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4
Basic feasible solutions
To summarize the main idea from the last section, a basis for an
m
by
n
matrix
A
is a list of numbers chosen from
{
1
,
2
, , .
. . , n
}
such that the matrix
A
B
with columns indexed by this list is invertible. The corresponding basic
solution of a system
Ax
=
b
is the unique solution of this system satisfying
x
j
= 0 for all indices
j
6∈
B
.
Consider now the constraints of a linear program in standard equality
form:
Ax
=
b
x
≥
0
.
A
basic feasible solution
of this system is a feasible solution that is also
basic. Such solutions play a special role in linear programming. To illustrate,
consider the simple example (2.1):
(4.1)
maximize
x
1
+
x
2
subject to
x
1
≤
2
x
1
+ 2
x
2
≤
4
x
1
,
x
2
≥
0
.
As before, we can sketch the feasible region:
If we transform the constraints of this linear program to standard equality
form, by introducing slack variables as we described in Section 2, we arrive
at the following system:
x
1
+
x
3
= 2
x
1
+ 2
x
2
+
x
4
= 4
x
1
,
x
2
,
x
3
,
x
4
≥
0
.
26
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View Full DocumentFeasible solutions of this system correspond onetoone with feasible solutions
of the linear program (4.1), in an obvious way. If we think of this new system
as
Ax
=
b
, the matrix
A
has ﬁve possible bases, not including permutations:
[1
,
2]
,
[1
,
3]
,
[1
,
4]
,
[2
,
3]
,
[3
,
4]. The ﬁve corresponding basic solutions are
x
1
x
2
x
3
x
4
=
2
1
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 Fall '08
 TODD

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