`fminsearch`

Algorithm

`fminsearch`

uses the Nelder-Mead simplex algorithm as described in
Lagarias et al. [57]. This algorithm uses a simplex of *n* + 1 points for
*n*-dimensional vectors *x*. The algorithm first
makes a simplex around the initial guess *x*_{0} by
adding 5% of each component
*x*_{0}(*i*) to
*x*_{0}, and using these *n*
vectors as elements of the simplex in addition to
*x*_{0}. (The algorithm uses 0.00025 as
component *i* if *x*_{0}(*i*) = 0.) Then, the algorithm modifies the simplex repeatedly according to the
following procedure.

**Note**

The keywords for the `fminsearch`

iterative
display appear in **bold** after the
description of the step.

Let

*x*(*i*) denote the list of points in the current simplex,*i*= 1,...,*n*+ 1.Order the points in the simplex from lowest function value

*f*(*x*(1)) to highest*f*(*x*(*n*+ 1)). At each step in the iteration, the algorithm discards the current worst point*x*(*n*+ 1), and accepts another point into the simplex. [Or, in the case of step 7 below, it changes all*n*points with values above*f*(*x*(1))].Generate the

*reflected*point*r*= 2*m*–*x*(*n*+ 1),where

*m*= Σ*x*(*i*)/*n*,*i*= 1...*n*,and calculate

*f*(*r*).If

*f*(*x*(1)) ≤*f*(*r*) <*f*(*x*(*n*)), accept*r*and terminate this iteration.**Reflect**If

*f*(*r*) <*f*(*x*(1)), calculate the expansion point*s**s*=*m*+ 2(*m*–*x*(*n*+ 1)),and calculate

*f*(*s*).If

*f*(*s*) <*f*(*r*), accept*s*and terminate the iteration.**Expand**Otherwise, accept

*r*and terminate the iteration.**Reflect**

If

*f*(*r*) ≥*f*(*x*(*n*)), perform a*contraction*between*m*and either*x*(*n*+ 1) or*r*, depending on which has the lower objective function value.If

*f*(*r*) <*f*(*x*(*n*+ 1)) (that is,*r*is better than*x*(*n*+ 1)), calculate*c*=*m*+ (*r*–*m*)/2and calculate

*f*(*c*). If*f*(*c*) <*f*(*r*), accept*c*and terminate the iteration.**Contract outside**Otherwise, continue with Step 7 (Shrink).

If

*f*(*r*) ≥*f*(*x*(*n*+ 1)), calculate*cc*=*m*+ (*x*(*n*+ 1) –*m*)/2and calculate

*f*(*cc*). If*f*(*cc*) <*f*(*x*(*n*+ 1)), accept*cc*and terminate the iteration.**Contract inside**Otherwise, continue with Step 7 (Shrink).

Calculate the

*n*points*v*(*i*) =*x*(1) + (*x*(*i*) –*x*(1))/2and calculate

*f*(*v*(*i*)),*i*= 2,...,*n*+ 1. The simplex at the next iteration is*x*(1),*v*(2),...,*v*(*n*+ 1).**Shrink**

The following figure shows the points that `fminsearch`

might
calculate in the procedure, along with each possible new simplex.
The original simplex has a bold outline. The iterations proceed until
they meet a stopping criterion.