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Global Optimization Approach to Unequal Global Optimization Approach to Unequal Sphere Packing Problems in 3D

Author

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  • A. Sutou

    (Central Research Laboratory, Hitachi)

  • Y. Dai

    (University of Illinois at Chicago)

Abstract

The problem of the unequal sphere packing in a 3-dimen-sional polytope is analyzed. Given a set of unequal spheres and a poly-tope, the double goal is to assemble the spheres in such a way that (i) they do not overlap with each other and (ii) the sum of the volumes of the spheres packed in the polytope is maximized. This optimization has an application in automated radiosurgical treatment planning and can be formulated as a nonconvex optimization problem with quadratic constraints and a linear objective function. On the basis of the special structures associated with this problem, we propose a variety of algorithms which improve markedly the existing simplicial branch-and-bound algorithm for the general nonconvex quadratic program. Further, heuristic algorithms are incorporated to strengthen the efficiency of the algorithm. The computational study demonstrates that the proposed algorithm can obtain successfully the optimization up to a limiting size.

Suggested Citation

  • A. Sutou & Y. Dai, 2002. "Global Optimization Approach to Unequal Global Optimization Approach to Unequal Sphere Packing Problems in 3D," Journal of Optimization Theory and Applications, Springer, vol. 114(3), pages 671-694, September.
    • Handle: RePEc:spr:joptap:v:114:y:2002:i:3:d:10.1023_a:1016083231326
    DOI: 10.1023/A:1016083231326
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    References listed on IDEAS

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    1. Jie Wang, 1999. "Packing of Unequal Spheres and Automated Radiosurgical Treatment Planning," Journal of Combinatorial Optimization, Springer, vol. 3(4), pages 453-463, December.
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    Cited by:

    1. Andreas Fischer & Igor Litvinchev & Tetyana Romanova & Petro Stetsyuk & Georgiy Yaskov, 2023. "Quasi-Packing Different Spheres with Ratio Conditions in a Spherical Container," Mathematics, MDPI, vol. 11(9), pages 1-19, April.
    2. Bortfeldt, Andreas & Wäscher, Gerhard, 2013. "Constraints in container loading – A state-of-the-art review," European Journal of Operational Research, Elsevier, vol. 229(1), pages 1-20.
    3. Hifi, Mhand & Yousef, Labib, 2019. "A local search-based method for sphere packing problems," European Journal of Operational Research, Elsevier, vol. 274(2), pages 482-500.
    4. Moslem Zamani, 2023. "New bounds for nonconvex quadratically constrained quadratic programming," Journal of Global Optimization, Springer, vol. 85(3), pages 595-613, March.
    5. Stoyan, Yu. & Chugay, A., 2009. "Packing cylinders and rectangular parallelepipeds with distances between them into a given region," European Journal of Operational Research, Elsevier, vol. 197(2), pages 446-455, September.
    6. Igor Litvinchev & Andreas Fischer & Tetyana Romanova & Petro Stetsyuk, 2024. "A New Class of Irregular Packing Problems Reducible to Sphere Packing in Arbitrary Norms," Mathematics, MDPI, vol. 12(7), pages 1-17, March.
    7. Jean-Thomas Camino & Christian Artigues & Laurent Houssin & Stéphane Mourgues, 2019. "Linearization of Euclidean norm dependent inequalities applied to multibeam satellites design," Computational Optimization and Applications, Springer, vol. 73(2), pages 679-705, June.
    8. S. P. Li & Ka-Lok Ng, 2003. "Study Of The Unequal Spheres Packing Problem: An Application To Radiosurgery Treatment," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 14(06), pages 815-823.

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