Given a continuous, convex single-variable function that I want to minimize over a bounded interval [a,b], what options do I have? I have access to the numerical derivative, but not the analytic derivative.
This is done inside a loop that will be run an arbitrarily large number of times so it really needs to be as quick as possible. Bisection is elegant and simple, but I suspect you're missing out on efficiency by not utilizing convexity and slopes.
For this setting, I'd go with Golden Section Search.
Convexity implies unimodality, which is needed by this method.
Conversely, this method does not need derivatives. You can find derivatives numerically, but that's another way of saying "multiple function assessments"; might as well use these for golden-section partitions.