Convex hull representations of special monomials of binary variables

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.