Updated doc for neldermead

15 files changed, 585 insertions, 102 deletions

diff --git a/scilab_doc/neldermead/macros.tex b/scilab_doc/neldermead/macros.tex
new file mode 100644
index 0000000..a205b7b
--- /dev/null
+++ b/scilab_doc/neldermead/macros.tex
@@ -0,0 +1,69 @@
+%% Good fonts for PDF
+\usepackage[cyr]{aeguill}
+
+%% Package for page headers
+\usepackage{fancyhdr}
+
+%% Package to include graphics
+%% Comment for DVI
+\usepackage[pdftex]{graphicx}
+
+%% Figures formats: jpeg or pdf
+%% Comment for DVI
+\DeclareGraphicsExtensions{.jpg,.pdf}
+
+%% Package to create Hyperdocuments
+%% Comment for DVI
+\usepackage[pdftex,colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}
+
+%% Package to control printed area size
+\usepackage{anysize}
+%% ...by defining margins {left}{right}{top}{bottom}
+\marginsize{22mm}{14mm}{12mm}{25mm}
+
+%% Package used to include a bibliography
+\usepackage{natbib}
+
+%% R for real numbers
+\usepackage{amssymb}
+
+%% User defined commands
+
+%% Figure reference
+\newcommand{\figref}[1]{figure~\ref{#1}}
+
+%% Equation reference
+\newcommand{\Ref}[1]{(\ref{#1})}
+
+%% Emphasize a word or a group of words
+\newcommand{\empha}[1]{\textit{\textbf{#1}}}
+
+%% Derivation operators
+\newcommand{\D}{\partial}
+\newcommand{\Dt}{\partial_t}
+\newcommand{\Dx}{\partial_x}
+\newcommand{\Dy}{\partial_y}
+
+\newcommand{\bv}{\mathbf{v}}
+\newcommand{\bx}{\mathbf{x}}
+\newcommand{\bl}{\mathbf{l}}
+
+\usepackage{url}
+
+% Scilab macros
+\newcommand{\scifunction}[1]{\textit{#1}}
+
+% To highlight source code
+\usepackage{listings}
+
+\usepackage{algorithmic}
+
+% To allow one bibliograph by chapter
+%\usepackage[sectionbib]{chapterbib}
+\usepackage{url}
+
+% Maths shortcuts 
+\newcommand{\RR}{\mathbb{R}}
+\newcommand{\CC}{\mathbb{C}}
+
+
diff --git a/scilab_doc/neldermead/neldermead-simplex-so.pdf b/scilab_doc/neldermead/neldermead-simplex-so.pdf
new file mode 100644
index 0000000..7d0151f
--- /dev/null
+++ b/scilab_doc/neldermead/neldermead-simplex-so.pdf
diff --git a/scilab_doc/neldermead/neldermead-simplex-so.tex b/scilab_doc/neldermead/neldermead-simplex-so.tex
new file mode 100644
index 0000000..84719ac
--- /dev/null
+++ b/scilab_doc/neldermead/neldermead-simplex-so.tex
@@ -0,0 +1,89 @@
+%
+% neldermead.tex --
+%   Some notes about Nelder-Mead algorithms.
+%
+% Copyright 2008-2009 Michael Baudin
+%
+\documentclass[12pt]{report}
+
+\include{macros}
+
+\begin{document}
+%% User defined page headers
+\pagestyle{fancyplain}
+\renewcommand{\chaptermark}[1]{\markboth{\chaptername\ \thechapter. #1}{}}
+\renewcommand{\sectionmark}[1]{\markright{\thesection. #1}}
+\lhead[]{\fancyplain{}{\bfseries\leftmark}}
+\rhead[]{\fancyplain{}{\bfseries\thepage}}
+\cfoot{}
+
+%% User defined figure legends
+\makeatletter
+\def\figurename{{\protect\sc \protect\small\bfseries Fig.}}
+\def\f@ffrench{\protect\figurename\space{\protect\small\bf \thefigure}\space}
+\let\fnum@figure\f@ffrench%
+\let\captionORI\caption
+\def\caption#1{\captionORI{\rm\small #1}}
+\makeatother
+
+%% First page
+\thispagestyle{empty}
+{
+\begin{center}
+%% Comment for DVI
+\includegraphics[height=40mm]{scilab_logo}
+\vskip4cm
+
+%% Empty space between the box and the text
+\fboxsep6mm
+%% Box thickness
+\fboxrule1.3pt
+\Huge
+$$\fbox{$
+  \begin{array}{c}
+  \textbf{Nelder-Mead}\\
+  \textbf{Toolbox Manual}\\
+  \textbf{-- Simplex Theory --}
+  \end{array}
+  $}
+$$
+\end{center}
+\vskip8cm
+
+\normalsize
+
+\begin{flushright}
+Version 0.2 \\
+September 2009
+\end{flushright}
+
+\begin{flushright}
+Micha\"el BAUDIN
+\end{flushright}
+
+\clearpage
+
+%% Table of contents
+\renewcommand{\baselinestretch}{1.30}\small \normalsize
+
+\tableofcontents
+
+\renewcommand{\baselinestretch}{1.18}\small \normalsize
+
+
+\include{section-simplex}
+
+
+\clearpage
+
+%% Appendix
+\appendix
+
+%% Bibliography
+
+\addcontentsline{toc}{chapter}{Bibliography}
+\bibliographystyle{plain}
+\bibliography{neldermead}
+
+\end{document}
+
diff --git a/scilab_doc/neldermead/neldermead.bib b/scilab_doc/neldermead/neldermead.bib
index 0ff90c5..d2835e9 100644
--- a/scilab_doc/neldermead/neldermead.bib
+++ b/scilab_doc/neldermead/neldermead.bib
@@ -253,9 +253,9 @@ publisher= {}}
 }
 
