On Convex Minimization over Base Polytopes

Abstract

This note considers convex optimization problems over base polytopes of polymatroids. We show that the decomposition algorithm for the separable convex function minimization problems helps us give simple sufficient conditions for the rationality of optimal solutions and that it leads us to some interesting properties, including the equivalence of the lexicographically optimal base problem, introduced by Fujishige, and the submodular utility allocation market problem, introduced by Jain and Vazirani. In addition, we develop an efficient implementation of the decomposition algorithm via parametric submodular function minimization algorithms. Moreover, we show that, in some remarkable cases, non-separable convex optimization problems over base polytopes can be solved in strongly polynomial time.