3 duality of linear programming 143 then its dual is

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Unformatted text preview: the Simplex Method gives the optimal value z∗ of the primal in the following way: for any feasible solution (x1 , . . . , xn ) of the primal we have n n+m j=1 i=1 ¯ ∑ c j x j = z∗ + ∑ cixi. z= Recall that, for all i ≤ n + m, ci is non-positive, and that it is null if xi is one of the basis ¯ variables. We set y∗ = −cn+i ¯ i for 1 ≤ i ≤ m. Then, by definition of the y∗ ’s and the xn+i ’s for 1 ≤ i ≤ m, we have i z= n ∑ c jx j j=1 n m ￿ n = z∗ + ∑ ci xi − ∑ y∗ bi − ∑ ai j x j ¯ i = ￿ i=1 i=1 ￿ m n z∗ − ∑ y∗ bi + ∑ i i=1 j=1 ￿ j=1 m ￿ ￿ c j + ∑ ai j y∗ x j . ¯ i i=1 Since this equation must be true whatever be the affectation of the xi ’s and since the ci ’s are ¯ non-positive, this leads to z∗ = m ∑ y∗bi i i=1 and m m i=1 i=1 c j = c j + ∑ ai j y∗ ≤ ∑ ai j y∗ for all 1 ≤ j ≤ n. ¯ i i Hence, (y∗ , . . . , y∗ ) defined as above is a feasible solution achieving the optimal value of the m 1 primal. By Lemma 9.8, this is an optimal solution of the dual. 9.3.4 Relation between primal and dual By the Duality Theorem and Lemma 9.7, a linear programme admits a solution if and only if its dual admits a solution. Moreover, according to Lemma 9.8, if a linear programme is unbounded, 9.3. DUALITY OF LINEAR PROGRAMMING 143 then its dual is not feasible. Reciprocally, if a linear programme admits no feasible solution, then its dual is unbounded. Finally, it is possible that both a linear programme and its dual have no feasible solution as shown by the following example. Maximize 2x1 − x2 Subject to: x1 − x2 ≤ 1 −x1 + x2 ≤ −2 x1 , x2 ≥0 Besides the fact it provides a certificate of optimality, the Duality Theorem has also a practical interest in the application of the Simplex Method. Indeed, the time-complexity of the Simplex Method mainly yields in the number of constraints of the considered linear programme. Hence, when dealing with a linear programme with few variables and many constraints, it will be more efficient to apply the Simplex Method on its dual. Another interesting application of the Duality Theorem is that it is possible to compute an optimal solution for the dual problem from an optimal solution of the primal. Doing so gives an easy way to test the optimality of a solution. Indeed, if you have a feasible solution of some linear programme, then a solution of the dual problem can be derived (as explained below). Then the initial solution is optimal if and only if the solution obtained for the dual is feasible and leads to the same value. More formally, the following theorems can be proved Theorem 9.11 (Complementary Slackness). Let (x1 , . . . , xn ) be a feasible solution of Problem 9.5 and (y1 , . . . , ym ) be a feasible solution of Problem 9.6. These are optimal solutions if and only if m ∑ ai j yi = c j , or x j = 0, or both for all 1 ≤ j ≤ n, and ∑ ai j x j = bi, or yi = 0, or both for all 1 ≤ i ≤ m. i=1 n j=1 Proof. First, we note that since x and y are feasible (bi − ∑n=1 ai j x j )yi ≥ 0 and (∑m 1 ai j yi − j i= c j )x j ≥ 0. Summing these inequalities over i and j, we obtain m ∑ i=1 n ￿ n bi − ∑ ai j x j yi ≥ 0 ￿ j=1 n ∑ ∑ ai j yi − c j j=1 ￿ i=1 ￿ (9.24) xj ≥ 0 (9.25) Adding Inequalities 9.24 and 9.25 and using the strong duality theorem,...
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This note was uploaded on 11/20/2013 for the course CS 101 taught by Professor Smith during the Fall '13 term at Mitchell Technical Institute.

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