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
j
is non
negative. We deduce
optimal value
≤
20
.
79
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By combining the constraints of the linear program in more complex
ways, we can obtain other upper bounds. For example, we could multiply
the second constraint by 3 and add the first constraint to deduce that any
feasible
x
satisfies the inequality
x
1

x
2
+ 7
x
3
≤
(2
x
1

x
2
+
x
3
) + 3(
x
1
+
x
2
+ 2
x
3
) = 16
.
Hence
optimal value
≤
16
.
This upper bound of 16 is smaller than the previous upper bound of 20, so
provides more precise information about the true optimal value: the smaller
the upper bound, the better.
Experimenting in the same way, we might notice that any feasible
x
satisfies the inequality
(16.2)
x
1

x
2
+ 7
x
3
≤
3(2
x
1

x
2
+
x
3
) + 2(
x
1
+
x
2
+ 2
x
3
) = 13
,
so
optimal value
≤
13
.
But now we have solved out linear program! Combining this inequality with
inequality (16.1), we learn that the optimal value is
exactly
13, and further
more that the feasible solution [0
,
1
,
2]
T
is optimal (since it attains the value
13).
In this case we have managed to solve the linear program without using
the simplex method. Furthermore, we could easily convince an observer that
our solution is indeed optimal.
We simply present them with the feasible
solution [0
,
1
,
2]
T
and the “multipliers” 3 and 2 that we used to multiply the
constraints to derive inequality (16.2). They then follow three simple steps:
•
Verify the solution is feasible by substituting it into the constraints.
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 Fall '08
 TODD
 Linear Programming, Optimization, Dual problem, optimality, feasible solution, linear program

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