There are several minimization methods available. The best choice of algorithm depends on the problem. All of the algorithms uses the value of the function and most of its gradient at each evaluation point, too.
After each iteration, the best vertex is returned. Note, that due to the nature of the algorithm (getting rid of the worst estimate), every iteration doesn't necessarily improve the current best parameter vector. Usually several iterations are required.
The routine calculates the minimizer specific characteristic size as the
average distance from the geometrical center of the simplex to all its
vertices. This size can be used as a stopping criteria, as the simplex
contracts itself near the minimum. The size is returned by the function
gsl_multimin_fminimizer_size
.