P q p q p q p q p z 60 6 28 10 45 15 12 20 to subjec

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P Q P Q P Q P Q P Z 0 , , 60 6 28 10 45 15 12 20 to subjec 1440 1440 1800 Minimize + + + + + + = C B A C B A C B A C B A X X X X X X X X X X X X w P =10, Q =10, Z =1050 X A = 6, X B = 0, X C = 0, w =10800
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11 Weak Duality Corollaries Z ( x optimal ) ≤ w ( y optimal ) If Primal is unbounded, Dual is infeasible If Dual is unbounded, Primal is infeasible However, it is possible that both primal and dual are infeasible Given solutions X to Primal and Y to Dual if Z ( X ) = w ( Y ) X and Y are the optimal solutions respectively
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12 Strong Duality Theorem. In a primal-dual LP pair, if either the Primal or the Dual has an optimal feasible bounded solution, then 1. The other problem also has an optimal feasible bounded solution, and 2. The two optimal objective values are equal
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13 Complementary Slackness n i x m j b x a x c z i j i n i ji n i i i ,..., 1 , 0 ,..., 1 , subject to Maximize 1 1 = = = = = m j y n i c y a y b w j i j m j ji m j j j ,..., 1 , 0 ,..., 1 , subject to Minimize 1 1 = = = = = Define Slack variables for both Primal ( s j ) and Dual ( e i ): m j b s x a j j n i i ji ,..., 1 , 1 = = + = n i c e y a i i j m j ji ,..., 1 , 1 = = - = Theorem . The solution ( x 1 , …, x n , s 1 , …, s m ) is optimal to the Primal and ( y 1 , …, y m , e 1 , …, e n ) is optimal to the Dual if and only if (1) s j y j = 0 for all j ; and (2) e i x i = 0 for all i .
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14 Interpretation and Example If in an optimal solution of the Primal, a constraint is not tight (slack variable > 0), then in the optimal solution of the Dual, the corresponding variable must be 0 0 , 0 1440 6 15 1440 28 12 1800 10 20 to subjec 60 45 Maximize + + + + = Q P Q P Q P Q P Q P Z 0 , , 60 6 28 10 45 15 12 20 to subjec 1440 1440 1800 Minimize + + + + + + = C B A C B A C B A C B A X X X X X
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