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hw4soln - IE521 Advanced Optimization HW4 Solution 1(a...

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IE521 Advanced Optimization HW4 Solution 1. (a) Primal Problem: max 6 x 1 + 10 x 2 + 9 x 3 + 20 x 4 s.t. 4 x 1 + 9 x 2 + 7 x 3 + 10 x 4 600 x 1 + x 2 + 3 x 3 + 40 x 4 400 3 x 1 + 4 x 2 + 2 x 3 + x 4 500 x 1 , x 2 , x 3 , x 4 0 Dual Problem: min 600 p 1 + 400 p 2 + 500 p 3 s.t. 4 p 1 + p 2 + 3 p 3 6 (1) 9 p 1 + p 2 + 4 p 3 10 (2) 7 p 1 + 3 p 2 + 2 p 3 9 (3) 10 p 1 + 40 p 2 + p 3 20 (4) p 1 , p 2 , p 3 0 (b) Optimal solution to this LP is z = 2800 / 3 , x 1 = 400 / 3 , x 4 = 20 / 3 , x 2 = x 3 = 0 and s 1 = s 2 = 0 , s 3 = 280 / 3. Since x 1 > 0 and x 4 > 0 then the constraints associated with x 1 , x 4 in the dual problem (Constraints 1 and 4) must be binding. Therefore: 4 p 1 + p 2 + 3 p 3 = 6 (5) 10 p 1 + 40 p 2 + p 3 = 20 (6) Also, since the slack variable for third constraint in primal, s 3 > 0, then the dual variable associated with the third constraint, p 3 must be zero. If we put p 3 = 0 into equations 5 and 6 and solve the equation, we get p 1 = 22 / 15, p 2 = 2 / 15. Therefore the optimal dual solution is p 1 = 22 / 15, p 2 = 2 / 15, p 3 = 0 and optimal solution value = 2800/3. 2. Primal Problem: max 5 x 1 + 3 x 2 + x 3 s.t. 2 x 1 + x 2 + x 3 6 x 1 + 2 x 2 + x 3 7 x 1 , x 2 , x 3 0 Dual Problem: min 6 p 1 + 7 p 2 s.t. 2 p 1 + p 2 5 (7) p 1 + 2 p 2 3 (8) p 1 + p 2 1 (9) p 1 , p 2 0 Dual problem is two-dimensional. Therefore we can draw the constraints 7, 8, and 9 on two dimension and find the feasible region. When we find the feasible region, we can see that there are three extreme points which are (0 , 5) , (3 , 0) , (7 / 3 , 2 / 6). It is a minimization and the optimal solution is at the point (7 / 3 , 2 / 6) with objective function value of 49/3. Using the complementary slackness condition, we can say that since p 1 > 0 and p 2 > 0, the constraints associated with p 1 and p 2 in primal problem must be binding. Therefore, 2 x 1 + x 2 + x 3 = 6 (10) x 1 + 2 x 2 + x 3 = 7 (11) (12) 1
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In order to determine which variables are zero in primal problem, we need to look at which constraints are not binding at dual problem. At the solution (7
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