I have a function which takes as inputs n-dimensional (say n=10) vectors whose components are real numbers varying from 0 to a large positive number A say 50,000, ends included. For any such vector the function outputs an integer from 1 to say B=100. I have this function and want to find its global minima. Which are the best techniques suggested to solve this problem? Problem arises because this function assumes ""constant on an (n-D) neighbourhood"" behavior as rightly pointed out by Peter. The function does not have any other interesting property other than that as far as i can tell.
Hi,
As a first trial, i would recommend to use a Differential Evolution algorithm or a PSO with a large population size: 15 x n , where n is the input variables.
An other method would be to use some surrogate optimization algorithm that will learn a local meta-model to approximate the local piecewise function. Indeed, the surrogate model should rather be a random forest or a linear regression.
In practice you never actually deal with real numbers. You always have an accuracy you are working to so you can always formulate the problem so that the real numbers are integers (so if you want two decimal places accuracy you can take 7.2 to be 72 in your formulation or something along these lines). Once you have done this you have a combinatorial optimization problem of some sort. There are a variety of algorithms for this sort of thing, not knowing much more I can't recommend one of them, but you could try some sort of simulated annealing.
Basically, the derivatives of the functions of real numbers aren't going to really help you, so anything that uses something like a gradient isn't going to work.
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