Trilinear Monomials Research Articles

On branching-point selection for trilinear monomials in spatial branch-and-bound: the hull relaxation

In Speakman and Lee (2017), we analytically developed the idea of using volume as a measure for comparing relaxations in the context of spatial branch-and-bound. Specifically, for trilinear monomials, we analytically compared the three possible "double-McCormick relaxations" with the tight convex-hull relaxation. Here, again using volume as a measure, for the convex-hull relaxation of trilinear monomials, we establish simple rules for determining the optimal branching variable and optimal branching point. Additionally, we compare our results with current software practice.

Quantifying Double McCormick

When using the standard McCormick inequalities twice to convexify trilinear monomials, as is often the practice in modeling and software, there is a choice of which variables to group first. For the important case in which the domain is a nonnegative box, we calculate the volume of the resulting relaxation, as a function of the bounds defining the box. In this manner, we precisely quantify the strength of the different possible relaxations defined by all three groupings, in addition to the trilinear hull itself. As a by-product, we characterize the best double-McCormick relaxation. We wish to emphasize that, in the context of spatial branch and bound for factorable formulations, our results do not only apply to variables in the input formulation. Our results apply to monomials that involve auxiliary variables as well. So, our results apply to the product of any three (possibly complicated) expressions in a formulation.

Convex underestimation for posynomial functions of positive variables

The approximation of the convex envelope of nonconvex functions is an essential part in deterministic global optimization techniques (Floudas in Deterministic Global Optimization: Theory, Methods and Application, 2000). Current convex underestimation algorithms for multilinear terms, based on arithmetic intervals or recursive arithmetic intervals (Hamed in Calculation of bounds on variables and underestimating convex functions for nonconvex functions, 1991; Maranas and Floudas in J Global Optim 7:143–182, (1995); Ryoo and Sahinidis in J Global Optim 19:403–424, (2001)), introduce a large number of linear cuts. Meyer and Floudas (Trilinear monomials with positive or negative domains: Facets of convex and concave envelopes, pp. 327–352, (2003); J Global Optim 29:125–155, (2004)), introduced the complete set of explicit facets for the convex and concave envelopes of trilinear monomials with general bounds. This study proposes a novel method to underestimate posynomial functions of strictly positive variables.

Trilinear Monomials with Mixed Sign Domains: Facets of the Convex and Concave Envelopes

Convex underestimators of nonconvex functions, frequently used in deterministic global optimization algorithms, strongly influence their rate of convergence and computational efficiency. A good convex underestimator is as tight as possible and introduces a minimal number of new variables and constraints. Multilinear monomials over a coordinate aligned hyper-rectangular domain are known to have polyhedral convex envelopes which may be represented by a finite number of facet inducing inequalities. This paper describes explicit expressions defining the facets of the convex and concave envelopes of trilinear monomials over a box domain with bounds of opposite signs for at least one variable. It is shown that the previously used approximations based on the recursive use of the bilinear construction rarely yield the convex envelope itself.