The problem of finding an unconstrained minimizer of an n-dimensional function, f, may be stated as follows:

given f: Rn R, find x* (an element of Rn) such that f(x*) is a minimum of f.

For example:    f (x) = (x0 – 3)4 + (x1 - 2)2
x* = [3, 2] In minimizing an n-dimensional function f, it is a necessary condition that the gradient at the minimizer x*, — f(x*), be the zero vector. Mathematically expressing this condition defines the following system of nonlinear equations. This relation might suggest that finding a minimizer is equivalent to solving a system of linear equations based on the gradient. In most cases, however, this is not true. It is just as likely that a solution, x*, of — f(x)=0 be a maximizer or a local minimizer of f. Thus the gradient alone does not provide sufficient information in determining the role of x*.

IDL provides two algorithms that do sufficiently determine the global minimizer of an n-dimensional function. IDL's DFPMIN routine is among a class of algorithms known as variable metric methods and requires a user-supplied analytic gradient of the function to be minimized. IDL's POWELL routine implements a direction-set method that does not require a user-supplied analytic gradient. The utility of the POWELL routine is evident as the function to be minimized becomes more complicated and partial derivatives become more difficult to calculate.
Routines for Optimization

See Optimization (in the functional category Mathematics) for a brief description of IDL routines for optimization. Detailed information is available in the IDL Reference Guide.


Source: NASA.gov