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MthSc 810 – Mathematical Programming
Solutions of the second midterm exam
Exercise 1 (25 pts.)
Prove that if a polyhedron
P
=
{
x
∈
R
n
:
A
x
≤
b
}
does
not contain an extreme point, then its recession cone is not
{
0
}
.
Then consider the following polyhedron, where
α >
0:
P
′
=
{
x
∈
R
2
+
:
αx
1

x
2
≥ 
1
,
αx
1

x
2
≤
α
}
.
1. Find the extreme rays of
P
′
by applying the de±nition.
2. Prove that
P
′
is
not
contained in its recession cone.
Solution.
If a polyhedron
P
⊆
R
n
contains no extreme point, then it contains a
line
{
x
∈
R
n
:
x
= ¯
x
+
λ
d
}
, where ¯
x
∈
R
n
,
d
∈
R
n
\
0
, and
λ
∈
R
n
. Therefore,
d
and

d
are two extreme rays and the recession cone cannot be
0
.
1.
C
=
{
d
∈
R
2
+
:
αd
1

d
2
≥
0
, αd
1

d
2
≤
0
}
=
{
d
∈
R
2
+
:
αd
1

d
2
= 0
}
.
Hence
C
is the hal²ine
{
λ
(1
, α
)
, λ
≥
0
} ⊂
R
2
+
.
2. ¯
x
= (0
,
1)
∈
P
′
since

1
≥ 
1 and

1
≤
α
for any
α >
0, but ¯
x
is not in
C
.
Exercise 2 (25 pts.)
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 Fall '08
 Staff
 Math, Cone

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