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Valid inequalities and convex hulls for multilinear functions. (English) Zbl 1274.90499
Haouari, M. (ed.) et al., ISCO 2010. International symposium on combinatorial optimization. Papers based on the presentations at the symposium, Hammamet, Tunesia, March 24–26, 2010. Amsterdam: Elsevier. Electronic Notes in Discrete Mathematics 36, 805-812 (2010).
Summary: We study the convex hull of the bounded, nonconvex set \(M_n=\{(x_1,\ldots,x_n,x_{n+1})\in\mathbb{R}^{n+1}:x_{n+1}=\prod_{i=1}^nx_i;\ell_i\leq x_i\leq u_i, i=1,\ldots,n+1\}\) for any \(n\geq 2\). 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.
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 \(M_2\) 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.
For the entire collection see [Zbl 1236.90011].

90C57 Polyhedral combinatorics, branch-and-bound, branch-and-cut
90C27 Combinatorial optimization
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