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Mathematical Programming

, Volume 38, Issue 3, pp 271–286 | Cite as

Sequential stopping rules for the multistart algorithm in global optimisation

  • Bruno Betrò
  • Fabio Schoen
Article

Abstract

In this paper a sequential stopping rule is developed for the Multistart algorithm. A statistical model for the values of the observed local maxima of an objective function is introduced in the framework of Bayesian non-parametric statistics. A suitablea-priori distribution is proposed which is general enough and which leads to computationally manageable expressions for thea-posteriori distribution. Sequential stopping rules of thek-step look-ahead kind are then explicitly derived, and their numerical effectiveness compared.

Key words

Global optimisation Monte Carlo method stopping rules Multistart algorithm Bayes methods 

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References

  1. F. Archetti and F. Schoen, “A survey on the global optimization problem: general theory and computational approaches,”Annals of Operations Research 1 (1984) 87–110.CrossRefGoogle Scholar
  2. B. Betrò and C. Vercellis, “Bayesian nonparametric inference and Monte Carlo optimization,” I.A.M.I. report 84.20 (Milano, 1984).Google Scholar
  3. B. Betrò, “The tails of distribution functions from homogeneous processes neutral to the right,” to appear (1984).Google Scholar
  4. C.G.E. Boender and A.H.G. Rinnooy Kan, “Bayesian, stopping rules for multistart optimization methods,”Mathematical Programming 37 (1987) 59–80.zbMATHMathSciNetGoogle Scholar
  5. C.G.E. Boender and R. Zieliński, “A sequential Bayesian approach to estimating the dimension of a multinomial distribution,” preprint n. 256, Institute of Mathematics, Polish Academy of Sciences (Warsaw, 1982).Google Scholar
  6. M.H. De Groot, Optimal Statistical Decisions (McGraw-Hill, New York, 1970).Google Scholar
  7. L.C.W. Dixon and G.P. Szegö (eds), Towards Global Optimisation 2 (North-Holland, Amsterdam, 1978).zbMATHGoogle Scholar
  8. K. Doksum, “Tailfree and neutral random probabilities and their posterior distributions,”Annals of Statistics 2 (1974) 183–201.zbMATHGoogle Scholar
  9. T.S. Ferguson, “A Bayesian analysis of some nonparametric problems,”Annals of Statistics 2 (1973) 209–230.MathSciNetGoogle Scholar
  10. T.S. Ferguson and E.G. Phadia, “Bayesian nonparametric estimation based on censored data,”Annals of Statistics 7 (1979) 163–186.zbMATHMathSciNetGoogle Scholar
  11. J.P. Gilbert and F. Mosteller, “Recognizing the maximum of a sequence,”Journal of the American Statistical Association 61 (1966) 35–73.CrossRefMathSciNetGoogle Scholar
  12. R. Zieliński, “A statistical estimate of the structure of multi-extremal problems,”Mathematical Programming 21 (1981) 348–356.CrossRefMathSciNetGoogle Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1987

Authors and Affiliations

  • Bruno Betrò
    • 1
  • Fabio Schoen
    • 1
  1. 1.CNR-IAMIMilanoItaly

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