Abstract
In this paper we are concerned with global optimization, which can be defined as the problem of finding points on a bounded subset of ℝn in which some real valued functionf assumes its optimal (maximal or minimal) value.
We present a stochastic approach which is based on the simulated annealing algorithm. The approach closely follows the formulation of the simulated annealing algorithm as originally given for discrete optimization problems. The mathematical formulation is extended to continuous optimization problems, and we prove asymptotic convergence to the set of global optima. Furthermore, we discuss an implementation of the algorithm and compare its performance with other well-known algorithms. The performance evaluation is carried out for a standard set of test functions from the literature.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
E.H.L. Aarts and P.J.M. van Laarhoven, “Statistical cooling: a general approach to combinatorial optimization problems,”Philips Journal of Research 40 (1985) 193–226.
E.H.L. Aarts and J.H.M. Korst,Simulated Annealing and Boltzmann Machines (Wiley, Chichester, 1988).
F. Aluffi-Pentini, V. Parisi and F. Zirilli, “Global optimization and stochastic differential equations,”Journal of Optimization Theory and Applications 47 (1985) 1–16.
I.O. Bohachevsky, M.E. Johnson and M.L. Stein, “Generalized simulated annealing for function optimization,”Technometrics 28 (1986) 209–217.
T.-S. Chiang, C.-R. Hwang and S.-J. Sheu, “Diffusion for global optimization in ℝn,”SIAM Journal on Control and Optimization 25 (1987) 737–753.
L. De Biase and F. Frontini, “A stochastic method for global optimization: its structure and numerical performance,” in: L.C.W. Dixon and G.P. Szegö, eds.,Towards Global Optimisation 2 (North-Holland, Amsterdam, 1978) pp. 85–102.
L.C.W. Dixon and G.P. Szegö, eds.,Towards Global Optimisation 2 (North-Holland, Amsterdam, 1978a).
L.C.W. Dixon and G.P. Szegö, “The global optimisation problem: an introduction,” in: L.C.W. Dixon and G.P. Szegö, eds.,Towards Global Optimisation 2 (North-Holland, Amsterdam, 1978b) pp. 1–15.
J.L. Doob,Stochastic Processes (Wiley, New York, 1953).
W. Feller,An Introduction to Probability Theory and its Applications, Vol 1 (Wiley, New York, 1957).
S. Geman and C.-R. Hwang, “Diffusions for global optimization,”SIAM Journal on Control and Optimization 24 (1986) 1031–1043.
J. Gomulka, “Deterministic versus probabilistic approaches to global optimisation,” in: L.C.W. Dixon and G.P. Szegö, eds.,Towards Global Optimisation 2 (North-Holland, Amsterdam, 1978a) pp. 19–30.
J. Gomulka, “A users experience with Törn's clustering algorithm,” in: L.C.W. Dixon and G.P. Szegö, eds.,Towards Global Optimisation 2 (North-Holland, Amsterdam, 1978b) pp. 63–70.
A. Khachaturyan, “Statistical mechanics approach in minimizing a multivariable function,”Journal of Mathematical Physics 27 (1986) 1834–1838.
S. Kirkpatrick, C.D. Gelatt Jr. and M.P. Vecchi, “Optimization by simulated annealing,”Science 220 (1983) 671–680.
H.J. Kushner, “Asymptotic global behavior for stochastic approximation and diffusions with slowly decreasing noise effects: global minimization via Monte Carlo,”SIAM Journal on Applied Mathematics 47 (1987) 169–185.
P.J.M. van Laarhoven and E.H.L. Aarts,Simulated Annealing: Theory and Applications (Reidel, Dordrecht, 1987).
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 21 (1953) 1087–1092.
W.L. Price, “A controlled random search procedure for global optimisation,” in: L.C.W. Dixon and G.P. Szegö, eds.,Towards Global Optimisation 2 (North-Holland, Amsterdam, 1978) pp. 71–84.
A.H.G. Rinnooy Kan and G.T. Timmer, “Stochastic methods for global optimization,”American Journal of Mathematical and Management Sciences 4 (1984) 7–40.
A.H.G. Rinnooy Kan and G.T. Timmer, “Stochastic global optimization methods. part I: clustering methods,”Mathematical Programming 39 (1987a) 27–56.
A.H.G. Rinnooy Kan and G.T. Timmer, “Stochastic global optimization methods. part II: multi level methods,”Mathematical Programming 39 (1987b) 57–78.
L.E. Scales,Introduction to Non-linear Optimization (Macmillan, London, 1985).
A.A. Törn, “A search-clustering approach to global optimization,” in: L.C.W. Dixon and G.P. Szegö, eds.,Towards Global Optimisation 2 (North-Holland, Amsterdam, 1978) pp. 49–62.
Tovey et al. (1989), unpublished manuscript.
D. Vanderbilt and S.G. Louie, “A Monte Carlo simulated annealing approach to optimization over continuous variables,”Journal of Computational Physics 56 (1984) 259–271.
A.J. Weir,Lebesgue Integration and Measure (Cambridge University Press, Cambridge, 1973).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Dekkers, A., Aarts, E. Global optimization and simulated annealing. Mathematical Programming 50, 367–393 (1991). https://doi.org/10.1007/BF01594945
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01594945