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Lecture7
Step Length Selection
Homework (Due 2/20/01)
• 3.1
• 3.2
• 3.5
• 3.6
• 3.7
•
3.9
• 3.10
• Show equation 3.44
• The last step in the proof of
Theorem 3.6. (see
slides)
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Sufficient condition
)
1
,
0
(
,
)
(
)
(
1
1
∈
∇
+
≤
+
c
p
f
c
x
f
p
x
f
k
T
k
k
k
k
a
a
)
1
,
0
(
,
)
(
)
(
1
1
∈
∇
≤

+
c
p
f
c
x
f
p
x
f
k
T
k
k
k
k
a
a
The reduction should be proportional to both the step length,
and directional derivative.
St line
)
1
,
0
(
,
)
(
)
(
1
1
∈
∇
+
≤
+
c
p
f
c
x
f
p
x
f
k
T
k
k
k
k
a
a
)
(
)
(
a
a
l
p
x
f
k
k
≤
+
4
1
10

=
c
Sufficient condition
)
(
)
(
a
a
l
p
x
f
k
k
≤
+
Problem:
The sufficient decrease
condition is satisfied for
all small values of step length
3
Curvature condition
)
1
,
(
,
)
(
)
(
1
2
2
c
c
p
x
f
c
p
p
x
f
k
k
T
k
k
T
k
k
∈
∇
≥
+
∇
a
The slope of
is greater than
times the gradient
.
)
0
(
f
′
)
(
k
a
f
2
c
Derivative
)
(
k
a
f
′
gradient
conjugate
1for
.
Newton

Quasi
and
Newton
for
9
.
2
2
=
=
c
c
Curvature condition
If the slope is strongly negative, that means we can reduce
f
further along the chosen direction
If the slope is positive, it indicates we can not decrease
f
further
in this direction.
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Wolfe conditions
)
1
,
(
,
)
(
)
(
1
2
2
c
c
p
x
f
c
p
p
x
f
k
k
T
k
k
T
k
k
∈
∇
≥
+
∇
a
)
1
,
0
(
,
)
(
)
(
1
1
∈
∇
+
≤
+
c
p
f
c
x
f
p
x
f
k
T
k
k
k
k
a
a
Sufficient
decrease
Curvature
Backtracking Line Search
with
Terminate
)
(
;
)
(
)
(
until
;
set
);
1
,
0
(
,
,
0
Choose
a
a
ra
a
a
a
a
a
r
a
=
←
∇
+
≤
+
←
∈
k
k
T
k
k
k
k
repeat
end
p
f
c
x
f
p
x
f
repeat
c
If line search method chooses its step length appropriately,
we can dispense with the second condition
This ensures that the step length is short enough to satisfy the
sufficient decrease condition, but not too short.
Newton
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This note was uploaded on 06/12/2011 for the course COT 6505 taught by Professor Shah during the Spring '07 term at University of Central Florida.
 Spring '07
 Shah

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