multimin.texi

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@cindex minimization, multidimensional

This chapter describes routines for finding minima of arbitrary
multidimensional functions.  The library provides low level components
for a variety of iterative minimizers and convergence tests.  These can
be combined by the user to achieve the desired solution, while providing
full access to the intermediate steps of the algorithms.  Each class of
methods uses the same framework, so that you can switch between
minimizers at runtime without needing to recompile your program.  Each
instance of a minimizer keeps track of its own state, allowing the
minimizers to be used in multi-threaded programs. The minimization
algorithms can be used to maximize a function by inverting its sign.

The header file @file{gsl_multimin.h} contains prototypes for the
minimization functions and related declarations.  

@menu
* Multimin Overview::       
* Multimin Caveats::        
* Initializing the Multidimensional Minimizer::  
* Providing a function to minimize::  
* Multimin Iteration::      
* Multimin Stopping Criteria::  
* Multimin Algorithms::     
* Multimin Examples::       
* Multimin References and Further Reading::  
@end menu

@node Multimin Overview
@section Overview

The problem of multidimensional minimization requires finding a point
@math{x} such that the scalar function,

@tex
\beforedisplay
$$
f(x_1, \dots, x_n)
$$
\afterdisplay
@end tex
@ifinfo
@example
f(x_1, @dots{}, x_n)
@end example
@end ifinfo
@noindent
takes a value which is lower than at any neighboring point. For smooth
functions the gradient @math{g = \nabla f} vanishes at the minimum. In
general there are no bracketing methods available for the
minimization of @math{n}-dimensional functions.  All algorithms
proceed from an initial guess using a search algorithm which attempts
to move in a downhill direction. 

All algorithms making use of the gradient of the function perform a
one-dimensional line minimisation along this direction until the lowest
point is found to a suitable tolerance.  The search direction is then
updated with local information from the function and its derivatives,
and the whole process repeated until the true @math{n}-dimensional
minimum is found.

The Nelder-Mead Simplex algorithm applies a different strategy.  It
maintains @math{n+1} trial parameter vectors as the vertices of a
@math{n}-dimensional simplex.  In each iteration step it tries to
improve the worst vertex by a simple geometrical transformation until
the size of the simplex falls below a given tolerance.

Several minimization algorithms are available within a single
framework. The user provides a high-level driver for the algorithms, and
the library provides the individual functions necessary for each of the
steps.  There are three main phases of the iteration.  The steps are,

@itemize @bullet
@item
initialize minimizer state, @var{s}, for algorithm @var{T}

@item
update @var{s} using the iteration @var{T}

@item
test @var{s} for convergence, and repeat iteration if necessary
@end itemize

@noindent
Each iteration step consists either of an improvement to the
line-minimisation in the current direction or an update to the search
direction itself.  The state for the minimizers is held in a
@code{gsl_multimin_fdfminimizer} struct or a
@code{gsl_multimin_fminimizer} struct.

@node Multimin Caveats
@section Caveats
@cindex Multimin, caveats

Note that the minimization algorithms can only search for one local
minimum at a time.  When there are several local minima in the search
area, the first minimum to be found will be returned; however it is
difficult to predict which of the minima this will be.  In most cases,
no error will be reported if you try to find a local minimum in an area
where there is more than one.

It is also important to note that the minimization algorithms find local
minima; there is no way to determine whether a minimum is a global
minimum of the function in question.

@node Initializing the Multidimensional Minimizer
@section Initializing the Multidimensional Minimizer

The following function initializes a multidimensional minimizer.  The
minimizer itself depends only on the dimension of the problem and the
algorithm and can be reused for different problems.

