First let provide an example:
Let's have
$$ f(x)=\sin(x)$$
$ x $ values that make this function are shown in the picture as $A=4.712$, $B=11$, $C=17.28$, $\dots$
I want to find the minimum $x$ value that minimizes $ f(x)=\sin(x)$ which is point $A$ with $x$ value of $x=4.712$.
how can I write it mathematically?
One idea is using $\operatorname{argmin}$ function $\operatorname{argmin} f(x)$.
another Idea is using bilevel optimization which finds the minimum value of $f(x)$.
Something like this:
$Z= \min \operatorname{argmin} \big(y=f(x)\big)$
but I don't know how to write it in a way that mathematically be reasonable and acceptable for publishing. I want to specifically define exactly indices and sets $t$ for $x$, $f(x)$ and $Z$ and the indices in lower part of $\operatorname{argmin}$ and other functions if required.
If you're just looking for an appropriate notation, then since $\textrm{arg min}_x f(x)$ is the set of $x$ values that minimise $f$, you could write $\min(\textrm{arg min}_x f(x))$ to represent the minimal element of that set.
Just remember that depending on the definition $f$ and its domain, this element might not exist - for example, for $f(x) = \sin x$ when the domain of $f$ is all real numbers, then $\textrm{arg min}_x f(x) = \{(2k - \frac{1}{2})\pi : k \in \mathbb{Z}\} = \{\ldots, -\frac{\pi}{2}, \frac{3 \pi}{2}, \ldots\}$ which has values going to infinity in both directions. Or $f(x) = 1 - H(x)$ where $H$ is the Heaviside step function has $\textrm{arg min}_x f(x) = \{x > 0\}$ whose infimum does not lie in the set, and it has no minimum element.