Fall 2010
Optimization I (ORIE 3300/5300)
Recitation 9 Clariﬁcation
We’ll illustrate the parallels between revised simplex method and the ’plain’ simplex method
using an example. Before that, we’ll derive the formula for the simplex tableau for a given basis
B
, which is as follows:
z

0
x
B

(
c
T
N

y
T
A
N
)
x
N
=
y
T
b
x
B
+
A

1
B
A
N
x
N
=
A

1
B
b.
(1)
Derivation of (1)
Given an LP in standard form:
min
c
T
x
s.t.Ax
=
b, x
≥
0
,
our tableau is initially as follows:
z

c
T
x
= 0
Ax
=
b
(2)
Suppose that we have a basis
B
, indicating which variables are basic. Then, let us rewrite the
tableau (2) where we group the basic variables together and the nonbasic together:
z

c
T
B
x
B

c
T
N
x
N
= 0
A
B
x
B
+
A
N
x
N
=
b.
(3)
Now, the coeﬃcients
c
B
in front of
x
B
are not necessarily zero, but recall that we want the
coeﬃcients in front of the basic variables, in the objective row, to be all zeros. So, we add some
linear combination of the constraint rows from the objective rows, to make the ﬁrst row all zeroes.
Let the vector
y
indicate this linear combination:
z

c
T
B
x
B

c
T
N
x
N
+
y
T
A
B
x
B
+
y
T
A
N
x
N
= 0 +
y
T
b
A
B
x
B
+
A
N
x
N
=
b.
(4)
Rewriting the ﬁrst row,
z

(
c
T
B

y
T
A
B
)
x
B

(
c
T
N

y
T
A
N
)
x
N
=
y
T
b
A
B
x
B
+
A
N
x
N
=
b.
(5)
We haven’t decided what
y
is so far, but we need to choose
y
such that
c
T
B

y
T
A
B
= 0.
Equivalently, we want
y
T
A
B
=
c
T
B
, or
y
T
=
c
T
B
A

1
B
.
Taking the transpose of both sides:
y
= (
A

1
B
)
T
c
B
= (
A
T
B
)

1
c
B
.
If we choose
y
as above, then we obtain
z

0
x
B

(
c
T
N

y
T
A
N
)
x
N
=
y
T
b
A
B
x
B
+
A
N
x
N
=
b.
(6)
1
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Optimization I (ORIE 3300/5300)
Almost there. Now, since we want the columns corresponding to the basic variables to be the
columns of the identity matrix, then we multiply the constraint rows by the inverse of
A
B
, such
that:
A

1
B
A
B
x
B
+
A

1
B
A
N
x
N
=
A

1
B
b,
or
x
B
+
A

1
B
A
N
x
N
=
A

1
B
b.
Hence, replacing the constraint rows in tableau (6) with the above equation, we get
z

0
x
B

(
c
T
N

y
T
A
N
)
x
N
=
y
T
b
x
B
+
A

1
B
A
N
x
N
=
A

1
B
b,
(7)
which is precisely the formula for the simplex tableau that we saw in (1).
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