Minimizing the object dimensions in circle and sphere packing problems

Given a fixed set of identical or different-sized circular items, the problem we deal with consists on finding the smallest object within which the items can be packed. Circular, triangular, squared, rectangular and also strip objects are considered. Moreover, 2D and 3D problems are treated. Twice-differentiable models for all these problems are presented. A strategy to reduce the complexity of evaluating the models is employed and, as a consequence, instances with a large number of items can be considered. Numerical experiments show the flexibility and reliability of the new unified approach.

[1]  AUT. SORfe,et al.  Computational Structure of the N-body Problem , 1989 .

[2]  D. Bacon,et al.  A fast algorithm for rendering space-filling molecule pictures , 1988 .

[3]  Nenad Mladenovic,et al.  Reformulation descent applied to circle packing problems , 2003, Comput. Oper. Res..

[4]  Leslie Greengard,et al.  A fast algorithm for particle simulations , 1987 .

[5]  Andrew W. Appel,et al.  An Efficient Program for Many-Body Simulation , 1983 .

[6]  José Mario Martínez,et al.  Algorithm 813: SPG—Software for Convex-Constrained Optimization , 2001, TOMS.

[7]  Marco Locatelli,et al.  Packing equal circles in a square: a deterministic global optimization approach , 2002, Discret. Appl. Math..

[8]  José Mario Martínez,et al.  Large-Scale Active-Set Box-Constrained Optimization Method with Spectral Projected Gradients , 2002, Comput. Optim. Appl..

[9]  R W Hockney,et al.  Computer Simulation Using Particles , 1966 .

[10]  Yu.G Stoyan,et al.  A mathematical model and a solution method for the problem of placing various-sized circles into a strip , 2004, Eur. J. Oper. Res..

[11]  José Mario Martínez,et al.  Method of sentinels for packing items within arbitrary convex regions , 2006, J. Oper. Res. Soc..

[12]  Y. Stoyan,et al.  Mathematical model and solution method of optimization problem of placement of rectangles and circles taking into account special constraints , 1998 .

[13]  K Schulten,et al.  VMD: visual molecular dynamics. , 1996, Journal of molecular graphics.

[14]  José Mario Martínez,et al.  Orthogonal packing of rectangular items within arbitrary convex regions by nonlinear optimization , 2006, Comput. Oper. Res..

[15]  Patric R. J. Östergård,et al.  Packing up to 50 Equal Circles in a Square , 1997, Discret. Comput. Geom..

[16]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[17]  David A. Kendrick,et al.  GAMS : a user's guide, Release 2.25 , 1992 .

[18]  José Mario Martínez,et al.  Numerical Comparison of Augmented Lagrangian Algorithms for Nonconvex Problems , 2005, Comput. Optim. Appl..

[19]  J. M. Martínez,et al.  A Box-Constrained Optimization Algorithm with Negative Curvature Directions and Spectral Projected Gradients , 2001 .

[20]  Panos M. Pardalos,et al.  New results in the packing of equal circles in a square , 1995, Discret. Math..

[21]  José Mario Martínez,et al.  Optimizing the packing of cylinders into a rectangular container: A nonlinear approach , 2005, Eur. J. Oper. Res..

[22]  David Kendrick,et al.  GAMS, a user's guide , 1988, SGNM.

[23]  Michael Goldberg.  The Packing of Equal Circles in a Square , 1970 .

[24]  Gerhard Wäscher,et al.  An improved typology of cutting and packing problems , 2007, Eur. J. Oper. Res..

[25]  José Mario Martínez,et al.  Packing optimization for automated generation of complex system's initial configurations for molecular dynamics and docking , 2003, J. Comput. Chem..

[26]  José Mario Martínez,et al.  Structured minimal-memory inexact quasi-Newton method and secant preconditioners for augmented Lagrangian optimization , 2008, Comput. Optim. Appl..

[27]  D. P. Ronconi,et al.  Method of Sentinels for Packing Objects within Arbitrary Regions , 2005 .

[28]  José Mario Martínez,et al.  On Augmented Lagrangian Methods with General Lower-Level Constraints , 2007, SIAM J. Optim..

[29]  Feng Zhao,et al.  An {\it bf O(N)} Algorithm for Three-Dimensional N-body Simulations , 1987 .

[30]  Piet Hut,et al.  A hierarchical O(N log N) force-calculation algorithm , 1986, Nature.

[31]  José Mario Martínez,et al.  Augmented Lagrangian methods under the constant positive linear dependence constraint qualification , 2007, Math. Program..

[32]  José Mario Martínez,et al.  Nonmonotone Spectral Projected Gradient Methods on Convex Sets , 1999, SIAM J. Optim..

[33]  Rimvydas Krasauskas.  Mathematical Methods in Geometric Design , 1997 .

[34]  V. Rokhlin,et al.  Rapid Evaluation of Potential Fields in Three Dimensions , 1988 .