Simplex circling

The effect of experimental error on the efficiency of simplex optimization is reduced if the number of the factor increases. Caused by error of experiment, a simplex movement to optimum may turn into a circle, which is called simplex circling. In such cases we suggest a replication of experiment, and then the movement to optimum is either continued or terminated. The cause of simplex circling may be both error of experiment and achieved optimum, Fig. 2.53D. With a large number of factors, to get a more reliable estimate whether optimum has been reached, it is necessary to continue movement to optimum until the number of simplexes with the same vertex does not surpass the value ... 

As can be seen in Fig. 6.29, point 8 is situated near to the optimum. All other new points overshoot the top of the response and therefore the simplexes circle around point 8. In practice one will not wait till point 12 is measured to decide that point 8 is near the optimum (or that an erroneous measurement was obtained). A third rule is applied "If a point is retained in three successive simplexes. it is replicated. If it is the highest result... 

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. 

In Figure 2.15, an example is given of the modified simplex procedure for the determination of fluticasone propionate with flow injection analysis (104). The initial simplex is formed by points 1, 2, and 3. Points 4-14 represent the sequentially selected vertices. Point 6 seems to be situated close to the optimum because it is maintained in many simplexes. It is observed that again, as in the classic procedure, the simplexes circle around the optimum, but here also their size decreases as the procedure continues. To optimize three or more factors, the simplex procedures can be generalized, as described in Reference 8. 

From Table V it can be seen that the last vertices in the simplex are not always the ones with the best response. This is due to the fact that once an optimum region is reached, the simplex may begin to step outside of the optimum and circle" it, sporadically moving back to the optimum or a near-optimum every few steps. This is precisely what is illustrated in Table V, where the boxed-in vertices represent the six best results. Statistical treatment of these data indicate that a consensus has more or less been reached for the optimum conditions, particularly the initial density and initial temperature. One should not infer, however, that the mean values for all 4 parameters should be used instead of the (best) values corresponding to vertex 13. We have yet to perform sufficient comparisons to draw such a conclusion. 

Figure 19.2 Schematic representation of the simplex algorithm. New points are denoted by closed circle , and the vector mean of all except the highest value is denoted by an open circle o a) a reflection from the point with the highest value through the vector-mean of the remaining points b) an expansion along the same line, taken if the resulting point yields a result that is lower than that seen at all other points c) a contraction along the same line, taken if the reflection point yields a result that is worse than that seen at all other points and d) a contraction among all dimensions toward the low point, taken if none of the actions taken yields a result that is better than than the highest value.

Applying rule 2 changes the direction of progression toward the optimum. This occurs most often in the region around the optimum. If a vertex in the vicinity of the optimum has been obtained, all new vertices are situated further from the optimum, and circle around it. This indicates that one is as near to the optimum as one can get with the initially chosen simplex size and starting from the initially chosen start conditions. Nevertheless, in practice, when the response surface is unknown, the optimum found may be only a local one. 

FIGURE 12.1. (a) Representation of a univariate optimization scheme. The concentric circles represent a surface response and the center is the maximum response. (1) The x-variable (or factor) value is fixed and variable y is optimized (2) y is fixed at best response while x is varied (3) during optimization of x, a better value is found, thus requiring new experiments varying y. According to this experimental setup, intersection of (2) and (3) would be the best response, (b) Representation of a bidimen-sional simplex BNW and the reflection R of the worse value W. Reprinted with permission from Reference 4. 

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