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A nonlinear programming model with implicit variables for packing ellipsoids

Author

Listed:
  • E. G. Birgin

    (University of São Paulo)

  • R. D. Lobato

    (University of São Paulo)

  • J. M. Martínez

    (State University of Campinas)

Abstract

The problem of packing ellipsoids is considered in the present work. Usually, the computational effort associated with numerical optimization methods devoted to packing ellipsoids grows quadratically with respect to the number of ellipsoids being packed. The reason is that the number of variables and constraints of ellipsoids’ packing models is associated with the requirement that every pair of ellipsoids must not overlap. As a consequence, it is hard to solve the problem when the number of ellipsoids is large. In this paper, we present a nonlinear programming model for packing ellipsoids that contains a linear number of variables and constraints. The proposed model finds its basis in a transformation-based non-overlapping model recently introduced by Birgin et al. (J Glob Optim 65(4):709–743, 2016). For solving large-sized instances of ellipsoids’ packing problems with up to 1000 ellipsoids, a multi-start strategy that combines clever initial random guesses with a state-of-the-art (local) nonlinear optimization solver is presented. Numerical experiments show the efficiency and effectiveness of the proposed model and methodology.

Suggested Citation

  • E. G. Birgin & R. D. Lobato & J. M. Martínez, 2017. "A nonlinear programming model with implicit variables for packing ellipsoids," Journal of Global Optimization, Springer, vol. 68(3), pages 467-499, July.
    • Handle: RePEc:spr:jglopt:v:68:y:2017:i:3:d:10.1007_s10898-016-0483-8
    DOI: 10.1007/s10898-016-0483-8
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    References listed on IDEAS

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    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. Yuriy Stoyan & Georgiy Yaskov, 2012. "Packing congruent hyperspheres into a hypersphere," Journal of Global Optimization, Springer, vol. 52(4), pages 855-868, April.
    3. Castillo, Ignacio & Kampas, Frank J. & Pintér, János D., 2008. "Solving circle packing problems by global optimization: Numerical results and industrial applications," European Journal of Operational Research, Elsevier, vol. 191(3), pages 786-802, December.
    4. Birgin, E. G. & Martinez, J. M. & Ronconi, D. P., 2005. "Optimizing the packing of cylinders into a rectangular container: A nonlinear approach," European Journal of Operational Research, Elsevier, vol. 160(1), pages 19-33, January.
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    7. E. G. Birgin & R. D. Lobato & J. M. Martínez, 2016. "Packing ellipsoids by nonlinear optimization," Journal of Global Optimization, Springer, vol. 65(4), pages 709-743, August.
    8. Yuriy Stoyan & Tatiana Romanova & Alexander Pankratov & Andrey Chugay, 2015. "Optimized Object Packings Using Quasi-Phi-Functions," Springer Optimization and Its Applications, in: Giorgio Fasano & János D. Pintér (ed.), Optimized Packings with Applications, edition 1, chapter 0, pages 265-293, Springer.
    9. Y G Stoyan & M V Zlotnik & A M Chugay, 2012. "Solving an optimization packing problem of circles and non-convex polygons with rotations into a multiply connected region," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 63(3), pages 379-391, March.
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    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. Tiago Montanher & Arnold Neumaier & Mihály Csaba Markót & Ferenc Domes & Hermann Schichl, 2019. "Rigorous packing of unit squares into a circle," Journal of Global Optimization, Springer, vol. 73(3), pages 547-565, March.
    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).

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