SIMPLEX local optimization procedure

The optimum found by sequential proceeding, both by Box-Wilson and simplex technique, is that local optimum situated nearest the starting point. It must not inevitably be identical with the global optimum. Therefore, it may be useful to repeat the optimization procedure one or several times. 

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. 

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. 

To determine the optimal parameters, traditional methods, such as conjugate gradient and simplex are often not adequate, because they tend to get trapped in local minima. To overcome this difficulty, higher-order methods, such as the genetic algorithm (GA) can be employed [31,32]. The GA is a general purpose functional minimization procedure that requires as input an evaluation, or test function to express how well a particular laser pulse achieves the target. Tests have shown that several thousand evaluations of the test function may be required to determine the parameters of the optimal fields [17]. This presents no difficulty in the simple, pure-state model discussed above. 

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. 

Simplex optimization has become very popular and has been widely used in chromatography, but there are three major disadvantages (1) the relationship between the factor to be optimized and the parameters involved is seldomly revealed in detail, and the procedure therefore does not lead to a better understanding of the separation process, (2) a local optimum may be found and the optimization process stops there, and (3) a larger number of experiments are required. In order to overcome these drawbacks, recently a computer-assisted mixture design simplex method has been introduced by Wang and co-workers [18]. 

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