hmk3_07_sol

hmk3_07_sol - MS&E 211 Linear and Nonlinear...

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Fall 2007 Linear and Nonlinear Optimization Oct 18, 2007 Prof. Yinyu Ye Homework Assignment 3: Sample Solution Problem 1. a) The dual of the problem: minimize 4 y 1 + 11 y 2 subject to 2 y 1 + 5 y 2 9 3 y 1 + 4 y 2 8 y 1 + 3 y 2 5 y 1 , y 2 0 b) After we change b to b 0 (6:11), then we need to check whether A - 1 B b 0 is greater or less than 0: A - 1 B b 0 = 3 - 1 - 5 2 6 11 = 7 - 8 Hence, the optimal basis will change. c) Since x 2 is a non-basic variable, we need to check whether r N + 4 c 2 e j is greater or less than 0: r N + 4 c 2 e j = 2 + ( - 3) * 1 0 Hence, the optimal basis will change. d) Since the optimal solution still satisfies the new constraint, the optimal solution won’t change after the new constraint is added. Problem 2. a) We need to use the complementary slackness conditions to find the dual solutions. First, note that based on the primal solutions, the constraints in the primal corre- sponding to λ 2 , λ 3 and λ 5 have slack. This indicates that λ 2 = 0, λ 3 = 0 and λ 5 = 0. Now, since x 1 , x 3 and x 4 are all greater than zero. This indicates that the first, third and fourth
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constraints of the dual will be binding. Now, just substitute in the value y = 0 . 7 and solve for the λ 0 s . We find ( λ 1 , λ 2 , λ 3 , λ 4 , λ 5 )=(0 . 2 , 0 , 0 , 0 . 5 , 0). b) The value of y represents the lowest ratio of p i r i of all the companies that you have invested in. It is basically your threshold ratio value for investing in a company. This value also represents the shadow price of your budget constraint. So, it indicates how much money you could make if your budget was increased by 1. c) The value of λ i represents the difference in return between an incremental amount of ownership in company i instead of your lowest ratio investment. If you were allowed to invest more in company i , the λ
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hmk3_07_sol - MS&E 211 Linear and Nonlinear...

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