 @TECHREPORT{Andersson01multiobjectiveoptimization,
-    author = {Johan Andersson and Linköpings Universitet},
-    title = {Multiobjective Optimization in Engineering Design: Applications to . . .},
-    institution = {Department of Mechanical Engineering, Linköping University},
+    author = {Johan Andersson and Link\"opings Universitet},
+    title = {Multiobjective Optimization in Engineering Design: Application to Fluid Power Systems},
+    institution = {Department of Mechanical Engineering, Link\"oping University},
     year = {2001},
   note={\url{https://polopoly.liu.se/content/1/c6/10/99/74/phdthesis.pdf}, 
   \url{http://www.iei.liu.se/machine/johan-olvander/home"l=en},
@@ -462,3 +462,12 @@ eprint = {http://comjnl.oxfordjournals.org/cgi/reprint/6/2/163.pdf}
  address = {New York, NY, USA},
  }
 
+@TECHREPORT{Fan2002,
+    author = {Ellen Fan},
+    title = {Global Optimization Of Lennard-Jones Atomic Clusters},
+    institution = {McMaster University},
+    year = {2002},
+    month = {February},
+    day={26}
+}
+
diff --git a/scilab_doc/neldermead/neldermead.pdf b/scilab_doc/neldermead/neldermead.pdf
index 856204e..c36eb3f 100644
--- a/scilab_doc/neldermead/neldermead.pdf
+++ b/scilab_doc/neldermead/neldermead.pdf
diff --git a/scilab_doc/neldermead/neldermead.tex b/scilab_doc/neldermead/neldermead.tex
index 951c8c5..1b927df 100644
--- a/scilab_doc/neldermead/neldermead.tex
+++ b/scilab_doc/neldermead/neldermead.tex
@@ -6,73 +6,7 @@
 %
 \documentclass[12pt]{report}
 
-%% Good fonts for PDF
-\usepackage[cyr]{aeguill}
-
-%% Package for page headers
-\usepackage{fancyhdr}
-
-%% Package to include graphics
-%% Comment for DVI
-\usepackage[pdftex]{graphicx}
-
-%% Figures formats: jpeg or pdf
-%% Comment for DVI
-\DeclareGraphicsExtensions{.jpg,.pdf}
-
-%% Package to create Hyperdocuments
-%% Comment for DVI
-\usepackage[pdftex,colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}
-
-%% Package to control printed area size
-\usepackage{anysize}
-%% ...by defining margins {left}{right}{top}{bottom}
-\marginsize{22mm}{14mm}{12mm}{25mm}
-
-%% Package used to include a bibliography
-\usepackage{natbib}
-
-%% R for real numbers
-\usepackage{amssymb}
-
-%% User defined commands
-
-%% Figure reference
-\newcommand{\figref}[1]{figure~\ref{#1}}
-
-%% Equation reference
-\newcommand{\Ref}[1]{(\ref{#1})}
-
-%% Emphasize a word or a group of words
-\newcommand{\empha}[1]{\textit{\textbf{#1}}}
-
-%% Derivation operators
-\newcommand{\D}{\partial}
-\newcommand{\Dt}{\partial_t}
-\newcommand{\Dx}{\partial_x}
-\newcommand{\Dy}{\partial_y}
-
-\newcommand{\bv}{\mathbf{v}}
-\newcommand{\bx}{\mathbf{x}}
-
-\usepackage{url}
-
-% Scilab macros
-\newcommand{\scimacro}[1]{\textit{#1}}
-\newcommand{\scicommand}[1]{\textit{#1}}
-
-% To highlight source code
-\usepackage{listings}
-
-\usepackage{algorithmic}
-
-% To allow one bibliograph by chapter
-%\usepackage[sectionbib]{chapterbib}
-\usepackage{url}
-
-% Maths shortcuts 
-\newcommand{\RR}{\mathbb{R}}
-\newcommand{\CC}{\mathbb{C}}
+\include{macros}
 
