The Structure of the Infinite Models in Integer Programming

Published Online:https://doi.org/10.1287/moor.2018.0977

References

  • [1] Aczél J, Dhombres JG (1989) Functional Equations in Several Variables, Encyclopedia of Mathematics and Its Applications, vol. 31 (Cambridge University Press, Cambridge, UK).CrossrefGoogle Scholar
  • [2] Basu A, Paat J (2015) Operations that preserve the covering property of the lifting region. SIAM J. Optim. 25(4):2313–2333.CrossrefGoogle Scholar
  • [3] Basu A, Conforti M, Di Summa M (2015) A geometric approach to cut-generating functions. Math. Programming 151(1):153–189.CrossrefGoogle Scholar
  • [4] Basu A, Dey SS, Paat J (2016) How to choose what you lift. Working paper, Johns Hopkins University, Baltimore.Google Scholar
  • [5] Basu A, Hildebrand R, Köppe M (2016) Light on the infinite group relaxation I: Foundations and taxonomy. 4OR 14(1):1–40.CrossrefGoogle Scholar
  • [6] Basu A, Hildebrand R, Köppe M (2016) Light on the infinite group relaxation II: Sufficient conditions for extremality, sequences, and algorithms. 4OR 14(2):1–25.CrossrefGoogle Scholar
  • [7] Basu A, Conforti M, Cornuéjols G, Zambelli G (2010) Maximal lattice-free convex sets in linear subspaces. Math. Oper. Res. 35(3):704–720.LinkGoogle Scholar
  • [8] Basu A, Conforti M, Di Summa M, Paat J (2017) The structure of the infinite models in integer programming. Proc. Inter. Conf. Integer Programming Combinatorial Optim. (Springer, New York), 63–74.CrossrefGoogle Scholar
  • [9] Conforti M, Cornuéjols G, Zambelli G (2010) Equivalence between intersection cuts and the corner polyhedron. Oper. Res. Lett. 38(3):153–155.CrossrefGoogle Scholar
  • [10] Conforti M, Cornuéjols G, Zambelli G (2014) Integer Programming, Graduate Texts in Mathematics, vol. 271 (Springer, New York).Google Scholar
  • [11] Conway JB (2013) A Course in Functional Analysis, Graduate Texts in Mathematics, vol. 96 (Springer Science & Business Media, New York).Google Scholar
  • [12] Cornuéjols G, Wolsey L, Yıldız S (2015) Sufficiency of cut-generating functions. Math. Programming 152(1-2):643–651.CrossrefGoogle Scholar
  • [13] Dash S, Günlük O (2006) Valid inequalities based on simple mixed-integer sets. Math. Programming 105(1):29–53.CrossrefGoogle Scholar
  • [14] Dey SS, Richard JPP, Li Y, Miller LA (2009) On the extreme inequalities of infinite group problems. Math. Programming 121(1):145–170.CrossrefGoogle Scholar
  • [15] Dickson LE (1913) Finiteness of the odd perfect and primitive abundant numbers withndistinct prime factors. Amer. J. Math. 35(4):413–422.CrossrefGoogle Scholar
  • [16] Gomory RE (1969) Some polyhedra related to combinatorial problems. Linear Algebra Appl. 2(4):451–558.CrossrefGoogle Scholar
  • [17] Gomory RE, Johnson EL (1972) Some continuous functions related to corner polyhedra. Math. Programming 3(1):23–85.CrossrefGoogle Scholar
  • [18] Gomory RE, Johnson EL (1972) Some continuous functions related to corner polyhedra, II. Math. Programming 3(1):359–389.CrossrefGoogle Scholar
  • [19] Hildebrand R (2013) Algorithms and cutting planes for mixed integer programs. Unpublished doctoral thesis, University of California, Davis, Davis.Google Scholar
  • [20] Hille E, Phillips R (1957) Functional Analysis and Semi-Groups (American Mathematical Society, Providence, RI).Google Scholar
  • [21] Johnson EL (1974) On the group problem for mixed integer programming. Math. Programming Stud. 2:137–179.CrossrefGoogle Scholar
  • [22] Kılınç-Karzan F (2015) On minimal valid inequalities for mixed integer conic programs. Math. Oper. Res. 41(2):477–510.LinkGoogle Scholar
  • [23] Kılınç-Karzan F, Steffy DE (2016) On sublinear inequalities for mixed integer conic programs. Math. Programming 159(1-2):585–605.CrossrefGoogle Scholar
  • [24] Kılınç-Karzan F, Yang B (2015) Sufficient conditions and necessary conditions for the sufficiency of cut-generating functions. Technical report, Carnegie Mellon University, Pittsburgh. Accessed April 10, 2017, http://www.andrew.cmu.edu/user/fkilinc/files/draft-sufficiency-web.pdf.Google Scholar
  • [25] Köppe M, Zhou Y (2015) An electronic compendium of extreme functions for the Gomory–Johnson infinite group problem. Oper. Res. Lett. 43(4):438–444.CrossrefGoogle Scholar
  • [26] Köppe M, Zhou Y (2018) All cyclic group facets inject. Working paper, University of California, Davis, Davis.Google Scholar
  • [27] Lemaréchal C, Hiriart-Urruty J (1996) Convex Analysis and Minimization Algorithms I, Grundlehren der Mathematischen Wissenschaften, vol. 305 (Springer, Berlin).Google Scholar
  • [28] Letchford AN, Lodi A (2002) Strengthening Chvátal–Gomory cuts and Gomory fractional cuts. Oper. Res. Lett. 30(2):74–82.CrossrefGoogle Scholar
  • [29] Meyer R (1974) On the existence of optimal solutions to integer and mixed-integer programming problems. Math. Programming 7(1):223–235.CrossrefGoogle Scholar
  • [30] Miller LA, Li Y, Richard JPP (2008) New inequalities for finite and infinite group problems from approximate lifting. Naval Res. Logist. 55(2):172–191.CrossrefGoogle Scholar
  • [31] Yıldız S, Cornuéjols G (2016) Cut-generating functions for integer variables. Math. Oper. Res. 41(4):1381–1403.LinkGoogle Scholar
  • [32] Zambelli G (2009) On degenerate multi-row Gomory cuts. Oper. Res. Lett. 37(1):21–22.CrossrefGoogle Scholar
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