NAME

       PDL::Opt::Simplex -- Simplex optimization routines

SYNOPSIS

         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 = $vec->slice('(0)');
           # The parabola (x+3)^2 - 5 has a minimum at x=-3:
           return (($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 = $vec->slice('(0)');
           # each vector element passed to log() has a min and max value.
           # ie: x=[6 0] -> vals=[76 4]
           # so, from above: f(6) == 76 and f(0) == 4
           print "$count [$ssize]: $x -> $vals\n";
         }
         my ($optimum, $ssize, $optval) = simplex(pdl(30), 3, 1e-6, 100, \&f, \&log);
         print "ssize=$ssize  opt=$optimum -> minimum=$optval\n";

DESCRIPTION

       This package implements the commonly used simplex optimization algorithm. The basic idea of the algorithm
       is to move a "simplex" 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.

       $init is a 1D vector holding the initial values of the N fitted parameters, $optimum is a vector holding
       the final values.  $optval is the evaluation of the final values.

       $initsize is the size of $init. It is only used if your supplied $init is a single point in your search
       space, to construct the simplex ("cloud") of N+1 points the algorithm uses, being the distance away from
       your single $init point along each dimension. This is done by the exportable function
       "make_simplex($init, $initsize)", e.g.:

         pdl> use PDL::Opt::Simplex
         pdl> p $t = make_simplex(pdl(0,0,0), pdl(0.12,0.12,0.12))
         [
          [    0 -0.06 -0.08]
          [ 0.12 -0.06 -0.08]
          [    0  0.06 -0.08]
          [    0     0  0.04]
         ]
         pdl> use PDL::Graphics::TriD
         pdl> spheres3d $t # spheres not points so can easily see

       $minsize is the convergence criterion, e.g. $minsize = 1e-6; the algorithm will terminate when all the
       values of $ssize are less than $minsize.

       The sub is assumed to understand more than 1 dimensions and broadcasting.  Its signature is
       "inp(nparams); [ret]out()". An example would be

               sub evaluate_func_at {
                       my($xv) = @_;
                       my ($x1, $x2) = $xv->using(0,1);
                       return $x1**4 + ($x2-5)**4 + $x1*$x2;
               }

       Here $xv is a vector holding the current values of the parameters being fitted which are then sliced out
       explicitly as $x1 and $x2.

       $ssize gives a very very approximate estimate of how close we might be - it might be miles wrong. It is
       the largest Euclidean distance between the first vertex and any other. If it is not very small, the
       algorithm has not converged.

FUNCTIONS

   simplex
       Simplex optimization routine

       Mutates its $init input if given as a full simplex (dims "n,n+1").

        ($optimum,$ssize,$optval) = simplex($init,$initsize,$minsize,
                        $maxiter,
                        sub {evaluate_func_at($_[0])},
                        sub {display_simplex($_[0])}
                        );

       See module "PDL::Opt::Simplex" for more information.

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.

       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).

SEE ALSO

       PDL::Opt::Simplex::Simple - Use names for Simplex-optimized values
       PDL::Opt::ParticleSwarm - A PDL implementation of Particle Swarm
       PDL::Opt::ParticleSwarm::Simple - Use names for Particle Swarm-optimized values
       <https://web.archive.org/web/19981206200518/http://chem1.nrl.navy.mil/~shaffer/chemoweb.html> - Ron
       Shaffer's chemometrics web page and references therein (archive from 1998)

       The demonstration (Examples/Simplex/tsimp.pl and tsimp2.pl).

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 COPYING in
       the PDL distribution. If this file is separated from the PDL distribution, the copyright notice should be
       included in the file.

perl v5.40.0                                       2025-01-17                                       Simplex(3pm)