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authorMichal Baudin <michael.baudin@scilab.org>2009-10-22 16:44:45 +0200
committerMichal Baudin <michael.baudin@scilab.org>2009-10-22 16:44:45 +0200
commitbb5aada5eba6a4a1a181e1589f82c2ec501ac75a (patch)
tree45f73034fae88d1ad9dc9fb6b3a56ea293cfc1a7 /scilab_doc
parent8a20667347f2a0a00adb84a944d68a0d26946f1a (diff)
downloadscilab-bb5aada5eba6a4a1a181e1589f82c2ec501ac75a.zip
scilab-bb5aada5eba6a4a1a181e1589f82c2ec501ac75a.tar.gz
fminsearch: updated validation in Manual
Diffstat (limited to 'scilab_doc')
-rw-r--r--scilab_doc/neldermead/Makefile1
-rw-r--r--scilab_doc/neldermead/fminsearch-so.pdfbin238829 -> 250095 bytes
-rw-r--r--scilab_doc/neldermead/fminsearch.tex86
-rw-r--r--scilab_doc/neldermead/implementations.tex4
-rw-r--r--scilab_doc/neldermead/introduction.tex2
-rw-r--r--scilab_doc/neldermead/macros.tex1
-rw-r--r--scilab_doc/neldermead/method-neldermead.tex8
-rw-r--r--scilab_doc/neldermead/method-spendley.tex5
-rw-r--r--scilab_doc/neldermead/neldermead.pdf16066
-rw-r--r--scilab_doc/neldermead/scripts/fminsearch_demo.m18
-rw-r--r--scilab_doc/neldermead/scripts/fminsearch_rosenbrock.sce5
-rw-r--r--scilab_doc/neldermead/scripts/outfun.m5
-rw-r--r--scilab_doc/neldermead/section-simplex.tex10
13 files changed, 8407 insertions, 7804 deletions
diff --git a/scilab_doc/neldermead/Makefile b/scilab_doc/neldermead/Makefile
index 85dc7ca..437e9cd 100644
--- a/scilab_doc/neldermead/Makefile
+++ b/scilab_doc/neldermead/Makefile
@@ -20,6 +20,7 @@ clean:
20 rm -f *.log 20 rm -f *.log
21 rm -f *.out 21 rm -f *.out
22 rm -f *.toc 22 rm -f *.toc
23 rm -f *.idx
23 24
24ortho: 25ortho:
25 ispell -t ${RAPPORT}.tex 26 ispell -t ${RAPPORT}.tex
diff --git a/scilab_doc/neldermead/fminsearch-so.pdf b/scilab_doc/neldermead/fminsearch-so.pdf
index 05a0864..9b0df46 100644
--- a/scilab_doc/neldermead/fminsearch-so.pdf
+++ b/scilab_doc/neldermead/fminsearch-so.pdf
Binary files differ
diff --git a/scilab_doc/neldermead/fminsearch.tex b/scilab_doc/neldermead/fminsearch.tex
index 4545f5e..c5960f2 100644
--- a/scilab_doc/neldermead/fminsearch.tex
+++ b/scilab_doc/neldermead/fminsearch.tex
@@ -9,6 +9,11 @@ to Matlab's \scifunction{fminsearch}.
9 9
10\section{\scifunction{fminsearch}'s algorithm} 10\section{\scifunction{fminsearch}'s algorithm}
11 11
12In this section, we analyse the specific choices used in
13\scifunction{fminsearch}'s algorithm. We detail what specific variant
14of the Nelder-Mead algorithm is performed, what initial simplex is used,
15the default number of iterations and the termination criteria.
16
12\subsection{The algorithm} 17\subsection{The algorithm}
13 18
14The algorithm used is the Nelder-Mead algorithm. This corresponds to the 19The algorithm used is the Nelder-Mead algorithm. This corresponds to the
@@ -17,6 +22,7 @@ The "non greedy" version is used, that is, the expansion point is
17accepted only if it improves over the reflection point. 22accepted only if it improves over the reflection point.
