multimin.texi

math library from gnu

new starting point.
@end deftypefun

@node Multimin Stopping Criteria
@section Stopping Criteria

A minimization procedure should stop when one of the following
conditions is true:

@itemize @bullet
@item
A minimum has been found to within the user-specified precision.

@item
A user-specified maximum number of iterations has been reached.

@item
An error has occurred.
@end itemize

@noindent
The handling of these conditions is under user control.  The functions
below allow the user to test the precision of the current result.

@deftypefun int gsl_multimin_test_gradient (const gsl_vector * @var{g}, double @var{epsabs})
This function tests the norm of the gradient @var{g} against the
absolute tolerance @var{epsabs}. The gradient of a multidimensional
function goes to zero at a minimum. The test returns @code{GSL_SUCCESS}
if the following condition is achieved,
@tex
\beforedisplay
$$
|g| < \hbox{\it epsabs}
$$
\afterdisplay
@end tex
@ifinfo

@example
|g| < epsabs
@end example

@end ifinfo
@noindent
and returns @code{GSL_CONTINUE} otherwise.  A suitable choice of
@var{epsabs} can be made from the desired accuracy in the function for
small variations in @math{x}.  The relationship between these quantities
is given by @c{$\delta{f} = g\,\delta{x}$}
@math{\delta f = g \delta x}.
@end deftypefun

@deftypefun int gsl_multimin_test_size (const double @var{size}, double @var{epsabs})
This function tests the minimizer specific characteristic
size (if applicable to the used minimizer) against absolute tolerance @var{epsabs}. 
The test returns @code{GSL_SUCCESS} if the size is smaller than tolerance,
otherwise @code{GSL_CONTINUE} is returned.
@end deftypefun

@node Multimin Algorithms with Derivatives
@section Algorithms with Derivatives

There are several minimization methods available. The best choice of
algorithm depends on the problem.  The algorithms described in this
section use the value of the function and its gradient at each
evaluation point.

@deffn {Minimizer} gsl_multimin_fdfminimizer_conjugate_fr
@cindex Fletcher-Reeves conjugate gradient algorithm, minimization
@cindex Conjugate gradient algorithm, minimization
@cindex minimization, conjugate gradient algorithm
This is the Fletcher-Reeves conjugate gradient algorithm. The conjugate
gradient algorithm proceeds as a succession of line minimizations. The
sequence of search directions is used to build up an approximation to the
curvature of the function in the neighborhood of the minimum.  

An initial search direction @var{p} is chosen using the gradient, and line
minimization is carried out in that direction.  The accuracy of the line
minimization is specified by the parameter @var{tol}.  The minimum
along this line occurs when the function gradient @var{g} and the search direction
@var{p} are orthogonal.  The line minimization terminates when
@c{$p\cdot g < tol |p| |g|$} 
@math{dot(p,g) < tol |p| |g|}.  The
search direction is updated  using the Fletcher-Reeves formula
@math{p' = g' - \beta g} where @math{\beta=-|g'|^2/|g|^2}, and
the line minimization is then repeated for the new search
direction.
@end deffn

@deffn {Minimizer} gsl_multimin_fdfminimizer_conjugate_pr
@cindex Polak-Ribiere algorithm, minimization
@cindex minimization, Polak-Ribiere algorithm
This is the Polak-Ribiere conjugate gradient algorithm.  It is similar
to the Fletcher-Reeves method, differing only in the choice of the
coefficient @math{\beta}. Both methods work well when the evaluation
point is close enough to the minimum of the objective function that it
is well approximated by a quadratic hypersurface.
@end deffn

@deffn {Minimizer} gsl_multimin_fdfminimizer_vector_bfgs2
@deffnx {Minimizer} gsl_multimin_fdfminimizer_vector_bfgs
@cindex BFGS algorithm, minimization
@cindex minimization, BFGS algorithm
These methods use the vector Broyden-Fletcher-Goldfarb-Shanno (BFGS)
algorithm.  This is a quasi-Newton method which builds up an approximation
to the second derivatives of the function @math{f} using the difference
between successive gradient vectors.  By combining the first and second
derivatives the algorithm is able to take Newton-type steps towards the
function minimum, assuming quadratic behavior in that region.

The @code{bfgs2} version of this minimizer is the most efficient
version available, and is a faithful implementation of the line
minimization scheme described in Fletcher's @cite{Practical Methods of
Optimization}, Algorithms 2.6.2 and 2.6.4.  It supercedes the original
@code{bfgs} routine and requires substantially fewer function and
gradient evaluations.  The user-supplied tolerance @var{tol}
corresponds to the parameter @math{\sigma} used by Fletcher.  A value
of 0.1 is recommended for typical use (larger values correspond to
less accurate line searches).

