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authorMichael Baudin <michael.baudin@scilab.org>2009-10-03 18:54:21 +0200
committerMichael Baudin <michael.baudin@scilab.org>2009-10-03 18:54:21 +0200
commitdc8c3a2df5cd66c22ed76ab0f3c348d6a825a164 (patch)
tree22224506d413b534720a2269521f4ff19d7ec1e4 /scilab_doc
parentf9ccc1320f38436d6a22bffd1121828c73ed0752 (diff)
downloadscilab-dc8c3a2df5cd66c22ed76ab0f3c348d6a825a164.zip
scilab-dc8c3a2df5cd66c22ed76ab0f3c348d6a825a164.tar.gz
Fixed Spendley's chapter. Transformed svg -> pdf for beauty of graphics.
Diffstat (limited to 'scilab_doc')
-rw-r--r--scilab_doc/neldermead/method-spendley.tex166
-rw-r--r--scilab_doc/neldermead/neldermead-spendley-so.pdfbin322977 -> 308835 bytes
-rw-r--r--scilab_doc/neldermead/neldermead.bib19
-rw-r--r--scilab_doc/neldermead/section-simplex.tex3
-rw-r--r--scilab_doc/neldermead/simplex_initialfixed.pdfbin0 -> 8287 bytes
-rw-r--r--scilab_doc/neldermead/simplex_initialfixed.pngbin48421 -> 0 bytes
-rw-r--r--scilab_doc/neldermead/simplex_initialfixed.svg203
-rw-r--r--scilab_doc/neldermead/spendley-steps-reflect.pdfbin0 -> 13762 bytes
-rw-r--r--scilab_doc/neldermead/spendley-steps-reflect.pngbin20971 -> 0 bytes
-rw-r--r--scilab_doc/neldermead/spendley-steps-reflect.svg38
-rw-r--r--scilab_doc/neldermead/spendley-steps-reflect2.pdfbin0 -> 14414 bytes
-rw-r--r--scilab_doc/neldermead/spendley-steps-reflect2.pngbin30169 -> 0 bytes
-rw-r--r--scilab_doc/neldermead/spendley-steps-reflect2.svg40
-rw-r--r--scilab_doc/neldermead/spendley-steps-shrink.pdfbin0 -> 14374 bytes
-rw-r--r--scilab_doc/neldermead/spendley-steps-shrink.pngbin33070 -> 0 bytes
-rw-r--r--scilab_doc/neldermead/spendley-steps-shrink.svg48
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-rw-r--r--scilab_doc/neldermead/spendley-steps.pngbin22149 -> 0 bytes
-rw-r--r--scilab_doc/neldermead/spendley-steps.svg384
19 files changed, 495 insertions, 406 deletions
diff --git a/scilab_doc/neldermead/method-spendley.tex b/scilab_doc/neldermead/method-spendley.tex
index b099261..06224dd 100644
--- a/scilab_doc/neldermead/method-spendley.tex
+++ b/scilab_doc/neldermead/method-spendley.tex
@@ -70,22 +70,54 @@ to the centroid $\overline{\bx}(j)$. The new vertex $\bx(\rho,j)$ is defined by
70 70
71\subsection{Algorithm} 71\subsection{Algorithm}
72 72
73In this section, we analyse Spendley's et al algorithm, which
74is presented in figure \ref{algo-spendley}.
