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Geometric proofs for convex hull defining formulations. (English) Zbl 1408.90193
Summary: A conjecture appeared recently in [V. Cacchiani et al., ibid. 41, No. 1, 74–77 (2013; Zbl 1266.90129)] that a proposed LP relaxation of a certain integer programming problem defines the convex hull of its integer points. We review a little known technique described in [the author, A set theoretic approach to lifting procedures for $$0, 1$$ integer programming. New York, NY: Columbia University (Ph.D. thesis) (2004)] that can be used to construct geometric proofs that an LP relaxation is convex hull defining. In line with this technique, we show that their conjecture is correct.

##### MSC:
 90C10 Integer programming 52B55 Computational aspects related to convexity 90C57 Polyhedral combinatorics, branch-and-bound, branch-and-cut
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##### References:
 [1] Balas, E.; Ceria, S.; Cornuéjols, G., A lift-and-project cutting plane algorithm for mixed 0-1 programs, Math. Program., 58, 295-324, (1993) · Zbl 0796.90041 [2] Bienstock, D.; Zuckerberg, M., Subset algebra lifting methods for 0-1 integer programming, SIAM J. Optim., 15, 63-95, (2004) · Zbl 1077.90041 [3] Cacchiani, V.; Caprara, A.; Maróti, G.; Toth, P., On integer polytopes with few nonzero vertices, Oper. Res. Lett., 41, 1, 74-77, (2013) · Zbl 1266.90129 [4] Deza, M.; Laurent, M., On integer polytopes with few nonzero vertices, (1997), Springer [5] Lasserre, J. B., An explicit exact sdp relaxation for nonlinear 0-1 programs, Lecture Notes in Comput. Sci., 293-303, (2001) · Zbl 1010.90515 [6] Lovász, L.; Schrijver, A., Cones of matrices and set-functions and 0-1 optimization, SIAM J. Optim., 1, 166-190, (1991) · Zbl 0754.90039 [7] Sherali, S.; Adams, W., A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems, SIAM J. Discrete Math., 3, 411-430, (1990) · Zbl 0712.90050 [8] Zuckerberg, M., A set theoretic approach to lifting procedures for 0,1 integer programming, (2004), Columbia University, (Ph.D. thesis)
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