The symmetry-controlled simplex optimization procedure

Development of the unlocking mechanisms for the complex method

“…The Raplex Method is compared with other improvements proposed in [14,15]. The main unlocking mechanism in [14] is Symmetry Control, and that in [15] is Degeneracy Constraint.…”

“…Van der Wiel proposed a measure of simplex distortion termed ''Symmetry'', which is defined by the Nth root of the absolute value of the determinant of the simplex divided by the radius of the sphere passing through all vertices of the simplex, where N represents the number of variables. If the Symmetry of the simplex is below a preset criterion, the simplex will be reshaped until it meets the Symmetry criterion [14]. Although it is mentioned that the applicable number of variables range from 2 to 8, no experiments were demonstrated to show that this mechanism works for N > 2, probably due to the limitations of computing capacities 20 years ago.…”

“…Van der Wiel introduced a symmetry restriction to control the degeneracy of the simplex to prevent it from repeatedly making steps that will create an elongated 0045-7949/$ -see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2005.02.027 and degenerate simplex [14]. A few years later, Hedlund adopted the idea and used degeneracy constraints to improve the converging ability of the Simplex Method [15].…”

A New Sequential Procedure for the Efficient and Automated Location of Optimum Conditions in High Performance Liquid Chromatography (HPLC)

“…This fact leads to the development of different modifications such as Super Modified Simplex [17] (SMS), Weighted Centroid method [18] (WCM). Each of these search methods includes symmetry criteria [19] by which the search algorithm may continue to decide the next search direction after each step in the search process. These limitations are especially important to select mobile phase composition for HPLC due to the limitations in the experimental modification of solvent percentages.…”

Application of chemometrics

“…In practice, modified Simplex procedures are used which perform better than the above mentioned symmetrical one (Ref. 2). Implicitely it has been assumed that optimization was wanted for one criterion only.…”