HW2-Sol - MS&E111 Introduction to Optimization Prof....

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HW-2 Solutions Introduction to Optimization Prof. Amin Saberi Homework 2 Solutions Problem 1. a) The feasible region for the given LP is: -5 -4 -3 -2 -1 0 1 2 3 4 5 -6 -4 -2 0 2 4 6 Feasible Region b) The dual of the LP is minimize - 4 y 1 - y 2 subject to - 2 y 1 - y 2 1 - 3 y 1 + y 2 ≥ - 1 y 1 0 , y 2 0 where there is no feasible solution for the given constraints. Problem 2. a) See Excel Fle on course web page.
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2 b) The dual is: min - 3 y 1 + 4 y 2 + 6 y 3 s.t. - y 1 - y 2 + y 3 ≥ - 1 - 2 y 1 + 2 y 2 + 7 y 3 ≥ - 1 y 1 , y 2 , y 3 0 The Shadow Prices in the Sensitivity Report in Excel are y 1 = 1 . 2, y 2 = 0 and y 3 = 0 . 2. All of them are non negative. - 1 . 2 - 0+0 . 2 = - 1 ≥ - 1, and - 2(1 . 2)+2(0)+7(0 . 2) = - 1 ≥ - 1. Therefore the given y is dual feasible. Moreover - 3(1 . 2) + 4(0) + 6(0 . 2) = - 2 . 4, which is equal to the primal optimal objective function value. Therefore, by strong duality, the latter shadow prices give the optimal dual solution. Problem 3. a) The primal problem we face is: min x 1 + 0 . 25 x 2 s.t. 40 x 1 + 200 x 2 400 5 x 1 + 40 x 2 40 100 x 1 + 20 x 2 200 x 1 , x 2 0 . Note that the dual of the dual is the primal problem. Thus, we can immediately write down the dual of the above problem by interpreting it as the dual to a symmetric form primal. Thus, we have the dual: max 400 y 1 + 40 y 2 + 200 y 3 s.t. 200 y 1 + 40 y 2 + 20 y 3 1 40 y 1 + 5 y 2 + 100 y 3 0 . 25 y 1 , y 2 , y 3 0 . b) The pharmaceutical company’s problem may be interpreted as one of maximizing its
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HW2-Sol - MS&E111 Introduction to Optimization Prof....

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