Valid Inequalities and Convex Hulls for Multilinear Functions

We study the convex hull of the bounded, nonconvex setMn = {(x1, . . . , xn, xn+1) ∈ R n+1 : xn+1 = ∏n i=1 xi; i ≤ xi ≤ ui, i = 1, . . . , n + 1} for any n ≥ 2. We seek to derive strong valid linear inequalities forMn; this is motivated by the fact that many exact solvers for nonconvex problems use polyhedral relaxations so as to compute a lower bound via linear programming solvers. We present a class of linear inequalities that, together with the well-known McCormick inequalities, defines the convex hull of M2. This class of inequalities, which we call lifted tangent inequalities, is uncountably infinite, which is not surprising given that the convex hull of M2 is not a polyhedron. This class of inequalities Electronic Notes in Discrete Mathematics 36 (2010) 805–812 1571-0653/$ – see front matter © 2010 Published by Elsevier B.V. www.elsevier.com/locate/endm doi:10.1016/j.endm.2010.05.102 generalizes directly to Mn for n > 2, allowing us to define strengthened relaxations for these higher dimensional sets as well.