75
76\begin{figure}[htbp]
77\begin{algorithmic}
78\STATE Compute an initial simplex $S_0$
79\STATE Sorts the vertices $S_0$ with increasing function values
80\STATE $S\gets S_0$
81\WHILE{$\sigma(S)>tol$}
82 \STATE $\overline{x}\gets \overline{\bx}(n+1)$ \COMMENT{Compute the centroid}
83 \STATE $\bx_r \gets \bx(\rho,n+1)$ \COMMENT{Reflect with respect to worst}
84 \STATE $f_r \gets f(\bx_r)$
85 \IF {$f_r<f_{n+1}$}
86 \STATE Accept $\bx_r$
87 \ELSE
88 \STATE $\overline{x}\gets \overline{\bx}(n)$ \COMMENT{Compute the centroid}
89 \STATE $\bx_r^\prime \gets \bx(\rho,n)$ \COMMENT{Reflect with respect to next-to-worst}
90 \STATE $f_r^\prime \gets f(\bx_r^\prime)$
91 \IF {$f_r^\prime<f_{n+1}$}
92 \STATE Accept $\bx_r^\prime$
93 \ELSE
94 \STATE Compute the points $\bx_i=\bx_1 + \sigma (\bx_i - \bx_1)$, $i=2,n+1$ \COMMENT{Shrink}
95 \STATE Compute $f_i = f(\bx_i), i=2,n+1$
96 \ENDIF
97 \ENDIF
98 \STATE Sort the vertices of $S$ with increasing function values
99\ENDWHILE
100\end{algorithmic}
101\caption{Spendley et al. algorithm}
102\label{algo-spendley}
103\end{figure}
104
73The first step of the algorithm is based on the centroid 105The first step of the algorithm is based on the centroid
74where the worst vertex $\bx_{n+1}$ has been excluded. This centroid 106$\overline{\bx} (n+1)$ where the worst vertex $\bx_{n+1}$
75$\overline{\bx} (n+1)$ is 107has been excluded. This centroid is
76\begin{eqnarray} 108\begin{eqnarray}
77\label{centroid-worst} 109\label{centroid-worst}
78\overline{\bx} (n+1) = \frac{1}{n} \sum_{i=1,n} \bx_i. 110\overline{\bx} (n+1) = \frac{1}{n} \sum_{i=1,n} \bx_i.
79\end{eqnarray} 111\end{eqnarray}
80This step is a reflection with respect to the worst vertex $\bx_{n+1}$ so that the 112We perform a reflection with respect to the worst vertex $\bx_{n+1}$,
81reflected point $\bx_r$ is defined by 113which creates the reflected point $\bx_r$ defined by
82\begin{eqnarray} 114\begin{eqnarray}
83\label{interpolate-worst} 115\label{interpolate-worst}
84\bx_r = \bx(\rho,n+1) = (1+\rho)\overline{\bx}(n+1) - \rho \bx_{n+1} 116\bx_r = \bx(\rho,n+1) = (1+\rho)\overline{\bx}(n+1) - \rho \bx_{n+1}
85\end{eqnarray} 117\end{eqnarray}
86 118
87We then compute the function value of the reflected 119We then compute the function value of the reflected
88point $f_r=f(\bx_r)$. If the function value $f_r$ is better than the worst function 120point as $f_r=f(\bx_r)$. If the function value $f_r$ is better than the worst function
89value $f_{n+1}$, i.e. if $f_r < f_{n+1}$, then the worst point $\bx_{n+1}$ is rejected from the 121value $f_{n+1}$, i.e. if $f_r < f_{n+1}$, then the worst point $\bx_{n+1}$ is rejected from the
90simplex and the reflected point $\bx_r$ is accepted. If the reflection point 122simplex and the reflected point $\bx_r$ is accepted. If the reflection point
91does not improve the function value $f_{n+1}$, we consider the centroid 123does not improve the function value $f_{n+1}$, we consider the centroid
@@ -103,34 +135,6 @@ Therefore, the algorithm shrinks the simplex toward the best vertex $\bx_1$.
103That last step uses the shrink coefficient $0<\sigma<1$. The standard 135That last step uses the shrink coefficient $0<\sigma<1$. The standard
104value for this coefficient is $\sigma=\frac{1}{2}$. 136value for this coefficient is $\sigma=\frac{1}{2}$.
105 137
106Spendley's et al. algorithm is presented in figure \ref{algo-spendley}.
107
108\begin{figure}[htbp]
109\begin{algorithmic}
110\STATE Compute an initial simplex $S_0$
111\STATE Sorts the vertices $S_0$ with increasing function values
112\STATE $S\gets S_0$
113\WHILE{$\sigma(S)>tol$}
114 \STATE $\overline{x}\gets \overline{x}(n+1)$
115 \STATE $x_r \gets x(\rho,n+1)$, $f_r \gets f(x_r)$ \COMMENT{Reflect with respect to worst}
116 \IF {$f_r<f_{n+1}$}
117 \STATE Accept $x_r$
118 \ELSE
119 \STATE $\overline{x}\gets \overline{x}(n)$
120 \STATE $x_r \gets x(\rho,n)$, $f_r \gets f(x_r)$ \COMMENT{Reflect with respect to next-to-worst}
121 \IF {$f_r<f_{n+1}$}
122 \STATE Accept $x_r$
123 \ELSE
124 \STATE Compute the points $x_i=x_1 + \sigma (x_i - x_1)$, $i=2,n+1$ \COMMENT{Shrink}
125 \STATE Compute the function values at the points $x_i, i=2,n+1$
126 \ENDIF
127 \ENDIF
128 \STATE Sort the vertices of $S$ with increasing function values
129\ENDWHILE
130\end{algorithmic}
131\caption{Spendley et al. algorithm}
132\label{algo-spendley}
133\end{figure}
134 138
135\subsection{Geometric analysis} 139\subsection{Geometric analysis}
136 140
@@ -139,9 +143,13 @@ moves of the Spendley et al. algorithm. It is obvious from the
139picture that the algorithm explores a pattern which is 143picture that the algorithm explores a pattern which is
140entirely determined from the initial simplex. 144entirely determined from the initial simplex.
