MATHEMATICS 116, FALL 2007
CONVEXITY AND OPTIMIZATION WITH APPLICATIONS
Assignment #9
Last modiﬁed: November 25, 2007
Due Thursday, Nov. 29 at class.
1. Luenberger, Problem 5.14.6. This is like example 2 of section 5.9.
Hint (based on discussions with Dara Blume and Ludovica Gazze)
The problem is like the door opener example in Luenberger. To guarantee
existence, call the space in which you are looking for the solution
X
*
and
the space with the constraints
X
. They will have diﬀerent norms!
You have two constraints, given by functionals
y
and
z
. The ﬁrst makes
sure that the car comes to rest at time
T
. The second makes sure that it
returns to the origin.
Take a linear combination
p
=
ay
+
bz
. (It is worth giving this functional
a name).
The solution
u
is aligned with this
p
. It has a constant magnitude, and like
p
it changes sign only once. One of the constraints lets you ﬁgure out when
that happens.
2. Luenberger, Problem 5.14.9. This is like example 3 of section 5.9. There is
a reference to a journal article by Neustadt at the end of the chapter.
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 Fall '09
 Math, Vector Space, Equals sign, normed vector space, Luenberger, door opener example

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