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A multi-start global minimization algorithm with dynamic search trajectories

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Abstract

A new multi-start algorithm for global unconstrained minimization is presented in which the search trajectories are derived from the equation of motion of a particle in a conservative force field, where the function to be minimized represents the potential energy. The trajectories are modified to increase the probability of convergence to a comparatively low local minimum, thus increasing the region of convergence of the global minimum. A Bayesian argument is adopted by which, under mild assumptions, the confidence level that the global minimum has been attained may be computed. When applied to standard and other test functions, the algorithm never failed to yield the global minimum.

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References

  1. Griewank, A. O.,Generalized Descent for Global Optimization, Journal of Optimization Theory and Applications, Vol. 34, pp. 11–39, 1981.

    Google Scholar 

  2. Dixon, L. C. W., Gomulka, J., andSzegö, G. P.,Toward a Global Optimization Technique, Toward Global Optimization, Edited by L. C. W. Dixon and G. P. Szegö, North-Holland Publishing Company, Amsterdam, Holland, pp. 29–54, 1975.

    Google Scholar 

  3. Dixon, L. C. W., Gomulka, J., andHersom, S. E.,Reflections on the Global Optimization Problem, Optimization in Action, Edited by L. C. W. Dixon, Academic Press, New York, New York, pp. 398–435, 1976.

    Google Scholar 

  4. Snyman, J. A.,A New and Dynamic Method for Unconstrained Minimization, Applied Mathematical Modelling, Vol. 6, pp. 449–462, 1982.

    Google Scholar 

  5. Snyman, J. A.,An Improved Version of the Original Leap-Frog Dynamic Method for Unconstrained Minimization: LFOP1(b), Applied Mathematical Modelling, Vol. 7, pp. 216–218, 1983.

    Google Scholar 

  6. Levitt, M.,Molecular Dynamics of Native Protein, I: Computer Simulation of Trajectories, Journal of Molecular Biology, Vol. 168, pp. 595–620, 1983.

    Google Scholar 

  7. Snyman, J. A. andSnyman, H. C.,Computed Epitaxial Monolayer Structures, III, Surface Science, Vol. 105, pp. 357–376, 1981.

    Google Scholar 

  8. Inomata, S., andCumada, M.,On the Golf Method, Denshi Gijutso Sogo Kenkyujo, Tokyo, Japan, Bulletin of the Electrotechnical Laboratory, Vol. 25, pp. 495–512, 1964.

    Google Scholar 

  9. Incerti, S., Parisi, V., andZirilli, F.,A New Method for Solving Nonlinear Simultaneous Equations, SIAM Journal on Numerical Analysis, Vol. 16, pp. 779–789, 1979.

    Google Scholar 

  10. Arnold, V. I., andAvez, A.,Ergodic Problems of Classical Mechanics, W. A. Benjamin, New York, New York, 1968.

    Google Scholar 

  11. Rinnooy Kan, A. H. G., andTimmer, G. T.,Stochastic Methods for Global Optimization, American Journal of Mathematical and Management Sciences, Vol. 4, pp. 7–40, 1984.

    Google Scholar 

  12. Box, G. E. P., andTiao, G. C.,Bayesian Inference in Statistical Analysis, Addison-Wesley, London, England, 1973.

    Google Scholar 

  13. Dixon, L. C. W., andSzegö, G. P.,The Global Optimization Problem: An Introduction, Toward Global Optimization 2, Edited by L. C. W. Dixon and G. P. Szegö, North-Holland Publishing Company, Amsterdam, Holland, pp. 1–15, 1978.

    Google Scholar 

  14. Price, W. L.,Global Optimization by Controlled Random Search, Journal of Optimization Theory and Applications, Vol. 40, pp. 333–348, 1983.

    Google Scholar 

  15. Anonymous, N. N.,International Mathematical and Statistical Libraries, Vol. 4, Edition 9, IMSL, Houston, Texas, 1982.

    Google Scholar 

  16. Boender, C. G. E., Rinnooy Kan, A. H. G., Timmer, G. T., andStougie, L.,A Stochastic Method for Global Optimization, Mathematical Programming, Vol. 22, pp. 125–140, 1982.

    Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Pretoria, Pretoria, South Africa

    J. A. Snyman (Professor)

  2. National Research Institute for Mathematical Sciences, CSIR, Pretoria, South Africa

    L. P. Fatti (Senior Specialist Researcher)

Authors
  1. J. A. Snyman
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  2. L. P. Fatti
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Communicated by L. C. W. Dixon

The first author wishes to thank Prof. M. Levitt of the Department of Chemical Physics of the Weizmann Institute of Science for suggesting this line of research and also Drs. T. B. Scheffler and E. A. Evangelidis for fruitful discussions regarding Conjecture 2.1. He also acknowledges the exchange agreement award received from the National Council for Research and Development in Israel and the Council for Scientific and Industrial Research in South Africa, which made possible the visit to the Weizmann Institute where this work was initiated.

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Snyman, J.A., Fatti, L.P. A multi-start global minimization algorithm with dynamic search trajectories. J Optim Theory Appl 54, 121–141 (1987). https://doi.org/10.1007/BF00940408

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