Computing the volume of the convex hull of the graph of a trilinear monomial using mixed volumes

Emily Speakman, Gennadiy Averkov

Abstract: Speakman and Lee (2017) gave a formula for the volume of the convex hull of the graph of a trilinear monomial, y=x1x2x3<math><mrow is="true"><mi is="true">y</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><msub is="true"><mrow is="true"><mi is="true">x</mi></mrow><mrow is="true"><mn is="true">1</mn></mrow></msub><msub is="true"><mrow is="true"><mi is="true">x</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msub><msub is="true"><mrow is="true"><mi is="true">x</mi></mrow><mrow is="true"><mn is="true">3</mn></mrow></msub></mrow></math>, over a box in the nonnegative orthant, in terms of the upper and lower bounds on the variables. This was done in the context of using volume as a measure for comparing alternative convexifications to guide the implementation of spatial branch-and-bound for mixed integer nonlinear optimization problems. Here, we introduce an alternative method for computing this volume, making use of the rich theory of mixed volumes. This new method may lead to a natural approach for considering extensions of the problem.