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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 primal-feasible solution, then the Weak duality theorem guarantees that this solution is optimal. The Strong Duality theorem guarantees the success of this approach: the dual-feasible solution we seek exists if and only if our primal-feasible 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 dual-feasible 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