Least Majorized Elements and Generalized Polymatroids

We prove that a bounded generalized polymatroid has a least weakly submajorized supermajorized vector. Such a vector simultaneously minimizes every nondecreasing nonincreasing, symmetric and quasi-convex function defined on the generalized polymatroid. The same result holds also for the set of integer vectors of a bounded integral generalized polymatroid. We then extend these results to more general sets, and discuss several computational aspects.

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