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@@ -223,16 +223,16 @@ Simulated annealing interprets the physical slow cooling as a
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slow decrease in the probability of accepting worse solutions as it
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explores the solution space.
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More precisely a state is a $d$-tuple of permutations, one for each dimension,
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-and the neighbourhood is the set of $d$-tuple of permutations which can be obtained
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-by application of exactly one transposition of one of the permutation of
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+and the neighborhood is the set of $d$-tuple of permutations which can be obtained
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+by application of exactly one transposition of one of the permutations of
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the current state.
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The selection phase is dependant on the current temperature:
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-after applying a random transposition on one of the current permutations, either
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+after selecting randomly a state in the neighborhood, either
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the discrepancy of the corresponding Halton set is decreased and the
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evolution is kept, either it does not but is still kept with
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a probability $e^{\frac{\delta}{T}}$ where $\delta$ is the difference
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between the old and new discrepancy, and $T$ the current temperature.
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-If the de discrepancy has decreased, the temperature $T$ is multiplied
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+If the discrepancy has decreased, the temperature $T$ is multiplied
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by a factor $\lambda$ (fixed to $0.992$ in all our simulations), hence
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is decreased.
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The whole algorithm is described in the flowchart~\ref{flow_rec}.
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@@ -247,8 +247,8 @@ The whole algorithm is described in the flowchart~\ref{flow_rec}.
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\subsubsection{Dependence on the temperature}
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First experiments were made to select the best initial temperature.
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-Results are compiled in graphs~\ref{temp_2},~\ref{temp3},\ref{temp3_z}.
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-Graphs~\ref{temp_2},~\ref{temp3} represent the results obtained respectively
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+Results are compiled in graphs~\ref{temp_2},~\ref{temp3}, and~\ref{temp3_z}.
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+Graphs~\ref{temp_2} and~\ref{temp3} represent the results obtained respectively
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in dimension 2 and 3 between 10 and 500 points. The curve obtained is
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characteristic of the average evolution of the discrepancy optimization
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algorithms for Halton points sets: a very fast decrease for low number of
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@@ -281,7 +281,7 @@ temperature is, the best the results are, with a threshold at $10^{-3}$.
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\subsubsection{Stability with regards to the number of iterations}
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As for the fully random search heuristic we investigated the stability
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-of the algorithm with regards to the number of iterations. We present here
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+of the algorithm with regards to the number of iterations. We present
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the result in dimension 3 in the graph~\ref{iter_sa}. Once again we
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restricted the window between 80 and 180 points were curves are split.
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An interesting phenomena can be observed: the error rates are somehow
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@@ -307,11 +307,11 @@ from which $\lambda$ new genes are derived. A gene is the set of parameters
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we are optimizing, i.e. the permutations.
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Each one is derived either from one gene applying a mutation
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(here a transposition of one of the permutations), or from two
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-genes applying a crossover : a blending of both genes (the
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+genes applying a crossover: a blending of both genes (the
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algorithm is described in details further). The probability of
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-making a mutation is $c$, the third parameter of the algorithm,
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+making a crossover rather than a mutation is $c$, the third parameter of the algorithm,
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among $\mu$ and $\lambda$. After that, only the $\mu$ best
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-genes are kept, according to their fitness, and the process
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+genes are kept, according to their fitness, and the evolutionary process
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can start again.
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Because making variations over $\mu$ or $\lambda$ does not change fundamentally
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@@ -323,10 +323,10 @@ each iteration and the size of the family.
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\subsubsection{Crossover algorithm}
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-We designed a crossover for permutations. The idea is simple: given two
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-permutations $A$ and $B$ of $\{1..n\}$, it constructs a new permutations
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+We designed an ad-hoc crossover for permutations. The idea is simple: given two
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+permutations $A$ and $B$ of $\{1..n\}$, it constructs a new permutation
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$C$ value after value, in a random order (we use our class permutation
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-for it). For each $i$, we take either $A_i$ or $B_i$. If exactly
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+for this). For each index $i$, we take either $A_i$ or $B_i$. If exactly
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one of those values is available (understand it was not already chosen)
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we choose it. If both are available, we choose randomly and we remember
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the
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@@ -398,7 +398,7 @@ The graph~\ref{res_gen2z} is a zoom of~\ref{res_gen2} in this window, and
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graphs~\ref{res_gen3} and~\ref{res_gen4} are focused directly into it too.
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We remark that in dimension 2, the results are better for $c$ close to $0.5$
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whereas for dimension 3 and 4 the best results are obtained for $c$ closer to
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-$0.1$.
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+$0.1$, that is a low probability of making a crossover.
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\begin{figure}
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@@ -444,13 +444,13 @@ like we get before.
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\section{Results and conclusions}
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Eventually we made extensive experiments to compare the three previously
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presented heuristics. The parameters chosen for the heuristics have been
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-chosen using the experiments conducted in the previous sections
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+guessed using the experiments conducted in the previous sections.
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Results are compiled in the last
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-figures~\ref{wrap2},~\ref{wrap2z},~\ref{wrap3z},~\ref{wrap4z}. The
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+figures~\ref{wrap2},~\ref{wrap2z},~\ref{wrap3z}, and~\ref{wrap4z}. The
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recognizable curve of decrease
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of the discrepancy is still clearly recognizable in the graph~\ref{wrap2},
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made for points ranged between 10 and 600. We then present the result
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-in the --- now classic --- window 80 points - 180 points ---.
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+in the --- now classic --- window 80 points - 180 points.
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For all dimensions, the superiority of non-trivial algorithms --- simulated
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annealing and genetic search --- is clear over fully random search.
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Both curves for these heuristics are way below the error band of random
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@@ -491,10 +491,10 @@ same for each heuristic}.
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\section*{Acknowledgments}
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We would like to thank Magnus Wahlstrom from the Max Planck Institute for Informatics
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for providing an implementation of the DEM algorithm.
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-We would also like to thank Christoff Durr and Carola Doerr
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+We would also like to thank Christoff D\"urr and Carola Doerr
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for several very helpful talks on the topic of this work.
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-Both Thomas Espitau and Olivier Marty supported by the French Ministry for
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-Research and Higher Education, trough the Ecole Normale Supérieure.
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+Both Thomas Espitau and Olivier Marty are supported by the French Ministry for
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+Research and Higher Education, trough the \'Ecole Normale Supérieure.
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\bibliographystyle{alpha}
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\bibliography{bi}
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\end{document}
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Olivier Marty