12
Termination
In order for the simplex method to be a reliable algorithm, we must be sure
that, starting from a feasible tableau, it terminates after a ﬁnite number of
iterations. In this section we shall see that the smallest subscript rule ensures
termination. We need one last ingredient.
Proposition 12.1 (uniqueness of tableau)
Two tableaux corresponding
to the same basis can diﬀer only in the order in which the equations are
written.
Proof
Consider two tableaux corresponding to a basis
B
:
(12.2)
z

X
j
6∈
N
¯
c
j
x
j
= ¯
v
x
i
+
X
j
6∈
N
¯
a
ij
x
j
=
¯
b
i
(
i
∈
B
)
and
(12.3)
z

X
j
6∈
N
c
*
j
x
j
=
v
*
x
i
+
X
j
6∈
N
a
*
ij
x
j
=
b
*
i
(
i
∈
B
)
.
These tableaux are equivalent systems of equations. For any nonbasic index
q
6∈
B
, setting
x
q
=
t
x
j
= 0
(
j
6∈
B, j
6
=
q
)
x
i
=
¯
b
i

¯
a
iq
t
(
i
∈
B
)
z
= ¯
v
+ ¯
c
q
t
satisﬁes tableau (12.2) for any number
t
, and hence must also satisfy tableau
(12.3). Therefore we have, for all
t
,
¯
v
+ ¯
c
q
t

c
*
q
t
=
v
*
¯
b
i

¯
a
iq
t
+
a
*
iq
t
=
b
*
i
(
i
∈
B
)
.
66
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View Full DocumentEquating the constant terms and the coeﬃcients of
t
on each side, we deduce
¯
v
=
v
*
, ¯
c
q
=
c
*
q
,
¯
b
i
=
b
*
i
, and ¯
a
iq
=
a
*
iq
for all basic indices
i
∈
B
. Since this
holds for all
q
6∈
B
, the two tableaux have identical coeﬃcients.
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 Fall '07
 BLAND
 Linear Algebra, Linear Programming, Optimization, Simplex algorithm, R.G. Bland

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