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Convergence of a global stochastic optimization algorithm with partial step size restarting

Part of: Limit theorems Sequential methods

Published online by Cambridge University Press:  19 February 2016

G. Yin
G. Yin*
Affiliation:
Wayne State University
*
∗ Postal address: Department of Mathematics, Wayne State University, Detroit, MI 48202, USA. Email address: [email protected]
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Abstract

This work develops a class of stochastic global optimization algorithms that are Kiefer-Wolfowitz (KW) type procedures with an added perturbing noise and partial step size restarting. The motivation stems from the use of KW-type procedures and Monte Carlo versions of simulated annealing algorithms in a wide range of applications. Using weak convergence approaches, our effort is directed to proving the convergence of the underlying algorithms under general noise processes.

Keywords

Global optimizationMonte Carlo methodsdiffusion; restartingweak convergence

MSC classification

Primary: 60F05: Central limit and other weak theorems 62L20: Stochastic approximation 65K10: Optimization and variational techniques
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 2 , June 2000 , pp. 480 - 498
Copyright
Copyright © Applied Probability Trust 2000 

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Footnotes

This research was supported in part by the National Science Foundation under grants DMS-9877090 and DMS-9971608.

References

[1] Cai, X., Kelly, P. and Gong, W. B. (1995). Digital diffusion network for image segmentation. Proc. IEEE Internat. Conf. Image Processing, Vol III, pp. 73–76.Google Scholar
[2] Chen, H.-F. (1998). Stochastic approximation with non-additive measurement noise. J. Appl. Prob. 35, 407–417.CrossRefGoogle Scholar
[3] Chiang, T. S., Hwang, C. R. and Sheu, S. J. (1987). Diffusion for global optimization in rrn . SIAM J. Control Optim. 25, 737–752.Google Scholar
[4] Dippon, J. (1998). Globally convergent stochastic optimization with optimal asymptotic distribution. J. Appl. Prob. 35, 395–406.CrossRefGoogle Scholar
[5] Dippon, J. and Fabian, V. (1994). Stochastic approximation of global minimum points. J. Statist. Plann. Inference 41, 327–347.Google Scholar
[6] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes, Characterization and Convergence. John Wiley, New York.CrossRefGoogle Scholar
[7] Eweda, E. and Macchi, O. (1983). Quadratic mean and almost sure convergence of unbounded stochastic approximation algorithm with correlated observations. Ann. Inst. Henri Poincaré 19, 235–255.Google Scholar
[8] Gelfand, S. B. and Mitter, S. K. (1991). Recursive stochastic algorithms for global optimization in rrd . SIAM J. Control Optim. 29, 999–1018.Google Scholar
[9] Geman, D. and Geman, S. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Machine Intelligence 6, 721–741.Google Scholar
[10] Gidas, B. (1985). Nonstationary Markov chains and convergence of the annealing algorithm. J. Statist. Physics 39, 73–131.Google Scholar
[11] Hwang, C.-R. (1980). Laplace's method revised: weak convergence of probability measures. Ann. Prob. 8, 1177–1182.Google Scholar
[12] Kirkpatrick, S., Gelatt, C. D. and Vecchi, M. P. (1983). Optimization by simulated annealing. Science 220, 671–680.Google Scholar
[13] Kushner, H. J. (1984). Approximation and Weak Convergence Methods for Random Processes, with Applications to Stochastic Systems Theory. MIT Press, Cambridge.Google Scholar
[14] Kushner, H. J. (1987). Asymptotic global behavior for stochastic approximation and diffusions with slowly decreasing noise effects: global minimization via Monte Carlo. SIAM J. Appl. Math. 47, 169–185.Google Scholar
[15] Kushner, H. J. and Yin, G. (1997). Stochastic Approximation Algorithms and Applications. Springer, New York.CrossRefGoogle Scholar
[16] L'Ecuyer, P. and Yin, G. (1998). Budget-dependent convergence rate of stochastic approximation. SIAM J. Optim. 8, 217–247.Google Scholar
[17] Manjunath, B., Simchony, T. and Chellappa, R. (1990). Stochastic and deterministic networks for texture segmentation. IEEE Trans. ASSP. 38, 1039–1049.Google Scholar
[18] Metropolis, M., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and Teller, E. (1953). Equations of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1091.Google Scholar
[19] Müller, H.-G. (1989). Adaptive nonparametric peak estimation. Ann. Statist. 17, 1053–1069.CrossRefGoogle Scholar
[20] Révész, P., (1977). How to apply the method of stochastic approximation in the non-parametric estimation of regression function. Matem. Operations Stat. Ser. Statist. 8, 119–126.Google Scholar
[21] Spall, J. C. (1992). Multivariate stochastic approximation using a simultaneous perturbation gradient approximation. IEEE Trans. Automat. Control AC-37, 331–341.CrossRefGoogle Scholar
[22] Wang, I. J., Chong, E. K. P. and Kulkarni, S. R. (1996). Equivalent necessary and sufficient conditions on noise sequences for stochastic approximation algorithms. Adv. Appl. Prob. 28, 784–801.Google Scholar
[23] Wong, E. (1991). Stochastic neural networks. Algorithmica 6, 466–478.CrossRefGoogle Scholar
[24] Yakowitz, S. (1993). A globally convergent stochastic approximation. SIAM J. Control Optim. 31, 30–40.Google Scholar
[25] Yin, G. (1999). Rates of convergence for a class of global stochastic optimization algorithms. SIAM J. Optim. 10, 99–120.CrossRefGoogle Scholar