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 .


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}.


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.


x1x2

x5

x4

x3

Polytope

Bounded polyhedron

a1

a2

a3a4

a5

Unbounded polyhedron

a1

a2

a3a4

a5


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?


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.


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}.


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?


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


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}.


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.


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


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


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)


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


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.


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?


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.


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.


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