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# hw3soln - IE521 Advanced Optimization Fall 2011 HW3...

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Fall 2011, HW3 Solutions 1. (a) g ( p 1 ,p 2 ,p 3 ) = p 1 + 3 p 2 + 4 p 3 + max { x 1 ,x 2 0 } (2 + p 1 - p 2 - p 3 ) x 1 + (1 - p 1 - p 2 + 2 p 3 ) x 2 (b) g ( p 1 ,p 2 ,p 3 ) = 4 p 1 + p 2 + 3 p 3 + min { y 1 ,y 2 0 } (1 - 2 p 1 - p 2 - p 3 ) y 1 + ( - 1 - p 1 - p 2 - 2 p 3 ) y 2 2. (a) min p 1 + 3 p 2 + 4 p 3 s.t. - p 1 + p 2 + p 3 2 p 1 + p 2 - 2 p 3 1 p 1 ,p 2 ,p 3 0 (b) max 4 p 1 + p 2 + 3 p 3 s.t. 2 p 1 + p 2 + p 3 1 p 1 + p 2 + 2 p 3 ≤ - 1 p 1 ,p 2 ,p 3 0 (c) min 5 p 1 + 7 p 2 + 6 p 3 + 4 p 4 s.t. p 1 + 2 p 2 + p 4 4 p 1 + p 2 + 2 p 3 = - 1 p 3 + p 4 = 2 p 1 ,p 2 0 ,p 4 is free ,p 3 0 (d) max 6 p 1 + 8 p 2 s.t. p 1 + p 2 4 2 p 1 - p 2 2 2 p 2 = - 1 p 1 0 ,p 2 is free 3. (a) (D1) min 6 p 1 + 5 p 2 s.t. 3 p 1 + 2 p 2 1 p 1 + p 2 2 p 2 is free, p 1 0 (b) (D2) min 6 y 1 + 5 y 2 + 5 y 3 s.t. 3 y 1 + 2 y 2 + 2 y 3 1 y 1 + y 2 + y 3 2 y 3 0 ,y 1 ,y 2 0 (c) In (D2), let y 2 = t 2 , y 3 = - t 3 where t 2 ,t 3 0. We obtain (D3):

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hw3soln - IE521 Advanced Optimization Fall 2011 HW3...

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