On unions and dominants of polytopes

A well-known result on unions of polyhedra in the same space gives an extended formulation whose projection is the convex hull of the union. Here in contrast we study the unions of polytopes in different spaces, giving a complete description of the convex hull without additional variables. When the underlying polytopes are monotone, the facets are described explicitly, generalizing results of Hong and Hooker on cardinality rules, and an efficient separation algorithm is proposed. These results are based on an explicit representation of the dominant of a polytope, and an algorithm for the separation problem for the dominant. For non-monotone polytopes, both the dominant and the union are characterized. We also give results on the unions of polymatroids both on disjoint ground sets and on the same ground set generalizing results of Conforti and Laurent. ∗The first author’s research was supported by the National Science Foundation through grant #DMI-0098423 and by the Office of Naval Research through contract #N00014-97-1-0196. The last three authors research was supported in part by the LISCOS, GROWTH Project GRDI–1999–10056 of the European Union. †Graduate School of Industrial Administration, Carnegie-Mellon University, Pittsburgh PA 15213, USA ‡Université Henri Poincaré, LORIA, B.P.239, F-54506 Vandoeuvre-lès-Nancy, France §Université Henri Poincaré, LORIA, B.P.239, F-54506 Vandoeuvre-lès-Nancy, France ¶CORE and INMA, Université Catholique de Louvain, 34 Voie du Roman Pays, Louvain-la-Neuve, B-1348 Belgium.