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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% La page de garde
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\begin{titlepage}
  \begin{center}
    {\large \bf \textcolor{bleu_fonce}{CEC 2002} }\\
    \bigskip
       { \large \bf  \textcolor{rouge_bordeau}
      {ASCHEA: New Results Using Adaptive Segregational \\
        Constraint Handling}}\\
    \vskip 1.2truecm
          {\bf Sana Ben Hamida$^1$  and Marc Schoenauer$^2$}\\~\\
          {\footnotesize $^1$CMAP, Ecole Polytechnique, Palaiseau, France}\\
          {\footnotesize $^2$FRACTALES, I.N.R.I.A. Rocquencourt, France}\\
   
    \vskip 0.4truecm
   {\sl May 20002}
  \end{center}
\end{titlepage}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\slideheading{Constrained Optimisation Problems}
\begin{center}
Optimize \textcolor{bleu_fonce}{$f(\vec{x}), \vec{x} = (x_1,\ldots,x_n) \in  {\cal F}
\subseteq {\cal S} \subseteq \RR^n$,}\\
such that:
 
{\psshadowbox[framearc=0.2,fillstyle=solid,fillcolor=white]{\bf \textcolor{bleu_fonce}{
$
\left\{\begin{array}{lll}
         g_i(\vec{x}) \leq 0,&  \mbox{for } i=1,\ldots,q & \mbox{\small  inequality constraints}\\
         h_j(\vec{x}) = 0, & \mbox{for } j=q+1,\ldots,m & \mbox{\small  equality constraints}
\end{array}\right.
$
}}}
\end{center}
where $f$, $g_i$ and $h_j$ are real-valued functions on the
search space ${\cal S}$.
The satisfaction of the set
of constraints ($g_i$, $h_j$) defines the feasible region ${\cal F}$.
\begin{center}
    \begin{tabular}{c}
      \psfig{width=12truecm,height=5.5truecm,figure=_Figures/espace_fais.ps}
    \end{tabular}
  \end{center}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\large
{\bf \textcolor{rouge_bordeau}{\underline{Penalty methods:}}  penalize infeasible solutions}\\

\normalsize

\begin{center}
{\psshadowbox[framearc=0.2,fillstyle=solid,fillcolor=vert_clair]{\bf \textcolor{bleu_fonce}{
$
eval(\vec{x}) = \left\{\begin{array}{ll} f(\vec{x}), & \mbox{if} \;
\vec{x} \in {\cal F}\\ f(\vec{x}) + penal(\vec{x}), & \mbox{otherwise}
\end{array}\right.
$
}}}
\end{center}

where $penal(\vec{x})$ is positive for minimizing and negative for
maximizing.

\medskip

\begin{center}

{\psshadowbox[framearc=0.2,fillstyle=solid,fillcolor=vert_clair]{\bf \textcolor{bleu_fonce}{
$
penal(\vec{x})= \sum_{j=1}^q \alpha_j g_j^{+}(\vec{x}) +
\sum_{j=q+1}^m \alpha_j |h_j|(\vec{x}), 
$
}}}
\end{center}

where $x^+$ denotes the positive part of $x$ and  the real numbers
$\alpha_j$, $j=1\ldots m$ are  the {\em penalty coefficients}.
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide} 
\slideheading{The Adaptive Segregational algorithm: ASCHEA}
\begin{center}
{\bf \textcolor{rouge_bordeau} {\underline{Goal}:}}
{\bf Maintain population diversity} \\
$\Longrightarrow$ maintain both feasible and infeasible
individuals in the population.

\normalsize
\begin{bundle}{\large \bf \textcolor{red}{ASCHEA}}
\setlength{\GapDepth}{1.5cm}
\setlength{\GapWidth}{0.6cm}
\chunk[(fitness computation)]{\bf Adaptive penalty}
\chunk[(recombination)]{\bf Seduction/Selection}
\chunk[(selection)]{\bf Segregated selection} 
\drawwith{\drawline}
\end{bundle}

\begin{tabular}{ccc}
\psfig{width=3.cm,figure=_Figures/penal.eps}&
\includegraphics[width=3.2cm]{_Figures/croisement.eps} &
\psfig{width=8.2cm,figure=_Figures/selection.eps}\\
%\psfig{width=3.5cm,figure=_Figures/select.eps}\\
\end{tabular}
\end{center}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\slideheading{ASCHEA+: Improving the Adaptive Penalty Function}
\medskip
{\textcolor{red}{ASCHEA}}: a single penalty c{\oe}fficient $\alpha(t)$ \\
$\Longrightarrow$ $\alpha(t)$  adapted  according to the proportion $\tau_t$ 
of feasible individuals in the population

\begin{center}
{\psshadowbox[framearc=0.2,fillstyle=solid,fillcolor=vert_clair]{\bf \textcolor{bleu_fonce}{  
$
\begin{array}{ll}
\mbox{if } (\tau_t > \tau_{target}) &  \alpha(t+1)(\vec{x})=\alpha(t)/fact\\
\mbox{otherwise} & \alpha(t+1)(\vec{x})=\alpha(t)*fact
\end{array}
$
}}}
 \end{center}

