How To Effectively Solve A Compound Cost Function Optimisation Problem?

I want to solve the following optimization problem with Python: I have a black box function f with multiple variables as input. The execution of the black box function is quite tim

Solution 1:

I’m still not 100% sure of your approach, but does this formula ring true to you:

A * max(f(x)) + B * max(standardDeviation(f(x)))

?

If it does, then I guess you may want to consider that maximizing f(x) may (or may not) be compatible with maximizing the standard deviation of f(x), which means you may be facing a multi-objective optimization problem.

Again, you haven’t specified what f(x) returns - is it a vector? I hope it is, otherwise I’m unclear on what you can calculate the standard deviation on.

The picture you posted is not so obvious to me. F(x) is the entire black curve, it has a maximum at the point v, but what can you say about the standard deviation? To calculate the standard deviation of you have to take into account the entire f(x) curve (including the point u), not just the neighborhood of u and v. If you only want to get the standard deviation in an interval around a maximum for f(x), then I think you’re out of luck when it comes to frameworks. The best thing that comes to my mind is to use a local (or maybe global, better) optimization algorithm to hunt for the maximum of f(x) - simulated annealing, differential evolution, tunnelling, and so on - and then, when you have found a maximum for f(x), sample a few points on the left and right of your optimum and calculate the standard deviation of these evaluations. Then you’ll have to decide if the combination of the maximum of f(x) and this standard deviation is good enough or not compared to any previous “optimal” point found.

This is all speculation, as I’m unsure that your problem is really an optimization one or simply a “peak finding” exercise, for which there are many different - and more powerful and adequate- methods.

Andrea.