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Unformatted text preview: 19 Complementary slackness Over the last few sections we have seen how we can use duality to verify the optimality of a feasible solution for a linear program. If we are able to find a feasible solution for the dual problem with dual objective value equal to the primal objective value of our primalfeasible solution, then the Weak duality theorem guarantees that this solution is optimal. The Strong Duality theorem guarantees the success of this approach: the dualfeasible solution we seek exists if and only if our primalfeasible solution is optimal. In this section we study how to make this approach to checking optimal ity more practical. Starting from a feasible solution for the primal linear program, we try to discover as much as we can about a dualfeasible solution verifying optimality. Consider a linear program in standard equality form. Denoting the columns of the constraint matrix by A 1 , A 2 , . . . , A m , we can write the problem as ( P ) maximize n X j =1 c j x j subject to n X j =1 A j x j = b x j ≥ 0 ( j = 1 , 2 , . . . , n ) The dual problem is ( D ) minimize b T y subject to A T j y ≥ c j ( j = 1 , 2 , . . . , n ) . We summarize our approach to checking optimality in the following result. Theorem 19.1 (Complementary slackness) A vector x * is optimal for the primal linear program ( P ) if and only if x * is feasible for ( P ) and some feasible solution y * for the dual problem ( D ) satisfies the following “comple mentary slackness” conditions (for j = 1 , 2 , . . . , n ): ( CS ) A T j y * = c j whenever x * j > ....
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This note was uploaded on 02/20/2009 for the course ORIE 3300 taught by Professor Todd during the Fall '08 term at Cornell University (Engineering School).
 Fall '08
 TODD

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