improved notes on trust region methods, corrected update rule as per Hauser notes and removed old links

release/4.3a0
Varun Agrawal 2019-05-31 11:47:40 -04:00
parent 72296f786b
commit c13bd50a1a
1 changed files with 46 additions and 19 deletions

View File

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@ -231,7 +241,7 @@ key "Hauser06lecture"
\end_inset
(in our /net/hp223/borg/Literature folder).
.
\end_layout
\begin_layout Standard
@ -465,22 +475,39 @@ where
\end_inset
.
A typical update rule [
\color blue
see where this came from in paper
\color inherit
] is
A typical update rule, as per Lec.
7-1.2 of
\begin_inset CommandInset citation
LatexCommand cite
key "Hauser06lecture"
\end_inset
is:
\begin_inset Formula
\[
\Delta\leftarrow\begin{cases}
\max\left(\Delta,3\norm{\delta x_{d}}\right)\text{,} & \rho>0.75\\
\Delta & 0.75>\rho>0.25\\
\Delta/2 & 0.25>\rho
\Delta_{k+1}\leftarrow\begin{cases}
\Delta_{k}/4 & \rho<0.25\\
\min\left(2\Delta_{k},\Delta_{max}\right)\text{,} & \rho>0.75\\
\Delta_{k} & 0.75>\rho>0.25
\end{cases}
\]
\end_inset
where
\begin_inset Formula $\Delta_{k}\triangleq\norm{\delta x_{d}}$
\end_inset
.
Note that the rule is designed to ensure that
\begin_inset Formula $\Delta_{k}$
\end_inset
never exceeds the maximum trust region size
\begin_inset Formula $\Delta_{max}.$
\end_inset
\end_layout
@ -641,7 +668,7 @@ Thus, mathematically, we can write the dogleg update
\begin_inset Formula
\[
\delta x_{d}^{\left(k\right)}=\begin{cases}
-\frac{\Delta}{\norm{g^{\left(k\right)}}}g^{\left(k\right)}\text{,} & \Delta<\norm{\delta x_{u}^{\left(k\right)}}\\
-\frac{\Delta}{\norm{\delta x_{u}^{\left(k\right)}}}\delta x_{u}^{\left(k\right)}\text{,} & \Delta<\norm{\delta x_{u}^{\left(k\right)}}\\
\left(1-\tau^{\left(k\right)}\right)\delta x_{u}^{\left(k\right)}+\tau^{\left(k\right)}\delta x_{n}^{\left(k\right)}\text{,} & \norm{\delta x_{u}^{\left(k\right)}}<\Delta<\norm{\delta x_{n}^{\left(k\right)}}\\
\delta x_{n}^{\left(k\right)}\text{,} & \norm{\delta x_{n}^{\left(k\right)}}<\Delta
\end{cases}