IDEAS home Printed from https://ideas.repec.org/a/inm/ormnsc/v64y2018i10p4721-4734.html
   My bibliography  Save this article

Sloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Author

Listed:
  • Juan Pablo Vielma

    (Sloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139)

Abstract

It is well known that selecting a good mixed-integer programming (MIP) formulation is crucial for effectively obtaining a solution with state-of-the art solvers. Although best practices and guidelines for constructing good formulations abound, there is rarely a single systematic construction approach that leads to the best possible formulation. Here, we introduce embedding formulations and complexity as a new MIP formulation paradigm for systematically constructing formulations for disjunctive constraints that are optimal with respect to size. More specifically, they yield the smallest possible ideal formulation (i.e., one whose LP relaxation has integral extreme points) among all formulations that only use 0-1 auxiliary variables. We use the paradigm to characterize optimal formulations for special ordered sets of type 2 and certain piecewise linear functions of two variables. We also show that the resultant formulations can provide a significant computational advantage over all known formulations for piecewise linear functions.

Suggested Citation

  • Juan Pablo Vielma, 2018. "Sloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139," Management Science, INFORMS, vol. 64(10), pages 4721-4734, October.
    • Handle: RePEc:inm:ormnsc:v:64:y:2018:i:10:p:4721-4734
    DOI: 10.287/mnsc.2017.2856
    as

    Download full text from publisher

    File URL: https://doi.org/10.287/mnsc.2017.2856
    Download Restriction: no
    File URL: https://libkey.io/10.287/mnsc.2017.2856?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
    ---><---

