IDEAS home Printed from https://ideas.repec.org/a/spr/annopr/v139y2005i1p95-12910.1007-s10479-005-3445-y.html
   My bibliography  Save this article

Depth-Optimized Convexity Cuts

Author

Listed:
  • Jonathan Eckstein
  • Mikhail Nediak

Abstract

This paper presents a general, self-contained treatment of convexity or intersection cuts. It describes two equivalent ways of generating a cut—via a convex set or a concave function—and a partial-order notion of cut strength. We then characterize the structure of the sets and functions that generate cuts that are strongest with respect to the partial order. Next, we specialize this analytical framework to the case of mixed-integer linear programming (MIP). For this case, we formulate two kinds of the deepest cut generation problem, via sets or via functions, and subsequently consider some special cases which are amenable to efficient computation. We conclude with computational tests of one of these procedures on a large set of MIPLIB problems. Copyright Springer Science + Business Media, Inc. 2005

Suggested Citation

  • Jonathan Eckstein & Mikhail Nediak, 2005. "Depth-Optimized Convexity Cuts," Annals of Operations Research, Springer, vol. 139(1), pages 95-129, October.
  • Handle: RePEc:spr:annopr:v:139:y:2005:i:1:p:95-129:10.1007/s10479-005-3445-y
    DOI: 10.1007/s10479-005-3445-y
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s10479-005-3445-y
    Download Restriction: Access to full text is restricted to subscribers.

    File URL: https://libkey.io/10.1007/s10479-005-3445-y?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. Balas, Egon & Jeroslow, Robert G., 1980. "Strengthening cuts for mixed integer programs," European Journal of Operational Research, Elsevier, vol. 4(4), pages 224-234, April.
    2. M. Raghavachari, 1969. "On Connections Between Zero-One Integer Programming and Concave Programming Under Linear Constraints," Operations Research, INFORMS, vol. 17(4), pages 680-684, August.
    3. Egon Balas & Sebastián Ceria & Gérard Cornuéjols, 1996. "Mixed 0-1 Programming by Lift-and-Project in a Branch-and-Cut Framework," Management Science, INFORMS, vol. 42(9), pages 1229-1246, September.
    4. Philip B. Zwart, 1973. "Nonlinear Programming: Counterexamples to Two Global Optimization Algorithms," Operations Research, INFORMS, vol. 21(6), pages 1260-1266, December.
    Full references (including those not matched with items on IDEAS)

    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. Sebastián Ceria, 2007. "A brief history of lift-and-project," Annals of Operations Research, Springer, vol. 149(1), pages 57-61, February.
    2. Egon Balas, 2005. "Projection, Lifting and Extended Formulation in Integer and Combinatorial Optimization," Annals of Operations Research, Springer, vol. 140(1), pages 125-161, November.
    3. Kent Andersen & Gérard Cornuéjols & Yanjun Li, 2005. "Reduce-and-Split Cuts: Improving the Performance of Mixed-Integer Gomory Cuts," Management Science, INFORMS, vol. 51(11), pages 1720-1732, November.
    4. Amitabh Basu & Robert Hildebrand & Matthias Köppe, 2016. "Light on the infinite group relaxation I: foundations and taxonomy," 4OR, Springer, vol. 14(1), pages 1-40, March.
    5. Mustafa Kılınç & Jeff Linderoth & James Luedtke & Andrew Miller, 2014. "Strong-branching inequalities for convex mixed integer nonlinear programs," Computational Optimization and Applications, Springer, vol. 59(3), pages 639-665, December.
    6. Amitabh Basu & Robert Hildebrand & Matthias Köppe, 2016. "Light on the infinite group relaxation II: sufficient conditions for extremality, sequences, and algorithms," 4OR, Springer, vol. 14(2), pages 107-131, June.
    7. Adam N. Letchford, 2001. "On Disjunctive Cuts for Combinatorial Optimization," Journal of Combinatorial Optimization, Springer, vol. 5(3), pages 299-315, September.
    8. Tao Tan & Yanyan Li & Xingsi Li, 2011. "A Smoothing Method for Zero–One Constrained Extremum Problems," Journal of Optimization Theory and Applications, Springer, vol. 150(1), pages 65-77, July.
    9. Drexl, Andreas & Nissen, Rudiger & Patterson, James H. & Salewski, Frank, 2000. "ProGen/[pi]x - An instance generator for resource-constrained project scheduling problems with partially renewable resources and further extensions," European Journal of Operational Research, Elsevier, vol. 125(1), pages 59-72, August.
    10. Vipul Jain & Ignacio E. Grossmann, 2001. "Algorithms for Hybrid MILP/CP Models for a Class of Optimization Problems," INFORMS Journal on Computing, INFORMS, vol. 13(4), pages 258-276, November.
    11. Egon Balas & Gérard Cornuéjols & Tamás Kis & Giacomo Nannicini, 2013. "Combining Lift-and-Project and Reduce-and-Split," INFORMS Journal on Computing, INFORMS, vol. 25(3), pages 475-487, August.
    12. Eva K. Lee, 2004. "Generating Cutting Planes for Mixed Integer Programming Problems in a Parallel Computing Environment," INFORMS Journal on Computing, INFORMS, vol. 16(1), pages 3-26, February.
    13. Peiping Shen & Kaimin Wang & Ting Lu, 2020. "Outer space branch and bound algorithm for solving linear multiplicative programming problems," Journal of Global Optimization, Springer, vol. 78(3), pages 453-482, November.
    14. André R. S. Amaral, 2008. "An Exact Approach to the One-Dimensional Facility Layout Problem," Operations Research, INFORMS, vol. 56(4), pages 1026-1033, August.
    15. Michele Conforti & Gérard Cornuéjols & Giacomo Zambelli, 2011. "A Geometric Perspective on Lifting," Operations Research, INFORMS, vol. 59(3), pages 569-577, June.
    16. Lawrence C. Leung & Yer Van Hui & Yong Wang & Gang Chen, 2009. "A 0--1 LP Model for the Integration and Consolidation of Air Cargo Shipments," Operations Research, INFORMS, vol. 57(2), pages 402-412, April.
    17. Brian Keller & Güzin Bayraksan, 2012. "Disjunctive Decomposition for Two-Stage Stochastic Mixed-Binary Programs with Generalized Upper Bound Constraints," INFORMS Journal on Computing, INFORMS, vol. 24(1), pages 172-186, February.
    18. G Appa & D Magos & I Mourtos, 2004. "A Branch & Cut algorithm for a four-index assignment problem," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 55(3), pages 298-307, March.
    19. Gérard Cornuéjols & Milind Dawande, 1999. "A Class of Hard Small 0-1 Programs," INFORMS Journal on Computing, INFORMS, vol. 11(2), pages 205-210, May.
    20. Stefano Lucidi & Francesco Rinaldi, 2010. "An Exact Penalty Global Optimization Approach for Mixed-Integer Programming Problems," DIS Technical Reports 2010-17, Department of Computer, Control and Management Engineering, Universita' degli Studi di Roma "La Sapienza".

    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:annopr:v:139:y:2005:i:1:p:95-129:10.1007/s10479-005-3445-y. 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.