Modified simplex optimization

A variation of the latter optimization procedure known as modified simplex optimization has evolved to eliminate the limitations imposed by the simplex method. The... 

Figure 8. Resolution of racemic Doxazosin on a Chiral-APG column (see also Table 2) using an optimization strategy of modified simplex procedure. Reprinted with permission from ref 86.

Fig. 12. The progress of the modified Simplex method for optimization. From P. J. Golden and S. N. Deming, Laboratory Microcomputer, i, 44 (1984). Reproduced by permission of Science Technology Letters, England...

The first results of optimization in chromatography were published in 1975 Since then a growing number of optimization experiments in HPLC using the Simplex procedure has been reported (table 9). The examples are mainly reversed-phase separations, in which the composition of the ternary or binary mobile phase composition is optimized. The factors optimized are usually a selection from flow rate, column temperature and length, the eluents constitution (e.g. organic modifier content, buffer concentration and pH), the gradient shape. Seven years after the first applications of Simplex optimization had appeared, the first fully automated optimization of HPLC separations was published by Berridge in 1982. This development coincid-... 

One way to include the less desirable variables of Table HI in the simplex optimization process would be to perform a simplex optimization for each of a limited number of user-selected combinations of the less suitable variables. The optimum results obtained for these selected combinations could then be compared, with the best overall result indicating which combination of fluid, modifier, stationary phase, etc. is best for that particular separation. A drawback of this approach, particularly if the individual optimization procedure is not automated, is its time-consuming, labor intensive nature. 

A systematic method development scheme is clearly desirable for SFC, and as shown in the present work, both the modified simplex algorithm and the window diagram method are promising approaches to the optimization of SFC separations. By using a short column and first optimizing the selectivity and retention, rapid... 

By far the most popular technique is based on simplex methods. Since its development around 1940 by DANTZIG [1951] the simplex method has been widely used and continually modified. BOX and WILSON [1951] introduced the method in experimental optimization. Currently the modified simplex method by NELDER and MEAD [1965], based on the simplex method of SPENDLEY et al. [1962], is recognized as a standard technique. In analytical chemistry other modifications are known, e.g. the super modified simplex [ROUTH et al., 1977], the controlled weighted centroid , the orthogonal jump weighted centroid [RYAN et al., 1980], and the modified super modified simplex [VAN DERWIEL et al., 1983]. CAVE [1986] dealt with boundary conditions which may, in practice, limit optimization procedures. 

Figure 5.8 Illustration of a two-dimensional optimization using a modified Simplex algorithm. A ternary mobile phase for RPLC is being optimized. The third component is acetonitrile. Figure taken from ref. [505]. Reprinted with permission.

Felinger and Guiochon [54] carried out a systematic investigation of the optimization of the experimental conditions for maximum production rate, using the same model as for their similar study on the optimization of elution [4] (competitive Langmuir isotherm, equilibrium-dispersive model [24], Knox equation [25], and super-modified simplex algorithm [34]). Their main conclusions are the following. 

The modified simplex methods have gained considerable popularity in analytical chemistry, especially for the optimization of instrumental methods. Applications in organic synthesis are, however, remarkably few. There are several reasons for this difference ... 

Optimization Using the Super-Modified Simplex Method... 

In the following, the basic and modified simplex procedures are discussed for the optimization of two factors. The simplex is then a triangle (f + 1 = 3 vertices). For the basic procedure, it is an equilateral triangle, while for the modified procedure, it does not necessarily have to be. 

In Figure 2.15, an example is given of the modified simplex procedure for the determination of fluticasone propionate with flow injection analysis (104). The initial simplex is formed by points 1, 2, and 3. Points 4-14 represent the sequentially selected vertices. Point 6 seems to be situated close to the optimum because it is maintained in many simplexes. It is observed that again, as in the classic procedure, the simplexes circle around the optimum, but here also their size decreases as the procedure continues. To optimize three or more factors, the simplex procedures can be generalized, as described in Reference 8. 

Zachariadis, G.A. and Stratis, J.A. (1991) Optimization of cold vapour atomic absorption spectrometric determination of mercury with and without amalgamation by subsequent use of complete and fractional designs with univariate and modified simplex methods. J. Anal. At. Spectrom., 6, 239-245. 

