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secn16

# secn16 - 16 Duality We have now seen in some detail how we...

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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 0 . 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

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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). In this case we have managed to solve the linear programming problem without using the simplex method. Furthermore, we could easily convince an observer that our solution is indeed optimal. We simply present them with the feasible solution [0 , 1 , 2] T and the “multipliers” 3 and 2 that we used to multiply the constraints to derive inequality (16.2). They then follow three simple steps: Verify the solution is feasible by substituting it into the constraints.
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