Assume any recursive function like: (just for example, my rekursive function is just too big to write)
$x_{n+1}=\frac{(x_{n}-3)^{5}x_{n}^{2}}{a\sqrt{x_{n}}}$
(or any other non-linear function)
Is there any optimization-algorithm, which allows me to find the optimal value for $a$ if I want to find the Mimimum of $x_{n+1}=\frac{(x_{n}-3)^{5}x_{n}^{2}}{a\sqrt{x_{n}}}$ for a fixed $x_{0}$?
For a parametric recursion like $x_{n+1} = f(x_n, a)$ consider simply the following nonlinear constraint program $$ \min x_{n+1} $$ such that \begin{align} x_1 &= f(x_0, a), \\ \vdots\; &= \quad\vdots \\ x_{n+1} &= f(x_n, a), \\ \end{align} where $a, x_1, \dotsc, x_{n+1}$ are the variables and $x_0$ some given constant.
It has the usual form of a nonlinear program. So, you can apply the known theories and methods.