Abstract
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 nγ+logO(1)(1/ɛ} 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.
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References
E.H.L. Aarts and J.H.M. Korst, Simulated Annealing and Boltzmann Machines: A Stochastic Approach, Wiley & Sons: New York, 1989.
E.H.L. Aarts and P.J.M. Laarhoven, “Statistical cooling: A general approach to combinatorial optimization problems,” Philips Journal of Research, vol. 40, pp. 193–226, 1985.
A. Albrecht, S.K. Cheung, K.C. Hui, K.S. Leung, and C.K. Wong, “Optimal placements of flexible objects (Part I: The unbounded case),” IEEE Transactions on Computers, vol. 46, no. 8, pp. 890–904, 1997.
A. Albrecht, S.K. Cheung, K.C. Hui, K.S. Leung, and C.K. Wong, “Optimal placements of flexible objects (Part II: A simulated annealing approach for the bounded case),” IEEE Transactions on Computers, vol. 46, no. 8, pp. 905–929, 1997.
A. Albrecht, S.K. Cheung, K.S. Leung, and C.K. Wong, “Stochastic simulations of two-dimensional composite packings,” Journal of Computational Physics, vol. 136, no. 2, pp. 559–79, 1997.
A. Albrecht, S.K. Cheung, K.S. Leung, and C.K. Wong, “Computing elastic moduli of two-dimensional random networks of rigid and nonrigid bonds by simulated annealing,” Mathematics and Computers in Simulation, vol. 44, no. 2, pp. 187–215, 1997.
S. Arbabi and M. Sahimi, “Mechanics of disordered solids I. Percolation on elastic networks with central forces,” Physical Review B, vol. 47, no. 2, pp. 695–702, 1993.
S. Azencott, “A common large deviations mathematical framework for sequential annealing and parallel annealing,” In Simulated Annealing: Parallelization Techniques, S. Azencott (Ed.), Wiley & Sons: New York, 1992, pp. 11–23.
O. Catoni, “Rates of convergence for sequential annealing: A large deviation approach,” In Simulated Annealing: Parallelization Techniques, S. Azencott (Ed.), Wiley & Sons: New York, 1992, pp. 25–35.
O. Catoni, “Rough large deviation estimates for simulated annealing: Applications to exponential schedules,” The Annals of Probability, vol. 20, no. 3, pp. 1109–1146, 1992.
V. Černy, A thermodynamical approach to the travelling salesman problem: An efficient simulation algorithm. Preprint, Inst. of Physics and Biophysics, Comenius Univ., Bratislava, 1982 (see also J. Optim. Theory Appl., vol. 45, pp. 41–51, 1985).
T.S. Chiang and Y. Chow, “On the convergence rate of annealing processes,” SIAM Journal on Control and Optimization, vol. 26, pp. 1455–1470, 1988.
P. Diaconis and D. Stroock, “Geometric bounds for eigenvalues of Markov chains,” Annals of Applied Probability, vol. 1, pp. 36–61, 1991.
E. Duering and D.J. Bergmann, “Scaling properties of the elastic stiffness moduli of a random rigid-nonrigid network near the rigidity threshold: Theory and simulations,” Physical Review B, vol. 37, no. 2, pp. 713–722, 1988.
B. Hajek, “Cooling schedules for optimal annealing,” Mathematics of Operations Research, vol. 13, pp. 311–329, 1988.
J. Helsing, “Thin bridges in isotropic electrostatics,” Journal of Computational Physics, vol. 127, pp. 142–151, 1996.
A. Jagota and G.W. Scherer, “Viscosities and sintering rates of a two-dimensional granular composite,” Journal of the American Ceramic Society, vol. 76, pp. 3123–3135, 1993.
V.B. Kashirin and E.V. Kozlov, “New approach to the dense random packing of soft spheres,” Journal of Non-Crystalline Solids, vol. 163, pp. 24–28, 1993.
S. Kirkpatrick, C.D. Gelatt, Jr., and M.P. Vecchi, “Optimization by simulated annealing,” Science, vol. 220, pp. 671–680, 1983.
P.J.M. Laarhoven and E.H.L. Aarts, Simulated Annealing: Theory and Applications, D.Reidel Publishing Company, 1988.
N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” The Journal of Chemical Physics, vol. 21, no. 6, pp. 1087–1092, 1953.
D. Mitra, F. Romeo, and A. Sangiovanni-Vincentelli, “Convergence and finite-time behaviour of simulated annealing,” Advances in Applied Probability, vol. 18, pp. 747–771, 1986.
F. Romeo and A. Sangiovanni-Vincentelli, “A theoretical framework for simulated annealing,” Algorithmica, vol. 6, pp. 302–345, 1991.
M. Sarrafzadeh and C.K. Wong, An Introduction to VLSI Physical Design, McGraw-Hill Companies., Inc.: New York, 1996.
M. Schulz and P. Reineker, “Dilute and dense systems of random copolymers in the equilibrium state,” Physical Review B, vol. 53, no. 18, pp. 12017–12023, 1996.
C. Sechen and A. Sangiovanni-Vincentelli, “The TimberWolf placement and routing package,” IEEE J. Solid-State Circuits, vol. 20, pp. 510–522, April 1985.
E. Seneta, Non-Negative Matrices and Markov Chains, Springer-Verlag: New York, 1981.
V.K.S. Shante and S. Kirkpatrik, “An introduction to percolation theory,” Advances in Physics, vol. 20, pp. 325–357, 1971.
A. Sinclair, Algorithms for Generation and Counting: A Markov Chain Approach, Birkhäuser: Boston, 1993.
A. Sinclair and M. Jerrum, “Approximate counting, uniform generation, and rapidly mixing markov chains,” Information and Computation, vol. 82, pp. 93–133, 1989.
A. Sinclair and M. Jerrum, “Polynomial-time approximation algorithms for the Ising model,” SIAM J. Comput., vol. 22, no. 5, pp. 1087–1116, 1993.
G. Sorkin, “Efficient simulated annealing on fractal energy landscapes,” Algorithmica, vol. 6, pp. 367–418, 1991.
W. Swartz and C. Sechen, “Time driven placement for large standard cell circuits,” in Proc. of the 32nd Design Automation Conference, 1995, pp. 211–215.
I.M. Ward, Mechanical Properties of Solid Polymers, John Wiley & Sons: New York, 1985.
D.F. Wong and C.L. Liu, “Floorplan design of VLSI circuits,” Algorithmica, vol. 4, pp. 263–291, 1989.
A.Z. Zinchenko, “Algorithms for random close packing of spheres with periodic boundary conditions,” Journal of Computational Physics, vol. 114. pp. 298–307, 1994.
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Albrecht, A., Cheung, S., Leung, K. et al. On the Convergence of Inhomogeneous Markov Chains Approximating Equilibrium Placements of Flexible Objects. Computational Optimization and Applications 19, 179–208 (2001). https://doi.org/10.1023/A:1011241620180
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DOI: https://doi.org/10.1023/A:1011241620180