I have an objective function which I want to maximize. Ordinarily, one would perform some sort of optimization process on the parameters of this objective function in order to find parameter assignments that maximize the value of the function.
The problem is that the parameter to my objective function is a function from one countable set to another countable set, and I am unaware of any method that addresses such a scenario.
How do I solve such an optimization problem? Under what set of assumptions can it be solved? Even just a pointer at relevant theory would be appreciated.
Edit: To clarify, my objective function $f : (A \to B) \to \mathbb{R}$ is a function of $s : A \to B$, where $A$ and $B$ are both countable sets. I want to choose $s$ so that it maximizes the value of $f (s)$.