Global optimization and stochastic differential equations. (English) Zbl 0549.65038

Let \({\mathbb{R}}^ n\) be the n-dimensional real Euclidean space, \(x=(x_ 1,x_ 2,...,x_ n)^{T}\in {\mathbb{R}}^ n\), and let f: \({\mathbb{R}}^ n\to {\mathbb{R}}\) be a real-valued function. We consider the problem of finding the global minimizers of f. 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.


65K05 Numerical mathematical programming methods
90C30 Nonlinear programming
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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[1] Powell, M. J. D., Editor,Nonlinear Optimization 1981, Academic Press, New York, New York, 1982.
[2] 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.
[3] Schuss, Z.,Theory and Applications of Stochastic Differential Equations, John Wiley and Sons, New York, New York, Chapter, 8, 1980. · Zbl 0439.60002
[4] Kirkpatrick, S., Gelatt, C. D., Jr., andVecchi, M. P.,Optimization by Simulated Annealing, Science, Vol. 220, pp. 671-680, 1983. · Zbl 1225.90162
[5] 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. · Zbl 0477.60057
[6] 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. · Zbl 0569.47017
[7] 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.
[8] Wolff, S., Private Communication, 1983.
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