 \begin{document}
 %% User defined page headers
@@ -118,8 +52,8 @@ $$
 \normalsize
 
 \begin{flushright}
-Version 0.1 \\
-March 2009
+Version 0.2 \\
+September 2009
 \end{flushright}
 
 \begin{flushright}
@@ -139,9 +73,11 @@ Micha\"el BAUDIN
 
 \include{overview}
 
-\include{installation}
+\include{section-simplex}
+
+\include{method-spendley}
 
-\include{validation}
+\include{method-neldermead}
 
 \include{conclusion}
 
@@ -155,13 +91,6 @@ Micha\"el BAUDIN
 
 \include{implementations}
 
-\include{section-simplex}
-
-\include{method-spendley}
-
-\include{method-neldermead}
-
-
 %% Bibliography
 
 \addcontentsline{toc}{chapter}{Bibliography}
diff --git a/scilab_doc/neldermead/nmbibliography.tex b/scilab_doc/neldermead/nmbibliography.tex
index a5191ab..3a23e03 100644
--- a/scilab_doc/neldermead/nmbibliography.tex
+++ b/scilab_doc/neldermead/nmbibliography.tex
@@ -177,7 +177,7 @@ satisfy the constraints.
 W. H. Press, Saul A. Teukolsky, William T. Vetterling and Brian P. Flannery, 
 1992
 
-An ANSI C implementation of the Nelder-Mead algorithm is given.
+In this book \cite{NumericalRecipes}, an ANSI C implementation of the Nelder-Mead algorithm is given.
 The initial simplex is based on the axis. 
 The termination criterion is based on the relative difference of the 
 function value of the best and worst vertices in the simplex.
@@ -331,10 +331,10 @@ function typical in penalty methods.
 
 No numerical experiment is presented.
 
-\section{Ölvander, 2001}
+\section{Andersson, 2001}
 
 "Multiobjective Optimization in Engineering Design - Application to fluid Power Systems"
-Johan Ölvander (formerly Andersson), 2001
+Johan Andersson, 2001
 
 This PhD thesis \cite{Andersson01multiobjectiveoptimization} gives a brief overview of the Complex method by Box in 
 section 5.1.
diff --git a/scilab_doc/neldermead/overview.tex b/scilab_doc/neldermead/overview.tex
index 7aba210..9e044d1 100644
--- a/scilab_doc/neldermead/overview.tex
+++ b/scilab_doc/neldermead/overview.tex
@@ -7,16 +7,16 @@ toolbox as well as an example of use.
 \section{How to use the Toolbox}
 
 The design of the toolbox is based on the creation of 
-a new token by the \scimacro{neldermead\_new} command.
+a new token by the \scifunction{neldermead\_new} command.
 The Nelder-Mead object associated with this token can then 
-be configured with \scimacro{neldermead\_configure} and queried 
-with \scimacro{neldermead\_cget}. To be more specific, the
-\scimacro{neldermead\_configure} command allows to configure the 
+be configured with \scifunction{neldermead\_configure} and queried 
+with \scifunction{neldermead\_cget}. To be more specific, the
+\scifunction{neldermead\_configure} command allows to configure the 
 number of variables, the objective function and the initial guess.
 
