Simplex algorithm modified

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... 

Fig. 3-2. Factorial experimental designs and sequential designs ys - next step in modified simplex algorithm y9 - next step in weighted simplex algorithm...

A more rewarding solution to this problem is the use of modified Simplex procedures, such as first described by Nelder and Mead [507], Such modified algorithms allow other operations besides reflecting the triangle, such as contraction or expansion. The manner in which such a modified Simplex algorithm proceeds is illustrated in figure 5.8 for a... 

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.

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. 

This last result has been verified by calculating numerical solutions of these models, fitting them to the equation of Wade et al. [46] for the Thomas model, and determining the best values of the three parameters Aq, Lf, and N a- A modified Simplex algorithm [1,59-61] was designed for the successive estimation of these three parameters. Typical results are shown in Figures 14.14 and 14.15, and in Table 14.1. 

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 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. 

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. 

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... 

The desire to restrict the number of variables when using the simplex algorithm introduces an interesting problem Of the variables listed in Table HI, how many should be included in the simplex procedure, and which ones Clearly, from our earlier discussion the variables in the second column of Table III can be excluded, but that still leaves 6 "ideal" parameters pressure (or density), temperature, modifier composition, and their respective gradients. How should one select from among these six parameters, since any of them may be important for a given sample ... 

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. 

Figure 20 Application of the dynamic simplex to the compensation of system-drift. An artificial example is considered here in which the temperature is ramped linearly with time and the simplex aims to compensate for the changes in the reaction temperature by modifying the flow rate accordingly. The plot compares the change in the peak wavelength when the flow rate is held fixed at its initial value of 12 llmin 1 and when it is adapted dynamically by the simplex algorithm. In the former case, the peak wavelength increases steadily with time due to the increasing temperature which increases the growth rate of the particles. In the latter case, the peak wavelength remains fairly close to its initial value of 508 nm.

In reality, an even-paced series of steps from starting point to optimisation, is neither to be expected nor desired, so the extent to which the simplex is modified is governed by a set of rules, which are shown in algorithmic form in Fig. 9, and whose operation is illustrated in Fig. 10. Even these are not sufficient, and the basic procedure has been modified by Denton 2l) to give a super modified simplex, in which it (a) is easier to adjust the size of the simplex, to take big steps to begin with... 

The different mixing rules and nomenclature used are described in table 1. The simplex algorithm modified by Nelder-Mead (10) is used to fit the model parameter to experimental solubility. 

Among chemometrical approaches, the Simplex algorithm [82] involving an evolutionary movement in the factor space to improve the response of interest was applied to analyze antibiotics in pharmaceutical formulations. The selected control variables included the concentrations of the organic modifier and the IPR in the eluent. The response variables were the peak area, resolution, asymmetry factor, and total number of chromatographic peaks. The mobile phase composition that gave optimum results was adopted [83]. A Simplex algorithm was successfully used to maximize the resolution of 14 cosmetic preservatives [84]. 

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)... 

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. 

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. 

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 application of pattern recognition to the recognition of chemical substructures on the basis of mass spectra did not reveal considerable advantages of the super-modified simplex over the modified simplex C1373. Extensive examinations of the behaviour of the simplex algorithm in pattern recognition have been made with artificial data C2423. 

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. 

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 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. 

Fig. 8.6. Displacement of the modified bidimensional simplex on a raising ridge. The points not included in the simplexes are vertexes that were rejected by the algorithm rules.

In the modified algorithm the initial simplex, BNW, can be submitted to five different operations refiection, expansion, contraction, contraction with direction change and massive contraction. In the supermodified simplex (Routh et al., 1977), the selection of options is amplified. 

With the variable-size simplex, the step width is changed by expansion and contraction of the reflected vertices. The algorithm is modified as follows (cf. Figure 4.18) ... 

Other methods of multidimensional search without using derivatives include Rosenbrock s method (1960) and the simplex method of Spendley et al. (1962), which was later modified by Nelder and Meade (1974). Although it has the same name, this simplex method is not the same algorithm as that used for linear progranuning it is a polytope algorithm that requires only functional evaluations and requires no smoothness assumptions. 

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