MATHEMATICS 116, FALL 2007
CONVEXITY AND OPTIMIZATION WITH APPLICATIONS
Outline #1 (Generalizing from two dimensions)
Last modiﬁed: September 10, 2007
Reading.
Luenberger, Chapter 1
Lecture topics.
1. Why such a long introduction?
The general approach in Luenberger, which has made the book a classic, is
this.
•
Identify techniques from algebra, elementary singlevariable calculus,
or elementary multivariable calculus that can be used to solve opti
mization problems.
•
Reformulate the solution geometrically.
•
Using geometry for inspiration, generalize the solution, typically to
inﬁnitedimension vector spaces and noneuclidean norms, and prove
(algebraically) that it is still valid.
All the ﬁnitedimensional problems in this outline should be familiar, though
they may be valuable review for some students. The inﬁnitedimensional
problems are just stated, not solved, and we will take quite a while to get
to them.
Important concepts are in CAPITAL LETTERS. These will appear in
Chapters 2, 3, and 5, generally in a context where there is no mention
of optimization, just some challenging mathematics.
My hope for these introductory lectures is to convince you that optimization
problems are fun and relevant, that some of the best ones can only be
formulated in inﬁnitedimensional vector spaces, and that it is worth your
while to learn quite a few new deﬁnitions and theorems in order to be able
to solve them.
1
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You are entering a student competition to draw up a business plan for a
company with
m
scientists and
n
other employees. Entries with
n
2
>
2
m
2
get rejected. You want to have the highest possible ratio of scientists to
other employees. Does this optimization problem have a solution?
As director of the state lottery, you are designing scratch tickets by assigning
probabilities to the possible payoﬀs from 0 through $4. Since a ticket sells
for $5, you want to be as generous as possible. Does this optimization
problem in
R
5
have a solution?
Change the problem so that any integer payoﬀ is allowed. Does this opti
mization problem in an inﬁnitedimensional vector space have a solution?
Can you remember the rule for when a realvalued function
f
is guaranteed
to have a maximum in a subset
X
⊂
R
2
?
3. An easy allocation problem
Your small bakery can produce only two products. A batch of frosted
cookies uses up 1 pound of ﬂour and 3 pounds of sugar. A batch of cake
uses up 2 pounds of ﬂour and 1 pound of sugar. Each day your suppliers
bring you 14 pounds of ﬂour and 17 pounds of sugar. Your optimization
problem is to look at the market price of cookies and cakes and decide what
to produce.
Let
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 Fall '09
 Math, Vector Space, Optimization, Luenberger

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