IDEAS home Printed from https://ideas.repec.org/a/spr/eurjco/v5y2017i4d10.1007_s13675-017-0085-3.html
   My bibliography  Save this article

Another pedagogy for mixed-integer Gomory

Author

Listed:
  • Jon Lee

    (University of Michigan)

  • Angelika Wiegele

    (Alpen-Adria-Universität Klagenfurt)

Abstract

We present a version of GMI (Gomory mixed-integer) cuts in a way so that they are derived with respect to a “dual form” mixed-integer optimization problem and applied on the standard-form primal side as columns, using the primal simplex algorithm. This follows the general scheme of He and Lee, who did the case of Gomory pure-integer cuts. Our input mixed-integer problem is not in standard form, and so our cuts are derived rather differently from how they are normally derived. A convenient way to develop GMI cuts is from MIR (mixed-integer rounding) cuts, which are developed from 2-dimensional BMI (basic mixed-integer) cuts, which involve a nonnegative continuous variable and an integer variable. The non-negativity of the continuous variable is not the right tool for us, as our starting point (the “dual form” mixed-integer optimization problem) has no non-negativity. So we work out a different 2-dimensional starting point, a pair of somewhat arbitrary inequalities in one continuous and one integer variable. In the end, we follow the approach of He and Lee, getting now a finitely converging primal simplex column-generation algorithm for mixed-integer optimization problems.

Suggested Citation

  • Jon Lee & Angelika Wiegele, 2017. "Another pedagogy for mixed-integer Gomory," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 5(4), pages 455-466, December.
  • Handle: RePEc:spr:eurjco:v:5:y:2017:i:4:d:10.1007_s13675-017-0085-3
    DOI: 10.1007/s13675-017-0085-3
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s13675-017-0085-3
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s13675-017-0085-3?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. M. L. Balinski, 1965. "Integer Programming: Methods, Uses, Computations," Management Science, INFORMS, vol. 12(3), pages 253-313, November.
    2. Hugues Marchand & Laurence A. Wolsey, 2001. "Aggregation and Mixed Integer Rounding to Solve MIPs," Operations Research, INFORMS, vol. 49(3), pages 363-371, June.
    3. MARCHAND, Hugues & WOLSEY, Laurence A., 2001. "Aggregation and mixed integer rounding to solve mips," LIDAM Reprints CORE 1513, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Michele Conforti & Marianna Santis & Marco Summa & Francesco Rinaldi, 2021. "Scanning integer points with lex-inequalities: a finite cutting plane algorithm for integer programming with linear objective," 4OR, Springer, vol. 19(4), pages 531-548, December.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. William Cook & Sanjeeb Dash & Ricardo Fukasawa & Marcos Goycoolea, 2009. "Numerically Safe Gomory Mixed-Integer Cuts," INFORMS Journal on Computing, INFORMS, vol. 21(4), pages 641-649, November.
    2. Chardy, M. & Costa, M.-C. & Faye, A. & Trampont, M., 2012. "Optimizing splitter and fiber location in a multilevel optical FTTH network," European Journal of Operational Research, Elsevier, vol. 222(3), pages 430-440.
    3. Wei-Kun Chen & Liang Chen & Mu-Ming Yang & Yu-Hong Dai, 2018. "Generalized coefficient strengthening cuts for mixed integer programming," Journal of Global Optimization, Springer, vol. 70(1), pages 289-306, January.
    4. Claudio Contardo & Andrea Lodi & Andrea Tramontani, 2023. "Cutting Planes from the Branch-and-Bound Tree: Challenges and Opportunities," INFORMS Journal on Computing, INFORMS, vol. 35(1), pages 2-4, January.
    5. Xiaoyi Gu & Santanu S. Dey & Jean-Philippe P. Richard, 2024. "Solving Sparse Separable Bilinear Programs Using Lifted Bilinear Cover Inequalities," INFORMS Journal on Computing, INFORMS, vol. 36(3), pages 884-899, May.
    6. Laszlo Ladanyi & Jon Lee & Robin Lougee-Heimer, 2005. "Rapid Prototyping of Optimization Algorithms Using COIN-OR: A Case Study Involving the Cutting-Stock Problem," Annals of Operations Research, Springer, vol. 139(1), pages 243-265, October.
    7. Agostinho Agra & Marielle Christiansen & Alexandrino Delgado, 2013. "Mixed Integer Formulations for a Short Sea Fuel Oil Distribution Problem," Transportation Science, INFORMS, vol. 47(1), pages 108-124, February.
    8. Amitabh Basu & Pierre Bonami & Gérard Cornuéjols & François Margot, 2011. "Experiments with Two-Row Cuts from Degenerate Tableaux," INFORMS Journal on Computing, INFORMS, vol. 23(4), pages 578-590, November.
    9. Ivo Nowak & Stefan Vigerske, 2008. "LaGO: a (heuristic) Branch and Cut algorithm for nonconvex MINLPs," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 16(2), pages 127-138, June.
    10. Quentin Louveaux & Laurence Wolsey, 2007. "Lifting, superadditivity, mixed integer rounding and single node flow sets revisited," Annals of Operations Research, Springer, vol. 153(1), pages 47-77, September.
    11. Alper Atamtürk & Martin Savelsbergh, 2005. "Integer-Programming Software Systems," Annals of Operations Research, Springer, vol. 140(1), pages 67-124, November.
    12. Christopher Hojny & Tristan Gally & Oliver Habeck & Hendrik Lüthen & Frederic Matter & Marc E. Pfetsch & Andreas Schmitt, 2020. "Knapsack polytopes: a survey," Annals of Operations Research, Springer, vol. 292(1), pages 469-517, September.
    13. Valentin Borozan & Gérard Cornuéjols, 2009. "Minimal Valid Inequalities for Integer Constraints," Mathematics of Operations Research, INFORMS, vol. 34(3), pages 538-546, August.
    14. Sanjeeb Dash, 2005. "Exponential Lower Bounds on the Lengths of Some Classes of Branch-and-Cut Proofs," Mathematics of Operations Research, INFORMS, vol. 30(3), pages 678-700, August.
    15. Andreas Eisenblätter & Hans-Florian Geerdes & Thorsten Koch & Alexander Martin & Roland Wessäly, 2006. "UMTS radio network evaluation and optimization beyond snapshots," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 63(1), pages 1-29, February.
    16. Matteo Fischetti & Domenico Salvagnin, 2013. "Approximating the Split Closure," INFORMS Journal on Computing, INFORMS, vol. 25(4), pages 808-819, November.
    17. Brian T. Denton & Andrew J. Miller & Hari J. Balasubramanian & Todd R. Huschka, 2010. "Optimal Allocation of Surgery Blocks to Operating Rooms Under Uncertainty," Operations Research, INFORMS, vol. 58(4-part-1), pages 802-816, August.
    18. Kianfar, Kiavash & Fathi, Yahya, 2010. "Generating facets for finite master cyclic group polyhedra using n-step mixed integer rounding functions," European Journal of Operational Research, Elsevier, vol. 207(1), pages 105-109, November.
    19. Ambros M. Gleixner & Timo Berthold & Benjamin Müller & Stefan Weltge, 2017. "Three enhancements for optimization-based bound tightening," Journal of Global Optimization, Springer, vol. 67(4), pages 731-757, April.
    20. Yaman, Hande & Sen, Alper, 2008. "Manufacturer's mixed pallet design problem," European Journal of Operational Research, Elsevier, vol. 186(2), pages 826-840, April.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:eurjco:v:5:y:2017:i:4:d:10.1007_s13675-017-0085-3. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.