The Modified Simplex Method

To compare this approach with the original simplex method, we will consider again the first initial simplex that we defined initially (see Table 2.29). 

We see that between R and W, the lead recovery (response) has improved to 71-50% =21%, so R should be the new B of simplex number 2. This suggests us that, perhaps, in the (W - R) direction we might obtain better responses. To assess this, a new point E (expanded) can be defined, whose coordinates are calculated as 

As Table 2.30 demonstrates, the experimental response obtained after the experiment in those conditions (Y = 78) indicates that indeed we have improved more. 

Simplex number Vertex number 7 F Order Times retained Rejected 

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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. 

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...

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. 

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

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 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. 

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. 

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... 

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. 

In the simplex procedures described above the step size was fixed. When the step size was taken too small it takes a large number of experiments to reach the optimum, and when it is taken too large the supposed optimum can be unacceptably far from the real one. To avoid this a so-called modified simplex method can be applied, in which the step size is variable throughout the procedure. The principles of the simplex method are maintained but rules for expansion or contraction of the simplexes are added. For a detailed description of these guidelines we refer to [27,831. 

To permit a more rapid convergence towards an optimum, several modified simplex methods have been suggested.[2] By these modified methods, the step-length of the next move is adjusted depending on the degree of improvement. If a new vertex should give a considerable improvement, it is rather natural to try to move further in that direction. One such modified simplex method is given in detail below, after a presentation of the basic simplex method. 

A modified simplex method is described below. The procedure is close to the modifications suggested by Nelder and Mead.[2a] Other modifications are described in the works given in the reference list. 

Optimization Using the Super-Modified Simplex Method... 

Numerous modifications have been made to the original simplex method. One of the more important modifications was made by Nelder and Mead l who modified the method to allow expansions in directions which are favorable and contractions in directions which are unfavorable. This modification increased the rate at which the optimum is found. Other important modifications were made by Brissey l who describes a high speed algorithm, and Keefert" who describes a high speed algorithm and methods dealing with bounds on the independent variables. 

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. 

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. 

Our simulation studies were extended to assess the precision and foundation of the calculated rate parameters by making use of a Modified Simplex Method (MSM). 

FIGURE 25. Modified simplex method. Instead of the worst vertex w one... 

A further variation of the simplex method - the super-modified simplex method - was developed by Denton et. al. C4323 and used in chemical pattern recognition by Kaberline and Wilkins C1373. For each... 

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. 

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. 

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