Journal of Global Optimization

, Volume 18, Issue 4, pp 385–398 | Cite as

Global Optimization Problems in Optimal Design of Experiments in Regression Models

  • E.P.J. Boer
  • E.M.T. Hendrix
Article

Abstract

In this paper we show that optimal design of experiments, a specific topic in statistics, constitutes a challenging application field for global optimization. This paper shows how various structures in optimal design of experiments problems determine the structure of corresponding challenging global optimization problems. Three different kinds of experimental designs are discussed: discrete designs, exact designs and replicationfree designs. Finding optimal designs for these three concepts involves different optimization problems.

Optimal design of experiments Nonconvex structure Mixed integer/continuous optimization Parameter estimation 

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References

  1. 1.
    Atkinson, A.C. (1996), The usefulness of optimum experimental designs, Journal of the Royal Statistical Society B 58(1): 59-76.Google Scholar
  2. 2.
    Atkinson, A.C. and Donev, A.N. (1992), Optimum experimental designs. Oxford University Press, Oxford.Google Scholar
  3. 3.
    Boer, E.P.J., Rasch, D.A.M.K. and Hendrix, E.M.T. (2000), Locally optimal designs in non-linear regression: A case study of the Michaelis-Menten function, in Balakrishnan, N., Ermakov, S.M. and Melas, V.B. (eds.), Advances in Stochastic Simulation Methods, Birkhauser, Boston.Google Scholar
  4. 4.
    Ermakov, S.M. and Zhiglijavsky, A.A. (1987), Matematitscheskaja teorija optimalnich experimentov. Nauka, Moskva.Google Scholar
  5. 5.
    Fedorov, V.V. (1972), Theory of optimal experiments. Academic Press, New York.Google Scholar
  6. 6.
    Fedorov, V.V. (1989), Optimal design with bounded density: Optimization algorithms of the exchange type, Journal of Statistical Planning and Inference 22: 1-13.Google Scholar
  7. 7.
    Gaffke, N. and Mathar, R. (1992), On a class of algorithms from experimental design theory, Optimization 24: 91-126.Google Scholar
  8. 8.
    Gaffke, N. and Heiligers, B. (1995), Algorithms for optimal design with application to multiple polynomial regression, Metrika 42: 173-190.Google Scholar
  9. 9.
    Jones, B. and Wang, J. (1999), Constructing optimal designs for fitting pharmacokinetic models, Computational Statistics 9: 209-218.Google Scholar
  10. 10.
    Kiefer, J.C. and Wolfowitz, J. (1960), The equivalence of two extremum problems, Canadian Journal of Mathematics 12: 363-366.Google Scholar
  11. 11.
    Ko, C., Lee, J. and Queyranne, M. (1995), An exact algorithm for maximum entropy sampling, Operations Research 43(4): 684-691.Google Scholar
  12. 12.
    Müller, W.G. (1998), Collecting spatial Data–Optimum design of experiments for random fields. Physica-Verlag, Heidelberg.Google Scholar
  13. 13.
    Müller, W.G. and Pázmann, A. (1998), Design measures and approximate information matrices for experiments without replications, Journal of Statistical Planning and Inference 71: 349-362.Google Scholar
  14. 14.
    Pukelsheim, F. (1993), Optimal design of experiments. Wiley, New York.Google Scholar
  15. 15.
    Pukelsheim, F. and Rieder, S. (1992), Efficient rounding of approximate designs, Biometrika 79(4): 763-770.Google Scholar
  16. 16.
    Rasch, D.A.M.K. (1990), Optimum experimental design in nonlinear regression, Commun. Statist.-Theory Meth. 19(12): 4789-4806.Google Scholar
  17. 17.
    Rasch, D.A.M.K., Hendrix, E.M.T. and Boer E.P.J. (1997), Replication-free optimal design in regresion analysis, Computational Statistics 12: 19-52.Google Scholar
  18. 18.
    Silvey, S.D. (1980), Optimal design. Chapman and Hall, London.Google Scholar
  19. 19.
    Vila, J.P. (1991), Local optimality of replications from a minimal D-optimal design in regression: A sufficient and quasi-necessary condition, Journal of Statistical Planning and Inference 29: 261-277.Google Scholar
  20. 20.
    Welch, W.J. (1982), Branch-and-Bound search for experimental designs based on D optimality and other criteria, Technometrics 24(1): 41-48.Google Scholar
  21. 21.
    White, L.V. (1973), An extension to the general equivalence theorem for nonlinear models, Biometrika 60: 345-348.Google Scholar
  22. 22.
    Zhigljavsky, A.A. (1991), Theory of Global Random Search. Kluwer Academic Publishers, Dordrecht.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • E.P.J. Boer
    • 1
  • E.M.T. Hendrix
    • 2
  1. 1.Sub-department of MathematicsWageningen UniversityWageningenThe Netherlands
  2. 2.Sub-department of MathematicsWageningen UniversityWageningenThe Netherlands

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