18
Interpreting the dual
In the previous two sections we showed how to associate with any maximiza
tion linear program a dual problem, derived by considering upper bounds on
the problem’s objective value. Duality is a powerful theoretical tool: optimal
solutions to the dual problem serve as “optimality certiﬁcates” for the primal
problem, giving a simple veriﬁcation of the optimality of a feasible solution.
In this section we see how the dual problem has a very important con
crete interpretation, fundamental in many practical applications of linear
programming. To develop this interpretation, we return to the linear pro
gram in standard equality form
(
P
)
maximize
c
T
x
subject to
Ax
=
b
x
≥
0
.
In many contexts, the variables
x
j
represent levels of activity involving cer
tain resources, and the availability of these resources is constrained by the
equations constituting the constraint
Ax
=
b
. Often, the components of the
rightand side vector
b
are simply available levels of resources. The objec
tive function
c
T
x
typically represents some measure of proﬁt that we wish to
maximize.
Practical modeling using linear programming almost invariably involves
questions of “sensitivity”: after solving the linear program, we wish to know
what happens to our optimal solution and optimal value if the data change
slightly. In particular, if the available level of one of the resources, represented
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 Fall '08
 TODD
 Optimization, Dual problem, linear program, objective value, optimal value

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