Cooling schedules for optimal annealing
Mathematics of Operations Research
On the convergence rate of annealing processes
SIAM Journal on Control and Optimization
Simulated annealing: theory and applications
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Approximate counting, uniform generation and rapidly mixing Markov chains
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Algorithms for random generation and counting: a Markov chain approach
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Polynomial-time approximation algorithms for the Ising model
SIAM Journal on Computing
Algorithm for random close packing of spheres with periodic boundary conditions
Journal of Computational Physics
Timing driven placement for large standard cell circuits
DAC '95 Proceedings of the 32nd annual ACM/IEEE Design Automation Conference
Thin bridges in isotropic electrostatics
Journal of Computational Physics
Stochastic simulations of two-dimensional composite packings
Journal of Computational Physics
Optimal Placements of Flexible Objects: Part I: Analytical Results for the Unbounded Case
IEEE Transactions on Computers
Optimal Placements of Flexible Objects: Part II: A Simulated Annealing Approach for the Bounded Case
IEEE Transactions on Computers
Mathematics and Computers in Simulation
An Introduction to VLSI Physical Design
An Introduction to VLSI Physical Design
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This paper deals with the minimization of local forces in two-dimensional placements of flexible objects within rigid boundaries. The objects are disks of the same size but, in general, of different materials. Potential applications include the design of new amorphous polymeric and related granular materials as well as the design of package cushioning systems. The problem is considered on a grid structure with a fixed step size w and for a fixed diameter of the discs, i.e., the number of placed disks may increase as the size of the placement region increases. The near-equilibrium configurations have to be calculated from uniformly distributed random initial placements. The final arrangements of disks must ensure that any particular object is deformed only within the limits of elasticity of the material. The main result concerns ϵ-approximations of the probability distribution on the set of equilibrium placements. Under a natural assumption about the configuration space, we prove that a run-time of n^{\gamma}+\log^{O(1)}{(1/\varepsilon)} is sufficient to approach with probability 1 – ϵ the minimum value of the objective function, where γ depends on the maximum Γ of the escape depth of local minima within the underlying energy landscape. The result is derived from a careful analysis of the interaction among probabilities assigned to configurations from adjacent distance levels to minimum placements. The overall approach for estimating the convergence rate is relatively independent of the particular placement problem and can be applied to various optimization problems with similar properties of the associated landscape of the objective function.