16
Duality
We have now seen in some detail how we can use the simplex method to
solve linear programs, and how, at termination, the simplex method provides
a proof of optimality. This proof, as we have seen, consists of a tableau
equivalent to the system of equations deﬁning the linear program, with no
strictly positive reduced costs.
In some contexts, we might be able to guess a good feasible solution
for a linear program
without
using the simplex method. For example, we
might have previously found an optimal solution of a related linear program,
perhaps identical except for a slightly diﬀerent objective function. How might
we judge how good this solution is? If this solution was in fact optimal, how
might we prove (or “certify”) that fact?
As a simple example, consider the linear program
(
*
)
maximize
x
1

x
2
+ 7
x
3
subject to 2
x
1

x
2
+
x
3
= 1
x
1
+
x
2
+ 2
x
3
= 5
x
1
,
x
2
,
x
3
≥
0
.
We might experiment by choosing feasible solutions:
[2
,
3
,
0]
T
has value

1
[1
,
2
,
1]
T
has value 6
[0
,
1
,
2]
T
has value 13
.
Notice that any of these objective values is a lower bound on the optimal
value of the linear program. In particular we conclude
(16.1)
optimal value
≥
13
.
After some more experiments, we might guess that [0
,
1
,
2]
T
is a good
feasible solution, and perhaps even optimal. To convince ourselves, we could
try to discover
upper
bounds on the optimal value. As a simple example,
notice that any feasible solution
x
satisﬁes the inequality
x
1

x
2
+ 7
x
3
≤
4(
x
1
+
x
2
+ 2
x
3
) = 20
,
as a consequence of the second constraint and the fact that each
x
i
is non
negative. We deduce
optimal value
≤
20
.
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