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Convex hull representations of special monomials of binary variables

  • Audrey DeVriesEmail author
  • Warren Adams
  • Boshi Yang
Original Paper

Abstract

This paper gives the convex hull representation of any monomial in n binary variables \({\mathbf {x}}\) wherein each variable is bounded above by an auxiliary binary variable y. The convex hull form is already known when the variable y is not present, but has not been considered for this more general case. Without y, the convex hull is obtained by replacing the monomial with a continuous variable, and then enforcing \((n+2)\) linear inequalities to ensure that the new variable equals the monomial value at all binary realizations. Specifically, these inequalities, together with the restrictions \({\mathbf {x}} \le {\mathbf {1}}\), give the convex hull of the corresponding set of \(2^n\) points in \({\mathbb {R}}^{n+1}\) that have the new variable equal to the monomial value. With y,  we show that for the case in which \(n=2\), an implementation of a special-structure RLT gives the convex hull, while for \(n \ge 3\), a different level-1 RLT implementation accomplishes the same task. In fact, the argument for \(n \ge 3\) allows us to obtain the convex hulls of various discrete and/or continuous sets, including those associated with certain supermodular functions, symmetric multilinear monomials in continuous variables over special box constraints, and the Boolean quadric polytope.

Keywords

Convex hull Binary program Linearization Multilinear 

Notes

Acknowledgements

The first and third authors are grateful for support from the Office of Naval Research under Award Number N00014-16-1-2168. We thank two anonymous referees for improving the presentation of this document.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematical and Statistical SciencesClemson UniversityClemsonUSA

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