Simplex fixed-size

Because the size of the simplex remains constant during the search, this algorithm is called a fixed-sized simplex optimization. Example 14.1 illustrates the application of these rules. 

Find the optimum response for the response surface in Figure 14.7 using the fixed-sized simplex searching algorithm. Use (0, 0) for the initial factor levels, and set the step size for each factor to 1.0. 

Progress of Fixed-Sized Simplex Optimization for Response Surface in Figure 14.10... 

Progress of a fixed-sized simplex optimization for the response surface of Example 14.1. The optimum response at (3, 7) corresponds to vertex 25. 

This experiment describes a fixed-size simplex optimization of a system involving four factors. The goal of the optimization is to maximize the absorbance of As by hydride generation atomic absorption spectroscopy using the concentration of HCl, the N2 flow rate, the mass of NaBH4, and reaction time as factors. 

A simplex is a geometric figure that has one more point than the number of factors. So, for two factors or independent variables, the simplex is represented by a triangle. Once the shape of a simplex has been determined, the method can employ a simplex of fixed size or of variable sizes that are determined by comparing the magnitudes of the responses after each successive calculation. Figure 5 represents the set of... 

The most common, and easiest to understand, method of simplex optimisation is called the fixed sized simplex. It is best described as a series of rules. 

The progress of a fixed sized simplex is illustrated in Figure 2.38. 

For the modified simplex, step 4 of the fixed sized simplex (Section 2.6.1) is changed as follows. A new response at point xlesl is determined, where the conditions are obtained as for fixed sized simplex. The four cases below are illustrated in Figure 2.39. 

Xnew = Xtest = C C X1 as in die normal (fixed-sized) simplex. 

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). 

Figure 4.17 Fixed-size simplex according to Nelder and Mead along an unknown response surface.

Hgure 5 Progress of a typical fixed size simplex. 

The progress of a fixed sized simplex is illustrated in Figure 5. Many elaborations have been developed over the years. One of the most important is the k + rule. If a vertex has remained part of the simplex for k + 1 steps, perform the experiment again. The reason for this is that response surfaces may be noisy, so... 

Then perform another experiment at x ew and keep this new experiment plus the k (=2) best experiments from the previous simplex to give a new simplex. Rule 5 of the fixed sized simplex still applies, if the value of the response at the new vertex is less than that of the remaining k responses, return to the original simplex and reject the second best response, repeating the calculation as above. 

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. 

If a point is retained in three consecutive simplexes, then it can be assumed that an optimum has been reached. (Note it may be that this optimum is not the true optimum, but that the simplex has been trapped at a false optimum. In this situation, it is necessary to start the simplex again, or use a modified simplex in which the step size is not fixed but variable, see Fig. 43.3.)... 

The Matlab Simulink Model was designed to represent the model stmctuie and mass balance equations for SSF and is shown in Fig. 6. Shaded boxes represent the reaction rates, which have been lumped into subsystems. To solve the system of ordinary differential equations (ODEs) and to estimate unknown parameters in the reaction rate equations, the inter ce parameter estimation was used. This program allows the user to decide which parameters to estimate and which type of ODE solver and optimization technique to use. The user imports observed data as it relates to the input, output, or state data of the SimuUnk model. With the imported data as reference, the user can select options for the ODE solver (fixed step/variable step, stiff/non-stiff, tolerance, step size) as well options for the optimization technique (nonlinear least squares/simplex, maximum number of iterations, and tolerance). With the selected solver and optimization method, the unknown independent, dependent, and/or initial state parameters in the model are determined within set ranges. For this study, nonlinear least squares regression was used with Matlab ode45, which is a Rimge-Kutta [3, 4] formula for non-stiff systems. The steps of nonlinear least squares regression are as follows ... 

In this example, the performance of the variable-size simplex is demonstrated for the enzyme determination based on the problem in Examples 4.4 and 4.7. For a fixed enzyme concentration of 13.6 mg-l ceruloplasmin (coded 0), the concentration of the substrate PPD and the pH value are sought for the maximum rate of the reaction, y. Since the simplex searches for a minimum, the rate as the objective criterion has to be... 

Initially we establish the working intervals (boundaries) of each factor Xi, between 0 and 60 x -10 (mol 1 ) and X2, between 5.0 and 9.0. Then, we define the initial simplex, i.e. we fix the initial levels of each factor Xi at 6.0 x -10 (mol 1 and X2 at 5.4. These values establish the initial vertex of the simplex (starting vertex). Next we define the variations that each factor may have from one experiment to another (step size) this difference must be large enough so that the response changes clearly between the experiments (usually about 10% and 20% working intervals). Here, we establish 5i = 9.0 x -10 (mol 1 ) for Xi and 52 = 0.6 for X2 see Table 3.21. 

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