16
Duality
We have now seen in some detail how we can use the simplex method to solve
linear programming problems, 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 programming problem,
with no strictly positive reduced costs.
In some contexts, we might be able to guess a good feasible solution for a
linear programming problem
without
using the simplex method. For example, we
might have previously found an optimal solution of a related linear programming
problem, 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 programming problem
(
*
)
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 programming problem. 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 nonneg
ative. We deduce
optimal value
≤
20
.
By combining the constraints of the linear programming problem in more
complex ways, we can obtain other upper bounds. For example, we could mul
tiply 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
.
81
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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 programming problem!
Combining this
inequality with inequality (16.1), we learn that the optimal value is
exactly
13,
and furthermore 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 programming problem
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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 '08
 TODD
 Linear Programming, Optimization, Dual problem, linear programming problem, optimality

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