I have the following optimization problem:
Minimize $\sum{C_i}{D_{i}^{x_{i}}}$
s.t. $\forall i \quad x_i \leq S_1$ $\quad$ $\sum{x_i * N_i} \leq S_2$
where $C_i,D_i,N_i,S_1,S_2$ are all know constants, $D_i$s are real numbers between 0 and 1. $N_i$s are positive integers. $C_i$s are positive real numbers. $x_s$ are the variables to solve and $x_i$s have to be natural numbers. I believe the objective function is convex, but because the variable $x_i$s are the exponents which make it hard to solve.