On convex relaxations of quadrilinear terms

The best known method to find exact or at least -approximate solutions to polynomial programming problems is the spatial Branch-and-Bound algorithm, which rests on computing lower bounds to the value of the objective function to be minimized on each region that it explores. These lower bounds are often computed by solving convex relaxations of the original program. Although convex envelopes are explicitly known (via linear inequalities) for bilinear and trilinear terms on arbitrary boxes, such a description is unknown, in general, for multilinear terms of higher order. In this paper, we study convex relaxations of quadrilinear terms. We exploit associativity to rewrite such terms as products of bilinear and trilinear terms. Using a general technique, we establish that, any relaxation for k-linear terms that employs a successive use of relaxing bilinear terms (via the bilinear convex envelope) can be improved by employing instead a relaxation of a trilinear term (via the trilinear conv...

• Type: Journal
• Year: 2010
• Where: JGO
• Authors: Sonia Cafieri, Jon Lee, Leo Liberti
• Keywords: Convex Envelope, Convex Relaxations, JGO 2010, Trilinear Term