12
Termination
In order for the simplex method to be a foolproof 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.
We consider the standard equalityform linear programming problem
maximize
c
T
x
=
z
subject to
Ax
=
b
x
≥
0
,
where
A
is an
m
by
n
matrix whose columns span (if they fail to do so, the
results below hold trivially).
Proposition 12.1 (uniqueness of tableau)
There is exactly one tableau cor
responding to each (ordered) basis.
Proof
Let
B
be a basis, and let
N
be a list of the indices not in
B
.
By definition, the basis matrix
A
B
with columns from
A
indexed by
B
is
invertible.
Recall from Section 5 that we also write
A
N
for the matrix with
columns from
A
indexed by
N
. Analogously, we write
x
B
, x
N
for vectors with
entries from
x
indexed by
B
and
N
respectively, with similar definitions for the
vectors
c
B
and
c
N
. Using this notation, we can rewrite the system of equations
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 '08
 TODD
 Linear Algebra, Linear Programming, Invertible matrix, Simplex algorithm

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