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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!

Sungmin Park
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  • I think you need to add more information. There are 2 problems which are very important. 1: Is your algorithm ensured to converge to a local minimal or just the stationary point? Generally, finding the local minimal is NP-hard, so I was wondering if your algorithm could find the local minimal. 2:Can you analyze the stationary point? I know it's hard to give a analytical solution, but analysis of the stationary point is possible. – Wenqing Ouyang May 05 '19 at 09:05
  • @foxell I better edit the question, I am using particle swarm optimization. It is a heuristic-based method without much analytical background. There is no solid evidence that the solution is a local minimum. But looking at the solution path of the algorithm you could say empirically that it is a local minimum? – Sungmin Park May 05 '19 at 11:05
  • There is some information about the total variation of a multivariable function at https://math.stackexchange.com/questions/4155191/computing-the-total-variation-for-a-multivariable-function – Gerry Myerson Jul 10 '23 at 09:10

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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.

Carl
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