03 - Industrial Engineering 634 Integer Programming Fall...

Industrial Engineering 634 Fall 2011
Integer Programming Lecturer: Nelson Uhan
Scribe: Nathan Hartje
August 26, 2011
Lecture 3. Faces of Polyhedra
1 Review
Up to now, we have been investigating mathematical programs. When we inspected the generic
linear program in the previous lecture, we began to conceptualize these programs in terms their
feasible sets. Diving deeper into this concept, we defined a polyhedron in R
n
as a set of type
P = { x ∈ R
n
: Ax ≤ b } for some matrix A ∈ R
m × n
and vector b ∈ R
m
.
We further described the structure of a polyhedron P = { x ∈ R
n
: Ax ≤ b } as follows.
Definition 1.1. If c ∈ R
n
is a nonzero vector such that δ = max { cx : x ∈ P } is finite, then
{ x ∈ R
n
: cx = δ } is a supporting hyperplane of P .
Definition 1.2. A face of P is P itself or the intersection of P with a supporting hyperplane of P .
Definition 1.3. A point x for which { x } is a face is called a vertex of P , or a basic solution of
Ax ≤ b .
In the last lecture, we began to discuss faces of P by establishing a proposition, and, today, we
continue our proof of this proposition.
2 Faces of polyhedra
Proposition 2.1. Let P = { x ∈ R : Ax ≤ b } be a polyhedron and F ⊆ P . Then the following
statements are equivalent:
(a) F is a face of P .
(b) There exists c such that δ = max { cx : x ∈ P } is finite and F = { x ∈ P : cx = δ } .
(c) F = { x ∈ P : A
0
x = b
0
} 6 = ∅ for some subsystem A
0
x ≤ b
0
of Ax ≤ b .
Proof. From the previous lecture, we concluded the following two statements.
F is a face of P if and only if F = { x ∈ P : A
0
x = b
0
} 6 = ∅ for some subsystem A
0
x ≤ b
0
of Ax ≤ b (i.e. the equivalence of ( a ) and ( c )).
If F = { x ∈ P : A
0
x = b
0
} 6 = ∅ for some subsystem A
0
x ≤ b
0
of Ax ≤ b , then there exists
c such that δ = max { cx : x ∈ P } is finite and F = { x ∈ P : cx = δ } (i.e. ( c ) implies
( b )).