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In order to find the quality indicators like Generational Distance, Inverted Generational Distance, Epsilon Indicator, and HyperVolume for a Pareto front I want to normalize the values of approximation front obtained on solving the algorithm based on reference front which I assume encloses the approximation front. Is it correct to do so?

Basically, rather than normalizing the approximation front and reference front. I have to normalize the approximate front between 0 to 1 based on max and min values of reference front?

Also how to compute reference point? Is it the maximum of each objective value in a reference front if the problem is a min-min?

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The reference front is another name for the pareto front of the problem. You find it either by solving the problem symbolically, constructing a problem around some pareto front or running and global optimizer long enough. If you want to normalize the any front it is sensible to do so using the reference front. Over what (hyper) cuboid you normalize the fronts is your decision, there is no right answer just answers that don't match your usecase. The maximum of each objective value might be meaningless if it is unbounded. Consider $$\min_{x,y} x,y \ \text{ subject to: } xy = 1$$

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  • $\begingroup$ Thank you. I have obtained a reference front by increasing the number of functional evaluations iteratively and then considering the non-dominated ones. If the decision variables are bounded, should reference point be the maximum of each objective function value in reference front or any front? $\endgroup$
    – vp_050
    Oct 28 '21 at 10:53
  • $\begingroup$ if i use reference point as maximum of objective function values in a approximation front obtained on solving an algorithm, then my hypervolume increases with increase in number of functional evaluations which is correct; however, my epsilon indicator increases with increase in number of functional evaluations which is again incorrect $\endgroup$
    – vp_050
    Oct 28 '21 at 10:58
  • $\begingroup$ You seem to be using a metric meant for comparing some fixed pareto front against it's approximation as a metric to make statements about iterates. First i would check that both point clouds you are comparing are actually pareto fronts by sorting out dominated points each iteration or ensuring that you are just comparing points of successive iterations. > should reference point be the maximum of each objective function value in reference front or any front? It depends on your application but in general assuming boundedness it's reasonable to pick the worst of objectives assumes on the rf $\endgroup$ Oct 28 '21 at 11:19

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