In this study, a modified Simplex method was used to regress the binary interaction parameter, fcy, using a packaged algorithm, DBCPOL (13), The objective function minimized by the optimization routine was the percent absolute average relative deviation (%AARD)... 

There exist several simplex methods. In this chapter, we will discuss three of them, in increasing order of complexity the basic simplex, the modified simplex and the super-modified simplex. The more sophisticated methods are able to adapt themselves better to the response surface studied. However, their construction requires a larger number of experiments. In spite of this, the modified and super-modified simplexes normally are able to come closer to the maximum (or minimum if this were of interest) with a total number of experiments that is smaller than would be necessary for the basic simplex. In this chapter, we will see examples with only two or three variables, so that we can graphically visualize the simplex evolution for instructive purposes. However, the efficiency of the simplex, in comparison with univariate optimization methods, increases with the number of factors. 

Application of the modified simplex to the optimization of Mo(VI) determination as a function of the H2O2, KI and H2SO4 concentrations. The observed response, that is to be maximized, is represented by AA... 

MORGAN, E. BURTON, K. W. C. and NICKLESS, G. Optimization using the super-modified simplex method. Chemom. Intell. Lab. Systems 8, 1990, pp. 97-107. 

A high speed algorithm for simplex optimization calculations in pattern recognition, similar to the super-modified simplex, has been described by Brissey et. al. C243. 

The simplex method was originally proposed by Spendley, Next, and Himsworth C4353 in 1962 and modified by Nelder and Mead C4253 in 1965. The simplex optimization has been successfully applied to several areas of analytical chemistry such as experimental optimization C189, 4143, data reduction C4263 and instrument control C4153. 

A weakness with the standard method for simplex optimization is a dependence on the initial step-size. Another method is called the modified simplex algorithm and allows the step size to be altered, reduced as the optimum is reach, or increased far from the optimum. 

Empirical techniques, simplex optimization, may be used to optimize the performance of modified electrodes for which no theoretical model is yet available (96). Improvements in micros-tructural design of electrodes modified with solids will make it possible to obtain mathematical descriptions similar to those that have been so successful in the case of polymer-modified electrodes as described above. 

Ghorbani, A.R., Momenbeik, F., Khorasani, J.H., and Amini, M.K., 2004. Simultaneous micellar liquid chromatographic analysis of seven water-soluble vitamins optimization using super-modified simplex. Analytical and... 

The basic simplex optimization method, first described by Spendley and co-workers in 1962 [ 11 ], is a sequential search technique that is based on the principle of stepwise movement toward the set goal with simultaneous change of several variables. Nelder and Mead [12] presented their modified simplex method, introducing the concepts of contraction and expansion, resulting in a variable size simplex which is more convenient for chromatography optimization. 

The rigidity that prevented an accurate optimal point from being obtained was solved by Nelder and Mead in 1965. They proposed a modification of the algorithm that allowed the size of the simplex to be varied to adapt it to the experimental response. It expanded when the experimental result was far from the optimum - to reach it more rapidly and it contracted when it approached a maximum value, so as to detect its position more accurately. This algorithm was termed the "modified simplex method . Deming and co-workers published the method in the journal Analytical Chemistry and in 1991 published a book on this method and its applications. 

Silva and Alvares-Ribeiro (2002) carried out a throughout optimization of the vanadate-based method. Experimental design was implemented to select the most important parameters (injection volume, flow rate, and vanadate concentration), followed by application of a modified simplex algorithm with a response function that included sensitivity, deviation from linearity at low concentrations, and residence time (used as an inverse measure of sampling rate) to optimize these variables. This system setup also comprised... 

Cimpoiu et al. [72] made a comparative study of the use of the Simplex and PRISMA methods for optimization of the mobile phase used for the separation of a group of drugs (1,4-benzodiazepines). They showed that the optimum mobile phase compositions by using the two methods were very similar, and in the case of polar compounds the composition of the mobile phase could be modified more precisely with the Simplex method than with the PRISMA. 

Future work. As mentioned earlier, use of the simplex algorithm for the systematic optimization of SFC separations is still in its early stages. The success already achieved, however, merits continued research along these lines. Research opportunities include (i) extension of the simplex method to less ideal variables and/or greater than 4 variables (ii) investigation of the benefits of the simplex method to packed columns and modified mobile phases and (iii) development of the capability to predict, for a given type of sample, the best combination of variables to optimize. 

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