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.