$\tau_{target}$ and $fact > 1$: user-defined parameters
({typically, $\tau_{target} \approx$  0.5})
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
{\textcolor{red}{ASCHEA+}}: Extension to multiple penalty coefficient\\

\begin{center}
{\psshadowbox[framearc=0.2,fillstyle=solid,fillcolor=vert_clair]{\bf \textcolor{bleu_fonce}{
$
penal(\vec{x})= \sum_{j=1}^q \alpha_j(t) g_j^{+}(\vec{x}) +
\sum_{j=q+1}^m \alpha_j(t) |h_j|(\vec{x}), 
$
}}}

\medskip
Adapt each $\alpha_i$ independently using the same strategy:
\medskip

{\psshadowbox[framearc=0.2,fillstyle=solid,fillcolor=vert_clair]{\bf \textcolor{bleu_fonce}{
$
\begin{array}{ll}
\mbox{si } (\tau_t(j) > \tau_{target}) &  \alpha_j(t+1)(\vec{x})=\alpha_j(t)/fact\\
\mbox{else} & \alpha_j(t+1)(\vec{x})=\alpha_j(t)*fact 
\end{array}
$
}}}

$\tau_t(i)$ is the proportion of individuals feasible for constraint $i$
\end{center}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\slideheading{ASCHEA+: Adding Niching}
\underline{Niching method}: Elitist clearing ({\em P\'etrowski, 96}) 
adapted to ASCHEA 
\begin{center}
%{\small
%\begin{minipage}[t]{.47\linewidth} 

  \centering\includegraphics[width=13truecm,height=8.5truecm]{_Figures/clearing.eps}\\
%\end{minipage}
%\hfil
%\begin{minipage}[t]{.50\linewidth}
 % \footnotesize
\begin{enumerate}
\item  Classify individuals on {\em \textcolor{blue}{leaders}} and 
      {\em \textcolor{bleu_clair}{followers}}. 
\item  Apply segregational selection to the
    {\em \textcolor{blue}{leaders}}.
\item If necessary , complete with  {\em \textcolor{bleu_clair}{followers}}.
\end{enumerate}
\end{center} 


\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\slidesubheading{Adaptive niching radius}
{\textcolor{rouge_bordeau}{Goal:}} adapt the niching radius (r) according 
to the proportion of {\em \textcolor{blue}{leaders}} ($\tau_{leaders}$) and 
{\em \textcolor{bleu_clair}{followers}} 
($\tau_{followers}$) in the population.
 
\begin{center} 
    \begin{tabular}{cc}
      \psfig{width=9truecm,height=7.5truecm,figure=_Figures/clearing2.eps}&
      \psfig{width=9truecm,height=7.5truecm,figure=_Figures/clearing3.eps}
\end{tabular}
\psshadowbox[framearc=0.2]{
$
\begin{array}{lll}  
\mbox{if }  \tau_{leaders} > \tau_{clear} & & r(t+1)=r(t)/fact\\
\mbox{else if } \tau_{followers} >  \tau_{clear} & & r(t+1)=r(t)*fact\\
\end{array}
$
}
\end{center}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide} 
\slideheading{ASCHEA+: Experimental Behavior}
{\bf Test 1 (G6)} $\Longrightarrow$  {\large optimum lies on the boundary of 
${\cal F}$}
  \begin{center} 
  \begin{tabular}{c}
  \psfig{width=10.2truecm,angle=-90,figure=_Figures/G6.eps}\\
  \end{tabular}
 \end{center}
  {\em {\bf G6  landscape}. \small The
  constraint boundaries have been artificially made visible.}  
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide} 
 {\bf Test 2 (G8)} $\Longrightarrow$  {\large two quasi-optima inside  ${\cal F}$}
  \begin{center} 
    \begin{tabular}{c}
      \psfig{width=11truecm,angle=-90,figure=_Figures/G8.eps}\\
  \end{tabular} 
\end{center}
  {\em {\bf G8 landscape}. Note that
  function G8 has huge peaks toward the $x_2$ axis that are not shown
   here.}  
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}  
\slidesubheading{Experimental conditions}

\begin{tabular}{ll}
{\bf \textcolor{rouge_bordeau}{Evol. engine}}:   &  (100+300)-ES segregational selection strategy\\
{\bf \textcolor{rouge_bordeau}{Mutation}}:    &  Gaussian mutation with self-adaptive standard deviations\\
             &  $\sigma_0 =0.03$, $\tau = 0.1$, 
                $\tau' = 1$, $p_m=0.9$\\
{\bf \textcolor{rouge_bordeau}{Recombination}}: & Arithmetical crossover, $p_c=0.9$\\
{\bf \textcolor{rouge_bordeau}{ASCHEA}}: & $\tau_{select}=0.3$, $\tau_{target}=0.5$, $fact=1.1$ \\
{\bf \textcolor{rouge_bordeau}{Niching}}: &$\tau_{clear}=0.4$, $niche\_capacity=8$\\
{\bf \textcolor{rouge_bordeau}{Max generations}}:  &5000\\
{\bf \textcolor{rouge_bordeau}{\# runs}}: & 31
\end{tabular}

