MMSE - How to minimize a complex error with respect to a set of real parameters Suppose there's a complex signal $X(k)$ (where $k \in \{0, 1, 2,...,N - 1\}$) corrupted by additive complex noise. Its estimate $\hat{X}(k)$ is a linear combination of a set of real parameters $A_r$ ($r \in \{1, 2, 3,..., R$})
$$\hat{X}(k) = \sum_{r=1}^R A_rZ_r(k)$$
where $Z_r(k)$ is complex (and known).

I wish to obtain the real values of $A_r$ for which the error is minimized. If I simply differentiate the mean squared error (MSE) $\frac{1}{N}\sum_{k = 0}^{N-1}(X(k) - \hat{X}(k))^2$ with respect to each $A_r$ and set the resulting derivatives equal to zero, the values of $A_r$ I'll obtain will be complex, so that's not a solution.

My question is: how do I obtain the optimal values of $A_r$ such that the MSE is minimized, under the constraint that each $A_r$ should be real?
A very common approach is to consider $X(k) \;\text{and} \;\hat{X}(k)$ as  elements vector space $C^R$, and consider the distance between the 2 vectors as a norm, which are real, positive or zero, and satisfy the triangle inequality.  

So using vector-matrix notation with the vectors as column vectors:

$$ \text{error}^2 =(\mathbf{X}-\mathbf{\hat{X}})^H
(\mathbf{X}-\mathbf{\hat{X}})
$$
where $H$ is conjugate transpose.

Constraint is satisfied when imaginary part of $A_r$ is zero.

$$
\frac{1}{2}(A_r - A_r^*)=0
$$
which is appended to the objective as a lagrange multiplier(s)
$$
\text{error}^2+\sum \lambda_r \frac{1}{2}(A_r - A_r^*)
$$

These sorts of problems are easier when you use Brandwood derivatives,

D. H. Brandwood, "A complex gradient operator and its application in adaptive array theory," in Communications, Radar and Signal Processing, IEE Proceedings F, vol. 130, no. 1, pp. 11-16, February 1983.
doi: 10.1049/ip-f-1.1983.0003
Abstract: The problem of minimising a real scalar quantity (for example array output power, or mean square error) as a function of a complex vector (the set of weights) frequently arises in adaptive array theory. A complex gradient operator is defined in the paper for this purpose and its use justified. Three examples of its application to array theory problems are given.
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&amp;arnumber=4645581&amp;isnumber=4645575

This is the convention used in 

Van Trees, Harry L. Optimum array processing: Part IV of detection, estimation and modulation theory. Vol. 1. New York, NY, USA: John Wiley &amp; Sons, 2002.