18 23
19\subsection{The initial simplex} 24\subsection{The initial simplex}
25\index{Pfeffer, L.}
20 26
21The fminsearch algorithm uses a special initial simplex, which is an 27The fminsearch algorithm uses a special initial simplex, which is an
22heuristic depending on the initial guess. The strategy chosen by 28heuristic depending on the initial guess. The strategy chosen by
@@ -532,16 +538,25 @@ Optimization terminated:
532 and F(X) satisfies the convergence criteria using OPTIONS.TolFun of 1.000000e-004 538 and F(X) satisfies the convergence criteria using OPTIONS.TolFun of 1.000000e-004
533\end{lstlisting} 539\end{lstlisting}
534 540
535A close inspection at the data reveals that the two softwares produces 541We check that the two softwares produces indeed the same intermediate results in terms
536indeed the same intermediate results. 542of iteration, function evaluations, function values and type of steps.
543The only difference is the iteration \#0, which is associated with
544function evaluation \#1 in Matlab and with function evaluation \#3 in Scilab.
545This is because Scilab calls back the output function once the initial
546simplex is computed, which requires 3 function evaluations.
537 547
538\subsection{Plot functions} 548\subsection{Output and plot functions}
539 549
540In this section, we check that the plotting features of the 550In this section, we check that the output and plot features of the
541\scifunction{fminsearch} function are the same. 551\scifunction{fminsearch} function are the same.
552We also check that the fields and the content of the \scivar{optimValues}
553data structure and the \scivar{state} variable are the same in
554both languages.
542 555
543The following output function plots in the current graphic 556The following output function plots in the current graphic
544window the value of the current parameter $\bx$. 557window the value of the current parameter $\bx$.
558It also unloads the content of the \scivar{optimValues}
559data structure and prints a message in the console.
545To let Matlab load that script, save the content in a 560To let Matlab load that script, save the content in a
546.m file, in a directory known by Matlab. 561.m file, in a directory known by Matlab.
547 562
@@ -552,6 +567,11 @@ function stop = outfun(x, optimValues, state)
552stop = false; 567stop = false;
553hold on; 568hold on;
554plot(x(1),x(2),'.'); 569plot(x(1),x(2),'.');
570fc = optimValues.funccount;
571fv = optimValues.fval;
572it = optimValues.iteration;
573pr = optimValues.procedure;
574disp(sprintf ( '%d %e %d -%s- %s\n' , fc , fv , it , pr , state ))
555drawnow 575drawnow
556\end{lstlisting} 576\end{lstlisting}
557 577
@@ -575,6 +595,29 @@ This produces the plot which is presented in figure \ref{fig-fminsearch-matlab-o
575\label{fig-fminsearch-matlab-outputfun} 595\label{fig-fminsearch-matlab-outputfun}
576\end{figure} 596\end{figure}
577 597
598Matlab also prints the following messages in the console.
599
600\lstset{language=matlabscript}
601\begin{lstlisting}
602% Matlab
6031 2.420000e+001 0 -- init
6041 2.420000e+001 0 -- iter
6053 2.005000e+001 1 -initial simplex- iter
6065 5.161796e+000 2 -expand- iter
6077 4.497796e+000 3 -reflect- iter
6089 4.497796e+000 4 -contract outside- iter
60911 4.381360e+000 5 -contract inside- iter
61013 4.245273e+000 6 -contract inside- iter
611[...]
612149 2.004302e-008 80 -contract inside- iter
613151 1.122930e-009 81 -contract inside- iter
614153 1.122930e-009 82 -contract outside- iter
615155 1.122930e-009 83 -contract inside- iter
616157 1.107549e-009 84 -contract outside- iter
617159 8.177661e-010 85 -contract inside- iter
618159 8.177661e-010 85 -contract inside- done
619\end{lstlisting}
620
578The following Scilab script sets the "OutputFcn" option and then calls 621The following Scilab script sets the "OutputFcn" option and then calls
579the \scifunction{fminsearch} in order to perform the optimization. 622the \scifunction{fminsearch} in order to perform the optimization.
580 623
@@ -583,6 +626,11 @@ the \scifunction{fminsearch} in order to perform the optimization.
583// Scilab 626// Scilab
584function outfun ( x , optimValues , state ) 627function outfun ( x , optimValues , state )
585 plot( x(1),x(2),'.'); 628 plot( x(1),x(2),'.');
629 fc = optimValues.funccount;
630 fv = optimValues.fval;
631 it = optimValues.iteration;
632 pr = optimValues.procedure;
633 mprintf ( "%d %e %d -%s- %s\n" , fc , fv , it , pr , state )
586endfunction 634endfunction
587opt = optimset ( "OutputFcn" , outfun); 635opt = optimset ( "OutputFcn" , outfun);
588[x fval] = fminsearch ( banana , [-1.2 1] , opt ); 636[x fval] = fminsearch ( banana , [-1.2 1] , opt );
@@ -602,6 +650,36 @@ in figure \ref{fig-fminsearch-scilab-outputfun}.
