From c13bd50a1adfecadd521612ff9145f11b657d18a Mon Sep 17 00:00:00 2001
From: Varun Agrawal <varagrawal@gmail.com>
Date: Fri, 31 May 2019 11:47:40 -0400
Subject: [PATCH] improved notes on trust region methods, corrected update rule
 as per Hauser notes and removed old links

---
 doc/trustregion.lyx | 65 ++++++++++++++++++++++++++++++++-------------
 1 file changed, 46 insertions(+), 19 deletions(-)

diff --git a/doc/trustregion.lyx b/doc/trustregion.lyx
index bc4393fdf..3296da512 100644
--- a/doc/trustregion.lyx
+++ b/doc/trustregion.lyx
@@ -1,10 +1,11 @@
-#LyX 2.0 created this file. For more info see http://www.lyx.org/
-\lyxformat 413
+#LyX 2.1 created this file. For more info see http://www.lyx.org/
+\lyxformat 474
 \begin_document
 \begin_header
 \textclass article
 \begin_preamble
-\usepackage{amssymb}
+\usepackage{url}
+\usepackage{hyperref}
 \end_preamble
 \use_default_options true
 \maintain_unincluded_children false
@@ -15,13 +16,13 @@
 \font_roman default
 \font_sans default
 \font_typewriter default
+\font_math auto
 \font_default_family default
 \use_non_tex_fonts false
 \font_sc false
 \font_osf false
 \font_sf_scale 100
 \font_tt_scale 100
-
 \graphics default
 \default_output_format default
 \output_sync 0
@@ -32,15 +33,24 @@
 \use_hyperref false
 \papersize default
 \use_geometry false
-\use_amsmath 1
-\use_esint 1
-\use_mhchem 1
-\use_mathdots 1
+\use_package amsmath 1
+\use_package amssymb 2
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 0
+\use_package stmaryrd 1
+\use_package undertilde 1
 \cite_engine basic
+\cite_engine_type default
+\biblio_style plain
 \use_bibtopic false
 \use_indices false
 \paperorientation portrait
 \suppress_date false
+\justification true
 \use_refstyle 1
 \index Index
 \shortcut idx
@@ -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}