@deftypefun {gsl_multimin_fdfminimizer *} gsl_multimin_fdfminimizer_alloc (const gsl_multimin_fdfminimizer_type *@var{T}, size_t @var{n})
@deftypefunx {gsl_multimin_fminimizer *} gsl_multimin_fminimizer_alloc (const gsl_multimin_fminimizer_type *@var{T}, size_t @var{n})
This function returns a pointer to a newly allocated instance of a
minimizer of type @var{T} for an @var{n}-dimension function.  If there
is insufficient memory to create the minimizer then the function returns
a null pointer and the error handler is invoked with an error code of
@code{GSL_ENOMEM}.
@end deftypefun

@deftypefun int gsl_multimin_fdfminimizer_set (gsl_multimin_fdfminimizer * @var{s}, gsl_multimin_function_fdf *@var{fdf}, const gsl_vector * @var{x}, double @var{step_size}, double @var{tol})
This function initializes the minimizer @var{s} to minimize the function
@var{fdf} starting from the initial point @var{x}.  The size of the
first trial step is given by @var{step_size}.  The accuracy of the line
minimization is specified by @var{tol}.  The precise meaning of this
parameter depends on the method used.  Typically the line minimization
is considered successful if the gradient of the function @math{g} is
orthogonal to the current search direction @math{p} to a relative
accuracy of @var{tol}, where @c{$p\cdot g < tol |p| |g|$} 
@math{dot(p,g) < tol |p| |g|}.

@deftypefunx int gsl_multimin_fminimizer_set (gsl_multimin_fminimizer * @var{s}, gsl_multimin_function *@var{f}, const gsl_vector * @var{x}, const gsl_vector * @var{step_size})
This function initializes the minimizer @var{s} to minimize the function
@var{f}, starting from the initial point
@var{x}. The size of the initial trial steps is given in vector
@var{step_size}. The precise meaning of this parameter depends on the
method used. 
@end deftypefun

@deftypefun void gsl_multimin_fdfminimizer_free (gsl_multimin_fdfminimizer *@var{s})
@deftypefunx void gsl_multimin_fminimizer_free (gsl_multimin_fminimizer *@var{s})
This function frees all the memory associated with the minimizer
@var{s}.
@end deftypefun

@deftypefun {const char *} gsl_multimin_fdfminimizer_name (const gsl_multimin_fdfminimizer * @var{s})
@deftypefunx {const char *} gsl_multimin_fminimizer_name (const gsl_multimin_fminimizer * @var{s})
This function returns a pointer to the name of the minimizer.  For example,

@example
printf ("s is a '%s' minimizer\n", 
        gsl_multimin_fdfminimizer_name (s));
@end example
@noindent
would print something like @code{s is a 'conjugate_pr' minimizer}.
@end deftypefun

@node Providing a function to minimize
@section Providing a function to minimize

You must provide a parametric function of @math{n} variables for the
minimizers to operate on.  You may also need to provide a routine which
calculates the gradient of the function and a third routine which
calculates both the function value and the gradient together.  In order
to allow for general parameters the functions are defined by the
following data type:

@deftp {Data Type} gsl_multimin_function_fdf
This data type defines a general function of @math{n} variables with
parameters and the corresponding gradient vector of derivatives,

@table @code
@item double (* f) (const gsl_vector * @var{x}, void * @var{params})
this function should return the result
@c{$f(x,\hbox{\it params})$}
@math{f(x,params)} for argument @var{x} and parameters @var{params}.

@item void (* df) (const gsl_vector * @var{x}, void * @var{params}, gsl_vector * @var{g})
this function should store the @var{n}-dimensional gradient
@c{$g_i = \partial f(x,\hbox{\it params}) / \partial x_i$}
@math{g_i = d f(x,params) / d x_i} in the vector @var{g} for argument @var{x} 
and parameters @var{params}, returning an appropriate error code if the
function cannot be computed.

@item void (* fdf) (const gsl_vector * @var{x}, void * @var{params}, double * f, gsl_vector * @var{g})
This function should set the values of the @var{f} and @var{g} as above,
for arguments @var{x} and parameters @var{params}.  This function provides
an optimization of the separate functions for @math{f(x)} and @math{g(x)} -- 
it is always faster to compute the function and its derivative at the
same time. 

@item size_t n
the dimension of the system, i.e. the number of components of the
vectors @var{x}.

