How to use a simplex for optimization

The basic principles are illustrated with two variables the simplex is a triangle. It will be evident how the principles can be extended to any number of variables. 

A contour plot of a response surface is shown in Fig. 11.3. The aim is to locate the optimum conditions. Neither the shape of the response surface, nor the location of the optimum domain is known when we begin the experimental studies. 

If three experiments are laid out so that the variable settings define the coordinates of the vertices of a simplex, one of the experiments is likely to give a poorer result than the other two. This experiment has been run under conditions which are more elongated from the optimum conditions than the conditions used for the two better experiments. We shall therefore move away from the poor conditions. The next step will be to run a new experiment so that the variables settings of the new experiment and the settings of the remaining two better experiments form a new simplex. The new simplex will be oriented away from the poorest conditions of the first simplex, see Fig. 11.4a. 

The results obtained in the vertices of the new simplex are compared with each other and the poorest vertex is discarded and replaced by another vertex oriented in such a way that it forms a third simplex with the remaining two better vertices of the second simplex, etc. After a series of experiments the simplex has approached the optimum domain by a systematic moving away from the poor experimental conditions, see Fig. 11.4b. 

In practice, this means that the series of simplex experiments will describe a zig-zag path along the direction of the steepest ascent. The decision in which direction to move is made on the basis of the last simplex run. Hence, the method does not involve any drastic extrapolations. The movement upwards will be adjusted to follow the direction of the steepest ascent, even if the slopes of the response surface is changed when we move away from the initial experimental domain. 

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