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Convex Hull Characterizations of Lexicographic Orderings

Warren Adams(wadams***at***clemson.edu)
Pietro Belotti(pietrobelotti***at***fico.com)
Ruobing Shen(rshen***at***g.clemson.edu)

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.

Keywords: convex hull, facets, knapsack problem

Category 1: Integer Programming ((Mixed) Integer Linear Programming )

Category 2: Combinatorial Optimization (Polyhedra )

Citation: Clemson University, Clemson SC, September 2015.

Download: [PDF]

Entry Submitted: 12/14/2015
Entry Accepted: 12/14/2015
Entry Last Modified: 12/14/2015

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