last pass : the return

main.tex

@@ -223,16 +223,16 @@ Simulated annealing interprets the physical slow cooling as a
 slow decrease in the probability of accepting worse solutions as it
 explores the solution space.
 More precisely a state is a $d$-tuple of permutations, one for each dimension,
-and the neighbourhood is the set of $d$-tuple of permutations which can be obtained
-by application of exactly one transposition of one of the permutation of
+and the neighborhood is the set of $d$-tuple of permutations which can be obtained
+by application of exactly one transposition of one of the permutations of
 the current state.
 The selection phase is dependant on the current temperature:
-after applying a random transposition on one of the current permutations, either
+after selecting randomly a state in the neighborhood, either
 the discrepancy of the corresponding Halton set is decreased and the
 evolution is kept, either it does not but is still kept with
 a probability $e^{\frac{\delta}{T}}$ where $\delta$ is the difference
 between the old and new discrepancy, and $T$ the current temperature.
-If the de discrepancy has decreased, the temperature $T$ is multiplied
+If the discrepancy has decreased, the temperature $T$ is multiplied
 by a factor $\lambda$ (fixed to $0.992$ in all our simulations), hence
 is decreased.
 The whole algorithm is described in the flowchart~\ref{flow_rec}.
@@ -247,8 +247,8 @@ The whole algorithm is described in the flowchart~\ref{flow_rec}.
 
 \subsubsection{Dependence on the temperature}
 First experiments were made to select the best initial temperature.
-Results are compiled in graphs~\ref{temp_2},~\ref{temp3},\ref{temp3_z}.
-Graphs~\ref{temp_2},~\ref{temp3} represent the results obtained respectively
+Results are compiled in graphs~\ref{temp_2},~\ref{temp3}, and~\ref{temp3_z}.
+Graphs~\ref{temp_2} and~\ref{temp3} represent the results obtained respectively
 in dimension 2 and 3 between 10 and 500 points. The curve obtained is
 characteristic of the average evolution of the discrepancy optimization
 algorithms for Halton points sets: a very fast decrease for low number of
@@ -281,7 +281,7 @@ temperature is, the best the results are, with a threshold at $10^{-3}$.
 \subsubsection{Stability with regards to the number of iterations}
 
 As for the fully random search heuristic we investigated the stability
-of the algorithm with regards to the number of iterations. We present here
+of the algorithm with regards to the number of iterations. We present 
 the result in dimension 3 in the graph~\ref{iter_sa}. Once again we
 restricted the window between 80 and 180 points were curves are split.
 An interesting phenomena can be observed: the error rates are somehow
@@ -307,11 +307,11 @@ from which $\lambda$ new genes are derived. A gene is the set of parameters
 we are optimizing, 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
+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,
+making a crossover rather than 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
+genes are kept, according to their fitness, and the evolutionary process
 can start again.
 
 Because making variations over $\mu$ or $\lambda$ does not change fundamentally
@@ -323,10 +323,10 @@ each iteration and the size of the family.
 
 \subsubsection{Crossover algorithm}
 
-We designed a crossover for permutations. The idea is simple: given two
-permutations $A$ and $B$ of $\{1..n\}$, it constructs a new permutations
+We designed an ad-hoc crossover for permutations. The idea is simple: given two
+permutations $A$ and $B$ of $\{1..n\}$, it constructs a new permutation
 $C$ value after value, in a random order (we use our class permutation
-for it). For each   $i$, we take either $A_i$ or $B_i$. If exactly
+for this). For each index  $i$, we take either $A_i$ or $B_i$. If exactly
 one of those values is available (understand it was not already chosen)
 we choose it. If both are available, we choose randomly and we remember 
 the
@@ -398,7 +398,7 @@ 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
-$0.1$.
+$0.1$, that is a low probability of making a crossover.
 
 
 \begin{figure}
@@ -444,13 +444,13 @@ like we get before.
 \section{Results and conclusions}
 Eventually we made extensive experiments to compare the three previously
 presented heuristics. The parameters chosen for the heuristics have been
-chosen using the experiments conducted in the previous sections
+guessed using the experiments conducted in the previous sections.
 Results are compiled in the last
-figures~\ref{wrap2},~\ref{wrap2z},~\ref{wrap3z},~\ref{wrap4z}. The
+figures~\ref{wrap2},~\ref{wrap2z},~\ref{wrap3z}, and~\ref{wrap4z}. The
 recognizable curve of decrease
 of the discrepancy is still clearly recognizable in the graph~\ref{wrap2},
 made for points ranged between 10 and 600. We then present the result
-in the --- now classic --- window 80 points - 180 points ---.
+in the --- now classic --- window 80 points - 180 points.
 For all dimensions, the superiority of non-trivial algorithms --- simulated
 annealing and genetic search --- is clear over fully random search.
 Both curves for these heuristics are way below the error band of random
@@ -491,10 +491,10 @@ same for each heuristic}.
 \section*{Acknowledgments}
 We would like to thank Magnus Wahlstrom from the Max Planck Institute for Informatics
 for providing an implementation of the DEM algorithm.
-We would also like to thank Christoff Durr and Carola Doerr
+We would also like to thank Christoff D\"urr and Carola Doerr
 for several very helpful talks on the topic of this work.
-Both Thomas Espitau and Olivier Marty  supported by the French Ministry for
-Research and Higher Education, trough the Ecole Normale Supérieure.
+Both Thomas Espitau and Olivier Marty are supported by the French Ministry for
+Research and Higher Education, trough the \'Ecole Normale Supérieure.
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