IDEAS home Printed from https://ideas.repec.org/a/pal/jorsoc/v57y2006i8d10.1057_palgrave.jors.2602052.html
   My bibliography  Save this article

A practical approach to approximate bilinear functions in mathematical programming problems by using Schur's decomposition and SOS type 2 variables

Author

Listed:
  • S A Gabriel

    (University of Maryland)

  • R García-Bertrand

    (Universidad de Castilla – La Mancha)

  • P Sahakij

    (University of Maryland)

  • A J Conejo

    (Universidad de Castilla – La Mancha)

Abstract

This paper provides a new methodology to solve bilinear, non-convex mathematical programming problems by a suitable transformation of variables. Schur's decomposition and special ordered sets (SOS) type 2 constraints are used resulting in a mixed integer linear or quadratic program in the two applications shown. While Beale, Tomlin and others developed the use of SOS type 2 variables to handle non-convexities, our approach is novel in two aspects. First, the use of Schur's decomposition as an integral part of the approximation step is new and leads to a numerically viable method to separate the variables. Second, the combination of our approach for handling bilinear side constraints in a complementarity or equilibrium problem setting is also new and opens the way to many interesting and realistic modifications to such models. We contrast our approach with other methods for solving bilinear problems also known as indefinite quadratic programs. From a practical point of view our methodology is helpful since no specialized procedures need to be created so that existing solvers can be used. The approach is illustrated with two engineering examples and the mathematical analysis appears in the Appendices.

Suggested Citation

  • S A Gabriel & R García-Bertrand & P Sahakij & A J Conejo, 2006. "A practical approach to approximate bilinear functions in mathematical programming problems by using Schur's decomposition and SOS type 2 variables," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 57(8), pages 995-1004, August.
    • Handle: RePEc:pal:jorsoc:v:57:y:2006:i:8:d:10.1057_palgrave.jors.2602052
    DOI: 10.1057/palgrave.jors.2602052
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1057/palgrave.jors.2602052
    File Function: Abstract
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1057/palgrave.jors.2602052?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. Wilson, J. M., 1990. "Generating cuts in integer programming with families of special ordered sets," European Journal of Operational Research, Elsevier, vol. 46(1), pages 101-108, May.
    2. De Wolf, D. & Smeers, Y., 1996. "Optimal dimensioning of pipe networks with application to gas transmission networks," LIDAM Reprints CORE 1249, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Richard P. O'Neill & Mark Williard & Bert Wilkins & Ralph Pike, 1979. "A Mathematical Programming Model for Allocation of Natural Gas," Operations Research, INFORMS, vol. 27(5), pages 857-873, October.
    4. Hummeltenberg, Wilhelm, 1984. "Implementations of special ordered sets in MP software," European Journal of Operational Research, Elsevier, vol. 17(1), pages 1-15, July.
    5. Daniel de Wolf & Yves Smeers, 1996. "Optimal Dimensioning of Pipe Networks with Application to Gas Transmission Networks," Operations Research, INFORMS, vol. 44(4), pages 596-608, August.
    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. S A Gabriel & P Sahakij & M Ramirez & C Peot, 2007. "A multiobjective optimization model for processing and distributing biosolids to reuse fields," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 58(7), pages 850-864, July.
    2. Ran, Cuiling & Zhang, Yanzi & Yin, Ying, 2021. "Demand response to improve the shared electric vehicle planning: Managerial insights, sustainable benefits," Applied Energy, Elsevier, vol. 292(C).
    3. Hammad, Ahmed W A & Akbarnezhad, Ali & Rey, David, 2017. "Sustainable urban facility location: Minimising noise pollution and network congestion," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 107(C), pages 38-59.
    4. S. Siddiqui & S. Gabriel, 2013. "An SOS1-Based Approach for Solving MPECs with a Natural Gas Market Application," Networks and Spatial Economics, Springer, vol. 13(2), pages 205-227, June.
    5. Zihao Jiao & Lun Ran & Xin Liu & Yuli Zhang & Robin G. Qiu, 2020. "Integrating Price-Incentive and Trip-Selection Policies to Rebalance Shared Electric Vehicles," Service Science, INFORMS, vol. 12(4), pages 148-173, December.
    6. E Bustamante-Cedeño & S Arora, 2008. "Stochastic and minimum regret formulations for transmission network expansion planning under uncertainties," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 59(11), pages 1547-1556, November.

