I have a problem that involves finding a bottleneck. It appears to me to be a linear bottleneck assignment problem, but recognizing (and solving) such problems is far outside my area of expertise. If anyone can help, I would appreciate it.
My problem is this: I have a family of $n$ functions $f_i(x_i)$, where $x_i = a_i f_{i-1}(x_{i-1})$ for $i > 0$ and $a_i$ is a known constant. I wish to solve for $x_0$ subject to the constraint that each $f_i(x_i)$ has a (different) maximum value $M_i$ such that $0 \le f_i(x_i) \le M_i$. This then will tell me the values of all of the other $x_i$.
I am doing this within a computer program and can solve it iteratively, of course, but I prefer to solve it directly if I can feasibly do so. However, my efforts so far always seem to contain errors.