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.