IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v184y2020i2d10.1007_s10957-019-01591-y.html
   My bibliography  Save this article

Granularity in Nonlinear Mixed-Integer Optimization

Author

Listed:
  • Christoph Neumann

    (Karlsruhe Institute of Technology (KIT))

  • Oliver Stein

    (Karlsruhe Institute of Technology (KIT))

  • Nathan Sudermann-Merx

    (BASF SE)

Abstract

We study a new technique to check the existence of feasible points for mixed-integer nonlinear optimization problems that satisfy a structural requirement called granularity. For granular optimization problems, we show how rounding the optimal points of certain purely continuous optimization problems can lead to feasible points of the original mixed-integer nonlinear problem. To this end, we generalize results for the mixed-integer linear case from Neumann et al. (Comput Optim Appl 72:309–337, 2019). We study some additional issues caused by nonlinearity and show how to overcome them by extending the standard granularity concept to an advanced version, which we call pseudo-granularity. In a computational study on instances from a standard test library, we demonstrate that pseudo-granularity can be expected in many nonlinear applications from practice, and that its explicit use can be beneficial.

Suggested Citation

  • Christoph Neumann & Oliver Stein & Nathan Sudermann-Merx, 2020. "Granularity in Nonlinear Mixed-Integer Optimization," Journal of Optimization Theory and Applications, Springer, vol. 184(2), pages 433-465, February.
    • Handle: RePEc:spr:joptap:v:184:y:2020:i:2:d:10.1007_s10957-019-01591-y
    DOI: 10.1007/s10957-019-01591-y
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-019-01591-y
    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/s10957-019-01591-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. C. Audet & P. Hansen & B. Jaumard & G. Savard, 1997. "Links Between Linear Bilevel and Mixed 0–1 Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 93(2), pages 273-300, May.
    2. Christoph Neumann & Oliver Stein & Nathan Sudermann-Merx, 2019. "A feasible rounding approach for mixed-integer optimization problems," Computational Optimization and Applications, Springer, vol. 72(2), pages 309-337, March.
    3. Pierre Bonami & João Gonçalves, 2012. "Heuristics for convex mixed integer nonlinear programs," Computational Optimization and Applications, Springer, vol. 51(2), pages 729-747, March.
    4. Michael R. Bussieck & Arne Stolbjerg Drud & Alexander Meeraus, 2003. "MINLPLib—A Collection of Test Models for Mixed-Integer Nonlinear Programming," INFORMS Journal on Computing, INFORMS, vol. 15(1), pages 114-119, February.
    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. Timo Berthold, 2018. "A computational study of primal heuristics inside an MI(NL)P solver," Journal of Global Optimization, Springer, vol. 70(1), pages 189-206, January.
    2. Alain Billionnet & Sourour Elloumi & Amélie Lambert, 2013. "An efficient compact quadratic convex reformulation for general integer quadratic programs," Computational Optimization and Applications, Springer, vol. 54(1), pages 141-162, January.
    3. Chunyi Wang & Fengzhang Luo & Zheng Jiao & Xiaolei Zhang & Zhipeng Lu & Yanshuo Wang & Ren Zhao & Yang Yang, 2022. "An Enhanced Second-Order Cone Programming-Based Evaluation Method on Maximum Hosting Capacity of Solar Energy in Distribution Systems with Integrated Energy," Energies, MDPI, vol. 15(23), pages 1-19, November.
    4. C. Audet & G. Savard & W. Zghal, 2007. "New Branch-and-Cut Algorithm for Bilevel Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 134(2), pages 353-370, August.
    5. Kai Zhou & Mustafa R. Kılınç & Xi Chen & Nikolaos V. Sahinidis, 2018. "An efficient strategy for the activation of MIP relaxations in a multicore global MINLP solver," Journal of Global Optimization, Springer, vol. 70(3), pages 497-516, March.
    6. Cao, Dong & Chen, Mingyuan, 2006. "Capacitated plant selection in a decentralized manufacturing environment: A bilevel optimization approach," European Journal of Operational Research, Elsevier, vol. 169(1), pages 97-110, February.
    7. Nair, Rahul & Miller-Hooks, Elise, 2014. "Equilibrium network design of shared-vehicle systems," European Journal of Operational Research, Elsevier, vol. 235(1), pages 47-61.
    8. Bhatt, Sneha Dhyani & Sinha, Ankur & Jayaswal, Sachin, 2024. "The capacitated r-hub interdiction problem with congestion: Models and solution approaches," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 185(C).
    9. S. Dempe & S. Franke, 2016. "On the solution of convex bilevel optimization problems," Computational Optimization and Applications, Springer, vol. 63(3), pages 685-703, April.
    10. Moon, Kyungduk & Lee, Kangbok & Chopra, Sunil & Kwon, Steve, 2022. "Bilevel integer programming on a Boolean network for discovering critical genetic alterations in cancer development and therapy," European Journal of Operational Research, Elsevier, vol. 300(2), pages 743-754.
    11. Sonia Cafieri & Claudia D’Ambrosio, 2018. "Feasibility pump for aircraft deconfliction with speed regulation," Journal of Global Optimization, Springer, vol. 71(3), pages 501-515, July.
    12. Cafieri, Sonia & Omheni, Riadh, 2017. "Mixed-integer nonlinear programming for aircraft conflict avoidance by sequentially applying velocity and heading angle changes," European Journal of Operational Research, Elsevier, vol. 260(1), pages 283-290.
    13. Juan S. Borrero & Oleg A. Prokopyev & Denis Sauré, 2019. "Sequential Interdiction with Incomplete Information and Learning," Operations Research, INFORMS, vol. 67(1), pages 72-89, January.
    14. Bo Zeng, 2020. "A Practical Scheme to Compute the Pessimistic Bilevel Optimization Problem," INFORMS Journal on Computing, INFORMS, vol. 32(4), pages 1128-1142, October.
    15. Christoph Neumann & Oliver Stein & Nathan Sudermann-Merx, 2019. "A feasible rounding approach for mixed-integer optimization problems," Computational Optimization and Applications, Springer, vol. 72(2), pages 309-337, March.
    16. Gabriel Lopez Zenarosa & Oleg A. Prokopyev & Eduardo L. Pasiliao, 2021. "On exact solution approaches for bilevel quadratic 0–1 knapsack problem," Annals of Operations Research, Springer, vol. 298(1), pages 555-572, March.
    17. Wei, Ningji & Walteros, Jose L., 2022. "Integer programming methods for solving binary interdiction games," European Journal of Operational Research, Elsevier, vol. 302(2), pages 456-469.
    18. Andrew Allman & Qi Zhang, 2021. "Branch-and-price for a class of nonconvex mixed-integer nonlinear programs," Journal of Global Optimization, Springer, vol. 81(4), pages 861-880, December.
    19. Meenarli Sharma & Prashant Palkar & Ashutosh Mahajan, 2022. "Linearization and parallelization schemes for convex mixed-integer nonlinear optimization," Computational Optimization and Applications, Springer, vol. 81(2), pages 423-478, March.
    20. Jing Yang & Juan S. Borrero & Oleg A. Prokopyev & Denis Sauré, 2021. "Sequential Shortest Path Interdiction with Incomplete Information and Limited Feedback," Decision Analysis, INFORMS, vol. 18(3), pages 218-244, September.

    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:joptap:v:184:y:2020:i:2:d:10.1007_s10957-019-01591-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.