section17 - 17 Duality for general linear programs In the...

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17 Duality for general linear programs In the previous section, we introduced the powerful idea of a “certificate of optimality” for a feasible solution x * of a linear program in standard equality form. A certificate of optimality is simply a feasible solution y * for the dual problem whose (dual) objective value equals the (primal) objective value of x * . This certificate constitutes a simple way to verify that x * is optimal: we simply need to check that x * is primal-feasible, that y * is dual-feasible, and that the two corresponding objective values are equal, and we then have a proof (by the Weak duality theorem) of optimality. The Strong duality theorem guarantees that any optimal solution of a linear program in standard equality form does indeed have a certificate of optimality. Many linear programs in practice are not in standard equality form. In this section we extend the idea of a certificate of optimality to general linear programs. We know how to transform such problems into standard equality form, so we could always resort to this technique before looking for a cer- tificate. However, a more appealing approach is to seek a certificate that reflects the structure of the original problem, so instead we retrace the steps of the last section, but in a more general context. Without much loss of generality, we can write any linear program in the form ( P ) maximize c T x + d T w subject to Ax + Ew = b Fx + Gw e x 0 , w free , for some given vectors c, d, b, e and matrices A, E, F, G of appropriate sizes. The variables are the vectors x and w . As in a standard equality-form prob- lem, each component of x is non-negative, whereas each component of w is “free”, or in other words, unrestricted in sign. We follow the same approach to construct the dual problem as before: we seek upper bounds on the optimal value by taking linear combinations of the constraints and comparing the result to the objective function. Beginning with the equality constraints, if we take the i th constraint, multiply it by any number y i , and add up the results, we arrive at the equation y T ( Ax + Ew ) = y T b. This equation must hold for any feasible pair of vectors ( x, w ). Turning to the inequality constraints, we can multiply inequality j by any number u j 0 to 93
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derive another inequality that any feasible pair ( x, w ) must satisfy. Adding these up, we obtain the inequality u T ( Fx + Gw ) u T e. Finally, adding the previous equation gives the inequality
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section17 - 17 Duality for general linear programs In the...

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