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tutorial:optimization [2013/08/26 16:55] – added example for lift-and-project pfetsch | user_guide:optimization [2019/01/25 09:27] – ↷ Links adapted because of a move operation oroehrig | ||
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There are several other tutorials that cover similar topics: | There are several other tutorials that cover similar topics: | ||
- | * [[tutorial: | + | * [[user_guide: |
- | * [[tutorial: | + | * [[user_guide: |
- | * [[tutorial: | + | * [[user_guide: |
- | * [[tutorial: | + | * [[user_guide: |
This tutorial is targeted towards the optimization community, since, surprisingly, | This tutorial is targeted towards the optimization community, since, surprisingly, | ||
+ | You can find files of the example LPs in the folder '' | ||
===== Input: lp2poly ===== | ===== Input: lp2poly ===== | ||
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</ | </ | ||
- | Thus, the file describes a 0/1-cube in three dimensions. It should be easy to adapt this format to other cases (If for example '' | + | Thus, the file describes a 0/1-cube in three dimensions. It should be easy to adapt this format to other cases (If for example '' |
- | + | ||
- | Now assume that this example is contained in file '' | + | |
< | < | ||
- | polytope > $f=lp2poly(' | + | polytope > $f=lp2poly(' |
</ | </ | ||
The polytope '' | The polytope '' | ||
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1 0 1 1 | 1 0 1 1 | ||
</ | </ | ||
- | This vertex corresponds to setting '' | + | This vertex corresponds to setting '' |
< | < | ||
polytope > print $p-> | polytope > print $p-> | ||
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polytope > $p-> | polytope > $p-> | ||
</ | </ | ||
- | {{ :tutorial: | + | {{ user_guide: |
The minimal and maximal faces can be visualized via | The minimal and maximal faces can be visualized via | ||
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polytope > $p-> | polytope > $p-> | ||
</ | </ | ||
- | {{ :tutorial: | + | {{ user_guide: |
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=== Bounded Polyhedra === | === Bounded Polyhedra === | ||
- | Let us illustrate the approach via the example of the //stable set problem//: Here one is given an (undirected) Graph G = (V,E) with node set V and edges E. The goal is to find a largest subset of node V' such that any two nodes in V' are not connected by an edge. | + | Let us illustrate the approach via the example of the //stable set problem//: Here one is given an (undirected) Graph G = (V,E) with node set V and edges E. The goal is to find a largest subset of nodes V' such that any two nodes in V' are not connected by an edge. |
For our example consider the 5-cycle, i.e., the graph C< | For our example consider the 5-cycle, i.e., the graph C< | ||
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We assume that the above information is contained in the file '' | We assume that the above information is contained in the file '' | ||
< | < | ||
- | polytope > $f=lp2poly(' | + | polytope > $f=lp2poly(' |
polytope > $p = new Polytope< | polytope > $p = new Polytope< | ||
</ | </ | ||
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End | End | ||
</ | </ | ||
- | {{ :tutorial: | + | {{ user_guide: |
We now assume that the example is contained in the file '' | We now assume that the example is contained in the file '' | ||
< | < | ||
- | polytope > $f = lp2poly(' | + | polytope > $f = lp2poly(' |
polytope > $pin = new Polytope< | polytope > $pin = new Polytope< | ||
</ | </ | ||
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</ | </ | ||
- | {{ :tutorial: | + | {{ user_guide: |
Note that the upper right part (including the red vertices) arises from truncation of the polyhedron for visualization. | Note that the upper right part (including the red vertices) arises from truncation of the polyhedron for visualization. | ||
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In this example there are two integral variables '' | In this example there are two integral variables '' | ||
< | < | ||
- | polytope > $m=lp2poly(' | + | polytope > $m=lp2poly(' |
polytope > $p = new Polytope< | polytope > $p = new Polytope< | ||
</ | </ | ||
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===== Integral Polytopes and Total Unimodularity ===== | ===== Integral Polytopes and Total Unimodularity ===== | ||
