Towards practical `neural' computation for combinatorial optimization problems
AIP Conference Proceedings 151 on Neural Networks for Computing
Sphere-packings, lattices, and groups
Sphere-packings, lattices, and groups
Computers and Operations Research
Global optimization approach to unequal sphere packing problems in 3D
Journal of Optimization Theory and Applications
Variable neighborhood search and local branching
Computers and Operations Research
European Journal of Combinatorics
Minimizing the object dimensions in circle and sphere packing problems
Computers and Operations Research
An Effective Hybrid Algorithm for the Circles and Spheres Packing Problems
COCOA '09 Proceedings of the 3rd International Conference on Combinatorial Optimization and Applications
Computers and Operations Research
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This paper addresses the NP hard optimization problem of packing identical spheres of unit radii into the smallest sphere (PSS). It models PSS as a non-linear program (NLP) and approximately solves it using a hybrid heuristic which couples a variable neighborhood search (VNS) with a local search (LS). VNS serves as the diversification mechanism whereas LS acts as the intensification one. VNS investigates the neighborhood of a feasible local minimum u in search for the global minimum, where neighboring solutions are obtained by shaking one or more spheres of u and the size of the neighborhood is varied by changing the number of shaken spheres, the distance and the direction each sphere is moved. LS intensifies the search around a solution u by subjecting its neighbors to a sequential quadratic algorithm with non-monotone line search (as the NLP solver). The computational investigation highlights the role of LS and VNS in identifying (near) global optima, studies their sensitivity to initial solutions, and shows that the proposed hybrid heuristic provides more precise results than existing approaches. Most importantly, it provides computational evidence that the multiple-start strategy of non-linear programming solvers is not sufficient to solve PSS. Finally, it gives new upper bounds for 29 out of 48 benchmark instances of PSS.