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