Mathematical Programming

, Volume 99, Issue 2, pp 223–239

# On unions and dominants of polytopes

• Egon Balas
• Alexander Bockmayr
• Nicolai Pisaruk
• Laurence Wolsey
Article

## Abstract.

A well-known result on unions of polyhedra in the same space gives an extended formulation in a higher-dimensional space 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.

### Keywords

unions of polytopes cardinality constraints separation convex hulls dominant matroids polymatroids

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## Authors and Affiliations

• Egon Balas
• 1
• Alexander Bockmayr
• 2
• Nicolai Pisaruk
• 2
• Laurence Wolsey
• 3