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OR320/520
9/13/07
Prof. Bland
The Simplex Method: Parts I and II
Part I: Standard Form Systems and Ordered Bases
Let
A
be an
m
×
n
matrix and let
b
be an
m
×
1 vector.
We will be interested in linear
programming problems in which the constraints take the form
Ax
=
b, x
≥
0. We observed
in the last class that the set
S
=
{
x
∈
R
I
n
:
Ax
=
b, x
≥
0
}
of feasible solutions to such l.p.
problems is a polyhedron.
This polyhedron may not be bounded (for example if
m
= 0),
but if it is nonempty, it has at least one extreme point. (We will give an intuitive argument
in class.)
For a little while we are going to ignore the nonnegativity constraints and focus on the linear
system
Ax
=
b
. For each index 1
≤
j
≤
n
let
A
j
denote the
jth
column of
A
. Now suppose
that for the list
B
= (
B
1
, ..., B
m
) of indices in
{
1
, ..., n
}
the
m
×
m
matrix
A
B
= [
A
B
1
, ..., A
B
m
]
is nonsingular. Then we say that
B
is
basic
for
A
, and that
B
is an
ordered basis
correspond
ing to the linear system
¯
Ax
=
¯
b
, where (
¯
A,
¯
b
) :=
A

1
B
(
A, b
) The variables
x
B
1
,
···
, x
B
m
are
basic variables
, and the other variables
x
j
, for
j
not in
B
, are called
nonbasic
. It is convenient
to denote by
N
the set of indices of the nonbasic variables.
For example let
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 Fall '07
 BLAND

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