IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v67y2017i1d10.1007_s10898-015-0348-6.html
   My bibliography  Save this article

Packing ellipsoids into volume-minimizing rectangular boxes

Author

Listed:
  • Josef Kallrath

    (BASF SE, Scientific Computing, GMC/MS-B009
    University of Florida)

Abstract

A set of tri-axial ellipsoids, with given semi-axes, is to be packed into a rectangular box; its widths, lengths and height are subject to lower and upper bounds. We want to minimize the volume of this box and seek an overlap-free placement of the ellipsoids which can take any orientation. We present closed non-convex NLP formulations for this ellipsoid packing problem based on purely algebraic approaches to represent rotated and shifted ellipsoids. We consider the elements of the rotation matrix as variables. Separating hyperplanes are constructed to ensure that the ellipsoids do not overlap with each other. For up to 100 ellipsoids we compute feasible points with the global solvers available in GAMS. Only for special cases of two ellipsoids we are able to reach gaps smaller than $$10^{-4}$$ 10 - 4 .

Suggested Citation

  • Josef Kallrath, 2017. "Packing ellipsoids into volume-minimizing rectangular boxes," Journal of Global Optimization, Springer, vol. 67(1), pages 151-185, January.
    • Handle: RePEc:spr:jglopt:v:67:y:2017:i:1:d:10.1007_s10898-015-0348-6
    DOI: 10.1007/s10898-015-0348-6
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10898-015-0348-6
    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/s10898-015-0348-6?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. Josef Kallrath & Steffen Rebennack, 2014. "Cutting ellipses from area-minimizing rectangles," Journal of Global Optimization, Springer, vol. 59(2), pages 405-437, July.
    2. 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.
    3. Y. Stoyan & T. Romanova & G. Scheithauer & A. Krivulya, 2011. "Covering a polygonal region by rectangles," Computational Optimization and Applications, Springer, vol. 48(3), pages 675-695, April.
    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. Romanova, Tatiana & Litvinchev, Igor & Pankratov, Alexander, 2020. "Packing ellipsoids in an optimized cylinder," European Journal of Operational Research, Elsevier, vol. 285(2), pages 429-443.
    2. Birgin, E.G. & Lobato, R.D., 2019. "A matheuristic approach with nonlinear subproblems for large-scale packing of ellipsoids," European Journal of Operational Research, Elsevier, vol. 272(2), pages 447-464.
    3. Josef Kallrath & Markus M. Frey, 2019. "Packing circles into perimeter-minimizing convex hulls," Journal of Global Optimization, Springer, vol. 73(4), pages 723-759, April.
    4. Frank J. Kampas & János D. Pintér & Ignacio Castillo, 2020. "Packing ovals in optimized regular polygons," Journal of Global Optimization, Springer, vol. 77(1), pages 175-196, May.
    5. A. Pankratov & T. Romanova & I. Litvinchev, 2019. "Packing ellipses in an optimized convex polygon," Journal of Global Optimization, Springer, vol. 75(2), pages 495-522, October.
    6. Ryu, Joonghyun & Lee, Mokwon & Kim, Donguk & Kallrath, Josef & Sugihara, Kokichi & Kim, Deok-Soo, 2020. "VOROPACK-D: Real-time disk packing algorithm using Voronoi diagram," Applied Mathematics and Computation, Elsevier, vol. 375(C).

    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. 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.
    2. Liang, Zheng & Liang, Yingzong & Luo, Xianglong & Chen, Jianyong & Yang, Zhi & Wang, Chao & Chen, Ying, 2022. "Superstructure-based mixed-integer nonlinear programming framework for hybrid heat sources driven organic Rankine cycle optimization," Applied Energy, Elsevier, vol. 307(C).
    3. Ricardo M. Lima & Ignacio E. Grossmann, 2017. "On the solution of nonconvex cardinality Boolean quadratic programming problems: a computational study," Computational Optimization and Applications, Springer, vol. 66(1), pages 1-37, January.
    4. Fränk Plein & Johannes Thürauf & Martine Labbé & Martin Schmidt, 2022. "A bilevel optimization approach to decide the feasibility of bookings in the European gas market," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 95(3), pages 409-449, June.
    5. Luo, Xianglong & Huang, Xiaojian & El-Halwagi, Mahmoud M. & Ponce-Ortega, José María & Chen, Ying, 2016. "Simultaneous synthesis of utility system and heat exchanger network incorporating steam condensate and boiler feedwater," Energy, Elsevier, vol. 113(C), pages 875-893.
    6. Artur M. Schweidtmann & Alexander Mitsos, 2019. "Deterministic Global Optimization with Artificial Neural Networks Embedded," Journal of Optimization Theory and Applications, Springer, vol. 180(3), pages 925-948, March.
    7. Birgin, E.G. & Lobato, R.D., 2019. "A matheuristic approach with nonlinear subproblems for large-scale packing of ellipsoids," European Journal of Operational Research, Elsevier, vol. 272(2), pages 447-464.
    8. Pavlo Muts & Ivo Nowak & Eligius M. T. Hendrix, 2020. "The decomposition-based outer approximation algorithm for convex mixed-integer nonlinear programming," Journal of Global Optimization, Springer, vol. 77(1), pages 75-96, May.
    9. Timo Berthold & Jakob Witzig, 2021. "Conflict Analysis for MINLP," INFORMS Journal on Computing, INFORMS, vol. 33(2), pages 421-435, May.
    10. A. Pankratov & T. Romanova & I. Litvinchev, 2019. "Packing ellipses in an optimized convex polygon," Journal of Global Optimization, Springer, vol. 75(2), pages 495-522, October.
    11. Huang, Xiaojian & Lu, Pei & Luo, Xianglong & Chen, Jianyong & Yang, Zhi & Liang, Yingzong & Wang, Chao & Chen, Ying, 2020. "Synthesis and simultaneous MINLP optimization of heat exchanger network, steam Rankine cycle, and organic Rankine cycle," Energy, Elsevier, vol. 195(C).
    12. Cafieri, Sonia & Conn, Andrew R. & Mongeau, Marcel, 2023. "Mixed-integer nonlinear and continuous optimization formulations for aircraft conflict avoidance via heading and speed deviations," European Journal of Operational Research, Elsevier, vol. 310(2), pages 670-679.
    13. Falk M. Hante & Martin Schmidt, 2019. "Complementarity-based nonlinear programming techniques for optimal mixing in gas networks," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 7(3), pages 299-323, September.
    14. Subramanian, Avinash S.R. & Gundersen, Truls & Adams, Thomas A., 2021. "Optimal design and operation of a waste tire feedstock polygeneration system," Energy, Elsevier, vol. 223(C).
    15. Adam, Alexandros & Fraga, Eric S. & Brett, Dan J.L., 2018. "A modelling study for the integration of a PEMFC micro-CHP in domestic building services design," Applied Energy, Elsevier, vol. 225(C), pages 85-97.
    16. 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.
    17. Stuart M. Harwood & Paul I. Barton, 2017. "How to solve a design centering problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 86(1), pages 215-254, August.
    18. Jaromił Najman & Alexander Mitsos, 2019. "On tightness and anchoring of McCormick and other relaxations," Journal of Global Optimization, Springer, vol. 74(4), pages 677-703, August.
    19. 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.
    20. Farough Motamed Nasab & Zukui Li, 2023. "Multistage Adaptive Robust Binary Optimization: Uncertainty Set Lifting versus Partitioning through Breakpoints Optimization," Mathematics, MDPI, vol. 11(18), pages 1-24, 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:jglopt:v:67:y:2017:i:1:d:10.1007_s10898-015-0348-6. 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.