Convex hull characterizations of lexicographic orderings

Abstract Given a p-dimensional nonnegative, integral vector α, this paper characterizes the convex hull of the set S of nonnegative, integral vectors x that is lexicographically less than or equal to α. To obtain a finite number of elements in S, the vectors x are restricted to be component-wise upper-bounded by an integral vector u. We show that a linear number of facets is sufficient to describe the convex hull. For the special case in which every entry of u takes the same value (n− 1) for some integer n ≥ 2, the convex hull of the set of n-ary vectors results. Our facets generalize the known family of cover inequalities for the n = 2 binary case. They allow for advances relative to both the modeling of integer variables using base-n expansions, and the solving of n-ary knapsack problems having weakly super-decreasing coefficients.