-The main command of the toolbox is the \scimacro{neldermead\_search} command, which
+The main command of the toolbox is the \scifunction{neldermead\_search} command, which
 solves the optimization problem. After an optimization has been performed,
-the \scimacro{neldermead\_get} command allows to retrieve the optimum $x^\star$,
+the \scifunction{neldermead\_get} command allows to retrieve the optimum $x^\star$,
 as well as other parameters, such as the number of iterations performed, the number 
 of evaluations of the function, etc...
 
diff --git a/scilab_doc/neldermead/scripts/initialsimplex_axes.sce b/scilab_doc/neldermead/scripts/initialsimplex_axes.sce
new file mode 100644
index 0000000..3400bf1
--- /dev/null
+++ b/scilab_doc/neldermead/scripts/initialsimplex_axes.sce
@@ -0,0 +1,29 @@
+// Scilab ( http://www.scilab.org/ ) - This file is part of Scilab
+// Copyright (C) 2009 - Digiteo - Michael Baudin
+//
+// This file must be used under the terms of the CeCILL.
+// This source file is licensed as described in the file COPYING, which
+// you should have received as part of this distribution.  The terms
+// are also available at
+// http://www.cecill.info/licences/Licence_CeCILL_V2-en.txt
+
+// Draw a simplex along the axes in 2D
+x = zeros(4,1)
+y = zeros(4,1)
+
+// First point is (0,0)
+// Second point is (1,0)
+x(2) = 1.0
+y(2) = 0
+// Third point is (0,2)
+x(3) = 0.0
+y(3) = 2.0
+// Fourth point is (0,0), just to make the loop in the simplex
+// Now plot the simplex
+plot(x,y)
+f = gcf()
+f.children.children.children.thickness = 4
+f.children.children.children.mark_style = 9
+f.children.children.children.mark_foreground = 3
+f.children.children.children.mark_size = 10
+
diff --git a/scilab_doc/neldermead/scripts/initialsimplex_regular.sce b/scilab_doc/neldermead/scripts/initialsimplex_regular.sce
new file mode 100644
index 0000000..ff97221
--- /dev/null
+++ b/scilab_doc/neldermead/scripts/initialsimplex_regular.sce
@@ -0,0 +1,33 @@
+// Scilab ( http://www.scilab.org/ ) - This file is part of Scilab
+// Copyright (C) 2009 - Digiteo - Michael Baudin
+//
+// This file must be used under the terms of the CeCILL.
+// This source file is licensed as described in the file COPYING, which
+// you should have received as part of this distribution.  The terms
+// are also available at
+// http://www.cecill.info/licences/Licence_CeCILL_V2-en.txt
+
+// Draw a regular simplex in 2D
+n = 2
+p  = (n - 1.0 + sqrt(n + 1))/(n * sqrt(2.0))
+q = (sqrt(n + 1) - 1.0)/(n * sqrt(2.0))
+
+x = zeros(4,1)
+y = zeros(4,1)
+
+// First point is (0,0)
+// Second point is (p,q)
+x(2) = p
+y(2) = q
+// Third point is (q,p)
+x(3) = q
+y(3) = p
+// Fourth point is (0,0), just to make the loop in the simplex
+// Now plot the simplex
+plot(x,y)
+f = gcf()
+f.children.children.children.thickness = 4
+f.children.children.children.mark_style = 9
+f.children.children.children.mark_foreground = 3
+f.children.children.children.mark_size = 10
+
diff --git a/scilab_doc/neldermead/section-simplex.tex b/scilab_doc/neldermead/section-simplex.tex
index b676b25..7b65b72 100644
--- a/scilab_doc/neldermead/section-simplex.tex
+++ b/scilab_doc/neldermead/section-simplex.tex
@@ -35,7 +35,6 @@ state a theoretical convergence result. In practical implementations, though, th
 ordering rules have no measurable influence.
 
 Let $V$ denote the $n\times n$ matrix of simplex directions 
-
 \begin{eqnarray}
 \label{simplex-directions}
 V(S) = (\bx_2 - \bx_1, \bx_3 - \bx_1 , \ldots , \bx_{n+1} - \bx_1) = (\bv_1, \ldots , \bv_n)
@@ -50,12 +49,11 @@ Several methods are available to compute the size of a simplex.
 In Kelley's book \cite{Kelley1999}, the author presents the diameter and the two oriented lengths.
 
