Construction of Constrained Optimal Designs
We consider the problem of finding an ‘approximate’ design maximising a criterion subject to an equality constraint. Initially the Lagrangian is formulated but the Lagrange parameter is removed through a substitution, using linear equation theory, in an approach which transforms the constrained optimisation problem to a problem of maximising two functions of the design weights simultaneously. They have a common maximum of zero which is simultaneously attained at the constrained optimal design weights. This means that established algorithms for finding optimising distributions can be considered. The approach can easily be extended to the case of several constraints, raising the ‘prospect’ of solving an expanded class of problem.
Keywordsconstrained optimal design multiplicative algorithms optimizing distributions directional derivatives Lagrangian theory equivalence theorem
Unable to display preview. Download preview PDF.
- Alahmadi, A. M. (1993). Algorithms for the Construction of Constrained and Unconstrained Optimal Designs. Ph.D. Thesis, University of Glasgow.Google Scholar
- Mandai, S. and Torsney, B. (2000). Algorithms for the construction of optimizing distributions. Communications in Statistics: Theory and Methods (to appear).Google Scholar
- Mandai, S. and Torsney, B. (2000). Construction of optimal designs using a clustering approach (in preparation).Google Scholar
- Silvey, S. D., Titterington, D. M. and Torsney, B. (1978). An algorithm for optimal designs on a finite design space. Communications in Statistics A 14, 1379–1389.Google Scholar
- Torsney, B. (1977). Contribution to discussion of ‘Maximum Likelihood estimation via the EM algorithm’ by Dempster et al. J. Roy. Statist. Soc. B 39, 26–27.Google Scholar
- Torsney, B. (1988) Computing optimizing distributions with applications in design, estimation and image processing. Optimal Design and Analysis of Experiments Eds Y. Dodge, V.V. Fedorov and H.P. Wynn, pp. 361–370. Amsterdam: Elsevier.Google Scholar
- Torsney, B. and Alahmadi, A. M, (1992). Further development of algorithms for constructing optimizing distributions. Model Oriented Data Analysis. Proc. 2nd IIASA Workshop in St. Kyrik, Bulgaria, Eds V.V. Fedorov, W.G. Müller and I. Vuchkov, pp. 121–129. Heidelberg: Physica-Verlag.Google Scholar
- Wu, C. F. J. (1978). Some iterative procedures for generating nonsingular optimal designs. Communications in Statistics A 14, 1399–1412.Google Scholar