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.IX Title "Simplex 3pm"
.TH Simplex 3pm "2023-06-17" "perl v5.36.0" "User Contributed Perl Documentation"
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.SH "NAME"
PDL::Opt::Simplex \-\- Simplex optimization routines
.SH "SYNOPSIS"
.IX Header "SYNOPSIS"
.Vb 1
\&  use PDL::Opt::Simplex;
\&
\&  ($optimum,$ssize,$optval) = simplex($init,$initsize,$minsize,
\&                 $maxiter,
\&                 sub {evaluate_func_at($_[0])},
\&                 sub {display_simplex($_[0])}
\&                 );
\&
\&  # more involved:
\&  use PDL;
\&  use PDL::Opt::Simplex;
\&
\&  my $count = 0;
\&  # find value of $x that returns a minimum
\&  sub f {
\&    my ($vec) = @_;
\&    $count++;
\&    my $x = $vec\->slice(\*(Aq(0)\*(Aq);
\&    # The parabola (x+3)^2 \- 5 has a minima at x=\-3:
\&    return (($x+3)**2 \- 5);
\&  }
\&
\&  sub log {
\&    my ($vec, $vals, $ssize) = @_;
\&    # $vec is the array of values being optimized
\&    # $vals is f($vec)
\&    # $ssize is the simplex size, or roughly, how close to being converged.
\&    my $x = $vec\->slice(\*(Aq(0)\*(Aq);
\&    # each vector element passed to log() has a min and max value.
\&    # ie: x=[6 0] \-> vals=[76 4]
\&    # so, from above: f(6) == 76 and f(0) == 4
\&    print "$count [$ssize]: $x \-> $vals\en";
\&  }
\&
\&  my $vec_initial = pdl [30];
\&  my ( $vec_optimal, $ssize, $optval ) = simplex($vec_initial, 3, 1e\-6, 100, \e&f, \e&log);
\&  my $x = $vec_optimal\->slice(\*(Aq(0)\*(Aq);
\&  print "ssize=$ssize  opt=$x \-> minima=$optval\en";
.Ve
.SH "DESCRIPTION"
.IX Header "DESCRIPTION"
This package implements the commonly used simplex optimization
algorithm. The basic idea of the algorithm is to move
a \*(L"simplex\*(R" of N+1 points in the N\-dimensional search space
according to certain rules. The main
benefit of the algorithm is that you do not need to calculate
the derivatives of your function.
.PP
\&\f(CW$init\fR is a 1D vector holding the initial values of the N fitted
parameters, \f(CW$optimum\fR is a vector holding the final solution.
\&\f(CW$optval\fR is the evaluation of the final solution.
.PP
\&\f(CW$initsize\fR is the size of \f(CW$init\fR (more...)
.PP
\&\f(CW$minsize\fR is some sort of convergence criterion (more...)
\&\- e.g. \f(CW$minsize\fR = 1e\-6
.PP
The sub is assumed to understand more than 1 dimensions and broadcasting.
Its signature is 'inp(nparams); [ret]\fBout()\fR'. An example would be
.PP
.Vb 6
\&        sub evaluate_func_at {
\&                my($xv) = @_;
\&                my $x1 = $xv\->slice("(0)");
\&                my $x2 = $xv\->slice("(1)");
\&                return $x1**4 + ($x2\-5)**4 + $x1*$x2;
\&        }
.Ve
.PP
Here \f(CW$xv\fR is a vector holding the current values of the parameters
being fitted which are then sliced out explicitly as \f(CW$x1\fR and \f(CW$x2\fR.
.PP
\&\f(CW$ssize\fR gives a very very approximate estimate of how close we might
be \- it might be miles wrong. It is the euclidean distance between
the best and the worst vertices. If it is not very small, the algorithm
has not converged.
.SH "FUNCTIONS"
.IX Header "FUNCTIONS"
.SS "simplex"
.IX Subsection "simplex"
Simplex optimization routine
.PP
.Vb 5
\& ($optimum,$ssize,$optval) = simplex($init,$initsize,$minsize,
\&                 $maxiter,
\&                 sub {evaluate_func_at($_[0])},
\&                 sub {display_simplex($_[0])}
\&                 );
.Ve
.PP
See module \f(CW\*(C`PDL::Opt::Simplex\*(C'\fR for more information.
.SH "CAVEATS"
.IX Header "CAVEATS"
Do not use the simplex method if your function has local minima.
It will not work. Use genetic algorithms or simulated annealing
or conjugate gradient or momentum gradient descent.
.PP
They will not really work either but they are not guaranteed not to work ;)
(if you have infinite time, simulated annealing is guaranteed to work
but only after it has visited every point in your space).
.SH "SEE ALSO"
.IX Header "SEE ALSO"
Ron Shaffer's chemometrics web page and references therein:
\&\f(CW\*(C`http://chem1.nrl.navy.mil/~shaffer/chemoweb.html\*(C'\fR.
.PP
Numerical Recipes (bla bla bla \s-1XXX\s0 ref).
.PP
The demonstration (Examples/Simplex/tsimp.pl and tsimp2.pl).
.SH "AUTHOR"
.IX Header "AUTHOR"
Copyright(C) 1997 Tuomas J. Lukka. 
All rights reserved. There is no warranty. You are allowed
to redistribute this software / documentation under certain
conditions. For details, see the file \s-1COPYING\s0 in the \s-1PDL\s0 
distribution. If this file is separated from the \s-1PDL\s0 distribution, 
the copyright notice should be included in the file.