211_midterm_2007_sol

# 211_midterm_2007_sol - MS&E 211 Midterm Examination...

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Unformatted text preview: MS&E 211 Midterm Examination SOLUTIONS - Autumn 2007 Page 1 of 10 M IDTERM E XAMINATION MS&E 211 INSTRUCTIONS a) Take alternate seating if possible. b) This is an open book and open notes exam. You are not permitted to use computers or any calculator programming functions. c) Answer all questions on the exam. d) The examination ends in 75 minutes. e) There are 4 questions on this exam plus one bonus question. The exam is worth 100 points with an additional 15 bonus points available. HONOR CODE In taking this examination, I acknowledge and accept the Stanford University Honor Code. NAME (signed) NAME (printed) MS&E 211 Midterm Examination SOLUTIONS - Autumn 2007 Page 2 of 10 Problem 1 – True or False (20 points) State whether each of the following italicized statements is True or False. Assume all other in- formation given is correct. For each part, only your answer, which should be one of “True” or “False”, will be graded. Explanations will not be read. a) Consider the linear program: Minmize c T x subject to Ax = b x ≥ where x ∈ < n , A ∈ < m × n and b ∈ < m . True or False: Assume that there are at least two different optimal solutions to this problem. Then there are at least two different basic feasible optimal solutions to this problem. False. consider the example: Minimize subject to x 1 = 2 x 1 ,x 2 ≥ All vectors (2 ,c ) ,c ≥ are the optimal solutions. There is only one solution (2,0) that is basic feasible optimal solution. b) Still consider the linear program above. True or False: Assume that there is at least one optimal solution to this problem and every optimal solution is a basic feasible solution. Then the problem has only one optimal solution. True. If there are two distinct optimal solutions. They are basic feasible according to our assumption. Then any point between them will be optimal too. But those points are not basic. This contradicts with the conditions of the problem. c) One can easily see that the linear program below is infeasible. Minmize- x 1- 2 x 2 subject to x 1 ≤ - 1- x 2 ≤ - 1 x 1 ,x 2 ≥ MS&E 211 Midterm Examination SOLUTIONS - Autumn 2007 Page 3 of 10 True or False: The dual problem is unbounded. False . The dual problem will be either infeasible or unbounded if the primal problem is infeasible. Check the dual of this problem. It is easy to see the dual is infeasible. d) Consider the following linear program: Maximize x 1 + x 2 subject to x 1 + x 2 ≥ 3 x 1 ≤ x 2 x 1 ≤ α x 1 ,x 2 ≥ True or False: The dual of the problem is infeasible for any α ≥ True. If α ≥ , then the system is always feasible and unbounded (by letting x 2 go to the infinity.). So its dual must be infeasible. MS&E 211 Midterm Examination SOLUTIONS - Autumn 2007 Page 4 of 10 Problem 2 – Optimal Currency Conversion (25 points) You are currently planning a vacation to Greece for the fall break and must therefore convert some of your earned U.S. dollars into Euros. Assume that 3 currencies are available for doing your con-of your earned U....
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211_midterm_2007_sol - MS&E 211 Midterm Examination...

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