In my studies I have come across formulations like the following:
$$\max_{f \in \mathcal{F}} f(x)$$
where $f$ is some function in a family of functions $\mathcal{F}$ and $x$ is some real-valued variable assumed fixed for the sake of this optimization. As far as I understand, $\max$ implies that we try to maximize $f(x)$, but I am uncertain as how to how I should read the text below ($f \in \mathcal{F}$). Does this mean I am maximizing under the assumption that $f \in \mathcal{F}$ (which seems an oddly unrelated piece of information), or does the text below specify what it is I am varying, i.e. that I maximize this by trying different $f$s (from my set $\mathcal{F}$)? Or does it mean something entirely different I haven't yet considered?
I apologize if this is a trivial question for someone with a more rigorous background in math, but I could not find a clear definition elsewhere.
Best Answer
The subscripted part describes how the item to be maximized can vary. So in $$\max_{f \in \mathcal{F}} f(x)$$ you are looking for the maximum value of $f(x)$ as $f$ varies over the collection $\mathcal F$.
Presumably, $x$ is fixed here, so this would determine a value dependent on $x$. If $\mathcal F$ is a finite collection, one might also write this as $$\max\{f_1(x), f_2(x),\ldots, f_n(x)\}.$$