 The simplex diameter $diam(S)$ is defined by 
-
 \begin{eqnarray}
 \label{simplex-diameter}
-diam(S) = \max_{i,j=1,n+1} \|\bx_i - \bx_j\|
+diam(S) = \max_{i,j=1,n+1} \|\bx_i - \bx_j\|_2,
 \end{eqnarray}
-
+where $\|.\|_2$ is the euclidian norm $\|x\|_2 = \sum_{i=1,n}\bx_i^2$.
 In practical implementations, computing the diameter requires two nested loops over the 
 vertices of the simplex, i.e. $(n+1)^2$ operations. This is why authors generally 
 prefer to use lengths which are less expensive to compute.
@@ -65,7 +63,7 @@ first vertex as the reference point and are defined by
 
 \begin{eqnarray}
 \label{simplex-sigma}
-\sigma_+(S) = \max_{i=2,n+1} \|\bx_i - \bx_1\| \qquad \textrm { and } \qquad \sigma_-(S) = \min_{i=2,n+1} \|\bx_i - \bx_1\|
+\sigma_+(S) = \max_{i=2,n+1} \|\bx_i - \bx_1\|_2 \qquad \textrm { and } \qquad \sigma_-(S) = \min_{i=2,n+1} \|\bx_i - \bx_1\|_2
 \end{eqnarray}
 
 The following inequalities are satisfied between the diameter and the maximum oriented length
@@ -76,24 +74,165 @@ The following inequalities are satisfied between the diameter and the maximum or
 \end{eqnarray}
 
 In Nash's book \cite{nla.cat-vn1060620}, the size of the simplex $s_N(S)$ is measured
-based on the $\l-1$ norm and is defined by 
-
+based on the $l1$ norm and is defined by 
 \begin{eqnarray}
 \label{simplex-sizenash}
-s_N(S) = \sum_{i=2,n+1} \|\bx_i - \bx_1\|
+s_N(S) = \sum_{i=2,n+1} \|\bx_i - \bx_1\|_1
 \end{eqnarray}
-
 where 
-
 \begin{eqnarray}
 \label{simplex-sizenash2}
-\|\bx_i - \bx_1\| = \sum_{i=1,n+1} |x_i^j - x_1^j|
+\|\bx_i - \bx_1\|_1 = \sum_{j=1,n} |x_i^j - x_1^j|
 \end{eqnarray}
-
 where $x_i^j\in\RR$ is the $j$-th coordinate of the $i$-th vertex of the simplex $S$.
 
