Valid Inequalities and Convex Hulls for Multilinear Functions

We study the convex hull of the bounded, nonconvex set M n = { ( x 1 , … , x n , x n + 1 ) ∈ R n + 1 : x n + 1 = ∏ i = 1 n x i ; ℓ i ⩽ x i ⩽ u i , i = 1 , … , n + 1 } for any n ⩾ 2 . We seek to derive strong valid linear inequalities for M n ; 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. sent a class of linear inequalities that, together with the well-known McCormick inequalities, defines the convex hull of M 2 . This class of inequalities, which we call lifted tangent inequalities, is uncountably infinite, which is not surprising given that the convex hull of M 2 is not a polyhedron. This class of inequalities generalizes directly to M n for n > 2 , allowing us to define strengthened relaxations for these higher dimensional sets as well.