Sequential simplex

Sangsila, S. Labinaz, G. Poland, J. S. et al. An Experiment on Sequential Simplex Optimization of an Atomic Absorption Analysis Procedure, /. Chem. Educ. 1989, 66, 351-353. 

Techniques used to find global and local energy minima include sequential simplex, steepest descents, conjugate gradient and variants (BFGS), and the Newton and modified Newton methods (Newton-Raphson). 

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. 

Phillips, G. R., and Eyring, E. M., Error Estimation Using the Sequential Simplex Method in Nonlinear Least Squares Data Analysis, Anal. Chem. 60, 1988, 738-741. 

F. Darvas, Application of the sequential simplex method in designing drug analogs. J. Med. Chem., 17 (1974) 99-804. 

The Sequential Simplex or simply Simplex method relies on geometry to create a heuristic rule for finding the minimum of a function. It is noted that the Simplex method of linear programming is a different method. 

Walters, F.H., Parker, J Llyod, R., Morgan, S.L., and S.N. Deming, Sequential Simplex Optimization A Technique for Improving Quality and Productivity in Research, Development, and Manufacturing, CRC Press Inc., Boca Raton, Florida, 1991. 

Because this proceeding is relatively expensive, an effective semi-quantitative method is widely used in optimization, the sequential simplex optimization. Simplex optimization is done without estimation of gradients and setting step widths. Instead of this, the progress of the optimization... 

For readers with no prior knowledge of optimization methods In the textbook of Box et.al. [14] the basic principles of optimization are also explained. The sequential simplex method is presented in Walters et.al. [20]. Multi-criteria optimization is presented in Chapter 4 on an introductory level. For those readers who want to know more about multicriteria optimization, see the references given in Section 1.3.4 and Chapter 4. 

The second approach to optimisation is a model independent one. One of these model independent methods is the sequential simplex [24,25] used by Shek et al. [26], The method is claimed to be ideally suited for the optimisation of formulations [27] because of the relatively low number of experiments to be performed. 

Figure 4.15 Illustration of the sequential simplex optimisation procedure in a mixture of three components. Point A is the result of situation A, point B is the result of situation B...

The three methods described in the preceding paragraphs each offer distinct advantages and disadvantages. The first and most obvious difference between the methods is the distinction between the sequential methods (sequential simplex and prisma method) and the simultaneous method (mixture design). With the sequential method some experiments are performed, these are evaluated, and on the basis of this evaluation new design points are selected, these are evaluated etc. With the simultaneous... 

Although application of chemometrics in sample preparation is very uncommon, several optimisation techniques may be used to optimise sample preparation systematically. Those techniques can roughly be divided into simultaneous and sequential methods. The main restrictions of a sequential simplex optimisation [6,7] find their origin in the complexity of the optimisation function needed. This function is a predefined function, often composed of several criteria. Such a composite criterion may lead to ambiguous results [8]. Other important disadvantages of simplex optimisation methods are that not seldom local optima are selected instead of global optima and that the number of experiments needed is not known beforehand. 

R. J. Stolzberg, Optimizing signal-to-noise ratio in flame atomic absorption spectrophotometry using sequential simplex optimisation, J. Chem. Educ., 76(6), 1999, 834-838. 

G. H. Lee, Development of new high temperature plasma sources for spectrochemical analysis multivariate optimisation by the modified sequential simplex method. Bull. Korean Chem. Soc., 14(2), 1993, 275-281. 

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). 

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. 

Mayur et al. (1970) formulated a two level dynamic optimisation problem to obtain optimal amount and composition of the off-cut recycle for the quasi-steady state operation which would minimise the overall distillation time for the whole cycle. For a particular choice of the amount of off-cut and its composition (Rl, xRI) (Figure 8.1) they obtained a solution for the two distillation tasks which minimises the distillation time of the individual tasks by selecting an optimal reflux policy. The optimum reflux ratio policy is described by a function rft) during Task 1 when a mixed charge (BC, xBC) is separated into a distillate (Dl, x DI) and a residue (Bl, xBi), followed by a function r2(t) during Task 2, when the residue is separated into an off-cut (Rl, xR2) and a bottom product (B2, x B2)- Both r2(t)and r2(t) are chosen to minimise the time for the respective task. However, these conditions are not sufficient to completely define the operation, because Rl and xRI can take many feasible values. Therefore the authors used a sequential simplex method to obtain the optimal values of Rl and xR which minimise the overall distillation time. The authors showed for one example that the inclusion of a recycled off-cut reduced the batch time by 5% compared to the minimum time for a distillation without recycled off-cut. 

RIGID PVC FORMULATION OPTIMISATION USING SEQUENTIAL SIMPLEX... 

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]. 

Once the right set of parameters has been identified, computer-aided optimization using modified sequential simplex or central composite design methods can be applied to further hne-tune the separation under investigation, as has been published for the optimization of reverse-phase HPLC [17-20] and chiral separations [21-23]. 

Second, the direct optimization method, the best known being the sequential simplex, is a rapid and powerful method for determining an experimental domain, best combined with experimental design for the optimization itself. 

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