I am trying to minimize a function and I was hoping that someone can point me in the right direction. I have tried using Lagrangian multipliers and experimenting with simple cases, but nothing has worked. I have also tried using Mathematica to no avail.
The function is this:
$$ \sum_{i}^n \prod_{j}^n \bigl ( 1 - b_{ij} x_{i} \bigr ) $$
with the following constraints:
$$ \sum_{i}^n xi = 1 $$ $$ 0 \lt x_{i} \lt 1 $$
If it's easier, I can work with this function instead:
$$ \prod_{i}^n \prod_{j}^n \bigl (1 - b_{ij} x_{i} \bigr ) $$
I would like to find a solution for $ x_i $ in terms of the constants $ b_{ij} $, but like I said, I am having some trouble. Since I already know the $ b_{ij} $, if I can just get it in a form that is easy to solve numerically, that would be great. In practice, $ n $ is roughly 10,000 - so it needs to be somewhat efficient.
I recently posted a similar question on Math StackExchange. I don't know best practices, so if you would like me to merge this question with that one, then I will try to do so. Thanks!
Update: I could also work with this function:
$$ \sum_i^n \prod_j^n \bigl ( 1 - b_{ij} x_j \bigr ) $$
Here, I substituted $x_j$ for $x_i$. Would this make things easier?