Trilinear Monomials with Positive or Negative Domains: Facets of the Convex and Concave Envelopes

Abstract

Approximations of the convex envelope of nonconvex functions play a central role in deterministic global optimization algorithms and the efficiency of these algorithms is highly infuenced by the tightness of these approximations. McCormick (1976), and AlKhayyal and Falk (1983) have shown how to construct the convex envelope of individual bilinear terms over a rectangular domain. Rikun (1997) has shown that the convex hull of multilinear monomials over a rectangular domain is polyhedral. Approximations of the convex envelope for higher order multilinear terms have been based on the recursive use of this bilinear construction. Only under very special circumstances, however, do these approximations yield the convex envelope itself. Explicit expressions defining the facets of the convex and concave envelopes for trilinear monomials, with positive or negative bounded domains for each variable, are derived in this paper.