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| HAL: hal-00547924, version 1 |
| DOI: 10.1016/j.endm.2010.05.102 |
| Detailed view |  Export this paper |
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| Electronic Notes in Discrete Mathematics 36 (2010) 805-812 |
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| Valid Inequalities and Convex Hulls for Multilinear Functions |
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| Pietro Belotti 1Andrew J. Miller 2, 3 |
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| (2010) |
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| 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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| 1: | Department of Industrial and Systems Engineering [Lehigh] (ISE) |
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Lehigh University, Bethlehem, USA |
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| 2: | Institut de Mathématiques de Bordeaux (IMB) |
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CNRS : UMR5251 – Université Sciences et Technologies - Bordeaux I – Université Victor Segalen - Bordeaux II |
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| 3: | RealOpt (INRIA Bordeaux - Sud-Ouest) |
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INRIA – Université Sciences et Technologies - Bordeaux I – Université Victor Segalen - Bordeaux II – CNRS : UMR – École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB) |
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| 4: | Department of Industrial and Systems Engineering [Wisconsin-Madison] (ISyE) |
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University of Wisconsin-Madison |
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| Subject | : | Computer Science/Operations Research |
| hal-00547924, version 1 | |
| http://hal.archives-ouvertes.fr/hal-00547924 | |
| oai:hal.archives-ouvertes.fr:hal-00547924 | |
| From: Andrew J. Miller | |
| Submitted on: Friday, 17 December 2010 17:25:51 | |
| Updated on: Friday, 17 December 2010 17:25:51 | |
