On the composition of convex envelopes for quadrilinear terms

Pietro Belotti (1) , Sonia Cafieri (2) , Jon Lee (3) , Leo Liberti (4) , Andrew Miller (5)

1 Department of Mathematical Sciences
2 ENAC - Ecole Nationale de l'Aviation Civile
3 Department of Mathematical Sciences
4 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
5 IMB - Institut de Mathématiques de Bordeaux

Within the framework of the spatial Branch-and-Bound algorithm for solving Mixed-Integer Nonlinear Programs, different convex relaxations can be obtained for multilinear terms by applying associativity in different ways. The two groupings ((x1x2)x3)x4 and (x1x2x3)x4 of a quadrilinear term, for example, give rise to two different convex relaxations. In [6] we prove that having fewer groupings of longer terms yields tighter convex relaxations. In this paper we give an alternative proof of the same fact and perform a computational study to assess the impact of the tightened convex relaxation in a spatial Branch-and-Bound setting.

Domaines: Optimisation et contrôle [math.OC]

Pietro Belotti, Sonia Cafieri, Jon Lee, Leo Liberti, Andrew Miller. On the composition of convex envelopes for quadrilinear terms. COSC 2011, International Conference on Optimization, Simulation and Control, Dec 2011, Berlin, Germany. pp xxxx. ⟨hal-00941976⟩