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CS205 – Class 8
Covered In Class
: 1, 3, 4, 5, 6
Reading
: Heath Chapter 6
1.
Optimization
– given an
objective function
f
, find relative maxima or minima. Note that since max
f
=
min
f
it is enough to only consider minima.
a. We’ll start with scalar functions f of one variable for now.
b.
unconstained
– any
x
n
R
is acceptable
c.
constrained
– minimize
f
on a subset
S
R
d. usually find
local minima,
since global minima are hard to find
i. one option is to find many local minima and compare them to find a global minimum
e.
Not equivalent
to solving for
f(x)
= 0. There might exist no such x or the minimum may be attained
somewhere
f(x)
< 0.
f.
poorly conditioned
since
'( )
0
f x
at a minimum, i.e. locally flat (similar to a multiple root) – error
tolerance should be more like
as opposed to
g. given a critical point where
'( )
0
f x
, we can use the sign of the second derivative to determine
whether we have a local minimum, a local maximum, or an inflection point
i. if
''( ) 0
f
x
, concave up, minimum
ii. if
''( ) 0
f
x
, concave down, maximum
iii. otherwise when the second derivative vanishes, we have an inflection point, i.e. neither a
minimum nor a maximum
2
0
2
2
0
2
15
10
5
0
2
0
2
2
0
2
0
5
10
15
5
2.5
0
2.5
5
5
2.5
0
2.5
5
20
0
20
Local maxima
Local minima
Saddle
h.
unimodal
– [ , *]
a x
is monotonically decreasing and [ *, ]
x b
is monotonically increasing
*
x
is the
minimum – most schemes need a unimodal interval in order to converge
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This document was uploaded on 05/25/2011.
 Spring '07

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