Formulation of the simplex matrix referred to n-D space, is presented in terms of the scalar
product of vectors, known from elementary algebra. The principles of a simplex
optimization procedure are presented on a simple example, with use of a target
function taken as a criterion of optimization, where accuracy and
precision are treated equally in searching optimal conditions of a gravimetric
analysis.
References
[1]
Anderson, V.L. and McLean, R.A. (1974) Design of Experiments: A Realistic Approach. Marcel Dekker, Inc., New York, 363.
[2]
Nelder, J.A. and Mead, R. (1965) A Simplex Method for Function Minimization. Computer Journal, 7, 308-313. http://dx.doi.org/10.1093/comjnl/7.4.308
Spendley, W., Hext, G.R. and Himsworth, F.R. (1962) Sequential Application of Simplex Designs in Optimisation and Evolutionary Operation. Technometrics, 4, 441-461. http://dx.doi.org/10.1080/00401706.1962.10490033
[5]
Fletcher, R. (1965) Function Minimization without Evaluating Derivatives—A Review. Computer Journal, 8, 33-41. http://dx.doi.org/10.1093/comjnl/8.1.33 http://folk.uib.no/ssu029/Pdf_file/Fletcher65.pdf
[6]
Olsson, D.M. and Nelson, L.S. (1975) The Nelder-Mead Simplex Procedure for Function Minimization. Technometrics, 17, 45-51. http://dx.doi.org/10.1080/00401706.1975.10489269
[7]
Fletcher, R. and Powell, M.J.D. (1963) A Rapidly Convergent Descent Method for Minimization. Computer Journal, 6, 163-168. http://dx.doi.org/10.1093/comjnl/6.2.163
[8]
Fletcher, R. and Reeves, C.M. (1964) Function Minimization by Conjugate Gradients. Computer Journal, 7, 149-154. http://dx.doi.org/10.1093/comjnl/7.2.149
[9]
Lagarias, J.C., Reeds, J.A., Wright, M.H. and Wright, P.E. (1998) Convergence Properties of the Nelder-Mead Simplex Algorithm in Low Dimensions. SIAM Journal of Optimization, 9, 112-147. http://dx.doi.org/10.1137/S1052623496303470 http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.120.6062&rep=rep1&type=pdf
[10]
Box, G.E.P. and Hunter, J.S. (1957) Multi-Factor Experimental Designs for Exploring Response Surfaces. Annals of Mathematical Statistics, 28, 195-241. http://dx.doi.org/10.1214/aoms/1177707047
[11]
Box, G.E.P. and Draper, N.R. (1969) Evolutionary Operation. John Wiley & Sons, Inc., New York.
[12]
Massart, D.L., Vanderginste, B.G.M., Deming, S.N., Michotte, Y. and Kaufman, L. (1988) Chemometrics: A Textbook. Elsevier, Amsterdam.
[13]
Massart, D.L., Vanderginste, B.G.M., Buydens, L.M.C., De Jong, S., Lewi, P.J. and Smeyers-Verbeke, J. (1997) Handbook of Chemometrics and Qualimetrics. In: Data Handling in Science and Technology, Vol. 22, Elsevier, Amsterdam.
[14]
Box, G.E.P. (1957) Evolutionary Operation: A Method for Increasing Industrial Productivity. Journal of the Royal Statistical Society. Series C (Applied Statistics), 6, 81-101. http://en.wikipedia.org/wiki/EVOP
[15]
Hahn, G.J. (1976) Process Improvement Using Evolutionary Operation. 204-206. http://rube.asq.org/statistics/2011/11/quality-tools/process-improvement-through-simplex-evop.pdf
[16]
James, F. (2004) MINUIT Tutorial, Function Minimization, Geneva. Reprinted from the Proceedings of the 1972 CERN Computing and Data Processing School, Pertisau, 10-24 September 1972 (CERN 72-21). http://seal.web.cern.ch/seal/documents/minuit/mntutorial.pdf
Michalowski, T., Rokosz, A. and Wójcik, E. (1980) Optimization of the Conventional Method for Determination of Zinc as 8-Oxyquinolate in Alkaline Tartrate Medium. Chemia Analityczna, 25, 563-566.
[20]
Palasota, J.A., Leonidou, I., Palasota, J.M., Chang, H.-L. and Deming, S.N. (1992) Sequential Simplex Optimization in a Constrained Simplex Mixture Space in Liquid Chromatography. Analytica Chimica Acta, 270, 101-106. http://dx.doi.org/10.1016/0003-2670(92)80096-P
[21]
Deming, S.N. and Morgan, S.L. (1973) Simplex Optimization of Variables in Analytical Chemistry. Analytical Chemistry, 45, 278A-283A.
[22]
Deming, S.N. and Morgan, S.L. (1983) Teaching the Fundamentals of Experimental Design. Analytica Chimica Acta, 150, 183-198. http://dx.doi.org/10.1016/S0003-2670(00)85470-7
[23]
Umeda, T. and Ichikawa, A. (1971) A Modified Complex Method for Optimization. Industrial & Engineering Chemistry Process Design and Development, 10, 229-236.