Advertisement

A constrained, globalized, and bounded Nelder–Mead method for engineering optimization

  • M.A. Luersen Email author
  • R. Le Riche
  • F. Guyon
Research paper

Abstract

One of the fundamental difficulties in engineering design is the multiplicity of local solutions. This has triggered much effort in the development of global search algorithms. Globality, however, often has a prohibitively high numerical cost for real problems. A fixed cost local search, which sequentially becomes global, is developed in this work. Globalization is achieved by probabilistic restarts. A spacial probability of starting a local search is built based on past searches. An improved Nelder–Mead algorithm is the local optimizer. It accounts for variable bounds and nonlinear inequality constraints. It is additionally made more robust by reinitializing degenerated simplexes. The resulting method, called the Globalized Bounded Nelder–Mead (GBNM) algorithm, is particularly adapted to tackling multimodal, discontinuous, constrained optimization problems, for which it is uncertain that a global optimization can be afforded. Numerical experiments are given on two analytical test functions and two composite laminate design problems. The GBNM method compares favorably with an evolutionary algorithm, both in terms of numerical cost and accuracy.

Keywords

global constrained optimization Nelder–Mead method composite laminated plates  

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bäck, T. 1996: Evolutionary Algorithms in Theory and Practice. Oxford: Oxford University Press Google Scholar
  2. 2.
    Barhen, J.; Protopopescu, V.; Reister, D. 1997: TRUST: a deterministic algorithm for global constrained optimization. Science 276, 1094–1097 Google Scholar
  3. 3.
    Berthelot, J.-M. 1999: Composite Materials: Mechanical Behavior and Structural Analysis, Mechanical Engineering Series. Berlin: Springer Google Scholar
  4. 4.
    Duda, O.R.; Hart, P.E.; Stork, D.G. 2001: Pattern Classification, 2nd edn. New York: John Wiley & Sons Google Scholar
  5. 5.
    Durand, N.; Alliot, J.-M. 1999: A combined Nelder–Mead simplex and genetic algorithm. Available at: http://www.recherche.enac.fr/opti/papers/ Google Scholar
  6. 6.
    Goldberg, D.E.; Voessner, S. 1999: Optimizing global-local search hybrids. In: GECCO 99 – Genetic and Evolutionary Computation Conference (held in Orlando), pp. 220–228 Google Scholar
  7. 7.
    Haftka, R.T.; Gürdal, Z. 1993: Elements of Structural Optimization, 3rd edn. Boston: Kluwer Academic Publishers Google Scholar
  8. 8.
    Hickernell, F.J.; Yuan, Y.-X. 1997: A simple multistart algorithm for global optimization. OR Trans. 1(2), 1–11 Google Scholar
  9. 9.
    Hu, X.; Shonkwiller, R.; Spruill, M.C. 1994: Random Restarts in Global Optimization. Technical Report, School of Mathematics, Georgia Institute of Technology, Atlanta Google Scholar
  10. 10.
    Le Riche, R.; Guyon, F. 2001: Dual evolutionary optimization. In: Collet, P.; Lutton, E.; Schoenauer, M.; Fonlupt, C.; Hao, J.-K. (eds.) Artificial Evolution, Lecture Notes in Computer Science, No. 2310, selected papers of the 5th International Conference on Artificial Evolution (held in Le Creusot), pp. 281–294 Google Scholar
  11. 11.
    Luersen, M.A.; Le Riche, R. 2001: Globalisation de l’Algorithme de Nelder–Mead : Application aux Composites. Technical Report, LMR, INSA de Rouen, France; in French Google Scholar
  12. 12.
    Michalewicz, Z.; Schoenauer, M. 1997: Evolutionary algorithms for constrained parameter optimization. Evolut. Comput. 4(1), 1–32 Google Scholar
  13. 13.
    Minoux, M. 1986: Mathematical Programming: Theory and Algorithms. New York: John Wiley & Sons Google Scholar
  14. 14.
    Moscato, P. 1989: On Evolution, Search, Optimization, Genetic Algorithms and Martial Arts: Towards Memetic Algorithms. Caltech Concurrent Computation Program, C3P Report 826 Google Scholar
  15. 15.
    Nelder, J.A.; Mead, R. 1965: A simplex for function minimization. Comput. J. 7, 308–313 Google Scholar
  16. 16.
    Okamoto, M.; Nonaka, T.; Ochiai, S.; Tominaga, D. 1998: Nonlinear numerical optimization with use of a hybrid genetic algorithm incorporating the modified Powell method. Appl. Math. Comput. 91, 63–72 Google Scholar
  17. 17.
    Rockafellar, R.T. 1976: Lagrange multipliers in optimization. In: Cottle R.W.; Lemke C.E. (eds.) Nonlinear Programming, Proc. SIAM-AMS, 9, 145–168 Google Scholar
  18. 18.
    Shang, Y.; Wan, Y.; Fromherz, M.P.J.; Crawford, L. 2001: Toward adaptive cooperation between global and local solvers for continuous constraint problems. In: CP’01 Workshop on Cooperative Solvers in Constraints Programming (held in Pahos) Google Scholar
  19. 19.
    Syswerda, G. 1991: A study of reproduction in generational and steady state genetic algorithms. In: Rawlins, G.J.E. (ed.) Foundations of Genetic Algorithms. San Mateo: Morgan Kaufmann Google Scholar
  20. 20.
    Törn, A.A. 1978: A search-glustering approach to global optimization. In: Towards Global Optimization 2, pp. 49–62 Google Scholar
  21. 21.
    Törn, A.A.; Zilinskas A. 1989: Global Optimization. Berlin: Springer-Verlag Google Scholar
  22. 22.
    Wright, M.H. 1996: Direct search methods: once scorned, now respectable. In: Dundee Biennial Conference in Numerical Analysis (held in Harlow), pp. 191–208Google Scholar

Copyright information

© Springer-Verlag 2003

Authors and Affiliations

  1. 1.Lab. de Mécanique de Rouen, France, and Mechanical Department, CEFET-PRCNRS UMR 6138Brazil
  2. 2.Ecole des Mines de Saint EtienneCNRS URA 1884 / SMSFrance
  3. 3.Lab. de Bio-statistiques Bio-mathématiquesUniv. Paris 7France

Personalised recommendations