The modified simplex

In the modified algorithm (Nelder and Mead, 1965), the simplex can change its size and form, and consequently adapt itself more efficiently to the response surface. This flexibility permits a more precise determination of the optimum point, because the simplex can shrink in its proximity. Besides this desirable characteristic, the modified method, compared to the basic simplex, can reduce the number of runs necessary to find the optimum, because it can stretch itself when it is far from the desired point, usually on a planar portion of the response surface. For this reason it approaches the experimental region of interest more rapidly. 

First case - R B. The new response is better than all those in the preceding simplex. This suggests assuming that the simplex is on the right path, and that we should continue our investigations in that direction. We then perform a new determination at point S, located on the WPR line in such a way that the PS distance is double the PR distance. Depending on the value of the response at point S we have two possibilities  

Ib-S R. The result is worse with the expansion. We should maintain the first unexpanded simplex, BNR. 

Second case - N R B. The observed response after reflection is worse than the best response of the initial simplex, but is better than the next-to-worst response. In this case, it is not worth performing another experiment in an attempt to expand. Neither expansion nor contraction is indicated and the BNR simplex is maintained. 

Third case - R N. The new response is less than the next-to-worst response of the starting simplex. We conclude that the direction of movement is not satisfactory, and that we need to correct the trajectory. We again have two possibilities  

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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...

It will not always be possible to make expansion movements because as we move closer to the optimum we must reduce the size of the simplex in order to locate the optimum accurately. This basic idea of adapting the size of the simplex to each movement is the one that sustains the modified simplex method proposed by Nelder and Mead [17]. Figure 2.15 displays the four possibilities to modify the size of the simplex and Table 2.32 gives their respective expressions for each factor. 

Table 2.33 Evolution of the modified simplex for the worked example. ... 

Table 2.33 summarises the evolution of the modified simplex until the optimum is reached in the worked example. 

Figure 2.17 summarises the evolution of the vertex B for each of the simplexes. Observe that the optimum (Y = 93%) was already obtained in simplex number 5 but we needed to continue until simplex number 11 to confirm it. In this example, 20 experiments were carried out, as in the previous original simplex method nevertheless, in general the modified simplex approaches the zone of the optimum faster (i.e. using fewer experiments). 

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. 

Fig. 9. Principle of the Modified Simplex method — 1. Schematic representation of the rules for expansion and contraction of the simplex...

A weakness with the standard mediod for simplex optimisation is a dependence on the initial step size, which is defined by the initial conditions. For example, in Figure 2.37 we set a very small step size for both variables this may be fine if we are sure we are near the optimum, but otherwise a bigger triangle would reach the optimum quicker, the problem being that the bigger step size may miss the optimum altogether. Another method is called the modified simplex algorithm and allows the step size to be altered, reduced as the optimum is reached, or increased when far from the optimum. 

For the modified simplex, step 4 of the fixed sized simplex (Section 2.6.1) is changed as follows. A new response at point xlesl is determined, where the conditions are obtained as for fixed sized simplex. The four cases below are illustrated in Figure 2.39. 

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 ... 

Under these circumstances, the modified simplex methods are very convenient. 

In the basic simplex method, the simplex thus can only be reflected to obtain the next experiment, and the simplex size remains the same throughout the procedure. In the modified simplex method, suggested by Nelder and Mead (100), the simplex can be reflected, expanded, or contracted to define the next experiment. Thus, in case the simplex is expanded or contracted, the simplex size changes. More information about the simplex procedures can be found in References 7,9,10, and 98-102. 

Let us now consider the variable-size or modified simplex procedure, proposed by Nelder and Mead (100). Whereas in the basic procedure, the size is fixed and determined by the initially chosen simplex, the size in the modified simplex procedure is variable. Besides the rules of the basic procedure, the modified procedure additionally allows expansion or contraction of simplexes. In favorable search directions, the simplex size is expanded to accelerate finding the optimum, while in other circumstances, the simplex size is contracted, for example, when approaching the optimum (Figure 2.14). 

Furthermore, rule 3 of the basic procedure is apphed if a certain point is retained in/ +1 successive simplexes. A difficulty is to define a criterion to stop the (modified) simplex procedure. In Reference 8, different possibilities are discussed. 

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. 

FIGURE 2.15. Example of the modified simplex proeedure for the determination of fluticasone propionate with flow injection analysis, based on Reference 104. 1, 2, 3 = initial simplex, and 4, 5,. .., 14 = sequentially selected vertices. 

A. P. Wade, Optimisation of Flow Injection Analysis and Polarography by the Modified Simplex Method. Anal. Proc., 20 (1983) 523. 

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. 

Fig. 8.4. Possible movements for the modified simplex. The BNR simplex is obtained from the initial simplex, BNW, by means of a simple reflection. The other three correspond to the following movements expansion (BNS), contraction (BNU) and contraction with direction change (BNT).

In the modified simplex, we have expansion and contraction displacements as we have just seen. For these movements the (P - W) vector (or, if it is the case, (P - N)) is multiplied by a factor that increases or decreases its contribution. In the expansion we have... 

In Figs. 8.5 and 8.6, we use the same response surface as the one in Figs. 8.1 and 8.2 to illustrate the application of the modified simplex. The corresponding numerical values of concentration and time for the various vertexes are given in Table 8.1. Note that the initial ABC simplex is the same as in the example of the basic simplex, permitting us to compare the efficiencies of the two algorithms. 

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... 

Equations (8.2)-(8.5), that govern the movement of the modified simplex, can be considered special cases of only one equation. 

Fig. 8.8. The supermodified simplex, (a) The responses observed at W and R would indicate an expansion of the modified simplex, (b) With this response surface expansion would not be the best movement. A P value of about 1.3 would produce a larger response, (c) For the concave surface the R vertex is maintained.

PoRt values less than —1 or larger than 3 represent larger extrapolations of the simplex than we would obtain with the modified algorithm, and this is considered excessive. In this case, the expansion (or contraction) determined by the modified simplex is adopted. 

In the modified simplex method C425 l further steps are necessary to compute an optimum new vertex and to find the maximum response with maximum speed and efficiency ... 

The modified simplex algorithm produces nearly optimum weight vectors even when linearly inseparable data sets are used for training but has the disadvantage of sometimes prohibitively large computer times C165I. 

The ability of the super-modified simplex algorithm to locate the position of the new vertex more precisely than is possible by the modified simplex method means that the super-modified simplex requires fewer iterations to converge and is therefore somewhat faster. 

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