Some NP-complete problems in quadratic and nonlinear programming
Mathematical Programming: Series A and B
Cooling schedules for optimal annealing
Mathematics of Operations Research
Simulated annealing: theory and applications
Simulated annealing: theory and applications
Simulated annealing and Boltzmann machines: a stochastic approach to combinatorial optimization and neural computing
On approximation algorithms for concave quadratic programming
Recent advances in global optimization
Polynomial-time approximation algorithms for the Ising model
SIAM Journal on Computing
Quadratic 0/1 optimization and a decomposition approach for the placement of electronic circuits
Mathematical Programming: Series A and B
Timing driven placement for large standard cell circuits
DAC '95 Proceedings of the 32nd annual ACM/IEEE Design Automation Conference
Optimal Placements of Flexible Objects: Part I: Analytical Results for the Unbounded Case
IEEE Transactions on Computers
Computational Optimization and Applications
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This paper is a continuation of the first part, where we considered regular arrangements of flexible objects for the unbounded case. The present part deals with a simulated annealing algorithm maximizing the number of flexible objects in equilibrium placements within rigid boundaries. The forces caused by the boundary are taken into account, i.e., the bounded case of placements is considered. The simulated annealing procedure makes use of the special structure of the underlying configuration space and relationships between deformations of flexible objects and resulting forces. This allows us to obtain tight bounds for the annealing parameters which result in n3/2· ln5/2n and n· ln2n time bounds, respectively, for the computation of equilibrium states by two different cooling schedules. The deformation/force formula is derived from a physical model of flexible discs and is based on numerical experiments which were performed for different materials and different sizes of objects. The algorithm was first implemented and tested for the unbounded case. The run-time is relatively short, even for large numbers of placed discs. These results are compared to the analytical ones obtained for regular placements in the first part of the paper, and agreement between these two sets of results are observed. Furthermore, several experiments for placements with boundary conditions were carried out and the resulting placements clearly show the effect of the forces from the rigid boundary. The specialized and provably efficient simulated annealing algorithm proposed in this paper is therefore a very effective tool for computing equilibrium states of placements and hence useful for the design of new amorphous polymeric materials and package cushioning systems as mentioned in Part I of this paper.