TY - JOUR
AU - Yang, Boshi
AB - This paper gives the convex hull representation of any monomial in n binary variables 
$${\mathbf {x}}$$


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 
$$(n+2)$$



(
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 
$${\mathbf {x}} \le {\mathbf {1}}$$



x
≤
1



, give the convex hull of the corresponding set of 
$$2^n$$



2
n



 points in 
$${\mathbb {R}}^{n+1}$$




R


n
+
1




 that have the new variable equal to the monomial value. With y,  we show that for the case in which 
$$n=2$$



n
=
2



, an implementation of a special-structure RLT gives the convex hull, while for 
$$n \ge 3$$



n
≥
3



, a different level-1 RLT implementation accomplishes the same task. In fact, the argument for 
$$n \ge 3$$



n
≥
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.
TI - Convex hull representations of special monomials of binary variables
JO - Optimization Letters
DO - 10.1007/s11590-019-01400-5
DA - 2019-02-13
UR - https://www.deepdyve.com/lp/springer-journals/convex-hull-representations-of-special-monomials-of-binary-variables-gSkh0HkNjN
SP - 977
EP - 992
VL - 13
IS - 5
DP - DeepDyve
ER -