Price-and-verify: a new algorithm for recursive circle packing using Dantzig–Wolfe decomposition
Packing rings into a minimum number of rectangles is an optimization problem which appears naturally in the logistics operations of the tube industry. It encompasses two major difficulties, namely the positioning of rings in rectangles and the recursive packing of rings into other rings. This proble...
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| Veröffentlicht in: | Annals of operations research Jg. 284; H. 2; S. 527 - 555 |
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| Abstract | Packing rings into a minimum number of rectangles is an optimization problem which appears naturally in the logistics operations of the tube industry. It encompasses two major difficulties, namely the positioning of rings in rectangles and the recursive packing of rings into other rings. This problem is known as the Recursive Circle Packing Problem (RCPP). We present the first dedicated method for solving RCPP that provides strong dual bounds based on an exact Dantzig–Wolfe reformulation of a nonconvex mixed-integer nonlinear programming formulation. The key idea of this reformulation is to break symmetry on each recursion level by enumerating one-level packings, i.e., packings of circles into other circles, and by dynamically generating packings of circles into rectangles. We use column generation techniques to design a “price-and-verify” algorithm that solves this reformulation to global optimality. Extensive computational experiments on a large test set show that our method not only computes tight dual bounds, but often produces primal solutions better than those computed by heuristics from the literature. |
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| AbstractList | Packing rings into a minimum number of rectangles is an optimization problem which appears naturally in the logistics operations of the tube industry. It encompasses two major difficulties, namely the positioning of rings in rectangles and the recursive packing of rings into other rings. This problem is known as the Recursive Circle Packing Problem (RCPP). We present the first dedicated method for solving RCPP that provides strong dual bounds based on an exact Dantzig–Wolfe reformulation of a nonconvex mixed-integer nonlinear programming formulation. The key idea of this reformulation is to break symmetry on each recursion level by enumerating one-level packings, i.e., packings of circles into other circles, and by dynamically generating packings of circles into rectangles. We use column generation techniques to design a “price-and-verify” algorithm that solves this reformulation to global optimality. Extensive computational experiments on a large test set show that our method not only computes tight dual bounds, but often produces primal solutions better than those computed by heuristics from the literature. |
| Author | Maher, Stephen J. Müller, Benjamin Gleixner, Ambros Pedroso, João Pedro |
| Author_xml | – sequence: 1 givenname: Ambros orcidid: 0000-0003-0391-5903 surname: Gleixner fullname: Gleixner, Ambros organization: Zuse Institute Berlin – sequence: 2 givenname: Stephen J. surname: Maher fullname: Maher, Stephen J. organization: Department of Management Science, Lancaster University – sequence: 3 givenname: Benjamin orcidid: 0000-0002-4463-2873 surname: Müller fullname: Müller, Benjamin email: benjamin.mueller@zib.de organization: Zuse Institute Berlin – sequence: 4 givenname: João Pedro orcidid: 0000-0003-1298-7191 surname: Pedroso fullname: Pedroso, João Pedro organization: Universidade do Porto |
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| CitedBy_id | crossref_primary_10_1016_j_cor_2025_107209 crossref_primary_10_1007_s12065_024_00931_w crossref_primary_10_1016_j_eswa_2023_121768 crossref_primary_10_1016_j_ejco_2023_100074 crossref_primary_10_3390_computation10070113 |
| Cites_doi | 10.1111/itor.12107 10.1023/A:1007606007369 10.1007/s10107-012-0561-8 10.1007/978-3-540-68279-0_13 10.1287/opre.13.1.94 10.1007/BF01300970 10.1023/A:1018346107246 10.1016/j.disopt.2005.01.002 10.1016/j.dam.2012.07.020 10.1007/s10898-015-0348-6 10.1287/opre.1050.0234 10.1007/s10107-014-0761-5 10.1287/opre.38.5.922 10.1287/opre.8.1.101 10.1007/BF00138693 10.1007/s10107-004-0559-y 10.1287/opre.9.6.849 10.1007/s10898-007-9274-6 10.1287/ijoc.1080.0283 10.1016/0167-6377(96)00033-8 |
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| DOI | 10.1007/s10479-018-3115-5 |
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| Keywords | Global optimization Recursive circle packing Mixed-integer nonlinear programming Dantzig–Wolfe decomposition Symmetry breaking |
