I want to find a set of coefficients ($n \in R$) that solve the following optimization problem,
maximize $\prod_{i=1}^k(1-p_i)^{n_i}$ s.t. $\sum_{i=1}^k n_i = N$.
The $p$'s are known positive constants ($p \in (0,1)$). $N$ is some integer ($N>0$).
So far, I've tried the Lagrangian method but for $\frac{\partial{F}}{\partial{n_1}}$ I get an inter-dependent expression, i.e. $n_1=f(n_2,...,n_k,\lambda)$. Hence, I don't see how to derive $\lambda$ and find tractable expressions for $n_1,...,n_k$ which are purely functions of $\{p_1,...,p_k\}$.
I've also tried to maximize the $\exp(ln(f))$ but I end up with,
$\exp(n_1ln(1-p_1)+n_2ln(1-p_2)+...+n_kln(1-p_k))$
which doesn't seem to improve compared to the previous approach.
Please advise of an analytic (preferable) or numeric method to solve that, thanks!
Let $r_i = \ln(1-p_i)$.
You want to maximize $\prod_i (1-p_i)^{n_i}$. $\ln$ is monotonic, so we can instead maximize $\ln(\prod_i (1-p_i)^{n_i})$.
This simplifies to $\sum_i n_i \ln(1-p_i)$. Using my definition of $r_i$, this is $\sum_i n_i r_i$.
You want to maximize this subject to $\sum_i n_i = N$.
This is an LP with a trivial solution: Choose the $i$ corresponding to the largest $r_i$.
(If you don't trust me, you can write the dual of this LP and the answer is clear.)
Thus, the optimum solution to your original problem is to set $n_i=N$ corresponding to the index with the largest value of $\ln(1-p_i)$.
This is also equivalent to the index of the smallest value of $p_i$.