Simplex modified method

The modified simplex methods have gained considerable popularity in analytical chemistry, especially for the optimization of instrumental methods. Applications in organic synthesis are, however, remarkably few. There are several reasons for this difference  

Instrumental analysis-. At the outset of the study, the experimental domain is not firmly settled. 

The variable settings can be fairly rapidly adjusted (currents in shim-coils to improve the resolution in NMR potentiometer setting to focus the mass spectrometers flow-rate in FIA analysis carrier gas flow, column temperature, temperature program in gas chromatography, etc). The response can be recorded directly as an output signal. Each experiment is rapidly run, and it is not very important if 25 or 40 iterations are necessary to establish the optimum conditions. 

Under these circumstances, the modified simplex methods are very convenient. 

Organic synthesis The experimental domain of interest is usually fairly well known. Each experiment is, however, time-consuming. The response must be evaluated by a separate analytical procedure. The number of experiments to achieve an optimum result is very important. 

---

Fig. 12. The progress of the modified Simplex method for optimization. From P. J. Golden and S. N. Deming, Laboratory Microcomputer, i, 44 (1984). Reproduced by permission of Science Technology Letters, England...

It will not always be possible to make expansion movements because as we move closer to the optimum we must reduce the size of the simplex in order to locate the optimum accurately. This basic idea of adapting the size of the simplex to each movement is the one that sustains the modified simplex method proposed by Nelder and Mead [17]. Figure 2.15 displays the four possibilities to modify the size of the simplex and Table 2.32 gives their respective expressions for each factor. 

G. A. Zachariadis and J. A. Stratis, Optimisation of cold vapour atomic absorption spectrometric determination of mercury with and without amalgamation by subsequent use of complete and fractional factorial designs with univariate and modified simplex methods, J. Anal. At. Speetrom., 6(3), 1991, 239-245. 

By far the most popular technique is based on simplex methods. Since its development around 1940 by DANTZIG [1951] the simplex method has been widely used and continually modified. BOX and WILSON [1951] introduced the method in experimental optimization. Currently the modified simplex method by NELDER and MEAD [1965], based on the simplex method of SPENDLEY et al. [1962], is recognized as a standard technique. In analytical chemistry other modifications are known, e.g. the super modified simplex [ROUTH et al., 1977], the controlled weighted centroid , the orthogonal jump weighted centroid [RYAN et al., 1980], and the modified super modified simplex [VAN DERWIEL et al., 1983]. CAVE [1986] dealt with boundary conditions which may, in practice, limit optimization procedures. 

MODIFIED SIMPLEX METHOD RULES FOR EXPANSION CONTRACTION OF SIMPLEX... 

Fig. 9. Principle of the Modified Simplex method — 1. Schematic representation of the rules for expansion and contraction of the simplex...

In the simplex procedures described above the step size was fixed. When the step size was taken too small it takes a large number of experiments to reach the optimum, and when it is taken too large the supposed optimum can be unacceptably far from the real one. To avoid this a so-called modified simplex method can be applied, in which the step size is variable throughout the procedure. The principles of the simplex method are maintained but rules for expansion or contraction of the simplexes are added. For a detailed description of these guidelines we refer to [27,831. 

To permit a more rapid convergence towards an optimum, several modified simplex methods have been suggested.[2] By these modified methods, the step-length of the next move is adjusted depending on the degree of improvement. If a new vertex should give a considerable improvement, it is rather natural to try to move further in that direction. One such modified simplex method is given in detail below, after a presentation of the basic simplex method. 

A modified simplex method is described below. The procedure is close to the modifications suggested by Nelder and Mead.[2a] Other modifications are described in the works given in the reference list. 

Optimization Using the Super-Modified Simplex Method... 

In the basic simplex method, the simplex thus can only be reflected to obtain the next experiment, and the simplex size remains the same throughout the procedure. In the modified simplex method, suggested by Nelder and Mead (100), the simplex can be reflected, expanded, or contracted to define the next experiment. Thus, in case the simplex is expanded or contracted, the simplex size changes. More information about the simplex procedures can be found in References 7,9,10, and 98-102. 

A. P. Wade, Optimisation of Flow Injection Analysis and Polarography by the Modified Simplex Method. Anal. Proc., 20 (1983) 523. 

In this study, a modified Simplex method was used to regress the binary interaction parameter, fcy, using a packaged algorithm, DBCPOL (13), The objective function minimized by the optimization routine was the percent absolute average relative deviation (%AARD)... 

MORGAN, E. BURTON, K. W. C. and NICKLESS, G. Optimization using the super-modified simplex method. Chemom. Intell. Lab. Systems 8, 1990, pp. 97-107. 

Our simulation studies were extended to assess the precision and foundation of the calculated rate parameters by making use of a Modified Simplex Method (MSM). 

In the modified simplex method C425 l further steps are necessary to compute an optimum new vertex and to find the maximum response with maximum speed and efficiency ... 

FIGURE 25. Modified simplex method. Instead of the worst vertex w one... 

A further variation of the simplex method - the super-modified simplex method - was developed by Denton et. al. C4323 and used in chemical pattern recognition by Kaberline and Wilkins C1373. For each... 

The ability of the super-modified simplex algorithm to locate the position of the new vertex more precisely than is possible by the modified simplex method means that the super-modified simplex requires fewer iterations to converge and is therefore somewhat faster. 

Aberg RA, Gustavson AGT (1982) Design and evaluation of modified simplex methods. Anal Chim Acta 144 39... 

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. 

The rigidity that prevented an accurate optimal point from being obtained was solved by Nelder and Mead in 1965. They proposed a modification of the algorithm that allowed the size of the simplex to be varied to adapt it to the experimental response. It expanded when the experimental result was far from the optimum - to reach it more rapidly and it contracted when it approached a maximum value, so as to detect its position more accurately. This algorithm was termed the "modified simplex method . Deming and co-workers published the method in the journal Analytical Chemistry and in 1991 published a book on this method and its applications. 

---