- | As explained in the previous example, the integral points in a polytope are of particular interest in discrete optimization. These points are called //lattice points// in polymake and the corresponding convex hull //lattice polytope//. The handling of such polytopes is explained in more detail in the [[tutorial: | + | As explained in the previous example, the integral points in a polytope are of particular interest in discrete optimization. These points are called //lattice points// in polymake and the corresponding convex hull //lattice polytope//. The handling of such polytopes is explained in more detail in the [[user_guide: |
Of particular interest for discrete optimization are properties of the orginal inequality system to define a lattice polytope, i.e., a polytope such that all of its vertices are integral (this can be tested by checking the property '' | Of particular interest for discrete optimization are properties of the orginal inequality system to define a lattice polytope, i.e., a polytope such that all of its vertices are integral (this can be tested by checking the property '' | ||
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In the second example, we reuse the file '' | In the second example, we reuse the file '' | ||
< | < | ||
- | polytope > $f=lp2poly(' | + | polytope > $f=lp2poly(' |
polytope > $p = new Polytope< | polytope > $p = new Polytope< | ||
</ | </ | ||
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Let us test whether the inequality system of this example is TDI. Thus, we first load the data as usual: | Let us test whether the inequality system of this example is TDI. Thus, we first load the data as usual: | ||
< | < | ||
- | polytope > $f = lp2poly(' | + | polytope > $f = lp2poly(' |
polytope > $p = new Polytope< | polytope > $p = new Polytope< | ||
</ | </ | ||
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*Note* that we need to take the inequalities instead of facets here, since facets are irredundant and thus might not be TDI, although the complete set of inequalities is TDI. | *Note* that we need to take the inequalities instead of facets here, since facets are irredundant and thus might not be TDI, although the complete set of inequalities is TDI. | ||
- | ===== 0/ | ||
===== Chvátal-Gomory Closure ===== | ===== Chvátal-Gomory Closure ===== | ||
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As before we read in the file using '' | As before we read in the file using '' | ||
< | < | ||
- | polytope > $f = lp2poly(' | + | polytope > $f = lp2poly(' |
polytope > $p = new Polytope< | polytope > $p = new Polytope< | ||
</ | </ | ||
- | The Chvátal-Gomory closure of a polytope can be computed with the function '' | + | The Chvátal-Gomory closure of a polytope can be computed with the function '' |
< | < | ||
polytope > $g = gc_closure($p); | polytope > $g = gc_closure($p); | ||
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==== Chvátal-Gomory Closure - Example 2 ==== | ==== Chvátal-Gomory Closure - Example 2 ==== | ||
- | Let us now consider the classical example of a polytope with the vertices of simplex in d dimensions and the point 1/2 times (1, ..., 1). It can be shown that such a polytope has rank at least log(d) - 1, see [Pokutta, 2011]. In our example, we use d = 4: | + | Let us now consider the classical example of a polytope with the vertices of simplex in d dimensions and the point 1/2 times (1, ..., 1). It can be shown that such a polytope has rank at least log(d) - 1, see [[http:// |
< | < | ||
polytope > $M = new Matrix< | polytope > $M = new Matrix< | ||
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my $p = shift; | my $p = shift; | ||
my $d = $p-> | my $d = $p-> | ||
- | my $q = new Polytope< | + | my $q = new Polytope< |
for (my $k = 0; $k < $d; $k = $k+1) | for (my $k = 0; $k < $d; $k = $k+1) | ||
{ | { | ||
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my $v1 = new Vector< | my $v1 = new Vector< | ||
my $v2 = new Vector< | my $v2 = new Vector< | ||
- | my $c1 = new Polytope< | ||
- | my $c2 = new Polytope< | ||
- | # intersect | + | # create intersection of corresponding polyhedra |
- | my $b1 = intersection($c1, $p); | + | my $b1 = new Polytope< |
- | my $b2 = intersection($c2, $p); | + | my $b2 = new Polytope< |
if ( ($b1-> | if ( ($b1-> | ||
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elsif ( ($b1-> | elsif ( ($b1-> | ||
{ | { | ||
- | my $c = conv($b1); | + | $q = intersection($q, |
- | $q = intersection($q, | + | |
} | } | ||
elsif ( ($b1-> | elsif ( ($b1-> | ||
{ | { | ||
- | my $c = conv($b2); | + | $q = intersection($q, |
- | $q = intersection($q, | + | |
} | } | ||
} | } | ||
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Thus, we have obtained the integral hull in a single step of the lift-and-project closure as opposed to two steps in the CG-closure. | Thus, we have obtained the integral hull in a single step of the lift-and-project closure as opposed to two steps in the CG-closure. | ||
- | ===== Other Software ===== | ||