Numerical Methods and Optimization in Finance

Figure 11.12 Detailed starting simplex and two reflections.

We now detail the rules according to which the simplex evolves toward the minimum. Consider a function depending on n variables. Then, at iteration k, the simplex is defined by the vertices x⁽ⁱ⁾, i = 1,…,n + 1 (left panel in following figure). These vertices are renamed such that $f(x^{(1)})\le f(x^{(2)})\le \cdots \le f(x^{(n+1)})$ (right panel). We then compute the mean over all vertices except the worst (the one with the highest function value):

$\overline{x}=\frac{1}{n}{\displaystyle \sum _{i=1}^n{x}^{(i)}}i=1,\dots ,n.$

The vertex x_n+1 with the worst function ...