I would like to search some function $f(x)$ "greedily" for a certain condition. Assume the condition $c = f(x_1)$ returned by this function is currently false and I can increase the input $x$ to the function until the function indicates that the condition is true. Let us assume the function is monotonic in the sense that once the condition becomes true, it will not become false again for any higher value of $x$.
I imagine finding (approximately) the critical point $x_\mathrm c$ where the condition becomes true in the following way:
- Calculate the function output $c = f(x_n)$ for exponentially increasing values of $x_n$: $x_n = x_1 + a^n,\ n=1,2,\ldots$ until $c$ becomes true.
- When $c$ becomes true for $n = n_\mathrm c$, start searching the interval between $n_\mathrm c - 1$ and $n_\mathrm c$ by using bisection until the desired precision is reached.
My rationale behind this is to quickly get to the point where I know an interval that $x_\mathrm c$ is contained in and then use bisection to narrow that interval becomes this seems like a fast solution.
If this is not a known algorithm, it's probably because it is stupid in some way. If so, please help me understand why.
It is not stupid. It is one of the basic line search methods that is called Bisection method, simple, but not the fastest to converge. Often in optimization one has to find the minimum of $F$, but many methods are actually searching for a stationary point, that is for a solution to $F'=0$. In your case, the equation $g(x)=f(x)-c=0$ can be interpreted as $F'=0$ for some other function, that you a trying to minimize. As the function $g(x)$ is very nice (monotonic), most line searches are going to work very well. You can try alternatively, Newton's method (may converge much faster), or quasi-Newton's method (if $f'$ is not available) or the parabolic interpolation method etc. The first step you described is known as the initial bracketing of the minimum.