Maximal lattice-free convex sets in linear subspaces

Mathematics of Operations Research

Volumn 35, Issue 3, 2010, Pages 704-720

  Basu, Amitabh ^a   Conforti, Michele ^b   Cornuéjols, Gérard ^a   Zambelli, Giacomo ^b  

^a CARNEGIE MELLON UNIVERSITY   (United States)

^b UNIVERSITY OF PADOVA   (Italy)

Author keywords

Geometry of numbers; Integer programming; Maximal lattice free convex sets; Minimal valid inequalities

Indexed keywords

AFFINE SUBSPACES; CONVEX SET; LINEAR SUBSPACE; VALID INEQUALITY;

DYNAMIC PROGRAMMING; INTEGER PROGRAMMING; NUMBER THEORY;

SET THEORY;

EID: 77956638760     PISSN: 0364765X     EISSN: 15265471     Source Type: Journal    
DOI: 10.1287/moor.1100.0461     Document Type: Article

References (25)

1. Andersen, K., Q. Louveaux, R. Weismantel. 2008. Certificates of linear mixed integer infeasibility. Oper. Res. Lett. 36 734-738.
2. Andersen, K., Q. Louveaux, R. Weismantel. 2010. An analysis of mixed integer linear sets based on lattice point free convex sets. Math. Oper. Res. 35 233-256.
3. Andersen, K., Q. Louveaux, R. Weismantel. 2010. Mixed-integer sets associated from two rows of two adjacent simplex bases. Math. Programming B 124 455-480.
4. Andersen, K., C. Wagner, R. Weismantel. 2009. On an analysis of the strength of mixed integer cutting planes from multiple simplex tableau rows. SIAM J. Optim. 20 967-982.
5. Andersen, K., Q. Louveaux, R. Weismantel, L. A. Wolsey. 2007. Cutting planes from two rows of a simplex tableau. Proc. IPCO XII, Lecture Notes in Computer Science 4513(June) 1-15.
6. Balas, E. 1971. Intersection cuts-A new type of cutting planes for integer programming. Oper. Res. 19 19-39.
7. Barvinok, A. 2002. A course in convexity. F. W. Gehring, P. R. Halmos, C. C. Moore, eds. Graduate Studies in Mathematics, Vol.54. American Mathematical Society, Providence, Rhode Island.
8. Basu, A., G. Cornuéjols, G. Zambelli. 2011. Convex sets and minimal sublinear functions. J. Convex Anal. 18(2), ePub ahead of print, http://www.heldermann.de/JCA/JCA18/JCA182/jca18027.htm.
9. Basu, A., P. Bonami, G. Cornuéjols, F. Margot. 2009. On the relative strength of split, triangle and quadrilateral cuts. Math. Programming, ePub ahead of print April 23, http://hal.archives-ouvertes.fr/hal-00421757/en/.
10. Borozan, V., G. Cornuéjols. 2009. Minimal valid inequalities for integer constraints. Math. Oper. Res. 34 538-546.
11. Cassels, J. W. S. 1959. An introduction to the geometry of numbers. Grundlehren Mathematischen Wissenschaften, Vol.99. Springer, Berlin.
12. Conway, J. B. 1990. A course in functional analysis. Graduate Texts in Mathematics. Springer, Berlin.
13. Cornuéjols, G., F. Margot. 2009. On the facets of mixed integer programs with two integer variables and two constraints. Math. Programming Ser. A 120 429-456.
14. Dey, S. S., L. A. Wolsey. 2008. Lifting integer variables in minimal inequalities corresponding to lattice-free triangles. IPCO 2008, Lecture Notes in Computer Science 5035(May) 463-475.
15. Dey, S. S., L. A. Wolsey. 2009. Constrained infinite group relaxations of MIPs. SIAM J. Optim. (March) Forthcoming.
16. Doignon, J. P. 1973. Convexity in crystallographic lattices. J. Geometry 3 71-85.
17. Espinoza, D. 2008. Computing with multi-row Gomory cuts. Proc. IPCO XIII, Lecture Notes in Computer Science, Vol.5035. Bertinoro, Italy, 214-224.
18. Gomory, R. G. 1969. Some polyhedra related to combinatorial problems. Linear Algebra Appl. 2 451-558.
19. Gomory, R. E. 2007. Thoughts about integer programming. 50th Anniversary Sympos. OR, University of Montreal, January, and Corner polyhedra and two-equation cutting planes. George Nemhauser Sympos., Atlanta, July.
20. Gomory, R. E., E. L. Johnson. 1972. Some continuous functions related to corner polyhedra. Math. Programming 3 23-85.
21. Hiriart-Urruty, J.-B., C. Lemaréchal. 2001. Fundamentals of Convex Analysis. Springer, Berlin.
22. Lenstra, Jr., A. H. 1983. Integer programming with a fixed number of variables. Math. Oper. Res. 8 538-548.
23. Lovász, L. 1989. Geometry of numbers and integer programming. M. Iri, K. Tanabe, eds. Mathematical Programming: Recent Developements and Applications. Kluwer, Dordrecht, The Netherlands, 177-210.
24. Schrijver, A. 1986. Theory of Linear and Integer Programming. Wiley, New York.
25. Zambelli, G. 2009. On degenerate multi-row gomory cuts. Oper. Res. Lett. 37 21-22.