Convex envelopes of bounded monomials on two-variable cones

TL;DR

This paper characterizes the convex envelopes of bounded monomials constrained by linear inequalities, providing conic inequalities for upper envelopes and volume calculations for convex hulls, aiding mixed-integer nonlinear optimization.

Contribution

It introduces explicit descriptions of convex envelopes of bounded monomials on two-variable cones, including conic inequalities and volume formulas, advancing optimization techniques.

Findings

Upper envelope characterized by conic inequality for n≥2

Lower envelope explicitly derived for n=2

Convex hull volume computed for n=2

Abstract

We consider an $n$-variate monomial function that is restricted both in value by lower and upper bounds and in domain by two homogeneous linear inequalities. Such functions are building blocks of several problems found in practical applications, and that fall under the class of Mixed Integer Nonlinear Optimization. We show that the upper envelope of the function in the given domain, for $n\ge 2$ is given by a conic inequality. We also present the lower envelope for $n=2$. To assess the applicability of branching rules based on homogeneous linear inequalities, we also derive the volume of the convex hull for $n=2$.