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--- tutorial:ilp_tutorial [2010/05/04 14:33] – joswig
+++ tutorial:ilp_tutorial [2017/07/31 18:07] (current) – migrated overview to tutorial overview page oroehrig
@@ Line 1: / Line 1: @@
-===== Integer Linear Programming =====
-This tutorial shall give an introduction to ILP in ''polymake''. It is primarily aimed at students of the course "Discrete Optimization" at TU Darmstadt and based on a demo shown in the lecture.
-==== A first example ====
-First we will construct a new rational polytope:
-<code>
-polytope > $p=new Polytope<Rational>;
-polytope > $p->POINTS=<<".";
-polytope (2)> 1 0 0 0
-polytope (3)> 1 1 0 0
-polytope (4)> 1 0 1 0
-polytope (5)> 1 1 1 0
-polytope (6)> 1 0 0 1
-polytope (7)> 1 1 0 1
-polytope (8)> 1 0 1 1
-polytope (9)> 1 1 1 1
-polytope (10)> .
-</code>
-Note that points in ''polymake'' are always given in homogenous coordinates. I.e., the point (a,b,c) in R<sup>3</sup> is represented as ''1 a b c'' in ''polymake''.
-Now we can examine some properties of ''$p''. For instance we can determine the number of facets or whether ''$p'' is simple:
-<code>
-polytope > print $p->N_FACETS;
-polytope > print $p->SIMPLE;
-</code>
-As you might already have noticed, our polytope is just a 3-dimensional cube. So there would have been an easier way to create it using the client ''cube'':
-<code>
-polytope > $c = cube(3,0);
-</code>
-(You can check out the details of any function in the [[http://wwwopt.mathematik.tu-darmstadt.de/polymake_doku/2.9.8/|''polymake'' documentation]].)
-And we can also verify that the two polytopes are actually equal:
-<code>
-polytope > print equal_polyhedra($p,$c);
-</code>
-==== Another example ====
-Now let us proceed with a somewhat more interesting example: The convex hull of 20 randomly chosen points on the 2-dimensional sphere.
-{{ :tutorial:ilp:rand_sphere.png?200|}}
-<code>
-polytope > $rs = rand_sphere(3,20);
-</code>
-''polymake'' can of course visualise this polytope:
-<code>
-polytope > $rs->VISUAL;
-</code>
-Now we will create yet another new polytope by scaling our random sphere by a factor lambda. (Otherwise there are rather few integral points contained in it.)
-To this end, we have to multiply every coordinate (except for the homogenising 1 in the beginning) of every vertex by lamda. Then we can create a new polytope by specifying its vertices.
-{{ :tutorial:ilp:rand_sphere_lattice.png?200|}}
-<code>
-polytope > $lambda=2;
-polytope > $s=new Matrix<Rational>([[1,0,0,0],[0,$lambda,0,0],[0,0,$lambda,0],[0,0,0,$lambda]]);
-polytope > print $s;
-0 0 0
-2 0 0
-0 2 0
-0 0 2
-polytope > $scaled_rs=new Polytope<Rational>(VERTICES=>($rs->VERTICES * $s));
-</code>
-''polymake'' can visualise the polytope together with its lattice points:
-<code>
-polytope > $scaled_rs->VISUAL->LATTICE_COLORED;
-</code>
-Now will construct the integer hull of ''$scaled_rs'' and visualise it:
-{{ :tutorial:ilp:ilp_lattice.png?200|}}
-<code>
-polytope > $integer_hull=new Polytope<Rational>(POINTS=>$scaled_rs->LATTICE_POINTS);
-polytope > $integer_hull->VISUAL->LATTICE_COLORED;
-</code>
-In order to obtain the integer hull we simply define a new polytope ''$integer_hull'' as the convex hull of all ''LATTICE_POINTS'' contained in ''$scaled_rs''.
-Note that if we give ''POINTS'' (in contrast to ''VERTICES'') ''polymake'' constructs a polytope that is the convex hull of the given points regardless of whether they are vertices or not. I.e., redundacies are allowed here.
-If you specify ''VERTICES'' you have to make sure yourself that your points are actually vertices since ''polymake'' does not check this.
-==== Linear Programming ====
-Now that we have constructed a nice integral polytope we want to apply some linear program to it.
-First we define a ''LinearProgram'' with our favourite ''LINEAR_OBJECTIVE''. The linear objective is an given as a vector of length d+1, d being the dimension of the space. The vector [c<sub>0</sub>,c<sub>1</sub>, ..., c<sub>d</sub>] corresponds to the linear objective c<sub>0</sub> + c<sub>1</sub>x<sub>1</sub> + ... + c<sub>d</sub>x<sub>d</sub>.
-<code>
-polytope > $objective=new LinearProgram<Rational>(LINEAR_OBJECTIVE=>[0,1,1,1]);
-</code>
-Then we define a new polytope, which is a copy of our old one (''$inter_hull'') with the LP as an additional property.
-<code>
-polytope > $ilp=new Polytope<Rational>(VERTICES=>$integer_hull->VERTICES, LP=>$objective);
-</code>
-{{ :tutorial:ilp:ilp_min_face.png?200|}}
-{{ :tutorial:ilp:ilp_max_face.png?200|}}
-And now we can perform some computations:
-<code>
-polytope > print $ilp->LP->MAXIMAL_VALUE;
-polytope > print $ilp->LP->MAXIMAL_FACE;
-{6 9 10}
-polytope > $ilp->VISUAL->MIN_MAX_FACE;
-</code>
-Hence the LP attains its maximal value 2 on the  2-face spanned by the vertices 6, 9 and 10.
-''polymake'' can visualise the polytope and highlight both its maximal and minimal face in a different (by default admittedly almost painful ;-) ) colour. Here you see the maximal face ''{6 9 10}'' in red and the minimal face ''{0 3}'' (on the opposite side of the polytope) in yellow.
-Note though that since we started out with a random polytope these results may vary if we perform the same computations another time on a different random polytope.
-<code>
-polytope > print $ilp->VERTICES;
--1 0 -1
--1 0 1
--1 1 0
-0 -1 -1
-0 -1 1
-0 1 -1
-0 1 1
-1 -1 0
-1 0 -1
-1 0 1
-1 1 0
-</code>
-==== Hilbert bases ====
-Finally, we can have ''polymake'' compute and print a Hilbert basis for the cone spanned by ''$ilp''.  Notice that this requires normaliz or 4ti2 to be installed in order to work.
-<code>
-polytope > print $ilp->HILBERT_BASIS;
-0 0 -1
--1 1 0
-1 0 0
-0 1 0
-0 1 -1
-1 1 0
-0 1 1
-1 0 -1
-1 0 1
-0 0 0
-0 0 1
-1 -1 0
--1 0 -1
--1 0 0
--1 0 1
-0 -1 -1
-0 -1 0
-0 -1 1
-</code>