    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. Daniel de Wolf, 2017. "Mathematical Properties of Formulations of the Gas Transmission Problem," Post-Print halshs-02396747, HAL.
    2. Steven A. Gabriel & Supat Kiet & Jifang Zhuang, 2005. "A Mixed Complementarity-Based Equilibrium Model of Natural Gas Markets," Operations Research, INFORMS, vol. 53(5), pages 799-818, October.
    3. 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.
    4. Conrado Borraz-Sánchez & Russell Bent & Scott Backhaus & Hassan Hijazi & Pascal Van Hentenryck, 2016. "Convex Relaxations for Gas Expansion Planning," INFORMS Journal on Computing, INFORMS, vol. 28(4), pages 645-656, November.
    5. Jesco Humpola & Felipe Serrano, 2017. "Sufficient pruning conditions for MINLP in gas network design," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 5(1), pages 239-261, March.
    6. Jesco Humpola & Armin Fügenschuh & Thorsten Koch, 2016. "Valid inequalities for the topology optimization problem in gas network design," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 38(3), pages 597-631, July.
    7. repec:cty:dpaper:10.1080/0013791x.2011.573615 is not listed on IDEAS
    8. Liang, Yingzong & Hui, Chi Wai, 2018. "Convexification for natural gas transmission networks optimization," Energy, Elsevier, vol. 158(C), pages 1001-1016.
    9. Thapalia, Biju K. & Crainic, Teodor Gabriel & Kaut, Michal & Wallace, Stein W., 2012. "Single-commodity network design with random edge capacities," European Journal of Operational Research, Elsevier, vol. 220(2), pages 394-403.
    10. repec:cty:dpaper:1464 is not listed on IDEAS
    11. Dieckhoener, Caroline, 2010. "Simulating security of supply effects of the Nabucco and South Stream projects for the European natural gas market," EWI Working Papers 2010-7, Energiewirtschaftliches Institut an der Universitaet zu Koeln (EWI), revised 21 Jan 2012.
    12. Daniel de Wolf & Yves Smeers, 2021. "Generalized derivatives of the optimal value of a linear program with respect to matrix coefficients," Post-Print halshs-02396708, HAL.
    13. Frédéric Babonneau & Yurii Nesterov & Jean-Philippe Vial, 2012. "Design and Operations of Gas Transmission Networks," Operations Research, INFORMS, vol. 60(1), pages 34-47, February.
    14. Zhou, Jun & Zhao, Yunxiang & Fu, Tiantian & Zhou, Xuan & Liang, Guangchuan, 2022. "Dimension optimization for underground natural gas storage pipeline network coupling injection and production conditions," Energy, Elsevier, vol. 256(C).
    15. Ralf Lenz & Kai Helge Becker, 2022. "Optimization of capacity expansion in potential-driven networks including multiple looping: a comparison of modelling approaches," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 44(1), pages 179-224, March.
    16. GABRIEL, Steven & SMEERS, Yves, 2005. "Complementarity problems in restructured natural gas markets," LIDAM Discussion Papers CORE 2005037, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    17. Olivier Massol, 2011. "A Cost Function for the Natural Gas Transmission Industry: Further Considerations," The Engineering Economist, Taylor & Francis Journals, vol. 56(2), pages 95-122.
    18. Jesco Humpola & Armin Fügenschuh, 2015. "Convex reformulations for solving a nonlinear network design problem," Computational Optimization and Applications, Springer, vol. 62(3), pages 717-759, December.
    19. Shiono, Naoshi & Suzuki, Hisatoshi, 2016. "Optimal pipe-sizing problem of tree-shaped gas distribution networks," European Journal of Operational Research, Elsevier, vol. 252(2), pages 550-560.
    20. Demierre, Jonathan & Bazilian, Morgan & Carbajal, Jonathan & Sherpa, Shaky & Modi, Vijay, 2015. "Potential for regional use of East Africa’s natural gas," Applied Energy, Elsevier, vol. 143(C), pages 414-436.
    21. Massol, Olivier & Tchung-Ming, Stéphane & Banal-Estañol, Albert, 2015. "Joining the CCS club! The economics of CO2 pipeline projects," European Journal of Operational Research, Elsevier, vol. 247(1), pages 259-275.
    22. Jingkuan Han & Yingjun Xu & Dingzhi Liu & Yanfang Zhao & Zhongde Zhao & Shuhui Zhou & Tianhu Deng & Mengying Xue & Junchi Ye & Zuo-Jun Max Shen, 2019. "Operations Research Enables Better Planning of Natural Gas Pipelines," Interfaces, INFORMS, vol. 49(1), pages 23-39, January.

    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:pal:jorsoc:v:57:y:2006:i:8:d:10.1057_palgrave.jors.2602052. 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.palgrave-journals.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.