March 2014
Intermediate to advanced
412 pages
11h 16m
English
Content preview from Numerical Methods and Optimization
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342 Numerical Methods and Optimization: An Introduction
Therefore,
β
k
=
∇
T
k
Qd
(k−1)
(d
(k−1)
)
T
Qd
(k−1)
≈
∇
T
k
(∇
k
−∇
k−1
)
(d
(k−1)
)
T
(∇
k
−∇
k−1
)
. (13.21)
The Polak-Ribiere formula. Here we start with the Hestenes-Stiefel for-
mula and observe that for a quadratic function
(d
(k−1)
)
T
∇
k
=0
and
(d
(k−1)
)
T
∇
k−1
=(−∇
k−1
+ β
k−1
d
(k−2)
)
T
∇
k−1
= −∇
T
k−1
∇
k−1
+ β
k−1
(d
(k−2)
)
T
∇
k−1
= −∇
T
k−1
∇
k−1
(since (d
(k−2)
)
T
∇
k−1
= 0). Therefore,
β
k
≈
∇
T
k
(∇
k
−∇
k−1
)
(d
(k−1)
)
T
∇
k
− (d
(k−1)
)
T
∇
k−1
≈
∇
T
k
(∇
k
−∇
k−1
)
∇
T
k−1
∇
k−1
. (13.22)
The Fletcher-Reeves formula. Note that for a quadratic function
∇
T
k
∇
k−1
= ∇
T
k
(−d
(k−1)
+ β
k−1
d
(k−2)
)=0.
So, starting with the Polak-Ribiere formula, we obtain
β
k
≈
∇
T
k
(∇
k
−∇
k−1
)
∇
T
k−1
∇
k−1
=
∇
T
k
∇
k
−∇
T
k
∇
k−1
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I'm always learning. So when I got on to O'Reilly, I was like a kid in a candy store. There are playlists. There are answers. There's on-demand training. It's worth its weight in gold, in terms of what it allows me to do.