Find a Black-Box Convex Function Minimum

You are given access to a black-box function F(x): each call returns the function value at a real number x. You are also given a search interval [a, b]. Assume F is convex or unimodal on this interval, so the global minimum lies within [a, b]. Design an algorithm to return an approximate minimizer x* or the corresponding minimum value F(x*) within precision epsilon. You may not use gradients, symbolic derivatives, or any information other than function evaluations. Discuss the continuous-domain case, the integer-domain case, and the time and space complexity.