Optimization by simplex

The alternative is to employ a multivariate optimization procedure such as Simplex. Simplex is an algorithm that seeks the vector of parameters that corresponds to the separation optimum within an n-dimensional experimental space. For example, a two-parameter CE separation optimized by Simplex would begin with three observations of the separation response at three different electrolyte conditions. These conditions are chosen by the analyst, often his or her best guess. From the evaluation of the response of each observation, the algorithm chooses the next experimental condition for investigation (4). As with the univariate method, the experiments continue until an optimal separation condition is determined. The disadvantage of such an approach is that it is unknown how many experiments are required to achieve an optimum, or if the optimum is local or global as the entire response surface is not known. 

After optimization by simplex, the plot from Figure 12.6 improved and yielded a linear equation (log[p] = -3.14 - 0.20 log[bp], = 0.998) suitable for the analysis of the 201-2036 bp size range. This equation was further used to determine the size of unknown DNA fragments (27). Thus, the simplex method was shown to be an efficient way to optimize an electrophoretic separation of DNA, since several variables could be simultaneously optimized. 

We stated above that the experimental domain is often thought of as being a centre of interest (point) around which extends the region of interest (experimental domain). We have also pointed out that the domain is relatively limited in extent, certainly with respect to the domains of other types of studies, factor studies, and especially screening and direct optimization by simplex (see chapter 6). We have also to suppose that the response is continuous, without steep variations, peaks or breaks. It is imagined as a curved surface, whether peak, valley or otherwise, with moderate slope or change of slope. There is an absence of sharp peaks, "wells", and "cliffs", as RSM will not work under such conditions. 

P. F. A. van der Wiel, B. G. M. Vandeginste, and G. Kateman, Chemo-metrical Optimization by Simplex (COPS). Elsevier Scientific Software, The Netherlands, (1986). 

The model [C] can be evaluated by comparing with [D] using a -criterion. The parameters of the model can be optimized by some optimization procedure (Simplex). 

Both the development and the optimization of simplex methods are still continuing. Several functions have been designed to test the performance of the simplex algorithms, one example is the famous ROSENBROCK valley. Other test functions have been reported by ABERG and GUSTAVSSON [1982]. Most analytical applications of simplex optimization are found in atomic spectroscopy [SNEDDON, 1990] and chromatography [BERRIDGE, 1990],... 

Water-soluble tertiary amines enhance signals and decrease polyatomic chloride interferences in the ICP-MS determination of As and Se in food samples [22]. Arsenic and Se ICP-MS determination parameters have been optimized by a simplex procedure with amines in Ar plasma. A simple, direct, quantitative procedure for As and Se determination in food samples was set up, that provides good accuracy and Rt-for-purpose LoDs. 

Fig. 1.22. Opiimisation of two operation parameters by simplex melhtxl. The dolled contour lines correspond to equal values of the optimisation criterion. ABC = original simplex. ACD = new simplex obtained by rejection of the point B with the worst value of ihe opiimisation criterion and reflecting the original simplex in the opposite field O = the combination of the iwo optimised operation parameters for the highest (optimal) value of the optimisation criterion.

Examples of optimizations in HPLC using the simplex approach can be found in [28,84]. In [28] the mobile phase composition for the chiral separation of (6/ )- and (65)-leucovorin on a BSA (bovine serum albumin) stationary phase is optimized by means of a variable-size simplex. Three factors were examined, the pH of the mobile phase buffer, the ionic strength of the buffer and the percentage of 1-propanol in the mobile phase. Table 6.19 shows the experimental origin, the initial step size and the acceptable limits for the factors. The criterion optimized is the valley-to-peak ratio (Section 6.2). The points selected and the results are pre.sented in Table 6.20 and... 

The general behaviour of simulated annealing in correcting for shift errors has been evaluated by comparing the p ormances of different optimization procedures simplex, steepest descent, and emulated aimealing in the resolution of two- and three-components overlapped synthetic band syst ns. 

Comparison of the Kalman filter resolution of the ESCA spectra of some binary and ternary lead compound mixtures after search for the optimal alignment by simplex, steepest descent and simulated annealing. Nr is the number of iterations of the optimisation until convergence n.c. no convergence.(From Fresenius J Anal. Chem (1993) 345 490, with permission). 

