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Economics 146 – Fall, 2007
Linear Programming – Part IIIB
Last revision: 10/26/07.
Geometry of Solutions.
Let
S
be the set of feasible solutions for an LP in standard
form. We can classify three possible cases for
S
. We ﬁrst establish some terminology.
Consider the feasible set given by
Ax
≤
b
(1)
and
x
≥
0
(2)
Another way to write the conditions is
a
11
x
1
+
a
12
x
2
+
···
+
a
1
n
x
n
≤
b
1
(3)
a
21
x
1
+
a
22
x
2
+
···
+
a
1
n
x
n
≤
b
2
(4)
.
.
.
.
.
.
(5)
a
m
1
x
1
+
a
m
2
x
2
+
···
+
a
mn
x
n
≤
b
m
(6)
A set in
R
n
of the form
{
x
∈
R
n

α
1
x
1
+
···
+
α
n
x
n
=
b
}
(7)
is called a
hyperplane
in
R
n
.
A set in
R
n
of the form
{
x
∈
R
n

α
1
x
1
+
···
+
α
n
x
n
≤
b
}
(8)
is called a
halfspace
in
R
n
, i.e. a halfspace consists of all the points on one side of a
hyperplane. Thus, the set
S
of feasible solutions is the intersection of a ﬁnite number of
halfspaces.
For example, a line such as
x
1
+
x
2
= 1 is a hyperplane in
R
2
and the inequality
x
1
+
x
2
≤
1 is a halfspace. The set of solutions to
x
1
+
x
2
≤
1
, x
1
≥
0
, x
2
≥
0 is the
intersection of three halfspaces. Similarly, a plane such as
x
1
+
x
2
+
x
3
= 1 is a hyperplane
in
R
3
, and the set of solutions to
x
1
+
x
2
+
x
3
≤
1
, x
1
≥
0
, x
2
≥
0
, x
3
≥
0 is the intersection
of four halfspaces.
There are two convenient names that we will introduce. These are multidimensional
versions of polygons in the plane. The intersection of a ﬁnite number of halfspaces is called
a
convex polytope
. A bounded convex polytope is called a
convex polyhedron
. We give three
examples (you should graph all three):
1. An example of an empty feasible set
x
1
+
x
2
≤
1

x
1
≤ 
2

x
2
≤ 
2
and
x
1
≥
0
,x
2
≥
0
1
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View Full Document2. An example of a convex polyhedron for the feasible set
x
1
+
x
2
≤
1
and
x
1
≥
0
,x
2
≥
0
3. An example of an unbounded convex polytope for the feasible set
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 Fall '07
 farmer
 Economics

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