5
The tableau
Solving simple linear programming problems graphically suggests the impor
tance of extreme points of the feasible region. In the last section, we related
the geometric idea of an extreme point to the algebraic idea of a basic feasi
ble solution. Our aim now is to capitalize on this algebraic idea to design a
solution algorithm for linear programming problems.
Consider a linear programming problem in standard equality form:
maximize
c
T
x
subject to
Ax
=
b
x
≥
0
.
By definition, a basis
B
for the matrix
A
is a list of indices such that the
matrix of corresponding columns
A
B
is invertible. We know from linear alge
bra 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 oper
ations to the system
Ax
=
b
. As we saw before, we can rewrite the original
system
(5.1)
A
B
x
B
+
A
N
x
N
=
b,
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
(5.2)
Ix
B
+
¯
Ax
N
=
¯
b,
or in other words
x
i
+
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, Linear Programming, Optimization, Invertible matrix

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