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sec18_Notes

# sec18_Notes - 18 Interpreting the dual In the previous two...

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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 right-and 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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sec18_Notes - 18 Interpreting the dual In the previous two...

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