141 145
146In Spendley's et al. original paper, the authors use a regular
147simplex, where the edges all have the same length. In practice,
148however, any non degenerate simplex can be used.
149
142\begin{figure} 150\begin{figure}
143\begin{center} 151\begin{center}
144\includegraphics[width=10cm]{spendley-steps.png} 152\includegraphics[width=8cm]{spendley-steps.pdf}
145\end{center} 153\end{center}
146\caption{Spendley et al. simplex moves} 154\caption{Spendley et al. simplex moves}
147\label{fig-spendley-moves} 155\label{fig-spendley-moves}
@@ -153,7 +161,7 @@ and \ref{fig-spendley-moves-shrink}.
153 161
154The basic move is the reflection step, presented in figure 162The basic move is the reflection step, presented in figure
155\ref{fig-spendley-moves-reflect} and \ref{fig-spendley-moves-reflect2}. 163\ref{fig-spendley-moves-reflect} and \ref{fig-spendley-moves-reflect2}.
156These two figures shows that the Spendley et al. 164These two figures show that Spendley's et al.
157algorithm is based on a discretization of the parameter space. 165algorithm is based on a discretization of the parameter space.
158The optimum is searched on that grid, which is based on regular simplices. 166The optimum is searched on that grid, which is based on regular simplices.
159When no move is possible to improve the situation on that grid, 167When no move is possible to improve the situation on that grid,
@@ -161,7 +169,7 @@ a shrink step is necessary, as presented in figure \ref{fig-spendley-moves-shrin
161 169
162\begin{figure} 170\begin{figure}
163\begin{center} 171\begin{center}
164\includegraphics[width=10cm]{spendley-steps-reflect.png} 172\includegraphics[width=10cm]{spendley-steps-reflect.pdf}
165\end{center} 173\end{center}
166\caption{Spendley et al. simplex moves -- Reflection with respect to highest point} 174\caption{Spendley et al. simplex moves -- Reflection with respect to highest point}
167\label{fig-spendley-moves-reflect} 175\label{fig-spendley-moves-reflect}
@@ -169,7 +177,7 @@ a shrink step is necessary, as presented in figure \ref{fig-spendley-moves-shrin
169 177
170\begin{figure} 178\begin{figure}
171\begin{center} 179\begin{center}
172\includegraphics[width=10cm]{spendley-steps-reflect2.png} 180\includegraphics[width=10cm]{spendley-steps-reflect2.pdf}
173\end{center} 181\end{center}
174\caption{Spendley et al. simplex moves -- Reflection with respect to next-to-highest point. 182\caption{Spendley et al. simplex moves -- Reflection with respect to next-to-highest point.
175It may happen that the next iteration is a shrink step.} 183It may happen that the next iteration is a shrink step.}
@@ -186,17 +194,46 @@ optimum is searched.
186 194
187\begin{figure} 195\begin{figure}
188\begin{center} 196\begin{center}
189\includegraphics[width=10cm]{spendley-steps-shrink.png} 197\includegraphics[width=10cm]{spendley-steps-shrink.pdf}
190\end{center} 198\end{center}
191\caption{Spendley et al. simplex moves -- Shrink.} 199\caption{Spendley et al. simplex moves -- The shrink step is the
200only possible move.}
192\label{fig-spendley-moves-shrink} 201\label{fig-spendley-moves-shrink}
193\end{figure} 202\end{figure}
194 203
195%% \subsection{Termination criteria} 204%%\subsection{Termination criteria}
196 205
197%% TODO... 206%%The original paper \cite{Spendley1962}
198 207
199%% \subsection{General features of the algorithm} 208\subsection{General features of the algorithm}
209
210From the performance point of viewn when a reflection step is performed,
211only 1 or 2 function evaluations are required. Instead, when a shrink
212step is performed, there are $n$ function evaluations required. In practice,
213reflection steps are performed when the simplex is away from the optimum.
