I am interested in optimizing a function $f(x)$ but evaluating the function for some $x$ is costly.
Each evaluation of the function takes a time $t$ and I have a total time $T$ in which to attempt to optimize this.
Has there been research into how to optimize a function with time constraints?
I realize that this might be problematic because we usually compare optimization methods by their asymptotic behaviour (as $T$ tends to infinity) and here I'm looking at the non-asymptotic behaviour.