\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%
%%ANIMATION
%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}  
\slideheading{Extensive experimental results}

\vspace{-0.3cm}


%The test cases G2 and G3 were run with respectively 20 and 10
% variables.\\

%Each equality constraint has been
 %turned into one inequality constraint (G3*, G5*, G11*).\\

\scriptsize
\begin{center}
\begin{tabular}{|c|c||c|c|c||c|c|c|}
\hline
Function & Opt. value &\multicolumn{3}{|c|}{Opt. found} & \multicolumn{3}{|c|}{After 5000 gen}\\
\cline{3-8}
         &            &   Hits    &  best    &   Avg.    & best         &  median       & Avg.  \\
         &            &           & (nb gen) &  (nb gen) &              &               &       \\
\hline
G1       & -15        &  29       & 262      &   330     & {\bf \textcolor{red}{-15}}    &  -15          &-14.84   \\
G2+       & 0.803619   &  --       &   --     &   --      & {\bf \textcolor{red}{0.803614}} & 0.794568       &  0.88949   \\
G3*      &  1.0       &  1        & 4712     &   4712    &  {\bf \textcolor{red}{1}}     &  0.999998     & 0.99989   \\
G4       & -30665.5   &  {\bf \textcolor{red}{31}}&  95     &   217     &{\bf \textcolor{red}{-30665.5}}&  -30665.5     & -30665.5 \\
G5*+      & 5126.4981  &  30       &   98    &   127      &   {\bf \textcolor{red}{5126.5}} &  5126.5      & 5126.53   \\
G6       & -6961.81   &   {\bf \textcolor{red}{31}}      &   92     &    150     &{\bf \textcolor{red}{-6961.81}}&  -6961.81     & -6961.81   \\ 
G7       & 24.3062    &   --      &  --      &   --     &  24.3323 &   24.6162  & 24.6636     \\
G8       & 0.095825   &   {\bf \textcolor{red}{31}}      &   12     &    32     &{\bf \textcolor{red}{0.095825}}&  0.095825     &  0.095825   \\
G9       & 680.630    &   1       &  4567     &   4567   &{\bf \textcolor{red}{680.630}} &  680.635      & 680.641     \\
G10+      & 7049.33    &   --      &  --      &   --      &   {\bf \textcolor{red}{7049.42}}    & 7272.19       &  7615.2   \\
G11*     & 0.75       &   {\bf \textcolor{red}{31}}      &  27      &    37     & {\bf \textcolor{red}{0.75}}   &   0.75        &  0.75      \\
\hline
\end{tabular}
{ \footnotesize Summary of results on 11 test cases (from G1 to G11).
The stars * denote that each equality constraint has been
 turned into one inequality constraint.
The '+' denote that ASCHEA+ was applied with niching.
( G2$\rightarrow n=20$,G3$\rightarrow n=10$). }  
\end{center}


\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}  
\slideheading{Handling Equality Constraints}
\begin{tabular}{ll}
{\bf \textcolor{rouge_bordeau}{Idea}}: &Start with a large feasible domain \\
 & reduce it gradually along evolution
\end{tabular}
~\\
\begin{center}
\begin{tabular}{c}
  \psfig{width=22truecm,figure=_Figures/reduction_f.eps}\\
\end{tabular}
\end{center}
~\\
{\bf \textcolor{rouge_bordeau}{Two approaches}}: dynamic vs adaptive adjustment
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\slidesubheading{Dynamic Adjustment}
\bigskip

\begin{center}

{\psshadowbox[framearc=0.2,fillstyle=solid,fillcolor=vert_clair]{\bf \textcolor{bleu_fonce}{
$
\begin{array}{ll}
 \mbox{if } (\tau_t(j) > \tau_{reduct})&  \epsilon_j(t+1)= \epsilon_j(t) /fact_{\epsilon}\\
 \mbox{otherwise }                     &  \epsilon_j(t+1)= \epsilon_j(t)\\
\end{array}
$
}}} 

\bigskip

where $\tau_{reduct}$ and $fact_{\epsilon} > 1$ are user-defined parameters \\
$\tau_{reduct} > \tau_{target}$

