MMSE - How to minimize a complex error with respect to a set of real parameters
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.