Simplex search method

Random Search / 6.1.2 Grid Search / 6.1.3 Univariate Search / 6.1.4 Simplex Search Method / 6.1.5 Conjugate Search Directions / 6.1.6 Summary... 

An alternative method of optimisation is the simplex search method. This is a model independent procedure in which the results of earlier experiments are used to define subsequent... 

The Simplex optimization method can also be used in the search for optimal experimental conditions (Walters et al. 1991). A starting simplex is usually formed from existing experimental information. Subsequently, the response that plays the... 

The techniques most widely used for optimization may be divided into two general categories one in which experimentation continues as the optimization study proceeds, and another in which the experimentation is completed before the optimization takes place. The first type is represented by evolutionary operations and the simplex method, and the second by the more classic mathematical and search methods. (Each of these is discussed in Sec. V.)... 

Optimization techniques may be classified as parametric statistical methods and nonparametric search methods. Parametric statistical methods, usually employed for optimization, are full factorial designs, half factorial designs, simplex designs, and Lagrangian multiple regression analysis [21]. Parametric methods are best suited for formula optimization in the early stages of product development. Constraint analysis, described previously, is used to simplify the testing protocol and the analysis of experimental results. 

The constrained optimization procedure, originally developed from the simplex method and first described by Box, is ideally suited to model refinement (.8). It is a search method that searches for the minimum of a multidimensional function within given intervals. It possesses all the advantages of search methods, among them that calculation of derivatives is not necessary, a test to assure the independence of variables can be omitted, and diverse variables can be easily included. These are exactly the requirements of model refinement where bond lengths, bond angles, torsion angles, and other parameters are used within experimentally defined limits. 

Statistical optimization methods other than the Simplex algorithm have only occasionally been used in chromatography. Rafel [513] compared the Simplex method with an extended Hooke-Jeeves direct search method [514] and the Box-Wilson steepest ascent path [515] after an initial 23 full factorial design for the parameters methanol-water composition, temperature and flowrate in RPLC. Although they concluded that the Hooke-Jeeves method was superior for this particular case, the comparison is neither representative, nor conclusive. 

Includes Simplex, Modified Simplex, Other statistical search methods... 

Various more-or-less efficient optimization strategies have been developed [46, 47] and can be classified into direct search methods and gradient methods. The direct search methods, like those of Powell [48], of Rosenbrock and Storey [49] and of Nelder and Mead ( Simplex ) [50] start from initial guesses and vary the parameter values individually or combined thereby searching for the direction to the minimum SSR. 

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]. 

The main disadvantage of the simplex method consists in the laige number of experiments required to find optimal working conditions. Further, the optimisation criterion characterises the separation of the sample mixture by a single number, so that the detailed information on the separation of the individual sample components is lost and because of the high probability that the search method will slide into a region with a local maximum of the optimisation criterion, the simplex optimisation method can be expected to be fully successful only with the separations of relatively simple samples. 

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. 

The method of steepest ascent and the simplex search can handle only one criterion, while the resportse surface methods allow simultaneous mapping of several responses. Response surface modelling can therefore be used to optimize several responses simultaneously. The problem of multiple responses is elegantly handled by PLS modelling. This technique is discussed in Chapter 17. 

In the preceding chapters it was discussed how a near-optimum experimental domain can be found by the method of steepest ascent or by a simplex search. However, these methods cannot be used to efficiently locate the optimum conditions. For this, response surface modelling is a far more efficient technique. 

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. 

Nonderivative methods include random search, grid search, simplex search, and conjugate directions (or Powell s method). The nonderivative methods use various patterns for generating new test points for decision variables, and then a comparison of the new objective function value against previous values. A subsequent test point is then generated, either based on the immediate comparison or using the previous history of test points. 

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