\end{center}

{\bf \textcolor{rouge_bordeau}{But}} all feasible individuals might become infeasible!
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\slidesubheading{Adaptive Adjustment}

\begin{center}
\begin{tabular}{c}
  \includegraphics[width=18truecm,height=6truecm]{_Figures/ajust_eps.eps}\\
    \end{tabular} 
\medskip

         The $\epsilon_j$ value is updated such as a 
             proportion $\tau_{equality}$ of the population (individuals having the 
             $V_1, V_2, \cdots, V_{indice}$ violations) still feasible.
\end{center}
~\\
{\bf \textcolor{rouge_bordeau}{But}} $\epsilon_j(t)$ might stay away from 0
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\slideheading{ASCHEA++: Experimental Behavior}
\underline{Three test-cases}: G3, G11 ({\small 1 equality constraint}), 
                               G5 ({\small 3 equality constraints}).\\
\underline{Parameters}:\\ 
   {\small - dynamic adjustment: $fact_{\epsilon}$=1.01, $\tau_{reduct}=0.6$,\\
   - adaptive adjustment: $\tau_{reduct}=0.7$, $\tau_{equality}=0.3$ }
\begin{center}
\begin{tabular}{cc}
      \psfig{width=10truecm,height=5.5truecm,file=_Figures/eps_dep.ps}& 
      \psfig{width=10truecm,height=5.5truecm,file=_Figures/eps_dep_TE03.ps}\\
       (a): Dynamic adjustment  & (b): Adaptive adjustment\\
\end{tabular}
\end{center}
G11: Evolution of median and maximum $\epsilon_j$ (dynamic adjustment),  $\epsilon^{+}_j$ and 
 $\epsilon^{-}_j$ (adaptive adjustment) values.
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\slidesubheading{Experimental results}

\vspace{0.3cm}
\begin{center}
{\scriptsize
\begin{tabular}{|c||c|c|c|c||c|c|c|c|c}
\hline
Approach      & Case & \multicolumn{3}{|c||}{Fitness} & \multicolumn{3}{|c|}{Violation}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\
\hline
              &          & Best     &  Med.  &   Avg.  &     Best       &      Med.      &      Avg.      \\
\cline{2-9}  
Dynamic       & G3       & 1       &  0.999993 & 0.9932 & {\bf \textcolor{red}{$9.18^{-9}$}}   & $1.98.10^{-7}$  & $4.37.10^{-5}$      \\
\cline{2-9}
$\tau_{reduct} 0.6$ &  G5         & 5126.52  &  5154.13  & 5168.9 & {\bf \textcolor{red}{$1.6.10^{-8}$}}   & $6.1.10^{-7}$  & $2.6.10^{-4}$  \\
\cline{2-9}  
          &   G11   & 0.75      &  0.75 & 0.7519 & {\bf \textcolor{red}{$3.1.10^{-12}$}} & $2.1.10^{-11}$ & {\bf \textcolor{red}{$7.4.10^{-11}$}} \\
\hline
\hline
Adaptive      & G3         & 0.99919    &  0.9335 &  0.8925 & {\bf \textcolor{red}{$0$}}   & {\bf \textcolor{red}{$2.2.10^{-16}$}}  & $1.8.10^{-16} $  \\
\cline{2-9} 
$\tau_{reduct} 0.7$ & G5         & 5126.53 & 5159.19 & 5159.24 & $1.1.10^{-6}$   & $2.2.10^{-4}$& $4.5.10^{-3}$    \\
\cline{2-9}
$\tau_{equality} 0.3$ & G11        & 0.75     & 0.75      & 0.7525 & {\bf \textcolor{red}{$0$}}  & {\bf \textcolor{red}{$5.4.10^{-20}$}}   & $1.8.10^{-11}$  \\
\hline
\end{tabular}
\\
}
~\\
{\em \small Results on G3, G5 and G11 --  dynamic and 
          adaptive approaches.}
\end{center}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide} 
\slideheading{Conclusion}
\begin{tabular}{lll}
\textcolor{bleu_fonce}{\bf  ASCHEA+}  & $\rightarrow$  & extended penalty function,  \\
             & $\rightarrow$  & niching method with adaptive radius,\\
             &  \textcolor{red}{$\Rightarrow$} & improved robustness and better results.\\
 & & \\
\textcolor{bleu_fonce}{\bf  ASCHEA++} &  $\rightarrow$ &  handling equality constraints with progressive\\
             &                & reduction of the feasible space\\
                     &  $\rightarrow$ &  two strategies: adaptive and dynamic adjustment\\
                     &  \textcolor{red}{$\Rightarrow$} &  satisfactory results\\
                     &                & need improvements to limit the risk of loss\\
                     &                & of feasible individuals.
\end{tabular}

\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
\end{document}
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