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Unformatted text preview: 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 . 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 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)....
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This note was uploaded on 10/31/2010 for the course ORIE 5300 at Cornell University (Engineering School).