co227_a4_soln

# co227_a4_soln - Assignment 4 Due Wednesday November 16 at...

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Unformatted text preview: Assignment 4 Due: Wednesday November 16 at the BEGINNING of class 1. Given the following linear programs, write down the corresponding dual linear program: (a) maximize 2 x 1 + x 2- x 3 + 3 x 4 + x 5 subject to x 1 + 2 x 2 + x 4 + x 5 = 3- x 1 + x 2- 2 x 3 + 2 x 5 = 2 x 1- x 2 + x 3 + x 4- x 5 = 1 x 1 ,x 2 ,x 3 ,x 4 ,x 5 ≥ Solution: minimize 3 y 1 + 2 y 2 + y 3 subject to y 1- y 2 + y 3 ≥ 2 2 y 1 + y 2- y 3 ≥ 1- 2 y 2 + y 3 ≥ - 1 y 1 + y 3 ≥ 3 y 1 + 2 y 2- y 3 ≥ 1 (b) maximize x 1- 3 x 2 + 2 x 3 + 2 x 4 + x 5 subject to x 1 + 2 x 2- 2 x 3 + x 4 ≥ 5 x 1- x 3 + x 4- x 5 = 4- x 1 + x 2 + x 3 + 2 x 4- x 5 ≤ 9 x 1 ≥ x 2 ,x 5 ≤ x 3 ,x 4 free Solution: minimize 5 y 1 + 4 y 2 + 9 y 3 subject to y 1 + y 2- y 3 ≥ 1 2 y 1 + y 3 ≤ - 3- 2 y 1- y 2 + y 3 = 2 y 1 + y 2 + 2 y 3 = 2- y 2- y 3 ≤ 1 y 1 ≤ y 2 free y 3 ≥ (c) minimize- 3 x 1 + x 2 + 2 x 3 subject to 2 x 1 + x 2- x 3 = 2 3 x 2 + 7 x 3 ≤ 1- x 1 + x 2 + x 3 ≥ 4 x 1 + x 2 + x 3 =- 1 x 1 free x 2 ≥ x 3 ≤ Solution: maximize 2 y 1 + y 2 + 4 y 3- y 4 subject to 2 y 1- y 3 + y 4 =- 3 y 1 + 3 y 2 + y 3 + y 4 ≤ 1- y 1 + 7 y 2 + y 3 + y 4 ≥ 2 y 1 ,y 4 free y 2 ≤ y 3 ≥ 2. For the following, you are given a feasible solution x * for the linear program and a fea- sible solution y * for the corresponding dual. Show if they are both optimal solutions or not using complementary slackness conditions....
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co227_a4_soln - Assignment 4 Due Wednesday November 16 at...

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