pass

main.tex

@@ -213,8 +213,7 @@ discrepancy and this heuristic.
 \label{rand_iter3}
 \end{figure}
 
-%# TODO sa n'est pas evolutionnaire
-\subsection{Evolutionary heuristic: Simulated annealing and local search}
+\subsection{Local search with simulated annealing}
 The second heuristic implemented is a randomized local search with
 simulated annealing. This heuristic is inspired by the physical
 process of annealing in metallurgy.
@@ -239,7 +238,7 @@ The whole algorithm is described in the flowchart~\ref{flow_rec}.
 \begin{figure}
  \begin{mdframed}
 \includegraphics[scale=0.4]{flow_recuit.pdf}
-\caption{Flowchart of the simulated annealing local search heuristic}
+\caption{Flowchart of the local search with simulated annealing heuristic}
 \label{flow_rec}
 \end{mdframed}
 \end{figure}
@@ -297,7 +296,26 @@ rates for fully random search with 400 iterations.
   \label{iter_sa}
 \end{figure}
 
-\subsection{Genetic (5+5) search}
+\subsection{Genetic $(\mu+\lambda)$ search}
+
+The third heuristic implemented is the $(\mu+\lambda)$ genectic search.
+This heuristic is inspired from the evolution of species: a family
+of $\mu$ genes is known (they are generated randomly at the beginning),
+from which $\lambda$ new genes are derived. A gene is the set of parameters
+we are optimising, i.e. the permutations.
+Each one is derived either from one gene applying a mutation
+(here a transposition of one of the permutations), or from two
+genes applying a crossover : a blending of both genes (the
+algorithm is described in details further). The probability of
+making a mutation is $c$, the third parameter of the algorithm,
+among $\mu$ and $\lambda$. After that, only the $\mu$ best
+genes are kept, according to their fitness, and the process
+can start again.
+
+Because variating over $\mu$ or $\lambda$ does not change fundamentaly
+the algorithm, we have chosen to fix $\mu=\lambda=5$ once and for all,
+which seemed to be a good trade-off between the running time of
+each iteration and the size of the family.
 
 
 \begin{figure}
@@ -308,23 +326,6 @@ rates for fully random search with 400 iterations.
 \end{mdframed}
 \end{figure}
 
-\subsubsection{Dependence on the parameter p}
-First experiments were made to select the value for the crossover parameter
-p. Results are compiled in graphs~\ref{res_gen2},~\ref{res_gen2z},\ref{res_gen3}
-and~\ref{res_gen4}.
-Graph~\ref{res_gen2}, represents the results obtained
-in dimension 2 between 10 and 500 points. The curve obtained is, with no
-surprise again,
-the characteristic curve of the average evolution of the discrepancy we already
-saw with the previous experiments.
-The most interesting part of these results are concentrated --- once again ---
-between 80 and 160 points were the different curves splits.
-The graph~\ref{res_gen2z} is a zoom  of~\ref{res_gen2} in this window, and
-graphs~\ref{res_gen3} and~\ref{res_gen4} are focused directly into it too.
-We remark that in dimension 2, the results are better for $p$ close to $0.5$
-whereas for dimension 3 and 4 the best results are obtained for $p$ closer to
-$0.1$.
-
 \subsubsection{Algorithm of crossover}
 
 We designed a crossover for permutations. The idea is simple: given two
@@ -381,32 +382,49 @@ permutations.
 \end{algorithm}
 
 
+\subsubsection{Dependence on the parameter $c$}
+First experiments were made to select the value for the crossover parameter
+$c$. Results are compiled in graphs~\ref{res_gen2},~\ref{res_gen2z},\ref{res_gen3}
+and~\ref{res_gen4}.
+Graph~\ref{res_gen2}, represents the results obtained
+in dimension 2 between 10 and 500 points. The curve obtained is, with no
+surprise again,
+the characteristic curve of the average evolution of the discrepancy we already
+saw with the previous experiments.
+The most interesting part of these results are concentrated --- once again ---
+between 80 and 160 points were the different curves splits.
+The graph~\ref{res_gen2z} is a zoom  of~\ref{res_gen2} in this window, and
+graphs~\ref{res_gen3} and~\ref{res_gen4} are focused directly into it too.
+We remark that in dimension 2, the results are better for $c$ close to $0.5$
+whereas for dimension 3 and 4 the best results are obtained for $c$ closer to
+$0.1$.
+
 
 \begin{figure}
 \includegraphics[scale=0.3]{Results/res_gen_2.png}
-\caption{Dependence on parameter p: D=2}
+\caption{Dependence on parameter $c$: D=2}
   \label{res_gen2}
 \end{figure}
 
 \begin{figure}
 \includegraphics[scale=0.3]{Results/res_gen_2_zoom.png}
-\caption{Dependence on parameter p (zoom): D=2}
+\caption{Dependence on parameter $c$ (zoom): D=2}
 \label{res_gen2z}
 \end{figure}
 \begin{figure}
 \includegraphics[scale=0.3]{Results/res_gen_3_zoom.png}
-\caption{Dependence on parameter p: D=3}
+\caption{Dependence on parameter $c$: D=3}
   \label{res_gen3}
 \end{figure}
 
 \begin{figure}
 \includegraphics[scale=0.3]{Results/res_gen_4_zoom.png}
-\caption{Dependence on parameter p: D=4}
+\caption{Dependence on parameter $c$: D=4}
   \label{res_gen4}
 \end{figure}
 
 Once again we investigated the stability
-of the algorithm with regards to the number of iterations.  Once again we
+of the algorithm with regards to the number of iterations. Once again we
 restricted the window between 80 and 180 points were curves are split.
 The results are compiled in graph~\ref{gen_iter}.
 An interesting phenomena can be observed: the error rates are getting really