Abstract
Let ℝn be then-dimensional real Euclidean space,x=(x 1,x 2, ...,x n)T ∈ ℝn, and letf:ℝn → R be a real-valued function. We consider the problem of finding the global minimizers off. A new method to compute numerically the global minimizers by following the paths of a system of stochastic differential equations is proposed. This method is motivated by quantum mechanics. Some numerical experience on a set of test problems is presented. The method compares favorably with other existing methods for global optimization.
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References
Powell, M. J. D., Editor,Nonlinear Optimization 1981, Academic Press, New York, New York, 1982.
Zirilli, F.,The Use of Ordinary Differential Equations in the Solution of Nonlinear Systems of Equations, Nonlinear Optimization 1981, Edited by M. J. D. Powell, Academic Press, New York, New York, 1982.
Schuss, Z.,Theory and Applications of Stochastic Differential Equations, John Wiley and Sons, New York, New York, Chapter, 8, 1980.
Kirkpatrick, S., Gelatt, C. D., Jr., andVecchi, M. P.,Optimization by Simulated Annealing, Science, Vol. 220, pp. 671–680, 1983.
Matkowsky, B. J., andSchuss, Z.,Eigenvalues of the Fokker-Planck Operator and the Approach to Equilibrium for Diffusions in Potential Fields, SIAM Journal on Applied Mathematics, Vol. 40, pp. 242–254, 1981.
Angeletti, A., Castagnari, C., andZirilli, F.,Asymptotic Eigenvalue Degeneracy for a Class of One-Dimensional Fokker-Planck Operators, Journal of Mathematical Physics, Vol. 26, pp. 678–690, 1985.
Levy, A. V., andMontalvo, A.,Algoritmo de Tunelización para la Optimizatión Global de Funciones, Universidad Nacional Autónoma de México, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Report No. 204, 1979.
Wolff, S., Private Communication, 1983.
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Communicated by R. A. Tapia
This research has been supported by the European Research Office of the US Army under Contract No. DAJA-37-81-C-0740.
The third author gratefully acknowledges Prof. A. Rinnooy Kan for bringing to his attention Ref. 4.
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Aluffi-Pentini, F., Parisi, V. & Zirilli, F. Global optimization and stochastic differential equations. J Optim Theory Appl 47, 1–16 (1985). https://doi.org/10.1007/BF00941312
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DOI: https://doi.org/10.1007/BF00941312