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final_2006Qu

# final_2006Qu - Fall2006 Final Exam Thurs ARE211 This is the...

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Fall2006 ARE211 Final Exam Thurs, Dec 14, 2006 This is the final exam for ARE211. As announced earlier, this is an open-book exam. However, use of computers, calculators, Palm Pilots, cell phones, Blackberries and other non-human aids is forbidden. Read all questions carefully before starting the test. Allocate your 180 minutes in this exam wisely. The exam has 180 points, so aim for 1 minute per point. Make sure that you first do all the easy parts, before you move onto the hard parts. Always bear in mind that if you leave a part-question completely blank, you cannot conceivably get any marks for that part. The questions are designed so that, to some extent, even if you cannot answer some parts, you will still be able to answer later parts. Even if you are unable to show a result, you are allowed to use the result in subsequent parts of the question. So don’t hesitate to leave a part out. You don’t have to answer questions and parts of questions in the order that they appear on the exam, provided that you clearly indicate the question/part-question you are answering.

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2 Problem 1 [32 points] A) Let f ( x ) = ( x - 2)( x - 1)( x + 1)( x + 2). This function can be rewritten as: f ( x ) = ( x 2 - 1)( x 2 - 4) or f ( x ) = x 4 - 5 x 2 + 4 Consider the NPP min x R 1 f ( x ) s.t. braceleftbigg x ≥ - 2 x 0 . 5 (a) [ 2 points ] Convert the problem to the standard format for an NPP that we have been using in this course. (b) [ 5 points ] Is the constraint qualification (CQ) satisfied at all the points in the con- straint set? (c) [ 5 points ] Find the set of all points that satisfy the KKT conditions. (d) [ 5 points ] At what point is the minimum attained? B) Now consider the problem max x R 2 h ( x ) s.t x 2 ( x 1 - 2)( x 1 - 1)( x 1 + 1)( x 1 + 2) x 1 - 2 x 1 1 where h ( x ) = x 1 + x 2 . Do not solve this optimization problem! (a) [ 7 points ] Carefully apply KKT (including checking the CQ) for ¯ x = (0 , 4) and ˜ x = (1 , 0). (b) [ 8 points ] Using only KKT, what can we conclude about ¯ x = (0 , 4) and ˜ x = (1 , 0) as potential solutions to the optimization problem?
3 Problem 2 [32 points] Consider the system of equations: F ( x , α ) = S x + G ( α ) = 0 where S is an invertible 2x2 matrix, G

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final_2006Qu - Fall2006 Final Exam Thurs ARE211 This is the...

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