objectiveVector -- computes an objective vector of a face of a polyhedron

Synopsis

• Usage:

  v = objectiveVector(P,Q)

• Inputs:
  • P, a convex polyhedron
  • Q, a convex polyhedron, which must be a face of P
• Outputs:
  • v, a matrix, one column vector over QQ representing a vector

Description

An objective vector v of a face Q of a polyhedron P is vector such that Q = {p in P | v*p = max over P} i.e. it is the face on which v attains its maximum.

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

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

i3 : v = objectiveVector(P,Q)

o3 = | 0 |
     | 0 |
     | 1 |

              3        1
o3 : Matrix ZZ  <--- ZZ

Since it is the face on which v attains its maximum it can be recovered with maxFace:

i4 : Q == maxFace(v,P)

o4 = true