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final_2006_b_sol - MS&E 211 Final Examination Autumn 2006...

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MS&E 211 Final Examination - Autumn 2006 Page 1 of 16 F INAL E XAMINATION S OLUTIONS MS&E 211 December 11, 2006 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) The examination ends in 180 minutes. d) There are 5 questions plus one bonus question on this exam. The exam is worth 180 points plus 20 possible bonus points. HONOR CODE In taking this examination, I acknowledge and accept the Stanford University Honor Code. NAME (signed) NAME (printed)
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MS&E 211 Final Examination - Autumn 2006 Page 2 of 16 Problem 1 - True or False (30 points) State whether each of the following italicized statements is True or False. Assume all other informa- tion given is correct. For a statement to be “True”, it must hold for all instances of the problem. 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 following nonlinear optimization problem. minimize g ( x 1 , x 2 ) subject to c ( x 1 , x 2 ) 0 x 1 0 Let λ be the Lagrange multiplier for the constraint c ( x 1 , x 2 ) 0 . True or False: The KKT conditions for this problem include the condition that x 2 ( g ( x 1 , x 2 ) - λ c ( x 1 , x 2 )) = 0 . False - there is no complementary condition for x 2 as it is a free variable b) Assume that the point ( x, λ ) (where x, λ 6 = 0 satisfies the KKT conditions for the following optimization problem: maximize f ( x ) subject to c ( x ) 0 True or False: When we convert the problem to our standard form, minimize - f ( x ) subject to c ( x ) 0 then ( x, λ ) will satisfy the new KKT conditions. False - the vector ( x, - λ ) will satisfy the KKT conditions of the problem in standard form c) In lecture notes # 6, we defined the core of the multi-firm alliance problem as the set of payments vectors z that satisfy the following two conditions: i I z i = V I i S z i V S True or False: The core is a convex set. True - see midterm
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MS&E 211 Final Examination - Autumn 2006 Page 3 of 16 d) True or False: In a Fisher aggregate social problem, the optimal Lagrange multiplier vector of the equality constraints is guaranteed to be the unique equilibrium price vector. False - the equilibrium price vector does not need to be unique e) Consider the following optimization problem: maximize f ( x ) subject to c ( x ) 0 Assume that f ( x ) is strictly concave and the point ( x, λ ) satisfies the KKT conditions. True or False: The point ( x, λ ) is guaranteed to be the unique global maximizer. False - consider Max - x 2 s.t. x 2 1 f) Consider the following optimization problem minimize 2 x 2 1 + 2 x 1 x 2 + x 2 2 - 10 x 1 - 10 x 2 subject to x 2 1 + x 2 2 5 3 x 1 + x 2 6 True or False: The point ( x 1 , x 2 , μ 1 , μ 2 ) = (1 , 2 , 1 , 0) is the global minimizer.
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