Simplex method optimum reached

Disadvantages of the simplex method are the number of experiments to reach an optimum is not known beforehand, this can lead to better but also to worse results compared to a simultaneous approach. If an optimum is reached nothing is known about that part of the response surface that has not been investigated, e.g. other, even higher optima can be present and, which is more important, the stability of the reached optimum against small variations of a criterion, is not known. 

Following the establishment of the initial simplex, the simplex begins moving away from the conditions that give the worst result (as described above) and systematically reaches the optimum set of conditions that yield the best separation (vertex 8). In contrast to the false optimum (point H) found by the univariate approach described earlier, the simplex method has located the true optimum density and temperature for this separation. 

Nine trials were done in simplex optimization and a top value of the yield was obtained in trial No. 5 or vertex C. The maximal yield by the method of steepest ascent was obtained in trial No. 8, which coincides with simplex optimization. It can be concluded that to reach the optimum by the method of steepest ascent, six trials were realized, while by the simplex method, five trials were realized. We should, however, remember that FUFE 22 has to be replicated once, so that the method of steepest ascent, in this case, requires 12 trials, which is considerably more than by the simplex method. 

Let us analyze the previous case by taking into account the third factor X,. The outcomes of FUFE 23 and the results of application of method of steepest ascent are given in Table 2.216. Thirteen trials were necessary to reach the maximal yield of 85.2%. The outcomes of the simplex method are in Table 2.217. Maximal yield after 14 trials is 85.0%. Approximately the same number of trials has been necessary by both methods to reach the optimum. It should be stressed once again that FUFE requires replications, so that to reach optimum by the method of steepest ascent, we need at least twice as many trials. Evidently, a half-replica instead of FUFE in the basic experiment may reduce the number of trials. However, there is a possibility of wrong direction of the movement to optimum due to the possible effects of interactions. 

While the preceding generalization is sufficient to allow for reaching a final solution ultimately, it can be very inefficient unless some sort of special method is used to permit generation of extreme-point solutions in an efficient manner to allow rapid and effective approach to the optimum condition. This is what the simplex method does.f... 

The basis for the simplex method is the generation of extreme-point solutions by starting at any one extreme point for which a feasible solution is known and then proceeding to a neighboring extreme point. Special rules are followed which cause the generation of each new extreme point to be an improvement toward the desired objective function. When the extreme point is reached where no further improvement is possible, this will represent the desired optimum feasible solution. Thus, the simplex algorithm is an iterative process that starts at one extreme-point feasible solution, tests this point for optimality, and... 

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. 

In previous work of the same group (72), the electrokinetic injection of DNA fragments was optimized as well by means of a simplex method. CGE-LIF was also used. In this case, BGE concentration, sample injection voltage, and time were the factors to be optimized. The optimum conditions were reached after only nine experiments. Figure 6.5 shows the spatial evolution of the simplex method used in this work (the initial tetrahedron (vertices 1-4) and the subsequent movements of reflection and contraction). Vertex 9 was considered as the optimum for injection of the Ikbp DNA ladder (l.OmM TTE buffer, 20s injection, 55V/cm electric field injection). 

Numerically, the most commonly adopted programs are based on the Simplex methods (Martin, 1999 Vanderbei, 2007), which seek to establish the optimum by moving on adjacent vertices. Clearly, however, many vertices must be examined before the solution is reached. 

The best known sequential method for optimization is the simplex method. This requires the experimenter to perform a series of experiments until he or she reaches an optimum. The response surface is not mathematically modeled, but the experimenter follows a series of rules until he or she cannot improve. A simplex is the simplest possible object in N dimensional space, e.g., a line in one dimension, and a triangle in two dimensions. Simplex optimization means that a series of experiments are performed on the corners of such a figure. 

The rigidity that prevented an accurate optimal point from being obtained was solved by Nelder and Mead in 1965. They proposed a modification of the algorithm that allowed the size of the simplex to be varied to adapt it to the experimental response. It expanded when the experimental result was far from the optimum - to reach it more rapidly and it contracted when it approached a maximum value, so as to detect its position more accurately. This algorithm was termed the "modified simplex method . Deming and co-workers published the method in the journal Analytical Chemistry and in 1991 published a book on this method and its applications. 

In a sequential strategy only a few experiments at a time are carried out and, on the basis of the results, the experiment to be carried out next is defined. These designs are used when the optimum of a single response is the only information desired. The best known sequential method is called the simplex method. However, it does not provide much information about a model relating the response with the factors studied. Despite the fact that it would be possible to map part of the response surface (the route followed to reach the optimum), the model obtained would not necessarily be a good one to describe the whole response surface and that is the reason why, when a model of the response surface is needed, one should prefer simultaneous methods. There are several situations where sequential designs are not suitable. One of them is when more than one response needs to be optimized. It also happens that these methods are trapped on a local optimum, and do not find the global optimum. 

A weakness with the standard mediod for simplex optimisation is a dependence on the initial step size, which is defined by the initial conditions. For example, in Figure 2.37 we set a very small step size for both variables this may be fine if we are sure we are near the optimum, but otherwise a bigger triangle would reach the optimum quicker, the problem being that the bigger step size may miss the optimum altogether. Another method is called the modified simplex algorithm and allows the step size to be altered, reduced as the optimum is reached, or increased when far from the optimum. 

When a point has been reached that is considered as optimal, the analyst can decide to stop the optimization, or try to fine-tune the method and to determine the real optimum more exactly. This can be done by starting in the provisional optimal point a new procedure with smaller simplexes (e.g. with step sizes that are 0.25 or 0.10 of the original ones). 

A screening experiment may suggest that a better domain is likely to be found outside the explored domain. By the method of Steepest ascent (Chapter 10), or by a simplex search (Chapter 11) an near-optimum domain can be reached by a limited number of experiments. 

A weakness with the standard method for simplex optimization is a dependence on the initial step-size. Another method is called the modified simplex algorithm and allows the step size to be altered, reduced as the optimum is reach, or increased far from the optimum. 

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