New results in the packing of equal circles in a square
Discrete Mathematics
A polyhedral branch-and-cut approach to global optimization
Mathematical Programming: Series A and B
Minimizing the object dimensions in circle and sphere packing problems
Computers and Operations Research
Cutting circles and polygons from area-minimizing rectangles
Journal of Global Optimization
An Effective Hybrid Algorithm for the Circles and Spheres Packing Problems
COCOA '09 Proceedings of the 3rd International Conference on Combinatorial Optimization and Applications
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The paper considers a problem of packing the maximal number of congruent nD hyperspheres of given radius into a larger nD hypersphere of given radius where n = 2, 3, . . . , 24. Solving the problem is reduced to solving a sequence of packing subproblems provided that radii of hyperspheres are variable. Mathematical models of the subproblems are constructed. Characteristics of the mathematical models are investigated. On the ground of the characteristics we offer a solution approach. For n 驴 3 starting points are generated either in accordance with the lattice packing of circles and spheres or in a random way. For n 3 starting points are generated in a random way. A procedure of perturbation of lattice packings is applied to improve convergence. We use the Zoutendijk feasible direction method to search for local maxima of the subproblems. To compute an approximation to a global maximum of the problem we realize a non-exhaustive search of local maxima. Our results are compared with the benchmark results for n = 2. A number of numerical results for 2 驴 n 驴 24 are given.