda ^*C^*(a)=2Re[\lambda C(a)]=2\lambda_rC_r(a)-2\lambda_iC_i(a)
$$
and letting $\mu_1=2\lambda_r$ and $\mu_2=-2\lambda_i$, then the two Lagrangians  are seen to be equivalent.

Note - to apply the gradient to a real valued function with respect to a complex parameter you should also be familiar with Brandwood "A complex gradient operator and its application in adaptive array theory" IEE Proceedings 130(1) 11-16, Feb, 1983. This essentially treats the variables $a$ and $a^*$ as two separate variables and you only have to take the derivative wrt to one of them to perform a minimization.