secn19 - 19 Complementary slackness Over the last few...

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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 programming problem. 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 programming problem, we try to discover as much as we can about a dual- feasible solution verifying optimality. Consider a linear programming problem in standard equality form. De- noting the columns of the constraint matrix by A 1 , A 2 , . . . , A n , we can write the problem as ( P ) maximize n j =1 c j x j subject to n 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 programming problem ( P ) if and only if x * is feasible for ( P ) and some feasible solution y * for the dual problem ( D ) satisfies the following “complementary slackness” conditions (for j = 1 , 2 , . . . , n ): ( CS ) A T j y * = c j whenever x * j > 0 .
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