Simplex method reflected vertex

Figure 4 A representative step m the downhill simplex method. The original simplex, a tetrahedron in this case, is drawn with solid lines. The point with highest energy is reflected through the opposite triangular plane (shaded) to form a new simplex. The new vertex may represent symmetrical reflection, expansion, or contractions along the same direction.

Step 3 to improve the response, the basic rule of the simplex method is reflect the rejected vertex (usually W) through the centroid (G) of the other f vertices (they are said to be retained ), obtaining the reflected vertex (R). Its experimental conditions are given by the following two expressions (apply them to each experimental factor) ... 

In the simplex method, die number of initial experiments conducted is one more than the number of parameters (temperature, gradient rate, etc.) to be simultaneously optimized. The conditions of the initial experiments constitute the vertices of a geometric figure (simplex), which will subsequently move through the parameter space in search of the optimum. Once the initial simplex is established, the vertex with the lowest value is rejected, and is replaced by a new vertex found by reflecting the simplex in the direction away from the rejected vertex. The vertices of the new simplex are then evaluated as before, and in this way the simplex proceeds toward the optimum set of conditions. 

Be flexible in the choice of simplex method. It can be difficult to state a priori which method will be the preferred in a given situation. If a reflected vertex should give a considerable improvement, it would be natural to try an experiment in an extended vertex. 

In previous work of the same group (72), the electrokinetic injection of DNA fragments was optimized as well by means of a simplex method. CGE-LIF was also used. In this case, BGE concentration, sample injection voltage, and time were the factors to be optimized. The optimum conditions were reached after only nine experiments. Figure 6.5 shows the spatial evolution of the simplex method used in this work (the initial tetrahedron (vertices 1-4) and the subsequent movements of reflection and contraction). Vertex 9 was considered as the optimum for injection of the Ikbp DNA ladder (l.OmM TTE buffer, 20s injection, 55V/cm electric field injection). 

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