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| References | BergnerMCapraraACeselliAFuriniFLübbeckeMEMalagutiETraversiEAutomatic Dantzig–Wolfe reformulation of mixed integer programsMathematical Programming20151491–239142410.1007/s10107-014-0761-5 DesaulniersGDesrosiersJSolomonMMColumn generation2006BerlinSpringer VanderbeckFrançoisWolseyLaurence A.Reformulation and Decomposition of Integer Programs50 Years of Integer Programming 1958-20082009Berlin, HeidelbergSpringer Berlin Heidelberg431502 PisingerDSigurdMThe two-dimensional bin packing problem with variable bin sizes and costsDiscrete Optimization20052215416710.1016/j.disopt.2005.01.002 Hifi, M., & M’Hallah, R. (2009). A literature review on circle and sphere packing problems: Models and methodologies. Advances in Operations Research, 150, 624:1–150, 624:22. http://dblp.uni-trier.de/db/journals/advor/advor2009.html#HifiM09. WächterABieglerLTOn the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programmingMathematical Programming20061061255710.1007/s10107-004-0559-y MUMPS, Multifrontal Massively Parallel sparse direct Solver. http://mumps.enseeiht.fr. Accessed 2011. KallrathJCutting circles and polygons from area-minimizing rectanglesJournal of Global Optimization200943229932810.1007/s10898-007-9274-6 LübbeckeMEDesrosiersJSelected topics in column generationOperations Research20055361007102310.1287/opre.1050.0234 VanderbeckFWolseyLAAn exact algorithm for IP column generationOperations Research Letters199619415115910.1016/0167-6377(96)00033-8 ILOG, Inc: ILOG CPLEX: High-performance software for mathematical programming and optimization. See http://www.ilog.com/products/cplex/. Accessed 2016. 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| References_xml | – reference: DrorMPolygon plate-cutting with a given orderIIE Transactions199931327127410.1023/A:1007606007369 – reference: Hales, T. C., Adams, M., Bauer, G., Dang, D. T., Harrison, J., Hoang, T. L., Kaliszyk, C., Magron, V., McLaughlin, S., Nguyen, T. T., Nguyen, T. Q., Nipkow, T., Obua, S., Pleso, J., Rute, J., Solovyev, A., Ta, A. H. T., Tran, T. N., Trieu, D. T., Urban, J., Vu, K. K., & Zumkeller, R. (2015). A formal proof of the Kepler conjecture. arXiv:1501.02155. – reference: COIN-OR: Ipopt, Interior point optimizer. http://www.coin-or.org/Ipopt. Accessed 2016. – reference: GilmorePGomoryREMultistage cutting stock problems of two and more dimensionsOperations Research19651319412010.1287/opre.13.1.94 – reference: PedrosoJPCunhaSTavaresJNRecursive circle packing problemsInternational Transactions in Operational Research2016231–235536810.1111/itor.12107 – reference: VancePHBranch-and-price algorithms for the one-dimensional cutting stock problemComputational Optimization and Applications19989321122810.1023/A:1018346107246 – reference: DesaulniersGDesrosiersJSolomonMMColumn generation2006BerlinSpringer – reference: Hifi, M., & M’Hallah, R. (2009). A literature review on circle and sphere packing problems: Models and methodologies. Advances in Operations Research, 150, 624:1–150, 624:22. http://dblp.uni-trier.de/db/journals/advor/advor2009.html#HifiM09. – reference: VanderbeckFrançoisWolseyLaurence A.Reformulation and Decomposition of Integer Programs50 Years of Integer Programming 1958-20082009Berlin, HeidelbergSpringer Berlin Heidelberg431502 – reference: GilmorePGomoryRA linear programming approach to the cutting stock problemOperations Research1961984985910.1287/opre.9.6.849 – reference: KallrathJCutting circles and polygons from area-minimizing rectanglesJournal of Global Optimization200943229932810.1007/s10898-007-9274-6 – reference: JansRSolving lot-sizing problems on parallel identical machines using symmetry-breaking constraintsINFORMS Journal on Computing200921112313610.1287/ijoc.1080.0283 – reference: MUMPS, Multifrontal Massively Parallel sparse direct Solver. http://mumps.enseeiht.fr. Accessed 2011. – reference: FarleyAAA note on bounding a class of linear programming problems, including cutting stock problemsOperations Research199038592292310.1287/opre.38.5.922 – reference: PisingerDSigurdMThe two-dimensional bin packing problem with variable bin sizes and costsDiscrete Optimization20052215416710.1016/j.disopt.2005.01.002 – reference: COIN-OR: CppAD, a package for differentiation of CPP Algorithms. http://www.coin-or.org/CppAD. Accessed 2014. – reference: CostaAlbertoHansenPierreLibertiLeoOn the impact of symmetry-breaking constraints on spatial Branch-and-Bound for circle packing in a squareDiscrete Applied Mathematics20131611-29610610.1016/j.dam.2012.07.020 – reference: Lenstra, J. K., & Kan, A. H. G. R. (1979). Complexity of packing, covering and partitioning problems. In A. Schrijver (Ed.), Packing and covering in combinatorics (pp. 275–291). 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| SubjectTerms | Algorithms Business and Management Combinatorics Dantzig-Wolfe decomposition Logistics Nonlinear programming Operations research Operations Research/Decision Theory Rectangles Recursive methods S.I.: Decomposition Methods for Hard Optimization Problems Theory of Computation |
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| Title | Price-and-verify: a new algorithm for recursive circle packing using Dantzig–Wolfe decomposition |
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