I have a (generalized) Linear Programming problem to solve. I anticipate exactly two equally valid optimizations of my objective function. I would be happy if I could receive both these points; it wouldn't be hard to figure out which one I really want. My question: can standard Linear Programming algorithms be modified to return both points instead of just one? Do they sacrifice their polynomial runtime to do so?
And for extra credit: how, exactly, would we modify these algorithms?
Edit: made the title more relevant to the question