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section18

section18 - 18 Interpreting the dual In the previous two...

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Unformatted text preview: 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 ≥ . In many contexts, the variables x i 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 by b i say, increases slightly, then the corresponding increase in the optimal value suggests how much we might be prepared to pay for that increase in resource. In other words, sensitivity to changes in the right-hand side gives “pricing” information for the resources. Remarkably, the information we seek about sensitivity to right-hand side perturbation is given precisely by the optimal solution to the dual problem. We summarize in the following result. Theorem 18.1 (Sensitivity to right-hand sides) Consider a nondegen- erate optimal basis B for the standard equality-form linear program ( P ) . Let y * be the dual-optimal solution returned by the revised simplex method with current basis B (so y * satisfies A T B y * = c B ). Then the rate of change of the optimal value with respect to small changes to the right-hand side b i is y * i ....
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section18 - 18 Interpreting the dual In the previous two...

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