Univariate methods optimization

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. 

Fig. 5.6. The sequential univariate method Starting at the point labelled 1 tzvo steps are made along one of the coordinates to give points 2 and 3 A parabola is fitted to these three points and the minimum located (point 4) The same procedure is then repeated along the next coordinate (points 5, 6 and 7). (Figure adapted from Schlegel H B 1987. Optimization of Equilibrium Geometries and Transition Structures In Lawley K P (Editor) Ab Initio Methods in Quantum Chenustry -1 Neio York, John Wiley, pp. 249-286 )...

The optimization of the variables is a critical step in the design of new analytical methods. Optimization involves the selection of the chemical and instrumental factors which may affect the analytical signal, and the choice of the values of the variables to obtain the best response from the chemical system. For this purpose, two different strategies can be used. In the traditional univariate optimization, all values of the different factors except one are constant, and this one is the object of the examination. The alternative to this strategy is the use of chemometric techniques based mainly on the use of experimental designs (Tarley, et al. 2009). 

More elaborate techniques have been published in the literature to obtain optimal or near optimal stepping parameter values. Essentially one performs a univariate search to determine the minimum value of the objective function along the chosen direction (Ak ) by the Gauss-Newton method. 

The advantage of the GA variable selection approach over the univariate approach discussed earlier is that it is a true search for an optimal multivariate regression solution. One disadvantage of the GA method is that one must enter several parameters before it... 

The complexity of the response surface is what makes the optimization of chromatographic selectivity stand out as a particular optimization problem rather than as an example to which known optimization strategies from other fields can be readily applied. This is illustrated by the application of univariate optimization. In univariate optimization (or univariate search) methods the parameters of interest are optimized sequentially. An optimum is located by varying a given parameter and keeping all other parameters constant. In this way, an optimum value is obtained for that particular parameter. From this moment on the optimum value is assigned to the parameter and another parameter is varied in order to establish its optimum value . 

Univariate optimization is a common way of optimizing simple processes, which are affected by a series of mutually independent parameters. For two parameters such a simple problem is illustrated in figure 5.3a. In this figure a contour plot corresponding to the three-dimensional response surface is shown. The independence of the parameters leads to circular contour lines. If the value of x is first optimized at some constant value of y (line 1) and if y is subsequently optimized at the optimum value observed for x, the true optimum is found in a straightforward way, regardless of the initial choice for the constant value of y. For this kind of optimization problem univariate optimization clearly is an attractive method. 

Univariate optimization (table 5.7a) is not a good method for the optimization of chromatographic selectivity. This will be clear from the table, since despite a fairly high number of experiments only a local optimum will be located on the kinds of response surfaces typically encountered in chromatography (see figure 5.3). Moreover, the local optimum may be of little value, because no overall impression of the response surface is obtained during the process. Once this has been established, the other (favourable) characteristics of this method are no longer relevant. 

It is perhaps useful to think of the pattern search method as an attempt to combine the certainty of the multivariate grid method with the ease of the univariate search method, in the sense that it seeks to avoid the enormous numbers of function evaluations inherent in the grid method, without getting involved in the (possibly fruitless and misleading) process of optimizing the variables separately. 

In an attempt to use milder acidic conditions the prereduction of Se(IV) to Se(VI) was carried out with a mixture of HC1 and HBr (10 percent v/v each) instead of HC1 alone (>50 percent v/v) [52]. Experimental parameters were selected by a univariate optimization method. The main advantage of the MW heating was that it allowed for a strict control over the heating power as well as over the time the heating was applied. Seven samples of orange juice were analyzed. Selenium was present in five of them as a mixture of Se(IV) and Se(VI), Se(IV) being the predominant species with concentrations ranging from 5.20 0.08 to 9.50 0.09 pg 1 1. 

Table 3.5 shows the three electrophoretic factors and levels selected in which experimental optimization, in terms of overall response (% conversion), could be performed. A design matrix was then generated for the Box-Behnken study (Table 3.6). It was found that voltage and mixing time, when combined, had a significant effect on % conversion. Here, the extent of contact between substrate and enzyme is dictated by the difference in electrophoretic mobilities, which is in turn dictated by mixing time and voltage. Such an interaction would not have been possible by use of classical univariate optimization methods.

MEKC separations. Traditionally, separation conditions have been optimized by simple univariate techniques, in which each factor is optimized individually and sequentially until the desired result is obtained (30-33). This method is generally time-consuming and labor-intensive. Relatively recently, chemomet-ric applications that have been used for optimizing chromatographic separations and standard CE separations have become more frequently used in MEKC. 

Whenever a new CE method is being developed, optimization strategies are usually applied to improve analysis speed, sensitivity, and resolution, using these three parameters or a combination of them as the monitored output (also called response or performance criteria). Very frequently, a step-by-step approach in which each factor is varied sequentially is followed. In this case, all parameters are kept constant, while the parameter of interest is varied and the response is measured. Depending on the problem (especially when the number of factors to optimize is very low) and on the performance criteria, univariate optimization can be useful, that is, the analysis of a single compound with only one component of the BGE. However, in most cases, a step-by-step optimization is laborious and tedious because it typically requires a high number of experiments. Furthermore, and more important, it does not consider possible interactions between factors. 

As noted in the introduction, energy-only methods are generally much less efficient than gradient-based techniques. The simplex method [9] (not identical with the similarly named method used in linear programming) was used quite widely before the introduction of analytical energy gradients. The intuitively most obvious method is a sequential optimization of the variables (sequential univariate search). As the optimization of one variable affects the minimum of the others, the whole cycle has to be repeated after all variables have been optimized. A one-dimensional minimization is usually carried out by finding the... 

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. 

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