rs

===== ILP and Hilbert bases ===== ==== A first example ==== First we will construct a new rational polytope:


> $p=new Polytope;
> $p->POINTS=<<".";
> 1 0 0 0
> 1 1 0 0
> 1 0 1 0
> 1 1 1 0
> 1 0 0 1
> 1 1 0 1
> 1 0 1 1
> 1 1 1 1
> .

Note that points in ''%%polymake%%'' are always given in homogenous coordinates. I.e., the point (a,b,c) in R³ 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:


> print $p->N_FACETS;
6
> print $p->SIMPLE;
true

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%%'':


> $c = cube(3,0);

(You can check out the details of any function in the [[documentation:latest:polytope|documentation]].) And we can also verify that the two polytopes are actually equal:


> print equal_polyhedra($p,$c);
true

==== 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.


> $rs = rand_sphere(3,20);

''%%polymake%%'' can of course visualise this polytope:


> $rs->VISUAL;

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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.


> $lambda=2;
> $s=new Matrix([[1,0,0,0],[0,$lambda,0,0],[0,0,$lambda,0],[0,0,0,$lambda]]);
> print $s;
1 0 0 0
0 2 0 0
0 0 2 0
0 0 0 2
> $scaled_rs=new Polytope(VERTICES=>($rs->VERTICES * $s), LINEALITY_SPACE=>[]);

''%%polymake%%'' can visualise the polytope together with its lattice points:


> $scaled_rs->VISUAL->LATTICE_COLORED;

scaled_rs

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Now will construct the integer hull of ''%%$scaled_rs%%'' and visualise it:


> $integer_hull=new Polytope(POINTS=>$scaled_rs->LATTICE_POINTS);
> $integer_hull->VISUAL->LATTICE_COLORED;

integer_hull

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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. You also need to specify the ''%%LINEALITY_SPACE%%'', see [[apps_polytope|Tutorial on polytopes]]. ==== 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 [c₀,c₁, ..., c_d] corresponds to the linear objective c₀ + c₁x₁ + ... + c_dx_d.


> $objective=new LinearProgram(LINEAR_OBJECTIVE=>[0,1,1,1]);

Then we define a new polytope, which is a copy of our old one (''%%$inter_hull%%'') with the LP as an additional property.


> $ilp=new Polytope(VERTICES=>$integer_hull->VERTICES, LP=>$objective);

{{:tutorials:release:4.8:ilp_and_hilbertbases:ilp_min_face.png|ilp_min_face.png}} {{:tutorials:release:4.8:ilp_and_hilbertbases:ilp_max_face.png|ilp_max_face.png}} And now we can perform some computations:


> print $ilp->LP->MAXIMAL_VALUE;
3
> print $ilp->LP->MAXIMAL_FACE;
{11}
> $ilp->VISUAL->MIN_MAX_FACE;

ilp

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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.


> print $ilp->VERTICES;
1 -1 -1 -1
1 -1 -1 0
1 -1 0 1
1 -1 1 0
1 -1 1 1
1 0 -1 0
1 0 -1 1
1 0 1 -1
1 1 0 0
1 1 0 1
1 1 1 -1
1 1 1 1

==== 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.


> print $ilp->HILBERT_BASIS;
1 -1 -1 -1
1 -1 -1 0
1 -1 0 0
1 -1 0 1
1 -1 1 0
1 -1 1 1
1 0 -1 0
1 0 -1 1
1 0 0 -1
1 0 0 0
1 0 0 1
1 0 1 -1
1 0 1 0
1 0 1 1
1 1 0 0
1 1 0 1
1 1 1 -1
1 1 1 0
1 1 1 1