Given the following sequences:
let value = $(b_0^{p_0})(b_1^{p_1})\cdots(b_n^{p_n})$
let productOfExponents = $p_0 \cdot p_1 \cdots p_n$
Where $p_i\geq 0$ and $p_i$ an element of $\mathbb{N}$ for all i
And $b_i < b_{i+1}$ and $b_i$ an element of $\mathbb{N}$ for all i
What is an efficient algorithm to minimize value given productOfExponents must be greater than anArbitraryNaturalNumber?
Update
The $b_i$ values are fixed and non-controllable. Only $p_i$ values can be modified.
Let $P$ be the given "anArbitraryNaturalNumber". To get a hold on the problem I propose to consider the $p_i$ as continuous variables and put $p_i\log b_i=:x_i>0$ $\ (1\leq i\leq n)$. Then we have to minimize $$\log v=\sum_{i=1}^n x_i$$ under the condition $\prod_{i=1}^n x_i= P'$ where $P':= P\ \prod_{i=1}^n\log b_i\ .$ Now it is well known that for a given value of the product $\prod_{i=1}^n x_i$ the sum $\sum_{i=1}^n x_i$ is minimal when all $x_i$ are equal. This implies that we should chose $$p_i:={\lambda \over \log b_i}\quad(1\leq i\leq n)$$ for a certain $\lambda$, and this $\lambda$ is determined by the condition $\prod_{i=1}^n p_i=P$. In terms of the original problem this means that all powers $b_i^{p_i}$ should have the same value $e^\lambda$.
Now this is only the beginning ...