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19
Complementary slackness
Over the last few sections we have seen how we can use duality to verify
the optimality of a feasible solution for a linear program. If we are able to
ﬁnd a feasible solution for the dual problem with dual objective value equal
to the primal objective value of our primalfeasible solution, then the Weak
duality theorem guarantees that this solution is optimal. The Strong duality
theorem guarantees the success of this approach: the dualfeasible solution
we seek exists if and only if our primalfeasible solution is optimal.
In this section we study how to make this approach to checking optimal
ity more practical. Starting from a feasible solution for the primal linear
program, we try to discover as much as we can about a dualfeasible solution
verifying optimality.
Consider a linear program in standard equality form.
Denoting the
columns of the constraint matrix by
A
1
, A
2
, . . . , A
m
, we can write the problem
as
(
P
)
maximize
n
X
j
=1
c
j
x
j
subject to
n
X
j
=1
x
j
A
j
=
b
x
j
≥
0 (
j
= 1
,
2
, . . . , n
)
The dual problem is
(
D
)
±
minimize
b
T
y
subject to
A
T
j
y
≥
c
j
(
j
= 1
,
2
, . . . , n
)
.
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 Fall '07
 BLAND

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