# Finding the minimum of a convex function with noisy evaluation

I want to find the argument of a function for which it is minimal. The function is expected to be convex but I cannot evaluate it exactly so I have to deal with the fact that there's noise on top. The noise is purely statistical and I roughly know its magnitude. Essentially, I run a Monte Carlo simulation for each evaluation of the function, so I can even control the error.

I don't need to know the minimum with too much precision. The higher the value of the function, the larger the yield of the simulation.

Thus my requirements are:

• really costly function evaluation
• 1D function
• no derivative available
• convexity up to statistical errors of know magnitude

Also, it's rather important that the whole procedure is not terribly complicated. I'm aware of this thread:

Finding a global minimum of a smooth, bounded, non-convex 2D function that is costly to evaluate

but that's just a little too much. I need something simple, yet reliable. I tried a simple bisection scheme but that really didn't handle the noise too well.

• since it's a 1d function, you can use golden section search, which finds the max of a unimodal function without derivatives. – Ronaldo Carpio Feb 3 '15 at 4:25
• Thanks for the pointer to golden section search. I previously tried a similar scheme but with equidistant spacing & occasional bisection. But using the golden ratio is a lot more elegant. However, I think I'll go with the surrogate model approach. – Jonas Greitemann Feb 3 '15 at 12:54