State-machine simplex minimizer

A simple minimizer of in-line coded functions, for teaching and experimentation.

This is a single M-file that implements a Nelder-Mead simplex minimizer. It makes use of MATLAB's persistent variables to create a "state machine" implementation. This allows entire minimization programs to be written as MATLAB scripts, as it does not require a function to be defined and passed to the minimization routine. In my opinion, this makes the fitting of functions to data much more fun and educational!

The traditional approach to optimization is to pass a function to an optimizer, which then takes control of the program until it is done. For example, the function 'fun' is minimized by a call to
x = fminsearch(fun,x0,options).
The disadvantage of this approach is that 'fun' must be defined in an M-file, and inserting code to observe the progress of the minimization is made difficult by the different scopes of the calling program and 'fun'.

Simplex.m does not take control of the program, but instead provides a new vector of parameters to "try" with each iteration. Here is a simple example, that minimizes the sixth power of the difference between two vectors:

% Example of use: minimize (p-q).^6

p=[2 2 1.5 2]'; % inital guess
q=[1 1 1 1]'; % true values

p=Simplex('init',p);

for i=1:200
y=sum((p-q).^6); % Evaluation of the function
p=Simplex(y); % Simplex provides a new vector p

if ~mod(i, 10) % every 10 iterations print out the fitted value
p'
end;

end;

p=Simplex('centroid'); % get the final best value.

The first call to Simplex is an initialization, where the starting guess of the parameter vector is supplied. The function sum(p-q).^6 is evaluated in-line, and its result is given to Simplex; in turn, a new parameter vector p is returned. The progress of the minimization is easily monitored because the function evaluation remains in the scope of the main program.