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A Stochastic Approach to Global Optimization

  • A. H. G. Rinnooy Kan
  • C. G. E. Boender
  • G. Th. Timmer
Part of the NATO ASI Series book series (volume 15)

Abstract

Letf : Rn → R be a real valued smooth objective function. The area of nonlinear programming is traditionally concerned with methods that find a local optimum (say local minimum) of f, i.e. a point x* e Rn such that there exists a neighbourhood B of x* with
$$f({{x}^{*}}) \leqslant f(x) \forall x \in B$$
(1)
.

Keywords

Sample Point Local Minimum Local Search Global Optimization Global Minimum 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Allgower, E. and K. Georg (1980), Simplicial and continuation methods for approximating fixed points and solutions of systems of equations. Siam Review 22, 28–84.CrossRefzbMATHMathSciNetGoogle Scholar
  2. Anderssen, R.S. (1972), Global optimization, In R.S. Anderssen, L.S. Jennings and D.M. Ryan (eds.). Optimization (University of Queensland Press).Google Scholar
  3. Beale, E.M.L. and J.J.H. Forrest (1-978), Global optimization as an extension of integer programming. In [Dixon & Szegö 1978].Google Scholar
  4. Becker, R.W. and G.V. Lago (1970), A global optimization algorithm. In Proceedings of the 8th Allerton Conference on Circuits and Systems Theory.Google Scholar
  5. Boender, C.G.E., A.H.G. Rinnooy Kan, L. Stougie and G.T. Timmer (1982), A stochastic method for global optimization. Mathematical Programming 22, 125–140.CrossRefzbMATHMathSciNetGoogle Scholar
  6. Boender, C.G.E. (1984), The Generalized Multinomial Distribution; A Bayesian Analysis and Applications. Ph.D. Dissertation, Erasmus Universiteit Rotterdam (Centrum voor Wiskunde en Informatica, Amsterdam).Google Scholar
  7. Branin, F.H. (1972), Widely convergent methods for finding multiple solutions of simultaneous nonlinear equations. IBM Journal of Research Developments, 504-522.Google Scholar
  8. Branin, F.H. and S.K. Hoo (1972), A method for finding multiple extrema of a function of n variables. In F.A. Lootsma (ed.), Numerical Methods of Nonlinear Optimization (Academic Press, London).Google Scholar
  9. Bremmerman, H. (1970), A method of unconstrained global optimization. Mathematical Biosciences 9, 1–15.CrossRefMathSciNetGoogle Scholar
  10. Brooks, S.H. (1958), A discussion of random methods for seeking maxima. Operations Research 6, 244–251.CrossRefMathSciNetGoogle Scholar
  11. De Biase, L. and F. Frontini (1978), A stochastic method for global optimization: its structure and numerical performance. In [Dixon & Szegö 1978].Google Scholar
  12. Devroye, L. (1978), Progressive global random search of continuous functions. Mathematical Programming 1, 330–342.CrossRefMathSciNetGoogle Scholar
  13. Dixon, L.C.W., J. Gomulka and G.P. Szegö (1975), Towards global optimization. In [Dixon & Szegö 1975].Google Scholar
  14. Dixon, L.C.W. (1978), Global optima without convexity. Technical Report, Numerical Optimisation Centre Hatfield Polytechnic, Hatfield, England.Google Scholar
  15. Dixon, L.C.W. and G.P. Szegö (eds.) (1978), Towards Global Optimization 2 (North-Holland, Amsterdam).Google Scholar
  16. Evtushenko, Y.P. (1971), Numerical methods for finding global extrema of a nonuniform mesh. U.S.S.R. Computing Machines and Mathematical Physics 11, 1390-1404.Google Scholar
  17. Falk, J.E. and R.M. Solund (1969), An algorithm for separable nonconvex programming. Management Science 15, 550-569.Google Scholar
  18. Goldstein, A.A. and J.F. Price (1971), On descent from local minima. Mathematics of Computation 25, 569-574.Google Scholar
  19. Hansen, E. (1980), Global optimization using interval analysis — the multidimensional case. Numerische Mathematik 34, 247–270.CrossRefzbMATHMathSciNetGoogle Scholar
  20. Kuhn, H.W., Z. Wang and S. Xu (1984), On the cost of computing roots of polynomials. Mathematical Programming 28, 156-164.Google Scholar
  21. Levy, A. and S. Gomez (1980), The tunneling algorithm for the global optimization problem of constrained functions. Technical Report, Universidad National Autonoma de Mexico.Google Scholar
  22. McCormick, G.P. (1976), Compatibility of global solutions to factorable non-convex programming, Part I — convex underestimating problems. Mathematical Programming 10, 147-175.Google Scholar
  23. Price, W.L. (1978), A controlled random search procedure for global optimization. In [Dixon & Szegö 1978a]. Rinnooy Kan, A.H.G. and G.T. Timmer (1984), Stochastic methods for global optimization. To appear in the American Journal of Mathematical and Management Sciences Google Scholar
  24. Rubinstein, R.Y. (1981), Simulation and the Monte Carlo Method (John Wiley & Sons, New York).Google Scholar
  25. Shubert, B.O. (1972), A sequential method seeking the global maximum of a function. Siam Journal on Numerical Analysis 9, 379–388.CrossRefzbMATHMathSciNetGoogle Scholar
  26. Solis, F.J. and R.J.E. Wets (1981), Minimization by random search techniques. Mathematics of Operations Research6, 19–30CrossRefzbMATHMathSciNetGoogle Scholar
  27. Solund, R.M. (1971), An algorithm for separable nonconvex programming problems 2. Management Science 17, 759–773.CrossRefGoogle Scholar
  28. Timmer, G.T. (1984), Global Optimization: A Stochastic Approach. Ph.D. Dissertation, Erasmus Universiteit Rotterdam (Centrum voor Wiskunde en Informatica, Amsterdam).Google Scholar
  29. Todd, M.J. (1976), The Computation of Fixed Points and Applications Springer Verlag, Berlin).Google Scholar
  30. Torn, A.A. (1976), Cluster analysis using seed points and density determined hyperspheres with an application to global optimization. In Proceeding of the third International Conference on Pattern Recognition, Coronado, California.Google Scholar
  31. Torn, A.A. (1978), A search clustering appraoch to global optimization. In [Dixon & Szegö, 1978].Google Scholar
  32. Treccani, G. (1975), On the convergence of Branin’s method: a counter example. In [Dixon & Szegö 1975].Google Scholar
  33. Zielinski, R. (1981), A stochastic estimate of the structure of multi-extremal problems. Mathematical Programming 21, 348–356.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • A. H. G. Rinnooy Kan
    • 1
  • C. G. E. Boender
    • 1
    • 2
  • G. Th. Timmer
    • 1
    • 2
  1. 1.Econometric InstituteErasmus University RotterdamThe Netherlands
  2. 2.Department of Mathematics and InformaticsDelft University of TechnologyThe Netherlands

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