Sequential simplex optimization

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. 

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. 

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. 

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. 

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. 

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. 

Unlike the other optimization methods described here, the sequential simplex method for optimization neither assumes nor determines a mathematical model for the phenomena studied. 

Walters, F. Sequential simplex optimization—an update. Anal. Lett. 1999, 32 (2), 193. 

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. 

In general, a simplex for/factors is a geometric figure in the /-dimensional factor space, defined by / + 1 points or vertices, that is, one more than the number of factors. During optimization, the simplex sequentially moves through the experimental domain in the direction of the optimum. The next simplex to be performed is based on the results of the previous, and is defined according to specific rules. 

The implicit approach of chapter 5 was to optimize the process or formulation by examining the response surface directly. But other methods are both useful and necessary when there are many factors (desirability) or when the optimum is outside the experimental region (steepest ascent and optimum path). Also there are direct optimization methods available (sequential simplex) which do not involve mapping the response surface at all. 

No mathematical model Direct optimization Single or multiple responses SEQUENTIAL SIMPLEX... 

F. Maynd, Optimization techniques and pharmaceutical formulation example of the sequential simplex in tableting, Proc. 1st Int. Conf. Pharm. Tech. (APGI), 5, 65-84 (1977). 

---