362 Nonlinear Programming II: Unconstrained Optimization Techniques
To find the minimizing step lengthλ∗ 1 alongS 1 , we minimizef(X 1 +λ 1 S 1 )=f({
0
0
}
+λ 1{
− 1
1
)}
=f (−λ 1 , λ 1 )=λ^21 − 2 λ 1with respect toλ 1. Since df/dλ 1 = at 0 λ∗ 1 = , we obtain 1X 2 =X 1 +λ∗ 1 S 1 ={
0
0
}
+ 1
{
− 1
1
}
=
{
− 1
1
}
Since∇f 2 = ∇ f(X 2 )={− 1
− 1}
and||∇f 2 || = 1. 4142 > ε, we proceed to update the
matrix [Bi] by computingg 1 = ∇f 2 − ∇f 1 ={
− 1
− 1
}
−
{
1
− 1
}
=
{
− 2
0
}
d 1 =λ∗ 1 S 1 = 1{
− 1
1
}
=
{
− 1
1
}
d 1 dT 1 ={
− 1
1
}
{−1 1} =
[
1 − 1
−1 1
]
dT 1 g 1 = {− 1 1 }{
− 2
0
}
= 2
d 1 gT 1 ={
− 1
1
}
{−2 0} =
[
2 0
−2 0
]
g 1 dT 1 ={
− 2
0
}
{−1 1} =
[
2 − 2
0 0
]
gT 1 [B 1 ]g 1 = {− 2 0 }[
1 0
0 1
]{
− 2
0
}
= {−2 0}
{
− 2
0
}
= 4
d 1 gT 1 [B 1 ]=[
2 0
−2 0
][
1 0
0 1
]
=
[
2 0
−2 0
]
[B 1 ]g 1 dT 1 =[
1 0
0 1
][
2 − 2
0 0
]
=
[
2 − 2
0 0
]
Equation (6.136) gives[B 2 ] =|
[
1 0
0 1
]
+
(
1 +
4
2
)
1
2
[
1 − 1
−1 1
]
−
1
2
[
2 0
−2 0
]
−
1
2
[
2 − 2
0 0
]
=
[
1 0
0 1
]
+
[ 3
2 −
3
2
−^3232]
−
[
1 0
−1 0
]
−
[
1 − 1
0 0
]
=
[ 1
2 −
1
2
−^1252