The computational approach described here, based on the combination of the Kalman filter algorithm and iterative optimization by the simulated annealing method, was able to find the optimal alignment of the pure component peaks with respect to the shifted components in the overlapped spectra, and hence, to correctly estimate the contributions of each component in the mixture. The simulated annealing demonstrated superior ability over the other optimization methods, simplex and steepest descent, in yielding more reliable convergences at the expense of not much more computer time, at least for resolving ternary shifted overlapped spectra. 

WEXPRED calculates weighted (1/square root y) sum of squared deviations for fitting % pharmacokinetic data (biexp) to a four parameter, biexponential decay model. This % allows demonstration of non-linear regression by simplex, simulated annealing, or other % optimization techniques. 

Parameter values were determined by simplex optimization method. 

Fig. 11.4 Optimization by a simplex search (a) Starting simplex and the first move from the poor conditions, (b) Progression of the simplex towards the optimum conditions.

In general, a simplex for/factors is a geometric figure in the /-dimensional factor space, defined by / + 1 points or vertices, that is, one more than the number of factors. During optimization, the simplex sequentially moves through the experimental domain in the direction of the optimum. The next simplex to be performed is based on the results of the previous, and is defined according to specific rules. 

The models for ISE, CSV, and FQ were fit to the titration curves by simplex optimization... 

In an experimental optimization by the simplex method, the yield of a process was determined as a function of two variables xi and X2- The following table shows the results of the initial simplex with three experiments ... 

Christian, N. P, Arnold, R. J., and Reily, J. P, Improved Calibration of Time-of-Flight Mass Spectra by Simplex Optimization of Electrostatic Ion Calculations, Anal. Chem., 11, 3317, 2000. 

Faller, R., Schmitz, H., Biermann, O., Miiller-Plathe, F. Automatic parameterization of force fields for liquids by simplex optimization. J. Comp. Chem. 20, 1009-1017 (1999)... 

The simplex method is a multivariate fitting procedure which, in our case, proceeds via a comparison between an experimental and a calculated curve. The fitting between experimental and computed data is optimized by minimizing the objective response function Rg ... 

Darvas [14] had previously taken an important conceptual leap and applied a mathematical technique for optimization— the simplex optimization method—to the lead optimization of natriuretic sulphonamides. By describing molecules by their Hansch parameters (crand it), a common molecule space could be created to describe the series and this space could be walked by the optimization algorithm (Figure 8.4) using the following steps ... 

The mechanical repetition of this procedure leads to the preparation of substances having outstanding activities as compared to their neighbors. This maximum is surrounded stepwise by simplexes, since the point with the highest effect in the triangle is always involved. If we wish to continue the optimization procedure, it is most preferable to include new substances in the optimization map around this maximum and to resume the optimization on a smaller scale. 

Christian, N.P., Arnold, R.J., and Reilly, J.P. (2000) Improved calibration of time-of-flight mass spectra by simplex optimization of electrostatic ion calculations. Anal. Chem., 71,... 

Dufour I, Bittoun J, Idy-Peretti I, Jolivet O, Darrasse L, Di Paola R (1993) Implementation and optimization by the simplex method of a 3D double echo sequence in steady-state free precession. Magn Reson Imaging 11 87-93... 

Optimization of Electrolyte Properties by Simplex Exemplified for Conductivity of Lithium Battery Electrolytes... 

Optimization of Electrolyte Properties by Simplex Exemplified for Conductivity of Lithium Battery Electrolytes, Fig. 1 The basic simplex method is based on simple reflections (R) of the simplex (shown in the lower right). The optimization begins at starting simplex (Xi,Yi,ZilX2,Y2,Z2lX3,Y3,Z3), soUd line. A new set of... 

To overcome this problem the mixed solvent approach is used. By blending at least one solvent with high permittivity (but high viscosity) and at least one solvent of low viscosity (but low permittivity), the conductivity of the electrolyte can be improved [27-29]. But it is a time-consuming task to find the best composition, if more than two solvents are used. Then, the composition of the blend and the concentration of the salt have to be optimized by methods like simplex to obtain an optimal conductivity. 

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