Sequential search simplex

The Simplex method (and related sequential search techniques) suffers mainly from the fact that a local optimum will be found. This will especially be the case if complex samples are considered. Simplex methods require a large number of experiments (say 25). If the global optimum needs to be found, then the procedure needs to be repeated a number of times, and the total number of experiments increases proportionally. A local optimum resulting from a Simplex optimization procedure may be entirely unacceptable, because only a poor impression of the response surface is obtained. 

Simplex optimization of the primary (program) parameters in programmed temperature GC analysis has been demonstrated [612]. A systematic sequential search [613] may be used as an alternative. The Simplex method may be used to optimize a limited number of program parameters, whereas the latter approach was developed for the optimization of multisegment gradients. The use of interpretive methods has so far only been suggested [614,615]. 

The basic simplex optimization method, first described by Spendley and co-workers in 1962 [ 11 ], is a sequential search technique that is based on the principle of stepwise movement toward the set goal with simultaneous change of several variables. Nelder and Mead [12] presented their modified simplex method, introducing the concepts of contraction and expansion, resulting in a variable size simplex which is more convenient for chromatography optimization. 

There are two basic types of unconstrained optimization algorithms (I) those reqmring function derivatives and (2) those that do not. The nonderivative methods are of interest in optimization applications because these methods can be readily adapted to the case in which experiments are carried out directly on the process. In such cases, an ac tual process measurement (such as yield) can be the objec tive function, and no mathematical model for the process is required. Methods that do not reqmre derivatives are called direc t methods and include sequential simplex (Nelder-Meade) and Powell s method. The sequential simplex method is quite satisfac tory for optimization with two or three independent variables, is simple to understand, and is fairly easy to execute. Powell s method is more efficient than the simplex method and is based on the concept of conjugate search directions. 

For the optimization of, for instance, a tablet formulation, two strategies are available a sequential or a simultaneous approach. The sequential approach consists of a series of measurements where each new measurement is performed after the response of the previous one is knovm. The new experiment is planned according to a direction in the search space that looks promising with respect to the quality criterion which has to be optimized. Such a strategy is also called a hill-climbing method. The Simplex method is a well known example of such a strategy. Textbooks are available that describe the Simplex methods [20]. 

Simplex Optimization. The sequential simplex method is an example of a sequential multivariate optimization procedure that uses a geometrical figure called a simplex to move through a user-specified of experimental conditions in search of the optimum. Various forms of the simplex have been successfully used in different modes of chromatography, particularly HPLC (40-42) and GC (43-46). 

Let us restrict ourselves to putting the numerous variants of sequential optimum search into some sort of order. Most of the methods mentioned in the following review are described by BUNDAY [1984 a], who also gives BASIC programs details of the simplex method and its programming may be found in BUNDAY [1984 b]. 

There are two types of unconstrained multivariable optimization techniques those requiring function derivatives and those that do not. An example of a technique that does not require function derivatives is the sequential simplex search. This technique is well suited to systems where no mathematical model currently exists because it uses process data directly. 

Sometimes it is not necessary to determine a response surface model tor locate the optimum conditions. Hill-climbing by direct search methods, e.g. search along the path of steepest ascent [8] or sequential simplex search [9], will lead to a point on the response surface near the optimum. The computations involved in these methods are rather trivial and do not require a computer and will for this reason not be discussed further in this chapter. Readers who require details of these direct search methods should consult Refs. [1,8,9]. 

If xi and X2 are varied one at a time, then the method is known as a univariate search and is the same as carrying out successive line searches. If the step length is determined so as to find the minimum with respect to the variable searched, then the calculation steps toward the optimum, as shown in Figure 1.15a. This method is simple to implement, but can be very slow to converge. Other direct methods include pattern searches such as the factorial designs used in statistical design of experiments (see, for example, Montgomery, 2001), the EVOP method (Box, 1957) and the sequential simplex method (Spendley et ah, 1962). 

With new synthetic methods, mechanistic details are still obscured. It is not likely that such details will be revealed until the preparative utility of the procedure has been demonstrated. This means that an optimization of the experimental conditions must generally precede a mechanistic understanding. Hence, the optimum conditions must be inferred from experimental observations. The common method of adjusting one-variable-at-a-time, is a poor strategy, especially in optimization studies (see below). It is necessary to use multivariate strategies also for determining the optimum experimental conditions. There are many useful, and very simple strategies for this sequential simplex search, the method of steepest ascent, response surface methods. These will be discussed in Chapters 9 - 12. 

As noted in the introduction, energy-only methods are generally much less efficient than gradient-based techniques. The simplex method [9] (not identical with the similarly named method used in linear programming) was used quite widely before the introduction of analytical energy gradients. The intuitively most obvious method is a sequential optimization of the variables (sequential univariate search). As the optimization of one variable affects the minimum of the others, the whole cycle has to be repeated after all variables have been optimized. A one-dimensional minimization is usually carried out by finding the... 

Figure 37 Sequential simplex procedure. A triangle is spanned on the biological activity values (z axis) of the H, 4-Cl, and 4-Ac analogs. The H analog is the least active one and a new analog is searched in the direction between and beyond the mean value of the other two analogs (left dashed line the right dashed line is the projection onto the n vs. ct plane) (reproduced from Figure 4 of ref. [635] with permission from the American Chemical Society, Washington, DC, USA).

Efficient experimentation is based on the methods of experimental design and its quantitative evaluation. The latter can be performed by means of mathematical models or graphical representations. Alternatively, sequential methods are apphed, such as the simplex method, instead of these simultaneous methods of experimental optimization. There, the optimum conditions are found by systematic search for the objective criterion, for example, the maximum yield of a chemical reaction, in the space of all experimental variables. 

To search for the optimum by sequential methods, that is, by means of the simplex method by Nelder and Mead. 

A database search specification. For structural databases, this may be a molecule, a substructure, or a data request. Sequential simplex... 

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