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On tackling reverse convex constraints for non-overlapping of unequal circles

Author

Listed:
  • Akang Wang

    (Carnegie Mellon University
    Carnegie Mellon University)

  • Chrysanthos E. Gounaris

    (Carnegie Mellon University
    Carnegie Mellon University)

Abstract

We study the unequal circle-circle non-overlapping constraints, a form of reverse convex constraints that often arise in optimization models for cutting and packing applications. The feasible region induced by the intersection of circle-circle non-overlapping constraints is highly non-convex, and standard approaches to construct convex relaxations for spatial branch-and-bound global optimization of such models typically yield unsatisfactory loose relaxations. Consequently, solving such non-convex models to guaranteed optimality remains extremely challenging even for the state-of-the-art codes. In this paper, we apply a purpose-built branching scheme on non-overlapping constraints and utilize strengthened intersection cuts and various feasibility-based tightening techniques to further tighten the model relaxation. We embed these techniques into a branch-and-bound code and test them on two variants of circle packing problems. Our computational studies on a suite of 75 benchmark instances yielded, for the first time in the open literature, a total of 54 provably optimal solutions, and it was demonstrated to be competitive over the use of the state-of-the-art general-purpose global optimization solvers.

Suggested Citation

  • Akang Wang & Chrysanthos E. Gounaris, 2021. "On tackling reverse convex constraints for non-overlapping of unequal circles," Journal of Global Optimization, Springer, vol. 80(2), pages 357-385, June.
  • Handle: RePEc:spr:jglopt:v:80:y:2021:i:2:d:10.1007_s10898-020-00976-y
    DOI: 10.1007/s10898-020-00976-y
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    References listed on IDEAS

    as
    1. Akang Wang & Christopher L. Hanselman & Chrysanthos E. Gounaris, 2018. "A customized branch-and-bound approach for irregular shape nesting," Journal of Global Optimization, Springer, vol. 71(4), pages 935-955, August.
    2. Stoyan, Yu. G. & Yas'kov, G., 2004. "A mathematical model and a solution method for the problem of placing various-sized circles into a strip," European Journal of Operational Research, Elsevier, vol. 156(3), pages 590-600, August.
    3. Donald Jones, 2014. "A fully general, exact algorithm for nesting irregular shapes," Journal of Global Optimization, Springer, vol. 59(2), pages 367-404, July.
    4. Mhand Hifi & Rym M'Hallah, 2009. "A Literature Review on Circle and Sphere Packing Problems: Models and Methodologies," Advances in Operations Research, Hindawi, vol. 2009, pages 1-22, July.
    5. Egon Balas, 1971. "Intersection Cuts—A New Type of Cutting Planes for Integer Programming," Operations Research, INFORMS, vol. 19(1), pages 19-39, February.
    6. 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.
    7. López, C.O. & Beasley, J.E., 2011. "A heuristic for the circle packing problem with a variety of containers," European Journal of Operational Research, Elsevier, vol. 214(3), pages 512-525, November.
    8. 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.
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