Function Optimization

Question

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 $x_{0}$?

@SergioParreiras: Yes, the maximum exist for my problem exists definitely — N8_Coder, Jul 25 '15 at 19:46
@Dr.SonnhardGraubner The sense is, if there are any optimization algorithms or other methods for non-linear rekursive functions. — N8_Coder, Jul 25 '15 at 19:47
introduce $x_1, \dotsc, x_{n+1}$ as variables with the recursion formula as equality constraints. then, minimize $x_{n+1}$. you can apply the usual theory and methods of nonlinear problems. — user251257, Jul 26 '15 at 02:29

score 1 · Answer 1 · answered Jul 26 '15 at 12:50

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.

Function Optimization

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