Maximal Lattice-Free Polyhedra: Finiteness and an Explicit Description in Dimension Three

Published Online:https://doi.org/10.1287/moor.1110.0510

References

  • Andersen K., Louveaux Q., Weismantel R. An analysis of mixed-integer linear sets based on lattice point free convex sets. Math. Oper. Res. (2010) 35(1):233–256LinkGoogle Scholar
  • Andersen K., Wagner C., Weismantel R. Maximal integral simplices with no interior integer points. (2009) . http://arxiv.org/abs/0904.2108Google Scholar
  • Andersen K., Wagner C., Weismantel R. On an analysis of the strength of mixed-integer cutting planes from multiple simplex tableau rows. SIAM J. Optim. (2009) 20(2):967–982CrossrefGoogle Scholar
  • Andersen K., Louveaux Q., Weismantel R., Wolsey L. A., Fischetti M., Williamson D. P. Inequalities from two rows of a simplex tableau. Proc. IPCO Conf. 2007, Vol. 4513 (2007) (Springer, Berlin) 1–15Lecture Notes in Computer ScienceCrossrefGoogle Scholar
  • Averkov G., Wagner C. Inequalities for the lattice width of lattice-free convex sets in the plane. Contributions Algebra Geometry (2011) . Forthcoming. http://arxiv.org/abs/1003.4365Google Scholar
  • Balas E. Intersection cuts—A new type of cutting planes for integer programming. Oper. Res. (1971) 19(1):19–39LinkGoogle Scholar
  • Barvinok A.A Course in Convexity (2002) 54(American Mathematical Society, Providence, RI) Graduate Studies in MathematicsCrossrefGoogle Scholar
  • Basu A., Cornuéjols G., Margot F. Intersection cuts with infinite split rank. (2011) . Working paper, http://wpweb2.tepper.cmu.edu/fmargot/PDF/splrk.pdfGoogle Scholar
  • Basu A., Bonami P., Cornuéjols G., Margot F. On the relative strength of split, triangle and quadrilateral cuts. Math. Programming (2011) 126(2, series A):281–314CrossrefGoogle Scholar
  • Borozan V., Cornuéjols G. Minimal valid inequalities for integer constraints. (2007) . Technical report, American Mathematical Society, Providence, RI. http://www.ams.org/mathscinet/search/publications.html?fmt=bibtex&pg1=MR&s1=2555335Google Scholar
  • Cornuéjols G., Margot F. On the facets of mixed-integer programs with two integer variables and two constraints. Proc. Latin American Theoretical Informatics Sympos. Conf. 2008: Theoretical Informatics, Vol. 4957 (2008) (Springer, Berlin) 317–328Lecture Notes in Computer ScienceCrossrefGoogle Scholar
  • Del Pia A., Weismantel R. On convergence in mixed-integer programming. Math. Programming, Ser. A. (2010) (Springer, Berlin/Heidelberg) . Published online July 19, 2010. http://www.springerlink.com/content/f424707441g41m2n/Google Scholar
  • Dey S. S., Louveaux Q. Split rank of triangle and quadrilateral inequalities. (2009) . http://arxiv.org/abs/0906.0887Google Scholar
  • Dey S. S., Richard J.-P. P. Facets of two-dimensional infinite group problems. Math. Oper. Res. (2008) 33(1):140–166LinkGoogle Scholar
  • Dey S. S., Wolsey L. A. Lifting integer variables in minimal inequalities corresponding to lattice-free triangles. Proc. IPCO Conf. 2008, Vol. 5035 (2008) (Springer, Berlin) 463–475Lecture Notes in Computer ScienceCrossrefGoogle Scholar
  • Espinoza D. G. Computing with multi-row Gomory cuts. Proc. IPCO Conf. 2008, Vol. 5035 (2008) (Springer, Berlin) 214–224Lecture Notes in Computer ScienceCrossrefGoogle Scholar
  • Gruber P. M.Convex and Discrete Geometry (2007) 336(Springer, Berlin) Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]Google Scholar
  • Gruber P. M., Lekkerkerker C. G.Geometry of Numbers (1987) 372nd ed.(North-Holland Mathematical Library, North-Holland, Amsterdam) Google Scholar
  • Grünbaum B. Convex Polytopes. Graduate Texts in Mathematics (2003) 2212nd ed.(Springer-Verlag, New York) Google Scholar
  • Hensley D. Lattice vertex polytopes with interior lattice points. Pacific J. Math. (1983) 105(1):183–191CrossrefGoogle Scholar
  • Hurkens C. A. J. Blowing up convex sets in the plane. Linear Algebra Appl. (1990) 134:121–128CrossrefGoogle Scholar
  • Lagarias J. C., Ziegler G. M. Bounds for lattice polytopes containing a fixed number of interior points in a sublattice. Canadian J. Math. (1991) 43(5):1022–1035CrossrefGoogle Scholar
  • Lovász L. Geometry of numbers and integer programming. Mathematical Programming (Tokyo, 1988) (1989) 6(SCIPRESS, Tokyo) 177–201Math. Appl. (Japanese Ser.)Google Scholar
  • Nill B., Ziegler G. M. Projecting lattice polytopes without interior lattice points. (2011) . http://arxiv.org/abs/1101.4292Google Scholar
  • Pikhurko O. Lattice points in lattice polytopes. Mathematika (2001) 48(1–2):15–24CrossrefGoogle Scholar
  • Rabinowitz S. A census of convex lattice polygons with at most one interior lattice point. Ars Combinatoria (1989) 28:83–96Google Scholar
  • Reznick B. Lattice point simplices. Discrete Math. (1986) 60:219–242CrossrefGoogle Scholar
  • Rockafellar R. T.Convex Analysis (1972) (Princeton University, NJ) Princeton Landmarks in MathematicsGoogle Scholar
  • Scarf H. E. Integral polyhedra in three space. Math. Oper. Res. (1985) 10(3):403–438LinkGoogle Scholar
  • Schrijver A.Theory of Linear and Integer Programming (1986) (A Wiley-Interscience Publication, John Wiley & Sons, West Sussex, UK) Wiley-Interscience Series in Discrete MathematicsGoogle Scholar
  • Sebő A. An introduction to empty lattice simplices. Proc. IPCO Conf. 1999, Vol. 1610 (1999) (Springer, Berlin) 400–414Lecture Notes in Computer ScienceCrossrefGoogle Scholar
  • Treutlein J. 3-dimensional lattice polytopes without interior lattice points. (2008) . http://arxiv.org/abs/0809.1787Google Scholar
  • Zambelli G. On degenerate multi-row Gomory cuts. Oper. Res. Lett. (2009) 37(1):21–22CrossrefGoogle Scholar
INFORMS site uses cookies to store information on your computer. Some are essential to make our site work; Others help us improve the user experience. By using this site, you consent to the placement of these cookies. Please read our Privacy Statement to learn more.