Volume Formulae for the Convex Hull of the Graph of a Trilinear Monomial: A Complete Characterization for General Box Domains

TL;DR

This paper derives comprehensive volume formulae for the convex hull of a trilinear monomial's graph over general box domains, extending previous results limited to nonnegative bounds, thus enabling better relaxations in mixed-integer nonlinear programming.

Contribution

It provides a complete set of volume formulae for the convex hull of a trilinear monomial over arbitrary box domains, filling a key gap in the literature.

Findings

Six distinct volume formulae characterize all domain configurations.

Removing nonnegative bounds changes the convex hull structure.

The results improve convex relaxations for mixed-integer nonlinear programs.

Abstract

Solving difficult mixed-integer nonlinear programs via spatial branch-and-bound requires effective convex outer-approximations of nonconvex sets. In this framework, complex problem formulations are decomposed into simpler library functions, whose relaxations are then composed to build relaxations of the overall problem. The trilinear monomial serves as one such fundamental library function, appearing frequently as a building block across diverse applications. By definition, its convex hull provides the tightest possible relaxation and thus serves as a benchmark for evaluating alternatives. Mixed volume techniques have yielded a parameterized volume formula for the convex hull of the graph of a trilinear monomial; however, existing results only address the case where all six bounds of the box domain are nonnegative. This restriction represents a notable gap in the literature, as variables with mixed-sign domains arise naturally in practice. In this work, we close the gap by extending to the general case via an exhaustive case analysis. We demonstrate that removing the nonnegative domain assumption alters the underlying structure of the convex hull polytope, leading to six distinct volume formulae that together characterize all possible parameter configurations.