Valid Inequalities and Convex Hulls for Multilinear Functions

Abstract : We study the convex hull of the bounded, nonconvex set of a product of n variables for any n ≥ 2. We seek to derive strong valid linear inequalities for this set, which we call 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. We present 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_n 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.
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Contributor : Andrew J. Miller <>
Submitted on : Friday, December 17, 2010 - 5:25:51 PM
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Pietro Belotti, Andrew J. Miller, Mahdi Namazifar. Valid Inequalities and Convex Hulls for Multilinear Functions. Electronic Notes in Discrete Mathematics, Elsevier, 2010, 36, pp.805-812. ⟨10.1016/j.endm.2010.05.102⟩. ⟨hal-00547924⟩



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