602Except for the size of the dots (which can be configured in 650Except for the size of the dots (which can be configured in
603both softwares), the graphics are exactly the same. 651both softwares), the graphics are exactly the same.
604 652
653Scilab also prints the following messages in the console.
654
655\lstset{language=scilabscript}
656\begin{lstlisting}
657// Scilab
6583 2.420000e+001 0 -- init
6593 2.005000e+001 1 -initial simplex- iter
6605 5.161796e+000 2 -expand- iter
6617 4.497796e+000 3 -reflect- iter
6629 4.497796e+000 4 -contract outside- iter
66311 4.381360e+000 5 -contract inside- iter
66413 4.245273e+000 6 -contract inside- iter
665[...]
666149 2.004302e-008 80 -contract inside- iter
667151 1.122930e-009 81 -contract inside- iter
668153 1.122930e-009 82 -contract outside- iter
669155 1.122930e-009 83 -contract inside- iter
670157 1.107549e-009 84 -contract outside- iter
671159 8.177661e-010 85 -contract inside- iter
672159 8.177661e-010 85 -- done
673\end{lstlisting}
674
675We see that the output produced by the two software are identical,
676expect for the two first lines and the last line.
677The lines \#1 and \#2 are different is because Scilab computes the
678function values of all the vertices before calling back the output function.
679The last line is different because Scilab considers that once the optimization
680is performed, the type of the step is an empty string. Instead, Matlab
681displays the type of the last performed step.
682
605\subsection{Predefined plot functions} 683\subsection{Predefined plot functions}
606 684
607Several pre-defined plot functions are provided with 685Several pre-defined plot functions are provided with
diff --git a/scilab_doc/neldermead/implementations.tex b/scilab_doc/neldermead/implementations.tex
index 2c15246..0cd7be6 100644
--- a/scilab_doc/neldermead/implementations.tex
+++ b/scilab_doc/neldermead/implementations.tex
@@ -22,6 +22,7 @@ It provides features such as
22\end{itemize} 22\end{itemize}
23 23
24\section{Kelley and the Nelder-Mead algorithm} 24\section{Kelley and the Nelder-Mead algorithm}
25\index{Kelley, C. T.}
25 26
26C.T. Kelley has written a book \cite{Kelley1999} on optimization method and devotes a 27C.T. Kelley has written a book \cite{Kelley1999} on optimization method and devotes a
27complete chapter to direct search algorithms, especially the Nelder-Mead 28complete chapter to direct search algorithms, especially the Nelder-Mead
@@ -168,6 +169,7 @@ based on a local axis by axis search for the optimality of the
168computed optimum. 169computed optimum.
169 170
170\section{Burkardt implementations} 171\section{Burkardt implementations}
172\index{Burkardt, John}
171 173
172John Burkardt gives several implementations of the Nelder-Mead 174John Burkardt gives several implementations of the Nelder-Mead
173algorithm 175algorithm
@@ -177,6 +179,7 @@ algorithm
177\end{itemize} 179\end{itemize}
178 180
179\section{NAG Fortran implementation} 181\section{NAG Fortran implementation}
182\index{NAG}
180 183
181The NAG Fortran library provides the E04CCF/E04CCA routines \cite{NAGE04CCF} which 184The NAG Fortran library provides the E04CCF/E04CCA routines \cite{NAGE04CCF} which
182implements the simplex optimization method. 185implements the simplex optimization method.
@@ -198,6 +201,7 @@ It can be used to print out the current values of any selection of its parameter
198but must not be used to change the values of the parameters. 201but must not be used to change the values of the parameters.
199 202
200\section{GSL implementation} 203\section{GSL implementation}
204\index{Gnu Scientific Library}
201 205
202The Gnu Scientific Library provides two Nelder-Mead implementations. 206The Gnu Scientific Library provides two Nelder-Mead implementations.
