Sequential optimization method

One of the drawbacks of sequential optimization methods is that optimizing two or more criteria at the same time is hard, if not impossible. If the two or more criteria are combined in one overall criterion, which is advocated sometimes, then ambiguous results are obtained. This is shown in Chapter 4. There are ways to overcome this ambiguity to some extent [21]. Another drawback of a sequential procedure is that it gives not much information on the dependence of the criterion on the design variables. In the context of robustness this is a very serious drawback. This is one of the reasons why the use of sequential optimization methods is not present in this book. 

The obvious alternative to the sequential optimization methods is the use of an interpretive optimization method. In such a method a limited number of experiments is performed and the results are used to estimate (predict) the retention behaviour of all individual solutes as a function of the parameters considered during the optimization (retention surfaces). Knowledge of the retention surfaces is then used to calculate the response surface, which in turn is searched for the global optimum (see the description of interpretive methods in section 5.5). For programmed temperature GC the framework of such an interpretive method has been described by Grant and Hollis [614] and by Bartu [615]. 

On the other hand, in situations where the experimental region containing the optimum is not a priori known, a sequential optimization method, for example, a simplex approach, can be applied. Then, the following steps are considered ... 

Optimum conditions. However, modeling a qualitative factor has no meaning because only discrete levels are possible and no intermediate values occur. Therefore, only mixture-related and quantitative factors are examined in the optimization step. Sequential optimization methods select successive experiments in the factor domain, which implies that again only mixture-related and quantitative factors can be examined. 

In this method optimization phase, response surface designs or sequential optimization methods are apphed.The main difference between the two is that for a response surface design the experimental domain enclosed by the design is expected to contain the optimum, while a sequential optimization method can be applied in situations where the experimental region containing the optimal result is not a priori known. Another difference is that the sequential methods allow optimization of only one response, while with response surface designs several responses can be considered simultaneously (see further). 

Different sequential optimization methods can be distinguished, of which the simplex approaches are most commonly applied. They can be further... 

Dejaegher, B. and Vander Heyden, Y. (2009) Chapter 17 Sequential optimization methods, in Comprehensive Chemometrics,Wo. 1 (eds. S. Brown, R.Tauler, and B. Walczak), Elsevier, Oxford, pp. 547-575. 

The most common sequential optimization method is based on the simplex method by Nelder and Mead. A simplex is a geometric figure having a number of vertices equal to one more than the number of factors. A simplex in one dimension is therefore a line, in two dimensions a triangle, in three dimensions a tetrahedron, and in multiple dimensions a hypertetrahedron. 

The sequential optimization method is used to solve the optimization problem. The maximal peak value of the bending moments in columns is chosen as the objective function. 

Because the algorithm of sequential optimization method is very simple in Example 1 and 3 all steps of this algorithm were done using Excel or manually except the calculation of the values of objective fimction. 

The aim of the fourth example is to illustrate the optimal position of dampers on stmcture when dampers are modelled using the fractional Kelvin model and the fractional Maxwell model. Moreover, it is shown that results obtained using the sequential optimization method (which is a heuristic method) and using the PSO method are very similar. It justifies that it is possible to find, using the sequential optimization method, a solution which is near the global optimum of the optimization problem at hand. 

A first solution to the optimization problem is obtained using the sequential optimization method. For every possible location of one damper, the values of fundamental frequency and non-dimensional damping ratios are calculated (see Figure 13 and 14). Next, the objective function is evaluated for the frame, taking into account every possible position of the damper. The results are presented in Figure 15. 

It can be concluded that the results obtained by both methods yield similar dampers distributions within the frame. Differences between the optimal values of damping coefficients, obtained as the result of optimization procedures, are partially affected by an incremental way of distribution of the damping coefficients in the sequential optimization method. Moreover, in the PSO method the values ofthe damping. parameters must be positive for every damper. Dining the iteration process, negative or zero values of parameters c. were substituted by c. andnormal-... 

The optimal damper distributions in buildings are found for various objective functions. The weighted sum of amplitudes of the transfer functions of interstorey drifts and the weighted sum of amplitudes of the transfer functions of displacements evaluated at the fundamental natural frequency of the frame with the dampers are most frequently used as the objective function. The optimization problem is solved using the sequential optimization method and the particle swarm optimization method. Several numerical solutions to the considered optimization problem are presented and discussed in detail. 

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