Abstract
In this paper Bayesian analysis and Wiener process are used in orderto build an algorithm to solve the problem of globaloptimization.The paper is divided in two main parts.In the first part an already known algorithm is considered: a new (Bayesian)stopping ruleis added to it and some results are given, such asan upper bound for the number of iterations under the new stopping rule.In the second part a new algorithm is introduced in which the Bayesianapproach is exploited not onlyin the choice of the Wiener model but also in the estimationof the parameter σ2 of the Wiener process, whose value appears to bequite crucial.Some results about this algorithm are also given.
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References
B. Betró, F. Schoen (1992), Optimal and sub-optimal stopping rules for the Multistart algorithm in global optimization, Mathematical Programming 57, 445–458.
J.M. Calvin (1991), Consistency of a myopic bayesian algorithm for global optimization, Technical Report,Georgia Institute of Technology.
M.H. DeGroot (1970), Optimal Statistical Decisions,McGraw-Hill, New York.
A. O’Hagan (1978), Curve fitting and optimal design for prediction, Journal of Royal Statistical Society B 40, 1–42.
A. O’Hagan (1991), Some bayesian numerical analysis, Technical Report, University of Nottingham.
H. Kushner (1962), A versatile stochastic model of a function of unknown and time varying form, Journal of Math.Anal.Appl. 5, 150–167.
M. Locatelli (1992), Algoritmi bayesiani per l’ottimizzazione globale, unpublished laurea thesis.
M. Locatelli, F. Schoen (1993), An adaptive stochastic global optimization algorithm for onedimensional functions, Proceedings of APMOD93,Budapest
J. Mockus (1988), Bayesian Approach to Global Optimization,Kluwer Academic Publishers, Dordrecht.
A.M. Mood, F.A. Graybill, D.C. Boes (1974), Introduction to the Theory of Statistics,McGraw-Hill, New York.
D. Revuz, M. Yor (1991), Continuous Martingales and Brownian Motion, Springer Verlag, Berlin.
E. Senkiene, A. Zilinskas (1978), On estimation of a parameter of Wiener process, Lithuanian Mathematical Journal 3, 59–62.
R.G. Strongin (1992), Algorithms for multi-extremal mathematical programming problems employing the set of joint space-filling curves, Journal of Global Optimization 2, 357–378.
A. Törn, A. Zilinskas (1987), Global Optimization,Springer Verlag, Berlin.
A.A. Zhigljavsky (1991), Theory of Global Random Search,Kluwer Academic Publishers, Dordrecht.
A. Zilinskas (1975), One-step Bayesianmethod of the search for extremum of an one-dimensional function, Cybernetics 1, 139–144.
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Locatelli, M. Bayesian Algorithms for One-Dimensional Global Optimization. Journal of Global Optimization 10, 57–76 (1997). https://doi.org/10.1023/A:1008294716304
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DOI: https://doi.org/10.1023/A:1008294716304