Chapter 28 Optimization

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Nelder-Mead (Simplex) Method

A totally different method that is quite commonly used in nonlinear regression programs is the Nelder-Mead or Simplex method. It is computationally quite simple, other than the calculation of the WSS. The method works with a number of rules. The starting point is used to construct a simplest, a shape with m+1 points, where m is the number of parameters. Thus for a two parameter problem there are three points, a triangle. The program calculates the WSS at each point of the simplex on the WSS surface.

The Rules

Reflect the point with the highest WSS through centroid (center) of the simplex
If this produces the lowest WSS (best point) expand the simplex and reflect further
If this is just a good point start at the top and reflect again
If this the highest WSS (worst point) compress the simplex and reflect closer

These rules are repeated until the convergence criteria are meet. The simplex moves over WSS surface and should contracts around minimum.

Figure 11.6.1 Contour map of kel versus V showing probable Path of Simplex Method

The simplex method is relatively robust and numerically less complicated but it can be inefficient (slow) for simple problems. This method is recommended as a starting point with Boomer. Actually the method I use most of the time involves starting with the simplex method and (automatically) continuing with the Damping-Gauss-Newton method. It seems work well much of the time.

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