Top

Published in:

2014 | OriginalPaper | Chapter

6. Intersection Cuts and Corner Polyhedra

Authors : Michele Conforti, Gérard Cornuéjols, Giacomo Zambelli

Published in: Integer Programming

Publisher: Springer International Publishing

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

In this chapter, we present two classical points of view for approximating a mixed integer linear set: Gomory’s corner polyhedron and Balas’ intersection cuts. It turns out that they are equivalent: the nontrivial valid inequalities for the corner polyhedron are exactly the intersection cuts. Within this framework, we stress two ideas: the best possible intersection cuts are generated from maximal lattice-free convex sets, and formulas for these cuts can be interpreted using the so-called infinite relaxation.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

previous chapter Split and Gomory Inequalities

next chapter Valid Inequalities for Structured Integer Programs

[12]

K. Andersen, Q. Louveaux, R. Weismantel, L.A. Wolsey, Inequalities from two rows of a simplex tableau, in Proceedings of IPCO XII, Ithaca, NY. Lecture Notes in Computer Science, vol. 4513 (2007), pp. 1–15CrossRefMathSciNet

[18]

G. Averkov, On maximal S-free sets and the Helly number for the family of S-convex sets. SIAM J. Discrete Math. 27(3), 1610–1624 (2013)CrossRefMATHMathSciNet

[19]

G. Averkov, A. Basu, On the unique lifting property, IPCO 2014, Bonn, Germany, Lecture Notes in Computer Science, 8494, 76–87 (2014)CrossRef

[22]

E. Balas, Intersection cuts—a new type of cutting planes for integer programming. Oper. Res. 19, 19–39 (1971)CrossRefMATHMathSciNet

[23]

E. Balas, Integer programming and convex analysis: intersection cuts from outer polars. Math. Program. 2 330–382 (1972)CrossRefMATHMathSciNet

[31]

E. Balas, R. Jeroslow, Strengthening cuts for mixed integer programs. Eur. J. Oper. Res. 4, 224–234 (1980)CrossRefMATHMathSciNet

[39]

A. Barvinok, A Course in Convexity. Graduate Studies in Mathematics, vol. 54 (American Mathematical Society, Providence, 2002)

[40]

A. Basu, M. Campelo, M. Conforti, G. Cornuéjols, G. Zambelli, On lifting integer variables in minimal inequalities. Math. Program. A 141, 561–576 (2013)CrossRefMATH

[41]

A. Basu, M. Conforti, G. Cornuéjols, G. Zambelli, Maximal lattice-free convex sets in linear subspaces. Math. Oper. Res. 35, 704–720 (2010)CrossRefMATHMathSciNet

[42]

A. Basu, M. Conforti, G. Cornuéjols, G. Zambelli, Minimal inequalities for an infinite relaxation of integer programs. SIAM J. Discrete Math. 24, 158–168 (2010)CrossRefMATHMathSciNet

[43]

A. Basu, R. Hildebrand, M. Köppe, M. Molinaro, A (k+1)-Slope Theorem for the k-Dimensional Infinite Group Relaxation. SIAM J. Optim. 23(2), 1021–1040 (2013)CrossRefMATHMathSciNet

[44]

A. Basu, R. Hildebrand, M. Köppe, Equivariant perturbation in Gomory and Johnson infinite group problem III. Foundations for the k-dimensional case with applications to the case k = 2. www.optimization-online.org (2014)

[45]

D.E. Bell, A theorem concerning the integer lattice. Stud. Appl. Math. 56, 187–188 (1977)MATH

[63]

V. Borozan, G. Cornuéjols, Minimal valid inequalities for integer constraints. Math. Oper. Res. 34, 538–546 (2009)CrossRefMATHMathSciNet

[76]

M. Conforti, G. Cornuéjols, A. Daniilidis, C. Lemaréchal, J. Malick, Cut-generating functions and S-free sets, Math. Oper. Res. http://dx.doi.org/10.1287/moor.2014.0670

[77]

M. Conforti, G. Cornuéjols, G. Zambelli, A geometric perspective on lifting. Oper. Res. 59, 569–577 (2011)CrossRefMATHMathSciNet

[78]

