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Heuristic optimization of experimental designs. (English) Zbl 1037.90058

Summary: We propose to integrate different algorithms for constructing \(D\)-optimum designs for linear models. Our emphasis is on efficiency gain and on applicability to larger models than those currently considered in the literature. We implement a one-exchange algorithm and use a generalized simulated annealing. This method does not require to construct or to enumerate each point of the candidate set, whose size grows exponentially with the number of variables. In order to handle more complex problems, we develop a procedure generating guided starting designs. A comparison of our results with those found in the literature shows that the simultaneous integration of these algorithms turns out to be very effective. As compared to results from the literature, our algorithmic process allows an increase in efficiency while, for larger models (up to 20 parameters), we attain a 90% \(D\)-efficiency level.

MSC:

90C59 Approximation methods and heuristics in mathematical programming
90C90 Applications of mathematical programming
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[1] Aarts, E.; Korst, J., Simulated Annealing and Boltzmann Machines (1989), John Wiley & Sons · Zbl 0674.90059
[2] Aragon, C.; Johnson, D.; McGeoch, L.; Schevon, C., Optimization by simulated annealing: an experimental evaluation-Part 1 (Graph partitioning), Operations Research, 37, 6, 865-892 (1992) · Zbl 0698.90065
[3] Atkinson, A.; Donev, A., Optimum Experimental Designs (1992), Clarendon Press: Clarendon Press Oxford, 329 pp · Zbl 0829.62070
[4] Bohachevsky, I.; Johnson, M.; Stein, M., Generalized simulated annealing for function optimization, Technometrics, 28, 3, 209-217 (1986) · Zbl 0609.65045
[5] Box, M.; Draper, N., Factorial designs-the \(X^′X\) criterion and some related matters, Technometrics, 13, 4, 731-742 (1971) · Zbl 0228.62048
[6] Dykstra, O., The augmentation of experimental data to maximize \(X^′X\), Technometrics, 13, 3, 682-688 (1971)
[7] Galil, Z.; Kiefer, J., Time- and space-saving computer methods, related to Mitchell’s DETMAX, for finding \(D\)-optimum designs, Technometrics, 22, 3, 301-313 (1980) · Zbl 0459.62060
[8] Haines, L., The application of the annealing algorithm to the construction of exact optimal designs for linear regression models, Technometrics, 29, 4, 439-447 (1987) · Zbl 0632.62071
[9] Johnson, M.; Nachtsheim, C., Some guidelines for constructing exact \(D\)-optimal designs on convex designs spaces, Technometrics, 25, 3, 271-277 (1983) · Zbl 0526.62070
[10] Lejeune, M. A., A coordinate-columnwise exchange algorithm for the construction of supersaturated, saturated and non-saturated experimental designs, American Journal of Mathematical and Management Sciences (2002)
[11] Meyer, R.; Nachtsheim, C., Constructing exact \(D\)-optimal experimental designs by simulated annealing, American Journal of Mathematical and Management Sciences, 8, 329-359 (1988) · Zbl 0676.65148
[12] Meyer, R.; Nachtsheim, C., The coordinate-exchange algorithm for constructing exact optimal experimental designs, Technometrics, 37, 1, 60-68 (1995) · Zbl 0825.62652
[13] Mitchell, T. J., An algorithm for the construction of “\(D\)-optimal” experimental designs, Technometrics, 16, 2, 203-210 (1974) · Zbl 0297.62055
[14] Mitchell, T. J., Computer construction of \(D\)-optimal first order designs, Technometrics, 16, 2, 211-220 (1974) · Zbl 0299.62040
[15] Pirlot, M., General local search heuristic in combinatorial optimization: a tutorial, Belgian Journal of Operations Research, Statistics and Computer Science, 32, 1-2 (1993)
[16] St. John, R. C.; Draper, N. R., \(D\)-optimality for regression designs: a review, Technometrics, 17, 1, 15-23 (1975) · Zbl 0295.62081
[18] Wynn, H., The sequential generation of \(D\)-optimum experimental designs, The Annals of Mathematical Statistics, 41, 1655-1664 (1970) · Zbl 0224.62038
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