diff --git a/doc/trustregion.bib b/doc/trustregion.bib index 7fcd509f4..7c87eef9b 100644 --- a/doc/trustregion.bib +++ b/doc/trustregion.bib @@ -8,7 +8,6 @@ %% Saved with string encoding Unicode (UTF-8) - @webpage{Hauser06lecture, Author = {Raphael Hauser}, Date-Added = {2011-10-10 15:21:22 +0000}, @@ -16,11 +15,5 @@ Title = {Lecture Notes on Unconstrained Optimization}, Url = {http://www.numerical.rl.ac.uk/nimg/oupartc/lectures/raphael/}, Year = {2006}, - Bdsk-Url-1 = {http://www.numerical.rl.ac.uk/nimg/oupartc/lectures/raphael/}, - Bdsk-File-1 = {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}, - 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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 @@ -593,9 +620,9 @@ To find the intersection of the line between \begin{align*} \Delta & =\norm{\left(1-\tau\right)\delta x_{u}+\tau\delta x_{n}}\\ \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}\\ -0 & =uu-2\tau uu+\tau^{2}uu+2\tau un-2\tau^{2}un+\tau^{2}nn-\Delta^{2}\\ -0 & =\left(uu-2un+nn\right)\tau^{2}+\left(2un-2uu\right)\tau-\Delta^{2}+uu\\ -\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)} +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}\\ +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}\\ +\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)} \end{align*} \end_inset @@ -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} diff --git a/doc/trustregion.pdf b/doc/trustregion.pdf index 31ca0c5b1..22dcac035 100644 Binary files a/doc/trustregion.pdf and b/doc/trustregion.pdf differ