$Is there general theory for solving optimization problem of the following kind
\begin{align} &\max _{x_1,x_2,...,x_N} N \\ \text{ s.t.}& \sum_{i=1}^N f(x_i) \le a\\ &\sum_{i=1}^N g(x_i) \le b\\ & \sum_{i=1}^N x_i \le c \end{align}
where $f()$ and $g()$ are given convex, non-negative functions. The 'weird' property of this problem is that we have to optimize over the argument of the summation. At this stage I am looking at reference and examples.
Thank you in advance.