I am working on an optimisation problem. I am trying to figure out if the objective function is convex/concave. I have taken derivative by simplifying the summations.
\begin{align} \sum_{n = 1}^{N} b_n \frac{1}{ 1 + \frac{k} {\sum_{i = 1}^m \sum_{j=1}^g a_{nij} x_{ij}}} \end{align}
where $a_{nij} \geq 0$ and $x_{ij}$ is a binary variable. Suppose $\sum_{i = 1}^m \sum_{j=1}^g a_{nij} x_{ij} = y_n$ then say, \begin{align} f(y_n) &= \frac{1}{1+k/y_n}\\ f'(y_n) &= \frac{k}{(1+k/y_n)^2 y_n^2}\\ f''(y_n) &= \frac{-2ky_n}{(1+k/y_n)^3 y_n^4} \end{align}
Since $a_{nij} \geq 0$ and $x_{ij} \in \{0,1\}$ then $y_n \geq 0$. Hence $f''(y_n) \leq 0$, therefore the objective function is concave.
Can you please help to understand if this approach is correct?