Variable-size simplex optimization method

The sequential simplex method of optimization was proposed by Nelder and Mead. With a number of improvements and enhancements the simplex method has found great utility in real situations in analytical laboratory experiments and process control situations. The simplex method is a hill-climbing method that seeks to climb the response surface depending on the features of the response surface in its immediate neighborhood. Only one new experiment is done for each step in the optimization sequence, and the location of this new experiment on the response surface is completely determined by the previous few experiments. The method of Nelder and Mead using a variable-size simplex is the most commonly used. A complete description of how the simplex method works is beyond the scope of this review, since the information is contained elsewhere.Many applications of simplex optimization have appeared, and a few examples follow. 

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 Matlab Simulink Model was designed to represent the model stmctuie and mass balance equations for SSF and is shown in Fig. 6. Shaded boxes represent the reaction rates, which have been lumped into subsystems. To solve the system of ordinary differential equations (ODEs) and to estimate unknown parameters in the reaction rate equations, the inter ce parameter estimation was used. This program allows the user to decide which parameters to estimate and which type of ODE solver and optimization technique to use. The user imports observed data as it relates to the input, output, or state data of the SimuUnk model. With the imported data as reference, the user can select options for the ODE solver (fixed step/variable step, stiff/non-stiff, tolerance, step size) as well options for the optimization technique (nonlinear least squares/simplex, maximum number of iterations, and tolerance). With the selected solver and optimization method, the unknown independent, dependent, and/or initial state parameters in the model are determined within set ranges. For this study, nonlinear least squares regression was used with Matlab ode45, which is a Rimge-Kutta [3, 4] formula for non-stiff systems. The steps of nonlinear least squares regression are as follows ... 

After optimization by simplex, the plot from Figure 12.6 improved and yielded a linear equation (log[p] = -3.14 - 0.20 log[bp], = 0.998) suitable for the analysis of the 201-2036 bp size range. This equation was further used to determine the size of unknown DNA fragments (27). Thus, the simplex method was shown to be an efficient way to optimize an electrophoretic separation of DNA, since several variables could be simultaneously optimized. 

This works in the same way as the extended method, except that the only operation is the reflection R. Thus the simplex remains regular (in terms of the coded variables) and the same size, throughout the optimization. Special rules apply when R is worse than W, and for stopping the simplex. See references (13) and (16) for the general rules and reference (17) for a pharmaceutical example, where it is used by Mayne, again to optimize a tablet formulation. 

The resulting set of differential equations was solved numerically by means of an improved Euler method with variable step size control [9]. Parameter optimization was accomplished by a Mead Nelder simplex algorithm [10]. 

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