203The authors are Tuomo Keskitalo, Ivo Alxneit and Brian Gough. 207The authors are Tuomo Keskitalo, Ivo Alxneit and Brian Gough.
diff --git a/scilab_doc/neldermead/introduction.tex b/scilab_doc/neldermead/introduction.tex
index e041af1..35677c0 100644
--- a/scilab_doc/neldermead/introduction.tex
+++ b/scilab_doc/neldermead/introduction.tex
@@ -5,6 +5,8 @@ We present the main features of the \scifunction{neldermead} component, and
5show how to use the component with a simple example. 5show how to use the component with a simple example.
6 6
7\section{Overview} 7\section{Overview}
8\index{Torczon, Virginia}
9\index{Wright, Margaret}
8 10
9The Nelder-Mead simplex algorithm \cite{citeulike:3009487}, published in 1965, is an enormously 11The Nelder-Mead simplex algorithm \cite{citeulike:3009487}, published in 1965, is an enormously
10popular search method for multidimensional unconstrained optimization. 12popular search method for multidimensional unconstrained optimization.
diff --git a/scilab_doc/neldermead/macros.tex b/scilab_doc/neldermead/macros.tex
index b713760..8c38808 100644
--- a/scilab_doc/neldermead/macros.tex
+++ b/scilab_doc/neldermead/macros.tex
@@ -62,6 +62,7 @@
62 62
63% Scilab macros 63% Scilab macros
64\newcommand{\scifunction}[1]{\textit{#1}} 64\newcommand{\scifunction}[1]{\textit{#1}}
65\newcommand{\scivar}[1]{\textit{#1}}
65 66
66% To highlight source code 67% To highlight source code
67\usepackage{listings} 68\usepackage{listings}
diff --git a/scilab_doc/neldermead/method-neldermead.tex b/scilab_doc/neldermead/method-neldermead.tex
index ea17976..397f2f7 100644
--- a/scilab_doc/neldermead/method-neldermead.tex
+++ b/scilab_doc/neldermead/method-neldermead.tex
@@ -323,6 +323,9 @@ shrink steps are rare.
323 323
324\section{Convergence properties on a quadratic} 324\section{Convergence properties on a quadratic}
325 325
326\index{Han, Lixing}
327\index{Neumann, Michael}
328
326In this section, we reproduce one result 329In this section, we reproduce one result
327presented by Han and Neumann \cite{HanNeumann2006}, which states 330presented by Han and Neumann \cite{HanNeumann2006}, which states
328the rate of convergence toward the optimum on a class of quadratic 331the rate of convergence toward the optimum on a class of quadratic
@@ -1034,6 +1037,8 @@ $10000.0$ & 167 & $3.0e-17$ & $(5.0e-11,-1.0e-10)$ \\
1034\end{figure} 1037\end{figure}
1035 1038
1036\subsection{Sensitivity to dimension} 1039\subsection{Sensitivity to dimension}
1040\index{Han, Lixing}
1041\index{Neumann, Michael}
1037 1042
1038In this section, we try to reproduce the result 1043In this section, we try to reproduce the result
1039presented by Han and Neumann \cite{HanNeumann2006}, which shows that the 1044presented by Han and Neumann \cite{HanNeumann2006}, which shows that the
@@ -1204,6 +1209,7 @@ depending on the number of variables}
1204\end{figure} 1209\end{figure}
1205 1210
1206\subsection{O'Neill test cases} 1211\subsection{O'Neill test cases}
1212\index{O'Neill, R.}
1207 1213
1208In this section, we present the results by O'Neill, who 1214In this section, we present the results by O'Neill, who
1209implemented a fortran 77 version of the Nelder-Mead algorithm 1215implemented a fortran 77 version of the Nelder-Mead algorithm
@@ -1315,6 +1321,7 @@ Baudin & 4 & 999 & 0 & 5.91 e-9 & 676 & - \\
1315 1321
1316\subsection{Convergence to a non stationnary point} 1322\subsection{Convergence to a non stationnary point}
1317\label{section-mcKinnon} 1323\label{section-mcKinnon}
1324\index{Mc Kinnon, K. I. M.}
1318 1325
1319In this section, we analyze the Mc Kinnon counter example 1326In this section, we analyze the Mc Kinnon counter example
1320from \cite{589109}. We show the behavior of the 1327from \cite{589109}. We show the behavior of the
@@ -1532,6 +1539,7 @@ They also reveal that the Nelder-Mead method can generate simplices which collap
1532into a degenerate simplex, by applying repeated inside contractions. 1539into a degenerate simplex, by applying repeated inside contractions.
