IDEAS home Printed from https://ideas.repec.org/a/spr/mathme/v84y2016i1d10.1007_s00186-016-0546-0.html
   My bibliography  Save this article

Computing tight bounds via piecewise linear functions through the example of circle cutting problems

Author

Listed:
  • Steffen Rebennack

    (Colorado School of Mines)

Abstract

This paper discusses approximations of continuous and mixed-integer non-linear optimization problems via piecewise linear functions. Various variants of circle cutting problems are considered, where the non-overlap of circles impose a non-convex feasible region. While the paper is written in an “easy-to-understand” and “hands-on” style which should be accessible to graduate students, also new ideas are presented. Specifically, piecewise linear functions are employed to yield mixed-integer linear programming problems which provide lower and upper bounds on the original problem, the circle cutting problem. The piecewise linear functions are modeled by five different formulations, containing the incremental and logarithmic formulations. Another variant of the cutting problem involves the assignment of circles to pre-defined rectangles. We introduce a new global optimization algorithm, based on piecewise linear function approximations, which converges in finitely many iterations to a globally optimal solution. The discussed formulations are implemented in GAMS. All GAMS-files are available for download in the Electronic supplementary material. Extensive computational results are presented with various illustrations.

Suggested Citation

  • Steffen Rebennack, 2016. "Computing tight bounds via piecewise linear functions through the example of circle cutting problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 84(1), pages 3-57, August.
    • Handle: RePEc:spr:mathme:v:84:y:2016:i:1:d:10.1007_s00186-016-0546-0
    DOI: 10.1007/s00186-016-0546-0
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s00186-016-0546-0
    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/s00186-016-0546-0?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. Stephen Frank & Steffen Rebennack, 2012. "A Primer on Optimal Power Flow: Theory, Formulation, and Practical Examples," Working Papers 2012-14, Colorado School of Mines, Division of Economics and Business.
    2. Frank, Stephen M. & Rebennack, Steffen, 2015. "Optimal design of mixed AC–DC distribution systems for commercial buildings: A Nonconvex Generalized Benders Decomposition approach," European Journal of Operational Research, Elsevier, vol. 242(3), pages 710-729.
    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. Ruth Misener & Christodoulos Floudas, 2014. "ANTIGONE: Algorithms for coNTinuous / Integer Global Optimization of Nonlinear Equations," Journal of Global Optimization, Springer, vol. 59(2), pages 503-526, July.
    5. Ahmet B. Keha & Ismael R. de Farias & George L. Nemhauser, 2006. "A Branch-and-Cut Algorithm Without Binary Variables for Nonconvex Piecewise Linear Optimization," Operations Research, INFORMS, vol. 54(5), pages 847-858, October.
    6. Björn Geißler & Oliver Kolb & Jens Lang & Günter Leugering & Alexander Martin & Antonio Morsi, 2011. "Mixed integer linear models for the optimization of dynamical transport networks," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 73(3), pages 339-362, June.
    7. Xiang Li & Asgeir Tomasgard & Paul I. Barton, 2011. "Nonconvex Generalized Benders Decomposition for Stochastic Separable Mixed-Integer Nonlinear Programs," Journal of Optimization Theory and Applications, Springer, vol. 151(3), pages 425-454, December.
    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. 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.
    2. Er-Rahmadi, Btissam & Ma, Tiejun, 2022. "Data-driven mixed-Integer linear programming-based optimisation for efficient failure detection in large-scale distributed systems," European Journal of Operational Research, Elsevier, vol. 303(1), pages 337-353.
    3. Lambert, Mathieu & Hassani, Rachid, 2023. "Diesel genset optimization in remote microgrids," Applied Energy, Elsevier, vol. 340(C).
    4. John Alasdair Warwicker & Steffen Rebennack, 2022. "A Comparison of Two Mixed-Integer Linear Programs for Piecewise Linear Function Fitting," INFORMS Journal on Computing, INFORMS, vol. 34(2), pages 1042-1047, March.
    5. Steffen Rebennack, 2022. "Data-driven stochastic optimization for distributional ambiguity with integrated confidence region," Journal of Global Optimization, Springer, vol. 84(2), pages 255-293, October.
    6. F. J. Hwang & Yao-Huei Huang, 2021. "An effective logarithmic formulation for piecewise linearization requiring no inequality constraint," Computational Optimization and Applications, Springer, vol. 79(3), pages 601-631, July.
    7. Akang Wang & Chrysanthos E. Gounaris, 2021. "On tackling reverse convex constraints for non-overlapping of unequal circles," Journal of Global Optimization, Springer, vol. 80(2), pages 357-385, June.

