Frank Dellaert
GitHub
commit
e439629349
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@ -8,7 +8,6 @@
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%% Saved with string encoding Unicode (UTF-8)
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@webpage{Hauser06lecture,
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Author = {Raphael Hauser},
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Date-Added = {2011-10-10 15:21:22 +0000},
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@ -16,11 +15,5 @@
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Title = {Lecture Notes on Unconstrained Optimization},
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Url = {http://www.numerical.rl.ac.uk/nimg/oupartc/lectures/raphael/},
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Year = {2006},
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howpublished = {\href{http://www.numerical.rl.ac.uk/nimg/oupartc/lectures/raphael/}{link}},
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}
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@ -1,10 +1,11 @@
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#LyX 2.0 created this file. For more info see http://www.lyx.org/
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\lyxformat 413
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#LyX 2.1 created this file. For more info see http://www.lyx.org/
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\lyxformat 474
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\begin_document
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\begin_header
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\textclass article
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\begin_preamble
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\usepackage{amssymb}
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\usepackage{url}
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\usepackage{hyperref}
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\end_preamble
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\use_default_options true
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\maintain_unincluded_children false
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@ -15,13 +16,13 @@
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\font_roman default
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\font_sans default
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\font_typewriter default
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\font_math auto
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\font_default_family default
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\font_sf_scale 100
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\font_tt_scale 100
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\graphics default
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\default_output_format default
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\output_sync 0
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@ -32,15 +33,24 @@
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\use_hyperref false
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\papersize default
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\use_geometry false
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\use_amsmath 1
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\use_esint 1
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\use_mhchem 1
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\use_mathdots 1
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\use_package amsmath 1
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\use_package amssymb 2
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\use_package cancel 1
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\use_package esint 1
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\use_package mathdots 1
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\use_package mathtools 1
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\use_package mhchem 1
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\use_package stackrel 0
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\use_package stmaryrd 1
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\use_package undertilde 1
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\cite_engine basic
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\cite_engine_type default
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\biblio_style plain
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\use_bibtopic false
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\paperorientation portrait
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\suppress_date false
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\justification true
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\use_refstyle 1
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\index Index
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\shortcut idx
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@ -231,7 +241,7 @@ key "Hauser06lecture"
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\end_inset
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(in our /net/hp223/borg/Literature folder).
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.
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\end_layout
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\begin_layout Standard
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@ -465,22 +475,39 @@ where
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\end_inset
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.
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A typical update rule [
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\color blue
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see where this came from in paper
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\color inherit
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] is
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A typical update rule, as per Lec.
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7-1.2 of
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\begin_inset CommandInset citation
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LatexCommand cite
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key "Hauser06lecture"
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\end_inset
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is:
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\begin_inset Formula
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\[
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\Delta\leftarrow\begin{cases}
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\max\left(\Delta,3\norm{\delta x_{d}}\right)\text{,} & \rho>0.75\\
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\Delta & 0.75>\rho>0.25\\
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\Delta/2 & 0.25>\rho
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\Delta_{k+1}\leftarrow\begin{cases}
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\Delta_{k}/4 & \rho<0.25\\
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\min\left(2\Delta_{k},\Delta_{max}\right)\text{,} & \rho>0.75\\
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\Delta_{k} & 0.75>\rho>0.25
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\end{cases}
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\]
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\end_inset
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where
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\begin_inset Formula $\Delta_{k}\triangleq\norm{\delta x_{d}}$
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\end_inset
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.
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Note that the rule is designed to ensure that
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\begin_inset Formula $\Delta_{k}$
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\end_inset
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never exceeds the maximum trust region size
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\begin_inset Formula $\Delta_{max}.$
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\end_inset
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\end_layout
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\begin{align*}
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\Delta & =\norm{\left(1-\tau\right)\delta x_{u}+\tau\delta x_{n}}\\
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\Delta^{2} & =\left(1-\tau\right)^{2}\delta x_{u}^{\t}\delta x_{u}+2\tau\left(1-\tau\right)\delta x_{u}^{\t}\delta x_{n}+\tau^{2}\delta x_{n}^{\t}\delta x_{n}\\
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0 & =uu-2\tau uu+\tau^{2}uu+2\tau un-2\tau^{2}un+\tau^{2}nn-\Delta^{2}\\
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0 & =\left(uu-2un+nn\right)\tau^{2}+\left(2un-2uu\right)\tau-\Delta^{2}+uu\\
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\tau & =\frac{-\left(2un-2uu\right)\pm\sqrt{\left(2un-2uu\right)^{2}-4\left(uu-2un+nn\right)\left(uu-\Delta^{2}\right)}}{2\left(uu-un+nn\right)}
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0 & =\delta x_{u}^{\t}\delta x_{u}-2\tau\delta x_{u}^{\t}\delta x_{u}+\tau^{2}\delta x_{u}^{\t}\delta x_{u}+2\tau\delta x_{u}^{\t}\delta x_{n}-2\tau^{2}\delta x_{u}^{\t}\delta x_{n}+\tau^{2}\delta x_{n}^{\t}\delta x_{n}-\Delta^{2}\\
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0 & =\left(\delta x_{u}^{\t}\delta x_{u}-2\delta x_{u}^{\t}\delta x_{n}+\delta x_{n}^{\t}\delta x_{n}\right)\tau^{2}+\left(2\delta x_{u}^{\t}\delta x_{n}-2\delta x_{u}^{\t}\delta x_{u}\right)\tau-\Delta^{2}+\delta x_{u}^{\t}\delta x_{u}\\
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\tau & =\frac{-\left(2\delta x_{u}^{\t}\delta x_{n}-2\delta x_{u}^{\t}\delta x_{u}\right)\pm\sqrt{\left(2\delta x_{u}^{\t}\delta x_{n}-2\delta x_{u}^{\t}\delta x_{u}\right)^{2}-4\left(\delta x_{u}^{\t}\delta x_{u}-2\delta x_{u}^{\t}\delta x_{n}+\delta x_{n}^{\t}\delta x_{n}\right)\left(\delta x_{u}^{\t}\delta x_{u}-\Delta^{2}\right)}}{2\left(\delta x_{u}^{\t}\delta x_{u}-\delta x_{u}^{\t}\delta x_{n}+\delta x_{n}^{\t}\delta x_{n}\right)}
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\end{align*}
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\end_inset
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@ -641,7 +668,7 @@ Thus, mathematically, we can write the dogleg update
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\begin_inset Formula
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\[
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\delta x_{d}^{\left(k\right)}=\begin{cases}
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-\frac{\Delta}{\norm{g^{\left(k\right)}}}g^{\left(k\right)}\text{,} & \Delta<\norm{\delta x_{u}^{\left(k\right)}}\\
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-\frac{\Delta}{\norm{\delta x_{u}^{\left(k\right)}}}\delta x_{u}^{\left(k\right)}\text{,} & \Delta<\norm{\delta x_{u}^{\left(k\right)}}\\
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\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)}}\\
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\delta x_{n}^{\left(k\right)}\text{,} & \norm{\delta x_{n}^{\left(k\right)}}<\Delta
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\end{cases}
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