user_guide:tutorials:ilp_and_hilbertbases

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user_guide:ilp_and_hilbertbases [2019/01/25 09:27] – ↷ Links adapted because of a move operation oroehriguser_guide:tutorials:ilp_and_hilbertbases [2019/02/04 22:55] (current) – external edit 127.0.0.1
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-===== ILP and Hilbert bases ===== +{{page>.:latest:@FILEID@}}
- +
-==== 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. +
-{{ user_guide: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. +
- +
-{{ user_guide: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; +
-1 0 0 0 +
-0 2 0 0 +
-0 0 2 0 +
-0 0 0 2 +
- +
-polytope > $scaled_rs=new Polytope<Rational>(VERTICES=>($rs->VERTICES * $s), LINEALITY_SPACE=>[]); +
-</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: +
- +
-{{ user_guide: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. You also need to specify the ''LINEALITY_SPACE'', see [[user_guide: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 [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> +
-{{ user_guide:ilp:ilp_min_face.png?200|}} +
-{{ user_guide: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 $ilp->VERTICES; +
-1 -1 0 -1 +
-1 -1 0 1 +
-1 -1 1 0 +
-1 0 -1 -1 +
-1 0 -1 1 +
-1 0 1 -1 +
-1 0 1 1 +
-1 1 -1 0 +
-1 1 0 -1 +
-1 1 0 1 +
-1 1 1 0 +
-</code> +
- +
-==== Hilbert bases ==== +
-Finally, we can have ''polymake'' compute and print a Hilbert basis for the cone spanned by ''$ilp'' Notice that this requires normaliz or 4ti2 to be installed in order to work. +
-<code> +
-polytope > print $ilp->HILBERT_BASIS; +
-1 0 0 -1 +
-1 -1 1 0 +
-1 1 0 0 +
-1 0 1 0 +
-1 0 1 -1 +
-1 1 1 0 +
-1 0 1 1 +
-1 1 0 -1 +
-1 1 0 1 +
-1 0 0 0 +
-1 0 0 1 +
-1 1 -1 0 +
-1 -1 0 -1 +
-1 -1 0 0 +
-1 -1 0 1 +
-1 0 -1 -1 +
-1 0 -1 0 +
-1 0 -1 1 +
-</code>+
  
  • user_guide/tutorials/ilp_and_hilbertbases.1548408447.txt.gz
  • Last modified: 2019/01/25 09:27
  • by oroehrig