    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. 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.
    2. Subramanian, Avinash S.R. & Gundersen, Truls & Barton, Paul I. & Adams, Thomas A., 2022. "Global optimization of a hybrid waste tire and natural gas feedstock polygeneration system," Energy, Elsevier, vol. 250(C).
    3. 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.
    4. Joey Huchette & Joey Huchette, 2019. "A Combinatorial Approach for Small and Strong Formulations of Disjunctive Constraints," Mathematics of Operations Research, INFORMS, vol. 44(3), pages 793-820, August.
    5. Emmanuel Ogbe & Xiang Li, 2019. "A joint decomposition method for global optimization of multiscenario nonconvex mixed-integer nonlinear programs," Journal of Global Optimization, Springer, vol. 75(3), pages 595-629, November.
    6. Bjarne Grimstad & Brage R. Knudsen, 2020. "Mathematical programming formulations for piecewise polynomial functions," Journal of Global Optimization, Springer, vol. 77(3), pages 455-486, July.
    7. 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.
    8. 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.
    9. Er-Rahmadi, Btissam & Ma, Tiejun, 2022. "Data-driven mixed-Integer linear programming-based optimisation for efficient failure detection in large-scale distributed systems," European Journal of Operational Research, Elsevier, vol. 303(1), pages 337-353.
    10. Liron Ravner & Yoni Nazarathy, 2017. "Scheduling for a processor sharing system with linear slowdown," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 86(1), pages 71-102, August.
    11. Hermann, Alexander & Jensen, Tue Vissing & Østergaard, Jacob & Kazempour, Jalal, 2022. "A complementarity model for electric power transmission-distribution coordination under uncertainty," European Journal of Operational Research, Elsevier, vol. 299(1), pages 313-329.
    12. Emmanuel Ogbe & Ali Almansoori & Michael Fowler & Ali Elkamel, 2023. "Optimizing Renewable Injection in Integrated Natural Gas Pipeline Networks Using a Multi-Period Programming Approach," Energies, MDPI, vol. 16(6), pages 1-24, March.
    13. Bei, Xiaoqiang & Zhu, Xiaoyan & Coit, David W., 2019. "A risk-averse stochastic program for integrated system design and preventive maintenance planning," European Journal of Operational Research, Elsevier, vol. 276(2), pages 536-548.
    14. Subramanian, Avinash S.R. & Kannan, Rohit & Holtorf, Flemming & Adams, Thomas A. & Gundersen, Truls & Barton, Paul I., 2023. "Optimization under uncertainty of a hybrid waste tire and natural gas feedstock flexible polygeneration system using a decomposition algorithm," Energy, Elsevier, vol. 284(C).
    15. Codas, Andrés & Camponogara, Eduardo, 2012. "Mixed-integer linear optimization for optimal lift-gas allocation with well-separator routing," European Journal of Operational Research, Elsevier, vol. 217(1), pages 222-231.
    16. Ogbe, Emmanuel & Li, Xiang, 2017. "A new cross decomposition method for stochastic mixed-integer linear programming," European Journal of Operational Research, Elsevier, vol. 256(2), pages 487-499.
    17. 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.
    18. Jia-Jiang Lin & Feng Xu & Xiong-Lin Luo, 2023. "Nonconvex sensitivity-based generalized Benders decomposition," Journal of Global Optimization, Springer, vol. 86(1), pages 37-60, May.
    19. Mengying Xue & Tianhu Deng & Zuo‐Jun Max Shen, 2019. "Optimizing natural gas pipeline transmission with nonuniform elevation: A new initialization approach," Naval Research Logistics (NRL), John Wiley & Sons, vol. 66(7), pages 547-564, October.
    20. Jaromił Najman & Dominik Bongartz & Alexander Mitsos, 2021. "Linearization of McCormick relaxations and hybridization with the auxiliary variable method," Journal of Global Optimization, Springer, vol. 80(4), pages 731-756, August.

    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:mathme:v:84:y:2016:i:1:d:10.1007_s00186-016-0546-0. 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.