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<head><title>objectiveVector -- computes an objective vector of a face of a polyhedron</title>
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<div><h1>objectiveVector -- computes an objective vector of a face of a polyhedron</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt> v = objectiveVector(P,Q)</tt></div>
</dd></dl>
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<li><div class="single">Inputs:<ul><li><span><tt>P</tt>, <span>a <a href="___Polyhedron.html">convex polyhedron</a></span></span></li>
<li><span><tt>Q</tt>, <span>a <a href="___Polyhedron.html">convex polyhedron</a></span>, which must be a face of <tt>P</tt></span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>v</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, one column vector over <a href="../../Macaulay2Doc/html/___Q__Q.html" title="the class of all rational numbers">QQ</a> representing a vector</span></li>
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<div class="single"><h2>Description</h2>
<div><p/>
An objective vector <tt>v</tt> of a face <tt>Q</tt> of a polyhedron <tt>P</tt> is vector
such that <tt>Q = {p in P | v*p = max over P}</tt> i.e. it is the face on which <tt>v</tt> attains
its maximum.<table class="examples"><tr><td><pre>i1 : P = hypercube 3
o1 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of polyhedron => 3
number of facets => 6
number of rays => 0
number of vertices => 8
o1 : Polyhedron</pre>
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<tr><td><pre>i2 : Q = convexHull matrix {{1,1,-1,-1},{1,-1,1,-1},{1,1,1,1}}
o2 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of polyhedron => 2
number of facets => 4
number of rays => 0
number of vertices => 4
o2 : Polyhedron</pre>
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<tr><td><pre>i3 : v = objectiveVector(P,Q)
o3 = | 0 |
| 0 |
| 1 |
3 1
o3 : Matrix QQ <--- QQ</pre>
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Since it is the face on which <tt>v</tt> attains its maximum it can be recovered with <a href="_max__Face.html" title="computes the face of a Polyhedron or Cone where a weight attains its maximum">maxFace</a>:<table class="examples"><tr><td><pre>i4 : Q == maxFace(v,P)
o4 = true</pre>
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<div class="waystouse"><h2>Ways to use <tt>objectiveVector</tt> :</h2>
<ul><li>objectiveVector(Polyhedron,Polyhedron)</li>
</ul>
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