Global response surface

Mountain-climbing analogy to using a searching algorithm to find the optimum response for a response surface. The path on the left leads to the global optimum, and the path on the right leads to a local optimum. 

However, the optima of X and x2 found in this way do not meet the global optimum of the response surface which is situated at x = 80 and x2 = 150. Because the global optimum is rarely found by such an obsolete proceeding, multivariate techniques of optimization should be applied. 

In addition to the basic methods such as SIMPLEX, other key methods for value chain management are the response surface methodology (RSM) to find a global optimum in a multi-dimensional simulation result surface (Merkuryeva 2005) or simulated annealing applied in the chemical production to find optima e.g. for reaction temperatures (Faber et al. 2005). 

Figure 2.7 Response surface exhibiting two local maxima, one at x, = 2, the other (the global maximum) at = 7.

Figure 2.8 Response surface exhibiting three local minima, one at x, = 3, one at Xj = 5, and the third (the global minimum) at x, = 8.

Figure 3.3. Examples of local and global optima, (a) One factor (pH) with the response (yield) giving a maximum, but with a wide acceptable range, (b) Function of two variables giving a number of maxima. The global maximum and one local maximum are shown, (c) The response surface of graph b as a contour map.

Figure 9. Contour plot of the response surface of Figure 8. Of the two most readily apparent optima, (2.15 10 3 K 1 191°C, 0.087 g/mL) and (2.65 10 3 K l 104°C, 0.19 g/mL), the latter is the global optimum over the range of experimental conditions we examined.

The Simplex method (and related sequential search techniques) suffers mainly from the fact that a local optimum will be found. This will especially be the case if complex samples are considered. Simplex methods require a large number of experiments (say 25). If the global optimum needs to be found, then the procedure needs to be repeated a number of times, and the total number of experiments increases proportionally. A local optimum resulting from a Simplex optimization procedure may be entirely unacceptable, because only a poor impression of the response surface is obtained. 

Because this optimization only concerned program parameters and not selectivity parameters, the response surface will have been relatively simple. Therefore, the probability that the Simplex procedure would arrive at the global optimum rather than at a local one was greater than it was in section 5.3, where we described the use of the Simplex method for selectivity optimization. 

The obvious alternative to the sequential optimization methods is the use of an interpretive optimization method. In such a method a limited number of experiments is performed and the results are used to estimate (predict) the retention behaviour of all individual solutes as a function of the parameters considered during the optimization (retention surfaces). Knowledge of the retention surfaces is then used to calculate the response surface, which in turn is searched for the global optimum (see the description of interpretive methods in section 5.5). For programmed temperature GC the framework of such an interpretive method has been described by Grant and Hollis [614] and by Bartu [615]. 

The response surface for the optimization of the primary (program) parameters in programmed temperature GC is less convoluted than a typical response surface obtained in selectivity optimization procedures (see section 5.1). This will increase the possibility of a Simplex procedure locating the global optimum. 

As stated earlier the model should be obtained for responses such as k (or log< ). These are also the responses that should be predicted from the models and only then responses such as or the global responses of Section 6.2 should be obtained. The optimum is typically derived by first obtaining isorespon.se contour plots from the response surfaces such as those of Fig. 6.20 or directly on the response surface and then visually deciding where the optimum is to be found. For the measured responses, the surfaces are often relatively simple (Fig. 6.21), but for the global responses they can be very complex (Fig. 6.22). If a threshold criterion is applied, then overlapping resolution maps can be obtained similar to those of Fig. 6.4. 

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