What is the Basin-Hopping Metropolis Temperature?

Question

I am setting up to use SciPy's basin-hopping global optimizer. Its documentation for parameter T states

T: float, optional

The “temperature” parameter for the accept or reject criterion. Higher “temperatures” mean that larger jumps in function value will be accepted. For best results T should be comparable to the separation (in function value) between local minima.

When it says "function value", does that mean the expected return value of the cost function func? Or the value passed to it? Or something else?

I read the source, and I see where T is passed to the Metropolis acceptance criterion, but I do not understand how it is used when converted to "beta".

Answer 1

I'm unfamiliar with the algorithm, but if you keep reading the documentation on the link you posted you'll find this:

Choosing T: The parameter T is the “temperature” used in the Metropolis criterion. Basinhopping steps are always accepted if func(xnew) < func(xold). Otherwise, they are accepted with probability:exp( -(func(xnew) - func(xold)) / T ). So, for best results, T should to be comparable to the typical difference (in function values) between local minima. (The height of “walls” between local minima is irrelevant.)

So I believe T should take on the value of the function which you are trying to optimize, func. This makes sense if you look at that probability expression -- you'd be comparing a difference in function values to what is meant to be a type of "upper bound" for the step. For example, if one local minima is func = 10 and another is func = 14, you might consider T = 4 to be an appropriate value.