8 Dec posting: simplex - fix #30

statocles-site/blog/2024/12/08/simplex/index.markdown

@@ -0,0 +1,121 @@
+---
+title: Day 8: Simplex optimisation with Perl Data Language
+disable_content_template: 1
+tags:
+    - optimisation
+    - simplex
+author: Ed J
+images:
+  banner:
+    src: './banner.jpg'
+    alt: 'Simplex optimisation'
+    data:
+      attribution: |-
+        <a href="https://commons.wikimedia.org/wiki/File:Simplex-description-en.svg">Graphical description of how the simplex method can solve linear programming problems</a> under Creative Commons Attribution-Share Alike 3.0 Unported
+data:
+  bio: mohawk
+  description: 'Simplex optimisation with PDL (Perl Data Language)'
+---
+
+If you want to solve linear programming problems, you need [PDL](https://pdl.perl.org/)!
+---
+
+This article discusses
+[PDL::Opt::Simplex](https://metacpan.org/pod/PDL::Opt::Simplex), and uses
+[PDL::Graphics::Simple](https://metacpan.org/pod/PDL::Graphics::Simple)
+to display output.
+
+The simplex algorithm finds the optimum "point" (coordinates) in a space
+you define, which can have any number (called `n` here) of dimensions.
+
+The algorithm takes either a fully-formed cloud of n+1 points, or a
+single starting point, in which case it constructs the cloud for you
+using the "initsize" parameter. It also takes a function that will take a
+series of points in your space, and returns the "value" at each of those
+points. From that, it works out which point of the simplex to move to be
+closer to the optimum point, which has the lowest value of your function.
+
+It also takes other, less important parameters, which you'll see,
+including a "logging" function which you can use to report progress,
+or plot data.
+
+Load the necessary modules, set up a plotting window:
+
+    use PDL::Graphics::Simple;
+    use PDL::Opt::Simplex;
+    $w = pgswin();
+
+Try a simple ellipsoid; the multiplier makes the algorithm prioritise
+the first (X) dimension, as you'll see on the plot.
+
+    $w->plot(with=>'lines', [0], [1], {xrange=>[-15,5],yrange=>[-15,5]});
+    my $mult = pdl 4,1;
+    sub func { (($mult * $_[0]) ** 2)->sumover }
+    sub logs {
+      $w->oplot(with=>'lines', $_[0]->glue(1,$_[0]->slice(",0"))->using(0,1));
+    }
+    simplex(pdl(-10,-10), 0.5, 0.01, 30,
+      \&func,
+      \&logs
+    );
+
+<img src="/../images/demos/simplex/vid-1.gif"/>
+
+Now the first of two examples contributed by Alison Offer.
+These values are for both:
+
+    $minsize = 1.e-6; # convergence: if simplex points are this far apart, stop
+    $maxiter = 100; # max number of iterations: if done these many, stop
+
+Now we minimise polynomial: (x-3)^2 + 2\*(x-3)\*(y-2.5) + 3\*(y-2.5)^2
+
+    $w->plot(with=>'lines', [0], [1], {xrange=>[-1,5],yrange=>[-1,4]}); # reset
+    $init = pdl [ 0 , 1 ];
+    $initsize = 2;
+    ($optimum,$ssize,$optval) = simplex($init,$initsize,$minsize,$maxiter,
+      sub {
+        my ($x, $y) = $_[0]->using(0,1);
+        ($x-3)**2 + 2*($x-3)*($y-2.5) + 3*($y-2.5)**2;
+      },
+      \&logs
+    );
+
+<img src="/../images/demos/simplex/vid-2.gif"/>
+
+Now to minimise least squares Gaussian fit to data + noise:
+32 *exp (-((x-10)/6)^2) + noise
+
+    $factor = 3; # noise factor
+
+data : gaussian + noise
+
+    $j = sequence(20); srandom(5);
+    $data = 32*exp(-(($j-10)/6)**2) + $factor * (random(20) - 0.5);
+    $init = pdl [ 33, 9, 12 ];
+    $initsize = 2;
+
+The plotting will flatten, i.e. ignore, the third dimension in the vectors.
+
+    $w->plot(with=>'lines', [0], [1], {xrange=>[30,36],yrange=>[7,12]}); # reset
+    ($optimum,$ssize,$optval) = simplex($init,$initsize,$minsize,$maxiter,
+      sub {
+        my ($x, $y, $z) = map $_[0]->slice($_), 0..2;
+        (($data - $x*exp(-(($j-$y)/$z)**2))**2)->sumover;
+      },
+      \&logs
+    );
+
+<img src="/../images/demos/simplex/vid-3.gif"/>
+
+## Further resources
+
+There are other optimisation modules for PDL, including:
+
+- [PDL::Opt::GLPK](https://metacpan.org/pod/PDL::Opt::GLPK)
+- [PDL::Opt::NonLinear](https://metacpan.org/pod/PDL::Opt::NonLinear)
+- [PDL::Opt::QP](https://metacpan.org/pod/PDL::Opt::QP)
+- [PDL::Opt::ParticleSwarm](https://metacpan.org/pod/PDL::Opt::ParticleSwarm)
+
+Take a look at
+[the Wikipedia page for the simplex algorithm](https://en.wikipedia.org/wiki/Simplex_algorithm)
+for more on the subject.

statocles-site/page/index.markdown

@@ -30,6 +30,9 @@ data:
     - date: 2018-12-06
     - date: 2018-12-07
     - date: 2018-12-08
+      title: Simplex optimisation
+      href: '/blog/2024/12/08/simplex/'
+      image: '/blog/2024/12/08/simplex/banner.jpg'
     - date: 2018-12-09
     - date: 2018-12-10
     - date: 2018-12-11