 \section{The initial simplex}
 
+While most of the theory can be developed without being very specific 
+about the initial simplex, the initial simplex plays a very important role in practice.
+All approaches are based on the initial guess $\overline{\bx}_0\in\RR^n$ and create a 
+geometric shape based on this point.
+
+In this section, we present the various approach to design the initial 
+simplex. In the first part, we emphasize the importance of the initial
+simplex in optimization algorithms. Then we present the regular simplex 
+approach by Spendley et al., the randomized bounds approach by Box and 
+Pfeffer's method.
+
+\subsection{Importance of the initial simplex}
+
+The initial simplex is particularily important in the case of Spendley's et al 
+method, where the shape of the simplex is fixed during the iterations.
+Therefore, the algorithm can only go through points which are on the pattern
+defined by the initial simplex. The pattern presented in figure \ref{fig-nm-simplex-fixedshape}
+is typical a fixed-shape simplex algorithm (see \cite{Torczon89multi-directionalsearch}, chapter 3, 
+for other patterns of a direct search method).
+If, by chance, the pattern is so that the optimum is close to one point 
+defined by the pattern, the number of iteration may be small. On the contrary, the 
+number of iterations may be high if the pattern does not come close to the 
+optimum.
+
+\begin{figure}
+\begin{center}
+\includegraphics[width=7cm]{simplex_initialfixed.png}
+\end{center}
+\caption{Typical pattern with fixed-shape Spendley's et al algorithm}
+\label{fig-nm-simplex-fixedshape}
+\end{figure}
+
+The variable-shape simplex algorithm designed by Nelder and Mead is also very
+sensitive to the initial simplex.
+One of the problems is that the initial simplex should be consistently scaled 
+with respect to the unknown $x$.
+In \cite{parkinson1972}, "An investigation into the efficiency of variants on the simplex method", 
+Parkinson and Hutchinson explored 
+several ways of improvement. First, they investigate the sensitivity
+of the algorithm to the initial simplex. Two parameters were investigated,
+i.e. the initial length and the orientation of the simplex. 
+The conclusion of their study with respect to the initial simplex is 
+the following. "The orientation of the initial simplex has a significant effect 
+on efficiency, but the relationship can be too sensitive for an automatic 
+predictor to provide sufficient accuracy at this time."
+
+Since no initial simplex clearly improves on the others, in practice, 
+it may be convenient to try different approaches.
+
+\subsection{Spendley's et al simplex}
+
+In their paper \cite{Spendley1962}, Spendley et al. use a regular 
+simplex with given size $\ell>0$. We define the parameters $p,q>0$ as 
+\begin{eqnarray}
+p &=& \frac{1}{n\sqrt{2}} \left(n-1 + \sqrt{n+1}\right), \\
+q &=& \frac{1}{n\sqrt{2}} \left(\sqrt{n+1} - 1\right).
+\end{eqnarray}
+We can now define the vertices of the simplex $S=\{\bx_i\}_{i=1,n+1}$.
+The first vertex of the simplex is the initial guess 
+\begin{eqnarray}
+\bx_1 &=& \overline{\bx}_0.
+\end{eqnarray}
+The other vertices are defined by $\bx_i = \left( x_i^1, \ldots x_i^n\right)\in\RR^n$
+where the coordinates $x_i^j$ are 
+\begin{eqnarray}
+x_i^j &=& 
+\left\{
+\begin{array}{l}
+\overline{x}_0^j + \ell p, \textrm{ if } j=i-1,\\
+\overline{x}_0^j + \ell q, \textrm{ if } j\neq i-1,\\
+\end{array}
+\right.
+\end{eqnarray}
+for $i=2,n+1$ where $\ell \in\RR$ is the length of the simplex ($\ell>0$). Notice that this 
+length is the same for all the vertices which keeps the simplex regular.
+
+The regular initial simplex is presented in figure \ref{fig-nm-simplex-regular}.
+
+\begin{figure}
+\begin{center}
+\includegraphics[width=10cm]{simplex_regular.png}
+\end{center}
+\caption{Regular simplex in 2 dimensions}
+\label{fig-nm-simplex-regular}
+\end{figure}
+
+\subsection{Simplex along the axes}
+
+A very efficient and simple approach leads to an axis-by-axis simplex.
+This simplex depends on a vector of positive lengths $\bl\in\RR^n$.
+The first vertex of the simplex is the initial guess 
+\begin{eqnarray}
+\bx_1 &=& \overline{\bx}_0.
+\end{eqnarray}
+The other vertices are defined by 
+\begin{eqnarray}
+x_i^j &=& 
+\left\{
+\begin{array}{l}
+\overline{x}_0^j + \bl_j, \textrm{ if } j=i-1,\\
+\overline{x}_0^j, \textrm{ if } j\neq i-1,\\
+\end{array}
+\right.
+\end{eqnarray}
+for $i=2,n+1$ 
+
+This kind of simplex is presented in figure \ref{fig-nm-simplex-axes}.
+The axis-by-axis approach is used in the very popular Nelder-Mead 
+algorithm provided in Numerical Recipes in C \cite{NumericalRecipes}.
+As stated in \cite{NumericalRecipes}, the length vector $\bl$ can 
+be used as a guess for the characteristic length scale of the problem.
+
+\begin{figure}
+\begin{center}
+\includegraphics[width=10cm]{simplex_axes.png}
+\end{center}
+\caption{Axis-based simplex in 2 dimensions}
+\label{fig-nm-simplex-axes}
+\end{figure}
+
+\subsection{Randomized bounds}
+
+Assume that the variable $\bx\in\RR^n$ is bounded so that 
+\begin{eqnarray}
+m^j \leq x^j \leq M^j,
+\end{eqnarray}
+for $j=1,n$, where $m_j,M_j\in\RR$ are minimum and maximum bounds and $m_j\leq M_j$.
+A method suggested by Box in \cite{Box1965} is based on the use of 
+pseudo-random numbers. Let $\{\theta_i^j\}_{i=1,n+1,j=1,n}\in[0,1]$ be 
+a sequence of random numbers uniform in the interval $[0,1]$.
+The first vertex of the simplex is the initial guess 
+\begin{eqnarray}
+\bx_1 &=& \overline{\bx}_0.
+\end{eqnarray}
+The other vertices are defined by 
+\begin{eqnarray}
+x_i^j &=& m^j + \theta_i^j (M^j - m^j),
+\end{eqnarray}
+for $i=2,n+1$.
+
+\subsection{Pfeffer's method}
+
+This initial simplex is used in the function \scifunction{fminsearch}
+and presented in \cite{Fan2002}. It is due to L. Pfeffer at Stanford.
+
 TODO...
 
 \section{The simplex gradient}
diff --git a/scilab_doc/neldermead/simplex_axes.png b/scilab_doc/neldermead/simplex_axes.png
new file mode 100644
index 0000000..5f4ae08
--- /dev/null
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