@item void * params
a pointer to the parameters of the function.
@end table
@end deftp
@deftp {Data Type} gsl_multimin_function
This data type defines a general function of @math{n} variables with
parameters,

@table @code
@item double (* f) (const gsl_vector * @var{x}, void * @var{params})
this function should return the result
@c{$f(x,\hbox{\it params})$}
@math{f(x,params)} for argument @var{x} and parameters @var{params}.

@item size_t n
the dimension of the system, i.e. the number of components of the
vectors @var{x}.

@item void * params
a pointer to the parameters of the function.
@end table
@end deftp
@noindent
The following example function defines a simple paraboloid with two
parameters,

@example
/* Paraboloid centered on (dp[0],dp[1]) */

double
my_f (const gsl_vector *v, void *params)
@{
  double x, y;
  double *dp = (double *)params;
  
  x = gsl_vector_get(v, 0);
  y = gsl_vector_get(v, 1);
 
  return 10.0 * (x - dp[0]) * (x - dp[0]) +
           20.0 * (y - dp[1]) * (y - dp[1]) + 30.0; 
@}

/* The gradient of f, df = (df/dx, df/dy). */
void 
my_df (const gsl_vector *v, void *params, 
       gsl_vector *df)
@{
  double x, y;
  double *dp = (double *)params;
  
  x = gsl_vector_get(v, 0);
  y = gsl_vector_get(v, 1);
 
  gsl_vector_set(df, 0, 20.0 * (x - dp[0]));
  gsl_vector_set(df, 1, 40.0 * (y - dp[1]));
@}

/* Compute both f and df together. */
void 
my_fdf (const gsl_vector *x, void *params, 
        double *f, gsl_vector *df) 
@{
  *f = my_f(x, params); 
  my_df(x, params, df);
@}
@end example
@noindent
The function can be initialized using the following code,

@example
gsl_multimin_function_fdf my_func;

double p[2] = @{ 1.0, 2.0 @}; /* center at (1,2) */

my_func.f = &my_f;
my_func.df = &my_df;
my_func.fdf = &my_fdf;
my_func.n = 2;
my_func.params = (void *)p;
@end example

@node Multimin Iteration
@section Iteration

The following function drives the iteration of each algorithm.  The
function performs one iteration to update the state of the minimizer.
The same function works for all minimizers so that different methods can
be substituted at runtime without modifications to the code.

@deftypefun int gsl_multimin_fdfminimizer_iterate (gsl_multimin_fdfminimizer *@var{s})
@deftypefunx int gsl_multimin_fminimizer_iterate (gsl_multimin_fminimizer *@var{s})
These functions perform a single iteration of the minimizer @var{s}.  If
the iteration encounters an unexpected problem then an error code will
be returned.
@end deftypefun
@noindent
The minimizer maintains a current best estimate of the minimum at all
times.  This information can be accessed with the following auxiliary
functions,

@deftypefun {gsl_vector *} gsl_multimin_fdfminimizer_x (const gsl_multimin_fdfminimizer * @var{s})
@deftypefunx {gsl_vector *} gsl_multimin_fminimizer_x (const gsl_multimin_fminimizer * @var{s})
@deftypefunx double gsl_multimin_fdfminimizer_minimum (const gsl_multimin_fdfminimizer * @var{s})
@deftypefunx double gsl_multimin_fminimizer_minimum (const gsl_multimin_fminimizer * @var{s})
@deftypefunx {gsl_vector *} gsl_multimin_fdfminimizer_gradient (const gsl_multimin_fdfminimizer * @var{s})
@deftypefunx double gsl_multimin_fminimizer_size (const gsl_multimin_fminimizer * @var{s})
These functions return the current best estimate of the location of the
minimum, the value of the function at that point, its gradient, 
and minimizer specific characteristic size for the minimizer @var{s}.
@end deftypefun

@deftypefun int gsl_multimin_fdfminimizer_restart (gsl_multimin_fdfminimizer *@var{s})
This function resets the minimizer @var{s} to use the current point as a
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