214When the simplex is closer to the optimum, or enters in a narrow valley, shrink
215steps are used.
216
217As stated in \cite{Singer:2009}, the main feature
218of Spendley's et al. algorithm is that the simplex can vary
219in size, but not in shape. As we are going to see in the numerical
220experiments, this leads to a slow convergence when a narrow
221valley is encountered. In that situation, the shrink
222steps are required, which leads to a large number
223of iterations and function evaluations.
224
225In fact, the Spendley's et al. algorithm is a pattern search
226algorithm \cite{Torczon98fromevolutionary}. This is a consequence
227of the fact that the search pattern used in the method is constant.
228Therefore, the design never degenerates.
229As stated in \cite{Torczon98fromevolutionary}, "under very mild
230assumptions on $f$, these simple heuristics
231provide enough structure to guarantee global convergence.
232This is not the case for the Nelder-Mead algorithm, which might
233converge to non-stationnary points \cite{589109, hanNeumann2003, Han2000, Torczon89multi-directionalsearch}.
234In all cases, the difficulty is that a sequence of simplices produced
235by the Nelder-Mead simplex method can come arbitrarily close to
236degeneracy.
200 237
201%% TODO... 238%% TODO...
202 239
@@ -204,22 +241,33 @@ optimum is searched.
204\section{Numerical experiments} 241\section{Numerical experiments}
205 242
206In this section, we present some numerical experiments 243In this section, we present some numerical experiments
207with the Spendley et al. algorithm. 244with Spendley's et al. algorithm.
245The first numerical experiments involves one quadratic function
246in 2 dimensions. The second experiment is based on a
247badly scaled quadratic in 2 dimension. In the third experiment,
248we analyse the behaviour of the algorithm with respect to the
249number of variables.
208 250
209\subsection{Quadratic function} 251\subsection{Quadratic function}
210 252
211The function we try to minimize is the following quadratic 253The function we try to minimize is the following quadratic
212in 2 dimensions 254in 2 dimensions
213
214\begin{eqnarray} 255\begin{eqnarray}
215f(x_1,x_2) = x_1^2 + x_2^2 - x_1 x_2 256f(x_1,x_2) = x_1^2 + x_2^2 - x_1 x_2.
216\end{eqnarray} 257\end{eqnarray}
217 258
218The stopping criteria is based on the relative size of the simplex 259The stopping criteria is based on the relative size of the simplex
219with respect to the size of the initial simplex 260with respect to the size of the initial simplex
220
221\begin{eqnarray} 261\begin{eqnarray}
222\sigma(S) < tol \times \sigma(S_0) 262\sigma_+(S) < tol \times \sigma_+(S_0).
263\end{eqnarray}
264The oriented length $\sigma_+(S)$ is defined by
265\begin{eqnarray}
266\sigma_+(S) = \max_{i=2,n+1} \|\bx_i - \bx_1\|_2
267\end{eqnarray}
268where $\|.\|_2$ is the euclidian norm defined by
269\begin{eqnarray}
270\|\bx\|_2 = \sum_{i=1,n}(x^j)^2.
223\end{eqnarray} 271\end{eqnarray}
224 272
225The initial simplex is a regular simplex with length unity. 273The initial simplex is a regular simplex with length unity.
@@ -253,7 +301,7 @@ The numerical results are presented in table \ref{fig-spendley-numexp1-table}.
253 301
254\begin{figure}[htbp] 302\begin{figure}[htbp]
255\begin{center} 303\begin{center}
256\begin{tiny} 304%\begin{tiny}
257\begin{tabular}{|l|l|} 305\begin{tabular}{|l|l|}
258\hline 306\hline
259Iterations & 49 \\ 307Iterations & 49 \\
@@ -265,7 +313,7 @@ Computed $x^\star$ & $(2.169e-10, 2.169e-10)$\\
265Computed $f(x^\star)$ & $4.706e-20$\\ 313Computed $f(x^\star)$ & $4.706e-20$\\
266\hline 314\hline
267\end{tabular} 315\end{tabular}
268\end{tiny} 316%\end{tiny}
269\end{center} 317\end{center}
270\caption{Numerical experiment with Spendley's et al. method on the quadratic function 318\caption{Numerical experiment with Spendley's et al. method on the quadratic function
271$f(x_1,x_2) = x_1^2 + x_2^2 - x_1 x_2$} 319$f(x_1,x_2) = x_1^2 + x_2^2 - x_1 x_2$}
@@ -363,7 +411,7 @@ function at optimum is very inacurate ($f(x^\star) \approx 0.08$).
