Polyhedra Leo Handout
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Transcript of Polyhedra Leo Handout
Polyhedra and Polytopes
Leo van Iersel
Eindhoven University of Technology
Optimization in Rn, lecture 6
Leo van Iersel (TUE) Polyhedra and Polytopes ORN4 1 / 22
Definition
Let x1, . . . , xk , z ∈ Rn. Then z is a convex combination of x1, . . . , xk ifthere exist λ1, . . . , λk ∈ R such that
z =∑i
λixi ,∑i
λi = 1, and λi ≥ 0 ∀i .
Lemma
Let C ⊆ Rn be a convex set.If x1, . . . , xk ∈ C , and z is a convex combination of the xi , then z ∈ C .
Leo van Iersel (TUE) Polyhedra and Polytopes ORN4 2 / 22
Definition
Let X ⊆ Rn. The convex hull of X is the set of all convex combinations ofx1, . . . , xk ∈ X :
conv.hull X := {∑i
λixi | xi ∈ X ,∑i
λi = 1, λi ∈ R and λi ≥ 0 ∀i}.
Lemma
Let X ⊆ Rn. Then conv.hull X is a convex set.
Lemma
Let X ⊆ Rn. Then conv.hull X =⋂{Y ⊆ Rn | X ⊆ Y ,Y convex}.
Leo van Iersel (TUE) Polyhedra and Polytopes ORN4 3 / 22
Definition
A set P ⊆ Rn is a polyhedron if there is a system of finitely manyinequalities Ax ≤ b such that
P = {x ∈ Rn | Ax ≤ b}.
Definition
A set P ⊆ Rn is a polytope if there is a finite set X ⊆ Rn such that
P = conv.hull X .
Theorem
Let P ⊆ Rn. Then
P is a bounded polyhedron ⇐⇒ P is a polytope.
Leo van Iersel (TUE) Polyhedra and Polytopes ORN4 4 / 22
x1x2
x5
x4
x3
Polytope
Bounded polyhedron
a1
a2
a3a4
a5
Unbounded polyhedron
a1
a2
a3a4
a5
Leo van Iersel (TUE) Polyhedra and Polytopes ORN4 5 / 22
Example
The hypercube is a polytope Qn := conv.hull {−1, 1}n
The simplex is a polytope Sn := conv.hull {e1, . . . , en+1}
(-1,-1,1) (1,-1,1)
(-1,1,1)
(1,1,1)
(-1,-1,-1)
(-1,1,-1) (1,1,-1)
(1,-1,-1)
3-dimensional hypercube
(0,0,1)
(1,0,0)
(0,1,0)
2-dimensional simplex
Question. Are the hypercube and simplex also polyhedra?
Leo van Iersel (TUE) Polyhedra and Polytopes ORN4 6 / 22
Example
The hypercube is a polytope Qn := conv.hull {−1, 1}n
The simplex is a polytope Sn := conv.hull {e1, . . . , en+1}
(-1,-1,1) (1,-1,1)
(-1,1,1)
(1,1,1)
(-1,-1,-1)
(-1,1,-1) (1,1,-1)
(1,-1,-1)
3-dimensional hypercube
(0,0,1)
(1,0,0)
(0,1,0)
2-dimensional simplex
Exercise. Find a polyhedral description of the hypercube and simplex.
Leo van Iersel (TUE) Polyhedra and Polytopes ORN4 7 / 22
Definition
Let P ⊆ Rn and c ∈ Rn and a d ∈ R. The hyperplane
Hc,d = {x ∈ Rn | cx = d}
is a supporting hyperplane of P if max{cx | x ∈ P} = d .
Definition
Let P ⊆ Rn. F is a face of P if F = P or F = P ∩ H for some supportinghyperplane H of P.
Observation
By definition, the face F = P ∩ Hc,d is exactly the set of optimal solutionsto max{cx | x ∈ P}.
Leo van Iersel (TUE) Polyhedra and Polytopes ORN4 8 / 22
Example. Supporting hyperplanes Hc,d and faces F .
Hc,dc
F=P \Hc,d
c
F=P \Hc,dc
Hc,d
Hc,d
Question. What are the faces of the hypercube?
Leo van Iersel (TUE) Polyhedra and Polytopes ORN4 9 / 22
Definition
Let P ⊆ Rn.
v is a vertex of P if {v} is a face of P.
e is an edge of P if e is a face of P, dim(e) = 1.
F is a facet of P if F is a face of P and dim(F ) = dim(P)− 1.
FacetVertex
Edge
Leo van Iersel (TUE) Polyhedra and Polytopes ORN4 10 / 22
Definition
Let P ⊆ Rn be a convex set. The lineality space of P is the linear space
lin(P) := {d ∈ Rn | {x + λd | λ ∈ R} ⊆ P for all x ∈ P}.
Definition
Let P ⊆ Rn be a convex set. The cone of directions of P is
dir(P) := {d ∈ Rn | {x + λd | λ ∈ R, λ ≥ 0} ⊆ P for all x ∈ P}.
Lemma
If P = {x ∈ Rn | Ax ≤ b} then
lin(P) = ker(A) = {d ∈ Rn | Ad = 0}
dir(P) = {d ∈ Rn | Ad ≤ 0}.
