# sol6 - CO350 Homework 6 Solutions – Winter 2007 1...

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Unformatted text preview: CO350 Homework 6 Solutions – Winter 2007 1. Exercise 4.8.7 (a) The dual of (P) is: min 3 y 1 + 2 y 2 + 7 y 3 + 4 y 4 y 1 + 2 y 3 ≥ 1 y 2 + 2 y 4 ≥ 5 y 1 + y 2- y 3 ≥ 3 3 y 1 + y 2- y 3 + y 4 ≥ 6 3 y 3 + 2 y 4 ≥ 6 y 1 , y 2 , y 3 , y 4 ≥ The complementary slackness conditions are: x 1 + x 3 + 3 x 4 = 3 or y 1 = 0 x 2 + x 3 + x 4 = 2 or y 2 = 0 2 x 1- x 3- x 4 + 3 x 5 = 7 or y 3 = 0 2 x 2 + x 4 + 2 x 5 = 4 or y 4 = 0 and x 1 = 0 or y 1 + + 2 y 3 = 1 x 2 = 0 or y 2 + 2 y 4 = 5 x 3 = 0 or y 1 + y 2- y 3 = 3 x 4 = 0 or 3 y 1 + y 2- y 3 + y 4 = 6 x 5 = 0 or 3 y 3 + 2 y 4 = 6 (b) Consider y * = (1 , 2 , , 3) T . From the first group of conditions we have: y 1 > 0 = ⇒ x 1 + x 3 + 3 x 4 = 3 y 2 > 0 = ⇒ x 2 + x 3 + x 4 = 2 y 4 > 0 = ⇒ 2 x 2 + x 4 + 2 x 5 = 4 From the second group of conditions we have: y 2 + 2 y 4 > 5 = ⇒ x 2 = 0 3 y 1 + y 2- y 3 + y 4 > 6 = ⇒ x 4 = 0 x * = (1 , , 2 , , 2) T is a solution to the above two sets of constraints. Substitution into (P) shows that...
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sol6 - CO350 Homework 6 Solutions – Winter 2007 1...

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