问题
What's the best way to initialize a simplex for use in a Nelder-Mead simplex search from a user's 'guess' vertex?
回答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
回答2:
I think there is no general rule to determine best the initial simplex of the Nelder-Mead optimization because this required at least a vague knowledge of the response surface.
However, it can be a reasonable policy to set the points in such a way that the simplex covers virtually the entire possible range. The algorithm of Nelder-Mead will shrink automatically the simplex and aproximate to the optimum. The practical advantage of this policy is that you will obtain a better overall-knowledge of the response-function.
We have done some tests with HillStormer("http://www.berkutec.com"). This program permits to test these policies on testfunctons and we found that this plicy works rather well.
Please remember that the first simplex-opereation is añways a reflection. If the starting simplex covers the whole permitted range the reflection necessarily will give a point off limits. But HillStormer allows to use linear constraints and can avoid this problem.
You can find some more information in the system-help of HillStormer.
B. Kühne
来源：https://stackoverflow.com/questions/17928010/choosing-the-initial-simplex-in-the-nelder-mead-optimization-algorithm