363 411
364\begin{figure}[h] 412\begin{figure}[h]
365\begin{center} 413\begin{center}
366\begin{tiny} 414%\begin{tiny}
367\begin{tabular}{|l|l|} 415\begin{tabular}{|l|l|}
368\hline 416\hline
369Iterations & 340 \\ 417Iterations & 340 \\
@@ -376,7 +424,7 @@ Computed $x^\star$ & $(0.001,0.2)$\\
376Computed $f(x^\star)$ & $0.08$\\ 424Computed $f(x^\star)$ & $0.08$\\
377\hline 425\hline
378\end{tabular} 426\end{tabular}
379\end{tiny} 427%\end{tiny}
380\end{center} 428\end{center}
381\caption{Numerical experiment with Spendley's et al. method on a badly scaled quadratic function} 429\caption{Numerical experiment with Spendley's et al. method on a badly scaled quadratic function}
382\label{fig-spendley-numexp2-table} 430\label{fig-spendley-numexp2-table}
@@ -408,7 +456,7 @@ when $a>10$
408 456
409\begin{figure}[htbp] 457\begin{figure}[htbp]
410\begin{center} 458\begin{center}
411\begin{tiny} 459%\begin{tiny}
412\begin{tabular}{|l|l|l|} 460\begin{tabular}{|l|l|l|}
413\hline 461\hline
414$a$ & Function evaluations & Computed $f(x^\star)$ \\ 462$a$ & Function evaluations & Computed $f(x^\star)$ \\
@@ -419,7 +467,7 @@ $1000.0$ & 400 & $30.3$ \\
419$10000.0$ & 400 & $56.08$ \\ 467$10000.0$ & 400 & $56.08$ \\
420\hline 468\hline
421\end{tabular} 469\end{tabular}
422\end{tiny} 470%\end{tiny}
423\end{center} 471\end{center}
424\caption{Numerical experiment with Spendley's et al. method on a badly scaled quadratic function} 472\caption{Numerical experiment with Spendley's et al. method on a badly scaled quadratic function}
425\label{fig-spendley-numexp2-scaling} 473\label{fig-spendley-numexp2-scaling}
@@ -557,7 +605,7 @@ We see that the algorithm mostly performs shrink steps.
557 605
558\begin{figure}[htbp] 606\begin{figure}[htbp]
559\begin{center} 607\begin{center}
560\begin{tiny} 608%\begin{tiny}
561\begin{tabular}{|l|l|l|l|l|} 609\begin{tabular}{|l|l|l|l|l|}
562\hline 610\hline
563$n$ & \#Iterations & \# Reflections & \# Reflection & \#Shrink\\ 611$n$ & \#Iterations & \# Reflections & \# Reflection & \#Shrink\\
@@ -585,7 +633,7 @@ $n$ & \#Iterations & \# Reflections & \# Reflection & \#Shrink\\
58520 & 32 & 2 & 0 & 29\\ 63320 & 32 & 2 & 0 & 29\\
586\hline 634\hline
587\end{tabular} 635\end{tabular}
588\end{tiny} 636%\end{tiny}
589\end{center} 637\end{center}
590\caption{Numerical experiment with Spendley et al method on a generalized quadratic function -- number 638\caption{Numerical experiment with Spendley et al method on a generalized quadratic function -- number
591and kinds of steps performed} 639and kinds of steps performed}
@@ -604,7 +652,7 @@ most iterations are shrink steps, most iterations improves the function value.