1533 1540
1534\subsection{Torczon's numerical experiments} 1541\subsection{Torczon's numerical experiments}
1542\index{Torczon, Virginia}
1535 1543
1536In her Phd Thesis \cite{Torczon89multi-directionalsearch}, Virginia Torczon 1544In her Phd Thesis \cite{Torczon89multi-directionalsearch}, Virginia Torczon
1537presents the multi-directional direct search algorithm. In order to analyze the 1545presents the multi-directional direct search algorithm. In order to analyze the
diff --git a/scilab_doc/neldermead/method-spendley.tex b/scilab_doc/neldermead/method-spendley.tex
index dcb183f..b6a4724 100644
--- a/scilab_doc/neldermead/method-spendley.tex
+++ b/scilab_doc/neldermead/method-spendley.tex
@@ -1,6 +1,9 @@
1\chapter{Spendley's et al. method} 1\chapter{Spendley's et al. method}
2\index{Spendley, W.}
3\index{Hext, G. R.}
4\index{Himsworth, F. R.}
2 5
3In this chapter, we present Spendley's et al. algorithm \cite{Spendley1962} for 6In this chapter, we present Spendley, Hext and Himsworth algorithm \cite{Spendley1962} for
4unconstrained optimization. 7unconstrained optimization.
5 8
6We begin by presenting a global overview of the algorithm. 9We begin by presenting a global overview of the algorithm.
diff --git a/scilab_doc/neldermead/neldermead.pdf b/scilab_doc/neldermead/neldermead.pdf
index 4a3c64b..f2b075a 100644
--- a/scilab_doc/neldermead/neldermead.pdf
+++ b/scilab_doc/neldermead/neldermead.pdf
@@ -328,7 +328,7 @@ endobj
328<< /S /GoTo /D (subsection.5.2.2) >> 328<< /S /GoTo /D (subsection.5.2.2) >>
329endobj 329endobj
330223 0 obj 330223 0 obj
331(Plot functions) 331(Output and plot functions)
332endobj 332endobj
333224 0 obj 333224 0 obj
334<< /S /GoTo /D (subsection.5.2.3) >> 334<< /S /GoTo /D (subsection.5.2.3) >>
@@ -665,17 +665,13 @@ endobj
665/ProcSet [ /PDF /Text /ImageC ] 665/ProcSet [ /PDF /Text /ImageC ]
666>> endobj 666>> endobj
667418 0 obj << 667418 0 obj <<
668/Length 1163 668/Length 1168
669/Filter /FlateDecode 669/Filter /FlateDecode
670>> 670>>
671stream 671stream
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673F q4!\I!+BrVL2) P P" ÀiXP4/@e4<;3,2<O2 673糈RGя5R`ى$2 A cjBQ,<U#.QLpt=.Tfh,ɐ:Rڜ7LH$nz&jSEXr.RL/oSw"~<x'0@ix
674Q@iw#CR>L/PO"S!v>}p\5=.| [>YF*pcBg:--e.G׏ӥ{>7J RH_q'rr0'uOgO3:9lE]{K 01׹Ejun 8ja--z_>%)) _bĒ,Um)æ]xO:*[ \8֌˘+bz5(`FK!t?^PUZN{yڔ A!;C]Ḵ7y_4+ 674nN"1ITK1teezeje Z,1ByH}Fឪf<cifn3t&Į)j&zXn3gxj9cމ31uew[zOHNJGð-rE7N^O̙ ^KrmG[X;77zB$;9ftmͯca֖S<4Bމs i~
675>@_S/Ixi@w xUXƖ;?qzY b*>I؁d
676XW2=țSrߊ&է[\q?õ''e|P
677`v2mM-H#O,HG3 )mi `/_fdL?Wm3[k-@zg|&}is;՝vOX;\9t!i.@9CCnFY6,\X.MSH&zM'&ؘGX;T.¡;*y;~W) JN5\F{pa夒z6{dp 8H ?vV
678y'|>=:J}
