Equivalence of convex minimization problems over base polytopes

Abstract

This paper considers convex optimization problems over base polytopes of polymatroids. We show that the decomposition algorithm for the separable convex function minimization problems helps us derive simple sufficient conditions for the rationality of optimal solutions, and leads us to some useful 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 deal with a class of non-separable convex objective functions. Moreover, we describe an algorithm for the lexicographically optimal base problem, which is a variant of the Fujishige–Wolfe algorithm.