If $q_{n}$ is positive real number, $P$ is a positive integer, $ h_{n}$ and $\sigma$ are positive random numbers (real). Is there any optimization technique that would ensure that the resultant $\log_{2}\bigg(1+\dfrac{q_{n}\mid h_{n}\mid^{2}}{\sigma^{2}}\bigg)$ would have integer values for all n?
\begin{align} \max_{q_{n}}\!imize\sum_{n=1}^N \log_{2}\bigg(1+\dfrac{q_{n}\mid h_{n}\mid^{2}}{\sigma^{2}}\bigg), \\ \text{subject to}\\ \sum_{n=1}^N q_{n}\leq P \end{align}
What you have to do is to constrain the argument to be a power of $2$, i.e., you add the constraint $1 + q_n \frac{|h_n|^2}{\sigma^2} = \sum_{i = 0}^N \delta_{ni}2^i$ where $\delta_n$ is a binary vector satisfying $\sum_i \delta_{ni} = 1$. Note that this effectively means that you can solve the problem as a MILP, as the nonlinear function value is given by the value of $\delta_n$.