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MATHEMATICS 116, FALL 20072008
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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 Fall '09
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