I am working on a minimax optimization problem where analytical solution to the problem is unavailable. The algorithm I am using seems to return a nearly optimal solution. However, the optimum varies slightly in each simulation depending on the initial value and random seed and what makes me wonder is that the point furnishing the numerical optimum keeps changing constantly. I am guessing that the function is very wiggly with multiple peaks and valleys.
Is there a standard method or measure to describe how wiggly my objective function is? I would have tried to draw an approximate plot of the function but this is a function with 10 variable so plotting is not suitable.
What first came to mind was total variation norm of a real function, but I do not know how to extend this concept to multivariate function nor have any idea how to produce a stable numerical approximation if the function is extremly wiggly in the first place.
I know this is a math page and I am sorry for too much words and too little mathematical formulation. Any help or guide in numerical optimization is welcome. Thank you!
One way to get an idea as to how wiggly a function is depends somewhat on what kine of function it is. If it is a smooth curve with a continuously smooth derivative everywhere, one can take that derivative and look at the bounds it has. For example, a straight line has a derivative that is a single constant, i.e., its slope. But a sine wave is wiggly, its derivative is cosine, which varies from -1 to +1. Thus, it is more wiggly than $y=x$ having only a single slope value, 1, for a not wiggly function.
However, I would not use the word wiggliness to describe noisy stochastic data. But I would try such things as noise reduction for you purposes. Topics to consider are signal processing, signal to noise ratio, and noise reduction with filters among others.