1 Examples
Why optimize?
1. Microeconomics: The consumer wants to choose the best bundle that she can afford.
2. Finance: Portfolio management (minimize risk subject to return)
3. Diet Problem: Find the cheapest combination of food that satisﬁes all the daily nutri
tional requirements.
4. Engineering: Design of aircrafts and aerospace structures of minimum weight given
constraints, or ﬁnd the optimal trajectories of space vehicles (dynamic optimization).
5. Physics: Thermodynamics (heat bath), quantum mechanics, etc.
2 Optimization
Optimize = Maximize or Minimize
2.1 Without Constraint
2.1.1 One Variable
Maximize
f
where
f
:
R
→
R
.
Definition
1
A stationary point
x
is a point such that
f
0
(
x
) = 0
.
If
x
*
is an optimizer, then it is a stationary point
. To ﬁnd an optimizer, we take
the ﬁrstorder condition and solve
f
0
(
x
) = 0
for
x
.
To determine whether an optimizer is a maximizer or a minimizer, we take the secondorder
condition; i.e. we look at the sign of
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 Fall '10
 Mathevet
 Optimization, Stationary point

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