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assign5.sol - CO 350 Assignment 5 Winter 2010 Due: Friday...

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Due: Friday February 26 at 10:25 a.m. Question # Max. marks Part marks 1 5 1 2 2 2 4 2 2 3 5 3 2 4 4 2 2 5 4 2 2 6 8 3 1 2 2 Total 30 1. Consider the LP problem maximize c T x subject to Ax = b, x 0, where c = - 1 1 - 1 - 1 - 1 1 , A = 1 0 0 1 0 6 3 1 - 4 0 0 2 1 0 2 0 1 2 , and b = 9 2 6 which was solved in Assignment 4.5. (a) Write the dual problem. (b) Use the optimal basis for the primal problem to find an optimal dual solution ˆ y . (c) Verify that ˆ y is optimal for the dual problem. Solution: (a) The dual is min 9 y 1 + 2 y 2 + 6 y 3 subj. to y 1 + 3 y 2 + y 3 ≥ - 1 y 2 1 - 4 y 2 + 2 y 3 ≥ - 1 y 1 ≥ - 1 y 3 ≥ - 1 6 y 1 + 2 y 2 + 2 y 3 1 (b) From the solution of the primal, we have the optimal basis B = { 3 , 6 , 2 } . We solve A T B y = c B for y , that is, 0 - 4 2 6 2 2 0 1 0 y 1 y 2 y 3 = c B = - 1 1 1 . Thus we get ˆ y = ( - 2 / 3 , 1 , 3 / 2) T . (c) A T ˆ y = (23 / 6 , 1 , - 1 , - 2 / 3 , 3 / 2 , 1) T ( - 1 , 1 , - 1 , - 1 , - 1 , 1) T = c T , so ˆ y is feasible for the dual. Also b T ˆ y = 5, which is equal to the objective value of the known feasible solution of the solution of the primal, so ˆ y is optimal to the dual problem. 2. Exercise 6 of Section 4.8. Solution:
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assign5.sol - CO 350 Assignment 5 Winter 2010 Due: Friday...

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