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Packing ellipsoids into volume-minimizing rectangular boxes

Author

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  • 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
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    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.
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    Cited by:

    1. 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).
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. 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.

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