5
The tableau
Solving simple linear programs graphically suggests the importance of ex
treme points of the feasible region. In the last section, we related the ge
ometric idea of an extreme point to the algebraic idea of a basic feasible
solution. Our aim now is to capitalize on this algebraic idea to design a
solution algorithm for linear programs.
Consider a linear programs in standard equality form:
maximize
c
T
x
subject to
Ax
=
b
x
≥
0
.
By deﬁnition, a basis for the matrix
A
is a list of indices such that the matrix
of corresponding columns
A
B
is invertible. We know from linear algebra that,
since
A
B
is invertible, some sequence of elementary row operations (adding
multiples of rows to other rows and interchanging rows) reduces
A
B
to the
identity matrix. Suppose we apply exactly that sequence of row operations
to the system
Ax
=
b
. As we saw before, we can rewrite the original system
A
B
x
B
+
A
N
x
N
=
b,
(5.1)
where the list
N
consists of the nonbasic indices (those indices not in
B
).
The vectors
x
B
and
x
N
have components respectively
x
i
for
i
∈
B
(the basic
variables) and
x
j
for
j
∈
N
(the nonbasic variables). Applying our sequence
of row operations, we must arrive at an equivalent system, having the form
Ix
B
+
¯
Ax
N
=
¯
b,
(5.2)
or in other words
x
i
+
X
j
∈
N
¯
a
ij
x
j
=
¯
b
i
(
i
∈
B
)
.
This form of the system of constraints is particularly wellsuited for un
derstanding the basis
B
. Given any values of the nonbasic variables
x
j
(for
j
∈
N
), we can easily read oﬀ the corresponding values of the basic variables
x
i
(for
i
∈
B
). In particular, if we set the nonbasic variables to zero, we
obtain the corresponding basic solution
x
B
=
¯
b
and
x
N
= 0. This basic solu
tion is feasible exactly when
¯
b
≥
0. Notice that each basic variable appears
exactly once in the system.
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 '08
 TODD
 Linear Algebra, Optimization, Invertible matrix

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