Abstract
We present an algorithm for finding the global maximum of a multimodal, multivariate function for which derivatives are available. The algorithm assumes a bound on the second derivatives of the function and uses this to construct an upper envelope. Successive function evaluations lower this envelope until the value of the global maximum is known to the required degree of accuracy. The algorithm has been implemented in RATFOR and execution times for standard test functions are presented at the end of the paper.
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References
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 Optimization 2 (North-Holland, Amsterdam, 1978) pp. 85–102.
H. Bremermann, “A method of unconstrained global optimization,”Mathematical Biosciences 9 (1970) 1–15.
R.P. Brent,Algorithms for Minimization Without Derivatives (Prentice-Hall, Englewood Cliffs, NJ, 1973).
A. Cutler, “Optimization methods in statistics,” Ph.D thesis, U.C. Berkeley (Berkeley, CA, 1988).
L.C.W. Dixon and G.P. Szegö, eds.,Towards Global Optimization (North-Holland, Amsterdam, 1975).
L.C.W. Dixon and G.P. Szegö, eds.,Towards Global Optimization 2 (North-Holland, Amsterdam, 1978).
L.C.W. Dixon and G.P. Szegö, “The global optimization problem: An introduction,” in: L.C.W. Dixon and G.P. Szegö, eds.,Towards Global Optimization 2 (North-Holland, Amsterdam, 1978) pp. 1–15.
D.A. Freedman and W. Navidi, “On the multistage model for carcinogenesis,” Technical Report No. 97, Statistics Department, U.C. Berkeley (Berkeley, CA, 1988).
A.O. Griewank, “Generalized descent for global optimization,”Journal of Optimization Theory and Applications 34(1) (1981) 11–39.
R.H. Mladineo, “An algorithm for finding the global maximum of a multimodal, multivariate function,”Mathematical Programming 34 (1986) 188–200.
S.A. Piyavskii, “An algorithm for finding the absolute extremum of a function,”USSR Computational Mathematics and Mathematical Physics 12(4) (1972) 57–67.
W.L. Price, “Global optimization by controlled random search,”Journal of Optimization Theory and Applications 40 (1983) 333–348.
A.H.G. Rinnooy Kan and G.T. Timmer, “Stochastic global optimization methods. Part I: Clustering methods,”Mathematical Programming 39 (1987) 27–56.
A.H.G. Rinnooy Kan and G.T. Timmer, “Stochastic global optimization methods. Part II: Multi level methods,”Mathematical Programming 39 (1987) 57–78.
B.O. Shubert, “A sequential method seeking the global maximum of a function,”SIAM Journal of Numerical Analysis 9(3) (1972) 379–388.
C.H. Slump and B.J. Hoenders, “The determination of the location of the global maximum of a function in the presence of several local extrema,”IEEE Transactions on Information Theory IT-31 (1985) 490–497.
J.A. Snyman and L.P. Fatti, “A multi-start global minimization algorithm with dynamic search trajectories,”Journal of Optimization Theory and Applications 54 (1987) 121–141.
A.A. Törn, “A search-clustering approach to global optimization,” in: L.C.W. Dixon and G.P. Szegö, eds.,Towards Global Optimization 2 (North-Holland, Amsterdam, 1978) pp. 49–62.
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.
D.R. Wingo, “Estimating the location of the Cauchy distribution by numerical global optimization,”Communications in Statistics Part B: Simulation and Computation 12(2) (1983) 201–212.
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Partially supported by NSF DMS-8718362.
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Breiman, L., Cutler, A. A deterministic algorithm for global optimization. Mathematical Programming 58, 179–199 (1993). https://doi.org/10.1007/BF01581266
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DOI: https://doi.org/10.1007/BF01581266