@end deffn

@deffn {Minimizer} gsl_multimin_fdfminimizer_steepest_descent
@cindex steepest descent algorithm, minimization
@cindex minimization, steepest descent algorithm
The steepest descent algorithm follows the downhill gradient of the
function at each step. When a downhill step is successful the step-size
is increased by a factor of two.  If the downhill step leads to a higher
function value then the algorithm backtracks and the step size is
decreased using the parameter @var{tol}.  A suitable value of @var{tol}
for most applications is 0.1.  The steepest descent method is
inefficient and is included only for demonstration purposes.
@end deffn

@node Multimin Algorithms without Derivatives
@section Algorithms without Derivatives

The algorithms described in this section use only the value of the function
at each evaluation point.

@deffn {Minimizer} gsl_multimin_fminimizer_nmsimplex2
@deffnx {Minimizer} gsl_multimin_fminimizer_nmsimplex
@cindex Nelder-Mead simplex algorithm for minimization
@cindex simplex algorithm, minimization
@cindex minimization, simplex algorithm
These methods use the Simplex algorithm of Nelder and Mead. They construct
@math{n} vectors @math{p_i} from the
starting vector @var{x} and the vector @var{step_size} as follows:
@tex
\beforedisplay
$$
\eqalign{
p_0 & = (x_0, x_1, \cdots , x_n) \cr
p_1 & = (x_0 + step\_size_0, x_1, \cdots , x_n) \cr
p_2 & = (x_0, x_1 + step\_size_1, \cdots , x_n) \cr
\dots &= \dots \cr
p_n & = (x_0, x_1, \cdots , x_n+step\_size_n) \cr
}
$$
\afterdisplay
@end tex
@ifinfo

@example
p_0 = (x_0, x_1, ... , x_n) 
p_1 = (x_0 + step_size_0, x_1, ... , x_n) 
p_2 = (x_0, x_1 + step_size_1, ... , x_n) 
... = ...
p_n = (x_0, x_1, ... , x_n+step_size_n)
@end example

@end ifinfo
@noindent
These vectors form the @math{n+1} vertices of a simplex in @math{n}
dimensions.  On each iteration the algorithm tries to improve
the parameter vector @math{p_i} corresponding to the highest
function value by simple geometrical transformations.  These
are reflection, reflection followed by expansion, contraction and multiple
contraction. Using these transformations the simplex moves through 
the parameter space towards the minimum, where it contracts itself.  

After each iteration, the best vertex is returned.  Note, that due to
the nature of the algorithm not every step improves the current
best parameter vector.  Usually several iterations are required.

The minimizer-specific characteristic size is calculated 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 @code{gsl_multimin_fminimizer_size}.

The @code{nmsimplex2} version of this minimiser is a new @math{O(N)}
implementation of the earlier @math{O(N^2)} @code{nmsimplex}
minimiser.  It calculates the size of simplex as the @sc{rms} distance
of each vertex from the center rather than the mean distance, which
has the advantage of allowing a linear update.
@end deffn

@node Multimin Examples
@section Examples

This example program finds the minimum of the paraboloid function
defined earlier.  The location of the minimum is offset from the origin
in @math{x} and @math{y}, and the function value at the minimum is
non-zero. The main program is given below, it requires the example
function given earlier in this chapter.

@smallexample
@verbatiminclude examples/multimin.c
@end smallexample  

@noindent
The initial step-size is chosen as 0.01, a conservative estimate in this
case, and the line minimization parameter is set at 0.0001.  The program
terminates when the norm of the gradient has been reduced below
0.001. The output of the program is shown below,

@example
@verbatiminclude examples/multimin.out
@end example

@noindent
Note that the algorithm gradually increases the step size as it
successfully moves downhill, as can be seen by plotting the successive
points.

@iftex
@sp 1
@center @image{multimin,3.4in}
@end iftex

@noindent
The conjugate gradient algorithm finds the minimum on its second
direction because the function is purely quadratic. Additional
iterations would be needed for a more complicated function.

Here is another example using the Nelder-Mead Simplex algorithm to
minimize the same example object function, as above.

@smallexample
@verbatiminclude examples/nmsimplex.c
@end smallexample

@noindent
The minimum search stops when the Simplex size drops to 0.01. The output is
shown below.

@example
@verbatiminclude examples/nmsimplex.out
@end example

@noindent
The simplex size first increases, while the simplex moves towards the
minimum. After a while the size begins to decrease as the simplex
contracts around the minimum.

@node Multimin References and Further Reading
@section References and Further Reading

The conjugate gradient and BFGS methods are described in detail in the
following book,

@itemize @asis
@item R. Fletcher,
@cite{Practical Methods of Optimization (Second Edition)} Wiley
(1987), ISBN 0471915475.
@end itemize

A brief description of multidimensional minimization algorithms and
more recent references can be found in,

@itemize @asis
@item C.W. Ueberhuber,
@cite{Numerical Computation (Volume 2)}, Chapter 14, Section 4.4
``Minimization Methods'', p.@: 325--335, Springer (1997), ISBN
3-540-62057-5.
@end itemize

@noindent
The simplex algorithm is described in the following paper, 

@itemize @asis
@item J.A. Nelder and R. Mead,
@cite{A simplex method for function minimization}, Computer Journal
vol.@: 7 (1965), 308--315.
@end itemize

@noindent