IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v84y2022i4d10.1007_s10898-022-01179-3.html
   My bibliography  Save this article

Discretization and global optimization for mixed integer bilinear programming

Author

Listed:
  • Xin Cheng

    (Queen’s University)

  • Xiang Li

    (Queen’s University)

Abstract

We consider global optimization of mixed-integer bilinear programs (MIBLP) using discretization-based mixed-integer linear programming (MILP) relaxations. We start from the widely used radix-based discretization formulation (called R-formulation in this paper), where the base R may be any natural number, but we do not require the discretization level to be a power of R. We prove the conditions under which R-formulation is locally sharp, and then propose an $$R^+$$ R + -formulation that is always locally sharp. We also propose an H-formulation that allows multiple bases and prove that it is also always locally sharp. We develop a global optimization algorithm with adaptive discretization (GOAD) where the discretization level of each variable is determined according to the solution of previously solved MILP relaxations. The computational study shows the computational advantage of GOAD over general-purpose global solvers BARON and SCIP.

Suggested Citation

  • Xin Cheng & Xiang Li, 2022. "Discretization and global optimization for mixed integer bilinear programming," Journal of Global Optimization, Springer, vol. 84(4), pages 843-867, December.
    • Handle: RePEc:spr:jglopt:v:84:y:2022:i:4:d:10.1007_s10898-022-01179-3
    DOI: 10.1007/s10898-022-01179-3
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10898-022-01179-3
    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/s10898-022-01179-3?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. João Teles & Pedro Castro & Henrique Matos, 2013. "Multi-parametric disaggregation technique for global optimization of polynomial programming problems," Journal of Global Optimization, Springer, vol. 55(2), pages 227-251, February.
    2. Scott Kolodziej & Pedro Castro & Ignacio Grossmann, 2013. "Global optimization of bilinear programs with a multiparametric disaggregation technique," Journal of Global Optimization, Springer, vol. 57(4), pages 1039-1063, December.
    3. Akshay Gupte & Shabbir Ahmed & Santanu S. Dey & Myun Seok Cheon, 2017. "Relaxations and discretizations for the pooling problem," Journal of Global Optimization, Springer, vol. 67(3), pages 631-669, March.
    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. Tiago Andrade & Fabricio Oliveira & Silvio Hamacher & Andrew Eberhard, 2019. "Enhancing the normalized multiparametric disaggregation technique for mixed-integer quadratic programming," Journal of Global Optimization, Springer, vol. 73(4), pages 701-722, April.
    2. Jianhui Xie & Qiwei Xie & Yongjun Li & Liang Liang, 2021. "Solving data envelopment analysis models with sum-of-fractional objectives: a global optimal approach based on the multiparametric disaggregation technique," Annals of Operations Research, Springer, vol. 304(1), pages 453-480, September.
    3. Teles, João P. & Castro, Pedro M. & Matos, Henrique A., 2013. "Univariate parameterization for global optimization of mixed-integer polynomial problems," European Journal of Operational Research, Elsevier, vol. 229(3), pages 613-625.
    4. Pedro Castro & Ignacio Grossmann, 2014. "Optimality-based bound contraction with multiparametric disaggregation for the global optimization of mixed-integer bilinear problems," Journal of Global Optimization, Springer, vol. 59(2), pages 277-306, July.
    5. Radu Baltean-Lugojan & Ruth Misener, 2018. "Piecewise parametric structure in the pooling problem: from sparse strongly-polynomial solutions to NP-hardness," Journal of Global Optimization, Springer, vol. 71(4), pages 655-690, August.
    6. Enayati, Shakiba & Özaltın, Osman Y., 2020. "Optimal influenza vaccine distribution with equity," European Journal of Operational Research, Elsevier, vol. 283(2), pages 714-725.
    7. Natashia Boland & Thomas Kalinowski & Fabian Rigterink, 2016. "New multi-commodity flow formulations for the pooling problem," Journal of Global Optimization, Springer, vol. 66(4), pages 669-710, December.
    8. Yifu Chen & Christos T. Maravelias, 2020. "Preprocessing algorithm and tightening constraints for multiperiod blend scheduling: cost minimization," Journal of Global Optimization, Springer, vol. 77(3), pages 603-625, July.
    9. G. Constante-Flores & A. J. Conejo & S. Constante-Flores, 2022. "Solving certain complementarity problems in power markets via convex programming," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(3), pages 465-491, October.
    10. Benhamou, Latifa & Giard, Vincent & Khouloud, Mehdi & Fenies, Pierres & Fontane, Frédéric, 2020. "Reverse Blending: An economically efficient approach to the challenge of fertilizer mass customization," International Journal of Production Economics, Elsevier, vol. 226(C).
    11. Tiago Andrade & Nikita Belyak & Andrew Eberhard & Silvio Hamacher & Fabricio Oliveira, 2022. "The p-Lagrangian relaxation for separable nonconvex MIQCQP problems," Journal of Global Optimization, Springer, vol. 84(1), pages 43-76, September.
    12. Teodora Dan & Andrea Lodi & Patrice Marcotte, 2020. "Joint location and pricing within a user-optimized environment," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 8(1), pages 61-84, March.
    13. Pedro A. Castillo Castillo & Pedro M. Castro & Vladimir Mahalec, 2018. "Global optimization of MIQCPs with dynamic piecewise relaxations," Journal of Global Optimization, Springer, vol. 71(4), pages 691-716, August.
    14. Fischetti, Matteo & Monaci, Michele, 2020. "A branch-and-cut algorithm for Mixed-Integer Bilinear Programming," European Journal of Operational Research, Elsevier, vol. 282(2), pages 506-514.
    15. Boukouvala, Fani & Misener, Ruth & Floudas, Christodoulos A., 2016. "Global optimization advances in Mixed-Integer Nonlinear Programming, MINLP, and Constrained Derivative-Free Optimization, CDFO," European Journal of Operational Research, Elsevier, vol. 252(3), pages 701-727.
    16. Escudero, Laureano F. & Garín, M. Araceli & Monge, Juan F. & Unzueta, Aitziber, 2020. "Some matheuristic algorithms for multistage stochastic optimization models with endogenous uncertainty and risk management," European Journal of Operational Research, Elsevier, vol. 285(3), pages 988-1001.
    17. Unai Aldasoro & María Merino & Gloria Pérez, 2019. "Time consistent expected mean-variance in multistage stochastic quadratic optimization: a model and a matheuristic," Annals of Operations Research, Springer, vol. 280(1), pages 151-187, September.
    18. Ahmadreza Marandi & Joachim Dahl & Etienne Klerk, 2018. "A numerical evaluation of the bounded degree sum-of-squares hierarchy of Lasserre, Toh, and Yang on the pooling problem," Annals of Operations Research, Springer, vol. 265(1), pages 67-92, June.
    19. Masaki Kimizuka & Sunyoung Kim & Makoto Yamashita, 2019. "Solving pooling problems with time discretization by LP and SOCP relaxations and rescheduling methods," Journal of Global Optimization, Springer, vol. 75(3), pages 631-654, November.
    20. Marcia Fampa & Jon Lee, 2021. "Convexification of bilinear forms through non-symmetric lifting," Journal of Global Optimization, Springer, vol. 80(2), pages 287-305, June.

    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:jglopt:v:84:y:2022:i:4:d:10.1007_s10898-022-01179-3. 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.