Variable-size simplex procedure

Chubb and co-workers applied simplex optimization to increasing the yield of the Bucherer-Berg reaction. In this reaction, a ketone reacts with NH3, COS, and HCN to give a complex, heterocyclic product. Eight variables are related to the yield obtained the initial concentrations of the four reactants, pH, temperature, time of the reaction, and the solvent used. A mixed solvent of ethanol and water was used, with the ratio varied as part of the optimization. Results were reported for several sets of experiments using cyclohexanone and adamantanone as the starting material. The yields were improved rapidly using a variable-size simplex procedure. In one experiment, the yield was improved from 49 to 88%. 

In the simplex procedures described above the step size was fixed. When the step size was taken too small it takes a large number of experiments to reach the optimum, and when it is taken too large the supposed optimum can be unacceptably far from the real one. To avoid this a so-called modified simplex method can be applied, in which the step size is variable throughout the procedure. The principles of the simplex method are maintained but rules for expansion or contraction of the simplexes are added. For a detailed description of these guidelines we refer to [27,831. 

It is observed that the simplexes circle around the optimum and point 8 is the closest the real optimum can be reached by the simplex used. The number of experiments or simplexes required to approach the optimum depends on the size of the simplex. A larger simplex will require fewer experiments than a smaller simplex. However, a smaller simplex will allow approaching the real optimum closer than a larger one. From this need to find a compromise between speed of moving through the domain and approachabUity of the optimum, the variable-size or modified simplex procedure has been developed. 

Let us now consider the variable-size or modified simplex procedure, proposed by Nelder and Mead (100). Whereas in the basic procedure, the size is fixed and determined by the initially chosen simplex, the size in the modified simplex procedure is variable. Besides the rules of the basic procedure, the modified procedure additionally allows expansion or contraction of simplexes. In favorable search directions, the simplex size is expanded to accelerate finding the optimum, while in other circumstances, the simplex size is contracted, for example, when approaching the optimum (Figure 2.14). 

To define the first simplex the multiplication factors shown in Table 6.18 are used. Suppose the simplex is a triangle since two variables are being optimized. The experimenter defines a first point (the first vertex, also called the experimental origin) and the step size for each variable, i.e. the maximum change one wants to apply for a variable at each step of the procedure. For instance, for the first point variable, vi = 10 and variable xi = 100 with step sizes of 5 and 10, respectively. The vertices of the initial triangle are obtained as Vertex 1... 

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