    References listed on IDEAS

    as
    1. Toriello, Alejandro & Vielma, Juan Pablo, 2012. "Fitting piecewise linear continuous functions," European Journal of Operational Research, Elsevier, vol. 219(1), pages 86-95.
    2. Roberto Roberti & Enrico Bartolini & Aristide Mingozzi, 2015. "The Fixed Charge Transportation Problem: An Exact Algorithm Based on a New Integer Programming Formulation," Management Science, INFORMS, vol. 61(6), pages 1275-1291, June.
    3. Juan Pablo Vielma & Shabbir Ahmed & George Nemhauser, 2010. "Mixed-Integer Models for Nonseparable Piecewise-Linear Optimization: Unifying Framework and Extensions," Operations Research, INFORMS, vol. 58(2), pages 303-315, April.
    4. Alan S. Manne, 1960. "On the Job-Shop Scheduling Problem," Operations Research, INFORMS, vol. 8(2), pages 219-223, April.
    5. Miles Lubin & Iain Dunning, 2015. "Computing in Operations Research Using Julia," INFORMS Journal on Computing, INFORMS, vol. 27(2), pages 238-248, May.
    6. Keely L. Croxton & Bernard Gendron & Thomas L. Magnanti, 2003. "A Comparison of Mixed-Integer Programming Models for Nonconvex Piecewise Linear Cost Minimization Problems," Management Science, INFORMS, vol. 49(9), pages 1268-1273, September.
    7. LEE, Jon & WILSON, Dan, 2001. "Polyhedral methods for piecewise-linear functions I: the lambda method," LIDAM Reprints CORE 1493, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    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. Jon Lee & Daphne Skipper & Emily Speakman & Luze Xu, 2023. "Gaining or Losing Perspective for Piecewise-Linear Under-Estimators of Convex Univariate Functions," Journal of Optimization Theory and Applications, Springer, vol. 196(1), pages 1-35, January.
    2. Steffen Rebennack & Vitaliy Krasko, 2020. "Piecewise Linear Function Fitting via Mixed-Integer Linear Programming," INFORMS Journal on Computing, INFORMS, vol. 32(2), pages 507-530, April.
    3. Lingxun Kong & Christos T. Maravelias, 2020. "On the Derivation of Continuous Piecewise Linear Approximating Functions," INFORMS Journal on Computing, INFORMS, vol. 32(3), pages 531-546, July.
    4. Silva, Thiago Lima & Camponogara, Eduardo, 2014. "A computational analysis of multidimensional piecewise-linear models with applications to oil production optimization," European Journal of Operational Research, Elsevier, vol. 232(3), pages 630-642.
    5. Christensen, Tue R.L. & Labbé, Martine, 2015. "A branch-cut-and-price algorithm for the piecewise linear transportation problem," European Journal of Operational Research, Elsevier, vol. 245(3), pages 645-655.
    6. Gambella, Claudio & Ghaddar, Bissan & Naoum-Sawaya, Joe, 2021. "Optimization problems for machine learning: A survey," European Journal of Operational Research, Elsevier, vol. 290(3), pages 807-828.
    7. Benjamin Beach & Robert Hildebrand & Joey Huchette, 2022. "Compact mixed-integer programming formulations in quadratic optimization," Journal of Global Optimization, Springer, vol. 84(4), pages 869-912, December.
    8. Kazda, Kody & Li, Xiang, 2024. "A linear programming approach to difference-of-convex piecewise linear approximation," European Journal of Operational Research, Elsevier, vol. 312(2), pages 493-511.
    9. Hua, Hao & Hovestadt, Ludger & Tang, Peng & Li, Biao, 2019. "Integer programming for urban design," European Journal of Operational Research, Elsevier, vol. 274(3), pages 1125-1137.
    10. Juan Pablo Vielma & Shabbir Ahmed & George Nemhauser, 2010. "Mixed-Integer Models for Nonseparable Piecewise-Linear Optimization: Unifying Framework and Extensions," Operations Research, INFORMS, vol. 58(2), pages 303-315, April.
    11. Xiaolin Huang & Jun Xu & Shuning Wang, 2012. "Exact Penalty and Optimality Condition for Nonseparable Continuous Piecewise Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 155(1), pages 145-164, October.
    12. Nasini, Stefano & Labbé, Martine & Brotcorne, Luce, 2022. "Multi-market portfolio optimization with conditional value at risk," European Journal of Operational Research, Elsevier, vol. 300(1), pages 350-365.
    13. Wu, Yaqing & Maravelias, Christos T., 2024. "Piecewise linear trees as surrogate models for system design and planning under high-frequency temporal variability," European Journal of Operational Research, Elsevier, vol. 315(2), pages 541-552.
    14. Zhang, Hongbin & Yang, Yu & Wu, Feng, 2024. "Scheduling a set of jobs with convex piecewise linear cost functions on a single-batch-processing machine," Omega, Elsevier, vol. 122(C).
    15. Archetti, Claudia & Bertazzi, Luca & Grazia Speranza, M., 2014. "Polynomial cases of the economic lot sizing problem with cost discounts," European Journal of Operational Research, Elsevier, vol. 237(2), pages 519-527.
    16. Fortz, Bernard & Gouveia, Luís & Joyce-Moniz, Martim, 2017. "Models for the piecewise linear unsplittable multicommodity flow problems," European Journal of Operational Research, Elsevier, vol. 261(1), pages 30-42.
    17. Hu, Qian & Lim, Andrew & Zhu, Wenbin, 2015. "The two-dimensional vector packing problem with piecewise linear cost function," Omega, Elsevier, vol. 50(C), pages 43-53.
    18. Zheng, Xiao-Xue & Chang, Ching-Ter, 2021. "Topology design of remote patient monitoring system concerning qualitative and quantitative issues," Omega, Elsevier, vol. 98(C).
    19. Srikrishna Sridhar & Jeffrey Linderoth & James Luedtke, 2014. "Models and solution techniques for production planning problems with increasing byproducts," Journal of Global Optimization, Springer, vol. 59(2), pages 597-631, July.
    20. Birolini, Sebastian & Jacquillat, Alexandre & Cattaneo, Mattia & Antunes, António Pais, 2021. "Airline Network Planning: Mixed-integer non-convex optimization with demand–supply interactions," Transportation Research Part B: Methodological, Elsevier, vol. 154(C), pages 100-124.

    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:inm:ormnsc:v:64:y:2018:i:10:p:4721-4734. 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: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    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.