A form of "generalized solution" which I saw in various areas like for combinatorial optimization problems, for diophanine equations, for computational complexity purposes, and others is "statistical physics relaxation". You regard your original problem as a "temperature 0" case of a more general problem and try to gain insight on the original problem based on statistical-physics insights for the generalized problem. I am not sure what is the general recipe for this apprach and I will be happy to see an edited version with further explanation and links.
Gil Kalai
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