First of all I am not mathematician.
I want to solve a very complex black box function with several constraints having to do with electrical power flows in electrical grids. At the moment I use the Nelder-Mead method.
Normally similar problems are solved using interior point (which by definition proves to find global optimums), but I have no interior point implementation or library available to use in the program that I developed for the purpose.
Since Nelder-Mead only finds local minimums, I would like somehow prove that the problem itself is convex, so when I find a local minimum, it is actually the global minimum.
I would like to know if such thing is possible or if it is absurd to try.
[Edit]
The function I optimize is the amount of overloading of branch elements in electrical grids plus other technical constraints that vary depending on the grid situation. The black-box function is the load flow problem, which is a set of non-linear equations. (So yes, it is not strctly a black-box function, but the formulation becomes too complex) Usually the same kind of problem (called Optimal Power Flow) is solved formulating the power flow equations into an interior point solver (which I cannot afford, and that is why I am using free implementation of other methods)
Best regards, Santi.
If your function is black-box one, then by definition you have no access to its explicit form and so no way to asses its convexity or concavity.
But since you mention interior-point methods (IPM), I wonder whether your function is really a black-box one: IPM are second-order methods, i.e. they require the Hessian. I guess some can use Hessian approximation and even gradient approximation, but then you loose all the advantages.
Moreover, when you mention global optimality, well this follows from convexity assumptions, not by IPM itself. In fact there are IPM designed for non-convex problem. For instance you can give a look to the Ipopt solver.
So when you say people usually use IPM, I suspect that you have a convex problem in closed form that you can solve using one of the many available solvers, for instance MOSEK (ok I am biased, it's the company I work for). Most of them provide trial or academic licenses.
If you stick with bloack-box or derivative free methods, I might suggest you to search for NOMAD.