M. Conforti, G. Cornuéjols, G. Zambelli, Equivalence between intersection cuts and the corner polyhedron. Oper. Res. Lett. 38, 153–155 (2010)CrossRefMATHMathSciNet

[80]

M. Conforti, G. Cornuéjols, G. Zambelli, Corner polyhedron and intersection cuts. Surv. Oper. Res. Manag. Sci. 16, 105–120 (2011)

[106]

S. Dash, S.S. Dey, O. Günlük, Two dimensional lattice-free cuts and asymmetric disjunctions for mixed-integer polyhedra. Math. Program. 135, 221–254 (2012)CrossRefMATHMathSciNet

[112]

A. Del Pia, R. Weismantel, Relaxations of mixed integer sets from lattice-free polyhedra. 4OR 10, 221–244 (2012)

[115]

S.S. Dey, Q. Louveaux, Split rank of triangle and quadrilateral inequalities. Math. Oper. Res. 36, 432–461 (2011)CrossRefMATHMathSciNet

[116]

S. S. Dey, D.A. Morán, On maximal S-free convex sets. SIAM J. Discrete Math. 25(1), 379–393 (2011)CrossRefMATHMathSciNet

[117]

S.S. Dey, J.-P.P. Richard, Y. Li, L.A. Miller, On the extreme inequalities of infinite group problems. Math. Program. A 121, 145–170 (2010)CrossRefMATHMathSciNet

[118]

S.S. Dey, L.A. Wolsey, Lifting Integer Variables in Minimal Inequalities Corresponding to Lattice-Free Triangles, IPCO 2008, Bertinoro, Italy. Lecture Notes in Computer Science, Springer, vol. 5035 (2008), pp. 463–475CrossRefMathSciNet

[119]

S.S. Dey, L.A. Wolsey, Constrained infinite group relaxations of MIPs. SIAM J. Optim. 20, 2890–2912 (2010)CrossRefMATHMathSciNet

[121]

J.-P. Doignon, Convexity in cristallographical lattices. J. Geom. 3, 71–85 (1973)CrossRefMATHMathSciNet

[178]

R.E. Gomory, Some polyhedra related to combinatorial problems. Linear Algebra Appl. 2, 451–558 (1969)CrossRefMATHMathSciNet

[179]

R.E. Gomory, E.L. Johnson, Some continuous functions related to corner polyhedra I. Math. Program. 3, 23–85 (1972)CrossRefMATHMathSciNet

[180]

R.E. Gomory, E.L. Johnson, T-space and cutting planes. Math. Program. 96, 341–375 (2003)CrossRefMATHMathSciNet

[201]

J.-B. Hiriart-Urruty, C. Lemaréchal. Fundamentals of Convex Analysis (Springer, New York, 2001)CrossRefMATH

[215]

E.L. Johnson, On the group problem for mixed integer programming. Math. Program. Study 2, 137–179 (1974)CrossRef

[216]

E.L. Johnson, Characterization of facets for multiple right-hand choice linear programs. Math. Program. Study 14, 112–142 (1981)CrossRefMATH

[261]

L. Lovász, Geometry of numbers and integer programming, in Mathematical Programming: Recent Developments and Applications, ed. by M. Iri, K. Tanabe (Kluwer, Dordrecht, 1989), pp. 177–201

[315]

J.-P.P. Richard, S.S. Dey (2010). The group-theoretic approach in mixed integer programming, in 50 Years of Integer Programming 1958–2008, ed. by M. Jünger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, L. Wolsey (Springer, New York, 2010), pp. 727–801

[316]

R.T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, 1969)

[322]

H.E. Scarf, An observation on the structure of production sets with indivisibilities. Proc. Natl. Acad. Sci. USA 74, 3637–3641 (1977)CrossRefMATHMathSciNet

Title: Intersection Cuts and Corner Polyhedra
Authors: Michele Conforti
Gérard Cornuéjols
Giacomo Zambelli
Publisher: Springer International Publishing
Book: Integer Programming
Print ISBN: 978-3-319-11007-3

Electronic ISBN: 978-3-319-11008-0

Copyright Year: 2014
DOI: https://doi.org/10.1007/978-3-319-11008-0_6

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"