Variational Calculus in the Space of Measures and Optimal Design

  • Ilya Molchanov
  • Sergei Zuyev
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 51)


The paper applies abstract optimisation principles in the space of measures within the context of optimal design problems. It is shown that within this framework it is possible to treat various design criteria and constraints in a unified manner providing a “universal” variant of the Kiefer-Wolfowitz theorem and giving a full spectrum of optimality criteria for particular cases. The described steepest descent algorithm uses the true direction of steepest descent and descends faster than the conventional sequential algorithms that involve renormalisation at every step.


design of experiments gradient methods optimal design regression space of measures 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Atkinson, A. C. and Donev, A. N. (1992). Optimum Experimental Designs. Oxford: Clarendon Press.zbMATHGoogle Scholar
  2. Atwood, C. L. (1973). Sequences converging to D-optimal designs of experiments. Ann. Statist 1, 342–352.MathSciNetCrossRefGoogle Scholar
  3. Atwood, C. L. (1976). Convergent design sequences, for sufficiently regular optimality criteria. Ann. Statist. 4, 1124–1138.MathSciNetzbMATHCrossRefGoogle Scholar
  4. Cook, D. and Fedorov, V. (1995). Constrained optimization of experimental design. Statistics 26, 129–178.MathSciNetzbMATHCrossRefGoogle Scholar
  5. Fedorov, V. V. (1972). Theory of Optimal Experiments. New York: Academic Press.Google Scholar
  6. Fedorov, V. V. and Hackl, P. (1997). Model-Oriented Design of Experiments, Volume 125 of Lect. Notes Statist. New York: Springer-Verlag.CrossRefGoogle Scholar
  7. Ford, I. (1976). Optimal Static and Sequential Design: a Critical Review. PhD Thesis, Department of Statistics, University of Glasgow.Google Scholar
  8. Hille, E. and Phillips, R. S. (1957). Functional Analysis and Semigroups, Volume XXXI of AMS Colloquium Publications. Providence, RI: American Mathematical Society.Google Scholar
  9. Molchanov, I. and Zuyev, S. (1997). Variational analysis of functionals of a Poisson process. Rapport de Recherche 3302, INRIA, Sophia-Antipolis. To appear in Math. Oper. Res. Google Scholar
  10. Molchanov, I. and Zuyev, S. (2000). Steepest descent algorithms in space of measures. To appear.Google Scholar
  11. Polak, E. (1997). Optimization. New York: Springer-Verlag.zbMATHCrossRefGoogle Scholar
  12. Wu, C.-F. (1978a). Some algorithmic aspects of the theory of optimal design. Ann. Statist. 6, 1286–1301.MathSciNetzbMATHCrossRefGoogle Scholar
  13. Wu, C.-F. (1978b). Some iterative procedures for generating nonsingular optimal designs. Comm. Statist. Theory Methods A7 14, 1399–1412.Google Scholar
  14. Wu, C.-F. and Wynn, H. P. (1978). The convergence of general step-length algorithms for regular optimum design criteria. Ann. Statist. 6, 1273–1285.MathSciNetzbMATHCrossRefGoogle Scholar
  15. Wynn, H. P. (1970). The sequential generation of D-optimum experimental designs. Ann. Math. Statist. 41, 1655–1664.MathSciNetzbMATHCrossRefGoogle Scholar
  16. Wynn, H. P. (1972). Results in the theory and construction of D-optimum experimental designs. J. Roy. Statist. Soc. B 34, 133–147.MathSciNetzbMATHGoogle Scholar
  17. Zowe, J. and Kurcyusz, S. (1979). Regularity and stability for the mathematical programming problem in Banach spaces. Appl. Math. Optim. 5, 49–62.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • Ilya Molchanov
    • 1
  • Sergei Zuyev
    • 2
  1. 1.Department of StatisticsUniversity of GlasgowUK
  2. 2.Department of Statistics and Modelling ScienceUniversity of StrathclydeUK

Personalised recommendations