The optimization problem is
$$I = \min_{N_i \in \mathbb N}\max_{1 \leq i \leq m} \frac{t_i}{N_i}$$
given the constraint
$$\sum_{i=1}^{n} N_i = N $$
$t_i$ are some known constants.
Now, intuitively, the $N_i$'s which most closely proportionately divides $N$ in the ration of $t_i$'s seems to be the answer, but how does one verify/refute this using proper optimization techniques?
You can perform binary search to find the value of $I$. Instead of your model consider $$I = \min_{N_i \in \mathbb N}x$$ given the constraints $$ \sum_{i=1}^{n} N_i = N \\ \forall i \frac{t_i}{N_i} \le x$$
Notice that the optimal solution of this model is the same as yours. Let's find the smallest value of $x$ such that there exist an answer to the problem. We can test for a fixed value of $x$ if there exist a solution. The following condition must hold: $$ \forall i N_i \ge \lceil \frac{t_i}{x}\rceil $$ It means that if $$ \sum \lceil \frac{t_i}{x}\rceil \le N$$ A solution to the problem clearly exist, just take $N_i = \lceil \frac{t_i}{x}\rceil$ and wathever it takes to $N_1$ so that first constraint holds. Since this function over $x$ is monotone, you can perform binary search to find it with as much precision as you want.
