Let’s suppose I have a function f(m): \mathbb{N}\rightarrow\mathbb{R}.

What method should I use to minimize it?

The function is supposed to be smooth if we continue it from \mathbb{N} to [1,+\infty).

Also, it’s continuation is either monotonically increasing (minimum for m=1) or has a single local minima which is also the global one (minimum is somewhere in (1,+\infty)).

May be I should adapt some derivative free method to integer input variables?

If it is indeed the right way to go, can anyone suggest a derivative free method which is particularly good for functions of **single** variable?

Thank you.