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I have a combinatorial optimization problem for which I have a genetic algorithm to approximate the global minima.

Given X elements find: min f(X)

Now I want to expand the search over all possible subsets X* of X and to find the one subset where its global minimum is maximal compared to all other subsets.

X* are the subsets of X, find: max min f(X*)

The example plot shows all solutions of three subsets (one for each color). The black dot indicates the highest value (maximum) of all three global minima.

image: solutions over three subsets

The main problem is that evaluating the fitness between subsets runs agains the convergence of the solution within a subset. Further the solution is actually a local minimum.

How can this problem be generally described? I couldn't find a similar problem in the literature so far. For example if its solvable with a multi-object genetic algorithm.

Any hint is much appreciated.

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