604 652
605\begin{figure}[htbp] 653\begin{figure}[htbp]
606\begin{center} 654\begin{center}
607\begin{tiny} 655%\begin{tiny}
608\begin{tabular}{|l|l|l|l|} 656\begin{tabular}{|l|l|l|l|}
609\hline 657\hline
610$n$ & Function & Iterations & $\rho(S_0,n)$\\ 658$n$ & Function & Iterations & $\rho(S_0,n)$\\
@@ -632,7 +680,7 @@ $n$ & Function & Iterations & $\rho(S_0,n)$\\
63220 & 660 & 30 & 0.511687 \\ 68020 & 660 & 30 & 0.511687 \\
633\hline 681\hline
634\end{tabular} 682\end{tabular}
635\end{tiny} 683%\end{tiny}
636\end{center} 684\end{center}
637\caption{Numerical experiment with Spendley et al. method on a generalized quadratic function} 685\caption{Numerical experiment with Spendley et al. method on a generalized quadratic function}
638\label{fig-sp-numexp3-dimension} 686\label{fig-sp-numexp3-dimension}
diff --git a/scilab_doc/neldermead/neldermead-spendley-so.pdf b/scilab_doc/neldermead/neldermead-spendley-so.pdf
index 70b6ca5..ae3fa22 100644
--- a/scilab_doc/neldermead/neldermead-spendley-so.pdf
+++ b/scilab_doc/neldermead/neldermead-spendley-so.pdf
Binary files differ
diff --git a/scilab_doc/neldermead/neldermead.bib b/scilab_doc/neldermead/neldermead.bib
index 384ad54..6f0ed09 100644
--- a/scilab_doc/neldermead/neldermead.bib
+++ b/scilab_doc/neldermead/neldermead.bib
@@ -468,3 +468,22 @@ eprint = {http://comjnl.oxfordjournals.org/cgi/reprint/6/2/163.pdf}
468 day={26} 468 day={26}
469} 469}
470 470
471@ARTICLE{Singer:2009,
472AUTHOR = {Singer, A. and Nelder, J. },
473TITLE = {Nelder-Mead algorithm},
474YEAR = {2009},
475JOURNAL = {Scholarpedia},
476VOLUME = {4},
477NUMBER = {7},
478PAGES = {2928}
479}
480
481@ARTICLE{Torczon98fromevolutionary,
482 author = {Virginia Torczon and Michael W. Trosset},
483 title = {From Evolutionary Operation to Parallel Direct Search: Pattern Search Algorithms for Numerical Optimization},
484 journal = {Computing Science and Statistics},
485 year = {1998},
486 volume = {29},
487 pages = {396--401}
488}
489
diff --git a/scilab_doc/neldermead/section-simplex.tex b/scilab_doc/neldermead/section-simplex.tex
index c56bc53..db3ed63 100644
--- a/scilab_doc/neldermead/section-simplex.tex
+++ b/scilab_doc/neldermead/section-simplex.tex
@@ -66,7 +66,6 @@ prefer to use lengths which are less expensive to compute.
66 66
67The two oriented lengths $\sigma_-(S)$ and $\sigma_+(S)$ are using the 67The two oriented lengths $\sigma_-(S)$ and $\sigma_+(S)$ are using the
68first vertex as the reference point and are defined by 68first vertex as the reference point and are defined by
69
70\begin{eqnarray} 69\begin{eqnarray}
71\label{simplex-sigma} 70\label{simplex-sigma}
72\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 71\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
@@ -123,7 +122,7 @@ optimum.
123 122
124\begin{figure} 123\begin{figure}
125\begin{center} 124\begin{center}
126\includegraphics[width=7cm]{simplex_initialfixed.png} 125\includegraphics[width=7cm]{simplex_initialfixed.pdf}
127\end{center} 126\end{center}
128\caption{Typical pattern with fixed-shape Spendley's et al algorithm} 127\caption{Typical pattern with fixed-shape Spendley's et al algorithm}
129\label{fig-nm-simplex-fixedshape} 128\label{fig-nm-simplex-fixedshape}
diff --git a/scilab_doc/neldermead/simplex_initialfixed.pdf b/scilab_doc/neldermead/simplex_initialfixed.pdf
new file mode 100644
index 0000000..05f45fd
--- /dev/null
+++ b/scilab_doc/neldermead/simplex_initialfixed.pdf
Binary files differ
diff --git a/scilab_doc/neldermead/simplex_initialfixed.png b/scilab_doc/neldermead/simplex_initialfixed.png
deleted file mode 100644
index 19744b1..0000000
--- a/scilab_doc/neldermead/simplex_initialfixed.png
+++ /dev/null
Binary files differ
diff --git a/scilab_doc/neldermead/simplex_initialfixed.svg b/scilab_doc/neldermead/simplex_initialfixed.svg
index 570e699..05fce7b 100644
--- a/scilab_doc/neldermead/simplex_initialfixed.svg
+++ b/scilab_doc/neldermead/simplex_initialfixed.svg
@@ -8,8 +8,8 @@
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