On the convex hull of the graph of a simple monomial

TL;DR

This paper characterizes the convex hull of a simple monomial's graph over a specific domain, providing linear inequalities and volume formulas to aid in spatial branch-and-bound analysis.

Contribution

It offers a new description of the convex hull of simple monomials and derives a volume formula, advancing mathematical understanding in this area.

Findings

Provides a linear inequality description of the convex hull.

Derives a formula for the volume of the convex hull.

Supports analysis in spatial branch-and-bound methods.

Abstract

Motivated by previous efforts toward mathematically analyzing the treatment of monomials in spatial branch-and-bound, we study the convex hull of the graph of a simple monomial on a nonnegative box domain in arbitrary dimension, where at most one of the variable lower bounds is positive. We give: (i) a description via linear inequalities, and (ii) a formula for the volume.