# Divide and conquer for optimizing weakly unimodal continuous function?

Is there a divide and conquer algorithm for optimizing weakly unimodal continuous functions?

My function has a flat line on the left and right and then there is a global minima in between but this global minima repeats for a while on what would seem like a straight line. To the left of the minima and right of the flat line on left, the behavior is monotonic decreasing. To the right of the minima and left of the flat line on right, the behavior is monotonic increasing. Also, the function is a positive function that is it's range is always >0. In addition, the function is defined only above zero. i.e, it's domain is always >0.

I see the main issue being that the initial bracket of three points may not be ideal. If the initial bracket was ideal, then does golden section search converge to optimal solution? Or is the issue bigger-even if the initial bracket was ideal?

• Could you add a few more details to your question (the exact definition of a weakly unimodal continuous function, problem size, any constraints -- explicit or implicit -- that are important to you)? – Christian Clason Apr 25 '16 at 7:37
• @ChristianClason I have added the information about the function! – user251385 Apr 26 '16 at 5:34

Suppose you had such an algorithm and ran it on a constant test function $f(x)=c$ for which it stopped after $k$ function evaluations (all of which had the same value.) Next run it on a function $g$ that has the same constant value at each of those $k$ points but also has a hidden bump that doesn't hit any of those points. Your algorithm would stop after the same $k$ iterations with an incorrect result.