Black-box optimization of a single parameter function with high cost evaluation

Question

I need to solve a series of single parameter black-box minimization problem. The underlying cost functions are quite simple. They always have the same shape: a global minimum inside a fixed interval (-15000; 15000).

The constraints are :

• The function is not differentiable;
• The function is slow to evaluate.

I can solve these problems using a coarse scan followed by a fine scan. But I need between 30 and 50 evaluations. I'm sure that there is a better way to do it, but I can't find how.

Two examples of these cost functions :

Answer 1

Given that your function is apparently unimodal (single local minimum, which is global), you might try golden section search. The first four function evaluations result in about a 40% reduction in the initial interval. Each additional function thereafter again reduces the remaining interval by about 40%.

Answer 2

I would advise Bayesian Optimization. The benefits imho are that they don’t require a gradient, work for a wide variety of optimization problems and are made for when we are dealing with functions that are hard or slow to evaluate.