Abstract
We study the convergence of a class of discrete-time continuous-state simulated annealing type algorithms for multivariate optimization. The general algorithm that we consider is of the formX k +1 =X k −a k (▽U(X k ) + ξk) +b k W k . HereU(·) is a smooth function on a compact subset of ℝd, {ξk} is a sequence of ℝd-valued random variables, {W k } is a sequence of independent standardd-dimensional Gaussian random variables, and {a k }, {b k } are sequences of positive numbers which tend to zero. These algorithms arise by adding decreasing white Gaussian noise to gradient descent, random search, and stochastic approximation algorithms. We show under suitable conditions onU(·), {ξ k }, {a k }, and {b k } thatX k converges in probability to the set of global minima ofU(·). A careful treatment of howX k is restricted to a compact set and its effect on convergence is given.
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References
Wasan, M. T.,Stochastic Approximation, Cambridge University Press, Cambridge, 1969.
Rubinstein, R. Y.,Simulation and the Monte Carlo Method, Wiley, New York, 1981.
Ljung, L., Analysis of recursive stochastic algorithms,IEEE Transactions on Automatic Control, Vol. 22, pp. 551–575, 1977.
Kushner, H., and Clark, D.,Stochastic Approximation Methods for Constrained and Unconstrained Systems, Applied Mathematical Sciences, Vol. 26, Springer Verlag, Berlin, 1978.
Metivier, M., and Priouret, P., Applications of a Kushner and Clark lemma to general classes of stochastic algorithm,IEEE Transactions on Information Theory, Vol. 30, pp. 140–151, 1984.
Grenender, U., Tutorial in Pattern Theory, Division of Applied Mathematics, Brown University, Providence, RI, 1984.
Geman, S., and Hwang, C. R., Diffusions for global optimization,SIAM Journal on Control and Optimization, Vol. 24, pp. 1031–1043, 1986.
Kirkpatrick, S., Gelatt, C. D., and Vecchi, M. P., Optimization by simulated annealing,Science, Vol. 220, pp. 621–680, 1983.
Gelfand, S. B., Analysis of Simulated Annealing Type Algorithms, Ph.D. Thesis, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA, 1987.
Gidas, B., Global optimization via the Langevin equation,Proceedings of the IEEE Conference on Decision and Control, pp. 774–778, Fort Lauderdale, FL, 1985.
Chiang, T. S., Hwang, C. R., and Sheu, S. J., Diffusion for global optimization in ℝn,SIAM Journal on Control and Optimization, Vol. 25, pp. 737–752, 1987.
Kushner, H. J., Asymptotic global behavior for stochastic approximation and diffusions with slowly decreasing noise effects: Global minimization via Monte Carlo,SIAM Journal on Applied Mathematics, Vol. 47, pp. 169–185, 1987.
Aluffi-Pentini, F., Parisi, V., and Zirilli, F., Global optimization and stochastic differential equations,Journal of Optimization Theory and Applications, Vol. 47, pp. 1–16, 1985.
Hwang, C.-R., Laplaces method revisited: weak convergence of probability measures,Annals of Probability, Vol. 8, pp. 1177–1182, 1980.
Gikhman, I.I., and Skorohod, A. V.,Stochastic Differential Equations, Springer-Verlag, Berlin, 1972.
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Communicated by Alberto Sangiovanni-Vincentelli.
Research supported by the Air Force Office of Scientific Research contract 89-0276B, and the Army Research Office contract DAAL03-86-K-0171 (Center for Intelligent Control Systems).
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Gelfand, S.B., Mitter, S.K. Simulated annealing type algorithms for multivariate optimization. Algorithmica 6, 419–436 (1991). https://doi.org/10.1007/BF01759052
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DOI: https://doi.org/10.1007/BF01759052