Choosing the initial simplex in the Nelder-Mead optimization algorithm

Question

What\'s the best way to initialize a simplex for use in a Nelder-Mead simplex search from a user\'s \'guess\' vertex?

Answer 1

I'm not sure if there is a best way to choose the initial simplex in the Nelder-Mead method, but the following is what is done in common practice.

The construction of the initial simplex S is obtained from generating n+1 vertices x0,..,xn around what you call a user's "guess" vertex xin in a N dimensional space. The most frequent choice is

x0=xin 

and the remaining n vertices are then generated so that

xj=x0+hj*ej 

where ej is the unit vector of the j-th coordinate axis in R^n and hj is a step-size in the direction of ej.

hj = 0.05    if (x0)j is non-zero
hj = 0.00025 if (x0)j=0

with (x0)j the j-th component of x0. Note that this is the choice in Matlab's fminsearch routine, which is based on the Nelder-Mead scheme.

You can find some more information in

F. Gao, L. Han, "Implementing the Nelder-Mead simplex algorithm with adaptive parameters", Comput. Optim. Appl., DOI 10.1007/s10589-010-9329-3