12-hw116

# 12-hw116 - MATHEMATICS 116 FALL 2007-2008 CONVEXITY AND...

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MATHEMATICS 116, FALL 2007-2008 CONVEXITY AND OPTIMIZATION WITH APPLICATIONS Assignment #12 Last modiﬁed: January 4, 2008 Due Monday, January 14 before 5PM, in Charles Chen’s mailbox on the 3rd ﬂoor. This deadline is ﬁrm, since solutions will be posted then. The ﬁrst four problems are all models for problems that might appear on the ﬁnal exam. 1. This problem is a special case of Luenberger, Exercise 7.14.19. Your aunt has leased a vacation home for three years, and you plan to help her decorate it by giving her a painting each year, starting immediately. The rate at which a painting gives visual pleasure is equal to the square root of the amount that you pay for it. Of course, the ﬁrst one will give visual pleasure for three years, the second for two years, the third for only one year. You have 21 units of money to spend on the three paintings. Thanks to skills that you have acquired in a Harvard investment club, you can double every year whatever you have left, so your constraint is that x 1 + 1 2 x 2 + 1 4 x 3 = 21. You want to maximize the total visual pleasure, which is g ( x ) = 3 x 1 + 2 x 2 + x 3 . (a) Use Lagrange multipliers to ﬁnd the maximum value of g ( x ), so that you will know what the correct answer is. (b) Take f = 0 and determine the conjugate set C * , which is a one- dimensional subspace of the set of triples ( y 1 ,y 2 ,y 3 ), speciﬁed by a parameter λ. What is f * ( λ )? (c) Calculate the conjugate functional g * ( y ). (d) Calculate the maximum visual pleasure by using the Fenchel duality theorem.

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12-hw116 - MATHEMATICS 116 FALL 2007-2008 CONVEXITY AND...

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