679endstream 675endstream
680endobj 676endobj
681417 0 obj << 677417 0 obj <<
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879/ProcSet [ /PDF /Text ] 875/ProcSet [ /PDF /Text ]
880>> endobj 876>> endobj
881456 0 obj << 877456 0 obj <<
882/Length 1423 878/Length 1441
883/Filter /FlateDecode 879/Filter /FlateDecode
884>> 880>>
885stream 881stream
886xZrH+- {dR5y' ![I8¯맟<$Re- hι,H;$ 882xZrH+- {̤jɘY8Y(٪H‘G맟<ge-@hι}Ip$
887AݝVpQA#,~}ni,IcO3?Up+gr&a|lt$ك;sƜn9=# VAUN=rGťQ;՟9dL]GF>I`xT~H肈K5y 883 
8885ȁ {HTqx1m 884#
889;\wS1!Glw=^DnyBb ;sZ֫8]-"F$}|z^w ň1?!)prOBYN-m+3 ~U+k_v}'5Qf^C#߀_:h|TWv/۹@Ⱦ 885 r5WrYe,!HÑj0S݁BPO\DTPy qX?0aKZ;EFnjw-
890ܦ$'Sc YLJ_=Eϭ Qf=3{3Bm;̬rsm߮6޸4o3ʏ-yۮ)V3TMڽ)GdQ݃g};lȢ>8҄d^v01 3ݨp7̕X@ (󖱪@|vpPpC6lq:Ox 886\~)IvG>4{~:l$1gbDȂ~UP-wهQ0j|<˜CTX'' z}1jƑ!xjh|WsLk`1!4Pc Y̖OR诼1Ns5(ԨPHް7Ri*/f`cϙsy[;oJ>FFVSը"8**OGygAّ-9hx$R yPIqcàA(=0yjWeF
887)l[48x%SBC¯4dڸU|nGܲܔtvʣ(։EsI%_f&͞5x$% +sÿi$lh`"e&@,*FQolnY MY$q@VTlU &~U[qA*vךsUכ\^n0O:-ǻ)h Bfyp~6cQ{u:}KM(:[TVV%=CC=B[O
888'^;&ZWfFs6mE'_PR٣<>AN, <+n*0t6yޫh]
889x!6#;s* \df ~G TfÛmyboyk<僈}r4.L~aNa^jo?䟂q,roGӱ/ WrND]cI,p*yen!Ų\^rLHyuwnճ|M1O% vdWI)xqց,̵\I@J
8906%GŀePe89F^\%}+/3~T+_r}GQfZCt#֍;/П5>>*\US^ͱ@ȡ
891ڥ$7SGc YŇW^Eϭ Qbѽ3{ 6Qwj_ٝٙmyFvbt]\|F^mk=70`] M8vz E_=LVؐEip 4A`&b&J @hS{zDqN靎/ 33g@<Md`
891endstream 892endstream
892endobj 893endobj
893455 0 obj << 894455 0 obj <<
@@ -1098,7 +1099,7 @@ endobj
1098/Type /Annot 1099/Type /Annot
1099/Subtype /Link 1100/Subtype /Link
1100/Border[0 0 0]/H/I/C[1 0 0] 1101/Border[0 0 0]/H/I/C[1 0 0]
1101/Rect [105.841 208.824 217.268 219.047] 1102/Rect [105.841 206.504 280.08 219.047]
1102/A << /S /GoTo /D (subsection.5.2.2) >> 1103/A << /S /GoTo /D (subsection.5.2.2) >>
1103>> endobj 1104>> endobj
1104449 0 obj << 1105449 0 obj <<
@@ -1137,13 +1138,24 @@ endobj
1137/ProcSet [ /PDF /Text ] 1138/ProcSet [ /PDF /Text ]
1138>> endobj 1139>> endobj
1139496 0 obj << 1140496 0 obj <<
1140/Length 1573 1141/Length 1582
1141/Filter /FlateDecode 1142/Filter /FlateDecode
1142>> 1143>>
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4995/D [1116 0 R /XYZ 146.393 541.253 null] 5083/D [1105 0 R /XYZ 158.097 438.977 null]
4996>> endobj 5084>> endobj
49971135 0 obj << 50851135 0 obj <<
4998/D [1116 0 R /XYZ 62.362 504.538 null] 5086/D [1105 0 R /XYZ 158.097 438.977 null]
4999>> endobj 5087>> endobj
50001136 0 obj << 50881136 0 obj <<
5001/D [1116 0 R /XYZ 62.362 504.538 null] 5089/D [1105 0 R /XYZ 62.362 401.93 null]
5002>> endobj 5090>> endobj
50031137 0 obj << 50911137 0 obj <<
5004/D [1116 0 R /XYZ 62.362 487.796 null] 5092/D [1105 0 R /XYZ 62.362 401.93 null]
5005>> endobj 5093>> endobj
