I would like to know the sort of penalty function used in NMinimize with differential evolution method in Mathematica.
I assume you mean the penalty function for constraints. I asked technical support that question a number of years ago and they didn’t answer. I’ve speculated that it is an augmented Lagrangian approach, since the documentation for the FindMinimum Interior Point Method talks about using an augmented Lagrangian penalty approach. The advantage of that approach is that accurate results can be obtained having to take the penalty factor to infinity. However, it’s just speculation on my part.
Not augmented Lagrangian. But NMinimize will call FindMinimum for post processing and one method does involve Interior Point. It may also have penalty terms from NMinimize itself, I don’t recall.
The penalty is something along the lines of adding absolute values of constraint violations, scaled by function value if it is large. Also scaled by iteration number.
Interesting. I have seen claims that global optimization for problems with constraints can be done with the augmented Lagrangian approach by doing global optimization at each iteration of the augmented Lagrangian.