Leo van Iersel (TUE) Polyhedra and Polytopes ORN4 11 / 22
Cone of directions
dd’
d
-d
x
d2lin(P)Lineality space
d,d’2dir(P)Leo van Iersel (TUE) Polyhedra and Polytopes ORN4 12 / 22
Lemma
Let P be a polyhedron P = {x ∈ R | Ax ≤ b}, let F ⊆ P be nonemptyand let a1, . . . an be the rows of A. Then the following are equivalent:
F is a face of P;
F = {x ∈ P | aix = bi for all i ∈ J} for some J ⊆ {1, . . . ,m}.
Lemma
Let P be a polyhedron and c ∈ R. If max{cx | x ∈ P} exists andlin(P) = {0} then the maximum is attained in a vertex of P.
Theorem
If P ⊆ Rn is a bounded polyhedron, then
P = conv.hull V (P).
Moreover, V (P) is finite and hence P is a polytope.
Leo van Iersel (TUE) Polyhedra and Polytopes ORN4 13 / 22
Simplex Method
The simplex method is applied to problems of the formmax{cx | Ax = b, x ≥ 0} and since x ≥ 0 we have lin(P) = {0}.Hence, if the optimization problem is feasible and bounded, themaximum is attained in a vertex;
This is why the simplex method walks from vertex to vertex.
Example
max x1 +2x2 +x3s.t. x1 +2x2 ≤ 2
x1 +x2 +x3 ≤ 2x1, x2, x3 ≥ 0
Leo van Iersel (TUE) Polyhedra and Polytopes ORN4 14 / 22
Example
max x1 +2x2 +x3s.t. x1 +2x2 ≤ 2
x1 +x2 +x3 ≤ 2x1, x2, x3 ≥ 0
(0,0,2)
(2,0,0)
(0,1,1)
(0,1,0)
x1
x2
x3
Leo van Iersel (TUE) Polyhedra and Polytopes ORN4 15 / 22
Example
max x1 +2x2 +x3s.t. x1 +2x2 ≤ 2
x1 +x2 +x3 ≤ 2x1, x2, x3 ≥ 0
(0,0,2)
(2,0,0)
(0,1,1)
(0,1,0)
x1
x2
x3
(0,0,0)
Leo van Iersel (TUE) Polyhedra and Polytopes ORN4 16 / 22
Example
max x1 +2x2 +x3s.t. x1 +2x2 ≤ 2
x1 +x2 +x3 ≤ 2x1, x2, x3 ≥ 0
(0,0,2)
(2,0,0)
(0,1,1)
(0,1,0)
x1
x2
x3
Leo van Iersel (TUE) Polyhedra and Polytopes ORN4 17 / 22
Example
max x1 +2x2 +x3s.t. x1 +2x2 ≤ 2
x1 +x2 +x3 ≤ 2x1, x2, x3 ≥ 0
(0,0,2)
(2,0,0)
(0,1,1)
(0,1,0)
x1
x2
x3
Leo van Iersel (TUE) Polyhedra and Polytopes ORN4 18 / 22
We want to prove:
Theorem
Let P ⊆ Rn. If P is a polytope, then P is a polyhedron.
We define:
Definition
Let X ⊆ Rn. An inequality cx ≤ d is valid for X if cx ≤ d holds for allx ∈ X .
Then we have:
Lemma
Let P be a polytope, then
P = {x ∈ Rn | x satisfies all valid inequalities for P}
However, the number of valid inequalities is infinite.
Leo van Iersel (TUE) Polyhedra and Polytopes ORN4 19 / 22
Definition
An inequality cx ≤ d is called facet-defining (or essential) for P ifP ∩ Hc,d is a facet of P.
a
·Ha,b
·Hc,d
c
·He,f
e
Question. Which of ax ≤ b, cx ≤ d and ex ≤ f are valid inequalities andwhich ones are facet-defining?
Leo van Iersel (TUE) Polyhedra and Polytopes ORN4 20 / 22
Definition
Let y , x1, . . . , xk ∈ Rn. We say that y is an affine combination ofx1, . . . , xk if there exist λ1, . . . , λk ∈ R such that
y = λ1x1 + . . .+ λkxk and λ1 + . . .+ λk = 1.
Definition
Let X ⊆ Rn. The affine hull of X is the set of all affine combinations ofelements of X .
Definition
A set X = {x1, . . . , xk} ⊆ Rn is affinely dependent if there existλ1, . . . , λk ∈ R not all zero such that
λ1x1 + . . .+ λkxk = 0 and λ1 + . . .+ λk = 0.
Definition
Let X ⊆ Rn. The dimension of X isdim(X ) := max{|Y | | Y ⊆ X ,Y affinely independent} − 1.
Leo van Iersel (TUE) Polyhedra and Polytopes ORN4 21 / 22
Definition
An inequality cx ≤ d is called facet-defining (or essential) for P ifP ∩ Hc,d is a facet of P.
Lemma
Let P be a polytope with dim(P) = n. If y /∈ P, then there is aninequality cx ≤ d that is facet-defining for P such that cx ≤ d, forall x ∈ P, and cy > d.
Theorem
Let P be a polytope with dim(P) = n. Let Ax ≤ b be a system of validinequalities such that for each facet F of P there is a row aix ≤ bi
of Ax ≤ b such that F = P ∩ Hai ,bi . Then P = {x ∈ Rn | Ax ≤ b}.
Theorem
Let P ⊆ Rn. Then
P is a polytope =⇒ P is a bounded polyhedron.
Leo van Iersel (TUE) Polyhedra and Polytopes ORN4 22 / 22