50061138 0 obj << 50941138 0 obj <<
5007/D [1116 0 R /XYZ 62.362 487.796 null] 5095/D [1105 0 R /XYZ 62.362 384.884 null]
5008>> endobj 5096>> endobj
50091139 0 obj << 50971139 0 obj <<
5010/D [1116 0 R /XYZ 62.362 473.069 null] 5098/D [1105 0 R /XYZ 62.362 384.884 null]
5011>> endobj 5099>> endobj
50121140 0 obj << 51001140 0 obj <<
5013/D [1116 0 R /XYZ 62.362 473.069 null] 5101/D [1105 0 R /XYZ 62.362 370.794 null]
5014>> endobj 5102>> endobj
50151141 0 obj << 51031141 0 obj <<
5016/D [1116 0 R /XYZ 277.695 473.069 null] 5104/D [1105 0 R /XYZ 62.362 370.794 null]
5017>> endobj 5105>> endobj
50181142 0 obj << 51061142 0 obj <<
5019/D [1116 0 R /XYZ 277.695 473.069 null] 5107/D [1105 0 R /XYZ 460.875 370.794 null]
5020>> endobj 5108>> endobj
50211143 0 obj << 51091143 0 obj <<
5022/D [1116 0 R /XYZ 370.275 456.023 null] 5110/D [1105 0 R /XYZ 460.875 370.794 null]
5023>> endobj 5111>> endobj
50241144 0 obj << 51121144 0 obj <<
5025/D [1116 0 R /XYZ 370.275 456.023 null] 5113/D [1105 0 R /XYZ 62.362 333.713 null]
5026>> endobj 5114>> endobj
50271145 0 obj << 51151145 0 obj <<
5028/D [1116 0 R /XYZ 158.097 438.977 null] 5116/D [1105 0 R /XYZ 62.362 333.713 null]
5029>> endobj 5117>> endobj
50301146 0 obj << 51181146 0 obj <<
5031/D [1116 0 R /XYZ 158.097 438.977 null] 5119/D [1105 0 R /XYZ 62.362 319.656 null]
5032>> endobj 5120>> endobj
50331147 0 obj << 51211147 0 obj <<
5034/D [1116 0 R /XYZ 62.362 401.93 null] 5122/D [1105 0 R /XYZ 62.362 319.656 null]
5035>> endobj 5123>> endobj
50361148 0 obj << 51241148 0 obj <<
5037/D [1116 0 R /XYZ 62.362 401.93 null] 5125/D [1105 0 R /XYZ 119.794 319.656 null]
5038>> endobj 5126>> endobj
50391149 0 obj << 51271149 0 obj <<
5040/D [1116 0 R /XYZ 62.362 384.884 null] 5128/D [1105 0 R /XYZ 119.794 319.656 null]
5041>> endobj 5129>> endobj
50421150 0 obj << 51301150 0 obj <<
5043/D [1116 0 R /XYZ 62.362 384.884 null] 5131/D [1105 0 R /XYZ 62.362 283.245 null]
5044>> endobj 5132>> endobj
50451151 0 obj << 51331151 0 obj <<
5046/D [1116 0 R /XYZ 62.362 370.794 null] 5134/D [1105 0 R /XYZ 62.362 283.245 null]
5047>> endobj
50481152 0 obj <<
5049/D [1116 0 R /XYZ 62.362 370.794 null]
5050>> endobj
50511153 0 obj <<
5052/D [1116 0 R /XYZ 460.875 370.794 null]
5053>> endobj
50541154 0 obj <<
5055/D [1116 0 R /XYZ 460.875 370.794 null]
5056>> endobj
50571155 0 obj <<
5058/D [1116 0 R /XYZ 62.362 333.713 null]
5059>> endobj 5135>> endobj
50601156 0 obj << 51361103 0 obj <<
5061/D [1116 0 R /XYZ 62.362 333.713 null] 5137/D [1105 0 R /XYZ 271.78 256.214 null]
5062>> endobj
50631157 0 obj <<
5064/D [1116 0 R /XYZ 62.362 319.656 null]