hw3solutions

hw3solutions - ORF 307 Homework 3 Solutions Exercise 1 3.1...

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ORF 307 Homework 3 Solutions Exercise 1. 3.1 The initial dictionary for the perturbed problem is: 0 +10 x 1 - 57 x 2 - 9 x 3 - 24 x 4 w 1 = ± 1 - . 5 x 1 +5 . 5 x 2 +2 . 5 x 3 - 9 x 4 w 2 = ± 2 - . 5 x 1 +1 . 5 x 2 + . 5 x 3 - x 4 w 3 = 1 + ± 3 - x 1 We pivot x 1 into the basis and w 2 out of the basis (noting that ± 1 ± ± 2 ). The resulting dictionary is: 20 ± 2 - 20 w 2 - 27 x 2 + x 3 - 4 x 4 w 1 = ± 1 - ± 2 + w 2 +4 x 2 +2 x 3 - 8 x 4 x 1 = 2 ± 2 - 2 w 2 +3 x 2 + x 3 - 2 x 4 w 3 = 1 - 2 ± 2 + ± 3 +2 w 2 - 3 x 2 - x 3 +2 x 4 We pivot x 3 into the basis and w 3 , the only leaving choice, out of the basis and get the resulting dictionary: 1 + 18 ± 2 + ± 3 - 18 w 2 - 30 x 2 - w 3 - 2 x 4 w 1 = 2 + ± 1 - 5 ± 2 + 2 ± 3 +5 w 2 - 2 x 2 - 2 w 3 - 4 x 4 x 1 = 1 + ± 3 x 3 = 1 - 2 ± 2 + ± 3 +2 w 2 - 3 x 2 - w 3 +2 x 4 This dictionary is optimal with objective value 1 and solution ( x 1 , x 2 , x 3 , x 4 ) = (1 , 0 , 1 , 0). Exercise 2. 3.4 Add slack variables and write the initial dictionary: Z= 0 + c 1 x 1 ... c n x n w i = 0 - a i 1 x 1 ... - a in x n for i = 1,. ..,m The initial feasible basic solution is (
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This note was uploaded on 04/30/2008 for the course ORF 307 taught by Professor Alexandrew.d'aspremont during the Spring '08 term at Princeton.

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hw3solutions - ORF 307 Homework 3 Solutions Exercise 1 3.1...

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