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.