On the composition of convex envelopes for quadrilinear terms

Abstract : 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 Cafieri et al. (J Global Optim 47:661-685, 2010) we prove that having fewer groupings of longer terms yields tighter convex relaxations. In this chapter 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.
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Pietro Belotti, Sonia Cafieri, Jon Lee, Leo Liberti, Andrew J. Miller. On the composition of convex envelopes for quadrilinear terms. Optimization, Simulation, and Control, Springer Verlag, pp 1-16, 2013, Springer Optimization and Its Applications, Volume 76, 978-1-4614-5130-3. ⟨10.1007/978-1-4614-5131-0_1⟩. ⟨hal-00769671⟩

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