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# HW6 - iterations If yes give the optimal solution Justify...

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IEOR 262A Mathematical Programming I Fall 2007 Homework 6 (Due Friday, Nov 2, Before Discussion) Solve Exercise 4.2, 4.4 and 4.8 in the text book Introduction to linear optimization by Bertsimas and Tsitsiklis. Problem 1. Write the duals of following problems Verify that the dual of the dual is the original problem. (1) max 4 x 1 - 3 x 2 + 8 x 3 s.t. - 2 x 1 6 4 x 2 14 - 12 x 3 - 8 (2) max n j =1 c 0 j x j s.t. n j =1 a 1 j x j = b 1 n j =1 a 2 j x j b 2 n j =1 a 3 j x j b 3 x j 0 for all j (3) min 3 x 1 - 7 x 2 +6 x 4 +5 x 5 - x 6 s.t. 5 x 1 +8 x 2 +3 x 3 +3 x 4 +2 x 5 +11 x 6 = 200 Problem 2. Suppose we solve the following LP by simplex method. min 3 x 1 +2 x 2 - 3 x 3 - 6 x 4 +10 x 5 - 5 x 6 s.t. x 1 +2 x 2 + x 4 - 6 x 6 = 11 x 2 + x 3 +3 x 4 - 2 x 5 - x 6 = 6 x 1 +2 x 2 + x 3 +3 x 4 - x 5 - 5 x 6 = 13 x 1 x 2 x 3 x 4 x 5 x 6 0 We get the optimal basis B = { 1 , 2 , 3 } and B - 1 = - 1 - 2 2 1 1 - 1 - 1 0 1 . For each of the following changes in the LP, can you get the optimal solution to the new problem without any further

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Unformatted text preview: iterations? If yes, give the optimal solution. Justify your answer. Each question is independent. 1 (1) The RHS of the ﬁrst constraint is changed from 11 to 13. (2) The coeﬃcient of x 1 in the objective function is changed from 3 to-1, and the coeﬃcient of x 4 in the objective function is changed from-6 to-4. (3) A new variable x 7 is added with cost coeﬃcient-7 and the corresponding column vector A 7 = (1 , 2 ,-3) (4) The coeﬃcient of x 5 in the second constraint is changed from-2 to-1. (5) A new constraint is added: x 1 + x 2-2 x 3 + x 4 ≥ 1. 2...
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