A function $f$ has to be minimized,
$$f= \sum_i^k \beta_i e^{\alpha_i n_i}$$
where $\alpha_i$'s and $\beta_i$'s are known. How can I find optimal values of $n_i$'s that minimises $f$, given the constraint $\sum_i^k {n_i} \leq N$.
Edit:
$n_i \geq 0$ and $\alpha_i$ are negative $\beta_i$ are positive.
Sorry for the confusion. I couldn't apply Lagrangian due to large number of constraints making it unsolvable. Is there some other way to do or program that could help me.