# Changeset 9887 for trunk/documentation/Test Functions/TestFunctions.tex

Ignore:
Timestamp:
08/20/13 19:02:23 (9 years ago)
Message:

updated template document for description of real valued test functions

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 r307 \usepackage[ansinew]{inputenc} \usepackage{epsfig} \usepackage{subcaption} \usepackage{amsmath} \usepackage{amssymb} \section*{Ackley Function} \begin{equation*} f(x) = 20 + e - 20 \cdot e^{-\frac{1}{5} \sqrt{\frac{1}{n} \sum_{i=1}^n x_i^2}} - e^{\frac{1}{n} \sum_{i=1}^n \cos(2 \pi x_i)} f(x) = 20 + e - 20e^{-\frac{1}{5} \sqrt{\frac{1}{n} \sum_{i=1}^n x_i^2}} - e^{\frac{1}{n} \sum_{i=1}^n \cos(2 \pi x_i)} \end{equation*} \end{tabbing} \begin{center} \includegraphics[width=0.45\textwidth]{Images/Ackley_large} \hfill \includegraphics[width=0.45\textwidth]{Images/Ackley_small} \end{center} \newpage \section*{Griewangk Function} \begin{equation*} f(x) = 1 + \sum_{i=1}^n \frac{x_i^2}{4000} - \prod_{i=1}^n cos(\frac{x_i}{\sqrt i}) \begin{figure}[ht] \begin{subfigure}{0.49\textwidth} \includegraphics[width=\linewidth]{Images/Ackley_large} \caption{[-32.768, 32.768]} \end{subfigure} \begin{subfigure}{0.49\textwidth} \includegraphics[width=\linewidth]{Images/Ackley_small} \caption{[-6.0, 6.0]} \end{subfigure} \caption{Ackley function plots.} \end{figure} \newpage \section*{Beale Function} \begin{equation*} f(x)=(1.5-x_1+x_1x_2)^2+(2.25-x_1+x_1x_2^2)^2+(2.625 - x_1 + x_1x_2^3)^2 \end{equation*} \begin{tabbing} \hspace{5cm}\=\kill \textbf{Dimensions:}     \> $2$ \\ \textbf{Domain:}         \> $-4.5 \leq x_i \leq 4.5$ \\ \textbf{Global Optimum:} \> $f(x) = 0.0$ at $x = (3.0, 0.5)$ \\ \textbf{Operator:}       \> BealeEvaluator \\ \textbf{Charts:}         \> \\ \end{tabbing} \begin{figure}[ht] \includegraphics[width=\textwidth]{Images/Beale} \caption{Beale function [-4.5, 4.5].} \end{figure} \newpage \section*{Booth Function} \begin{equation*} f(x)=(x_1+2x_2-7)^2+(2x_1+x_2-5)^2 \end{equation*} \begin{tabbing} \hspace{5cm}\=\kill \textbf{Dimensions:}     \> $2$ \\ \textbf{Domain:}         \> $-10.0 \leq x_i \leq 10.0$ \\ \textbf{Global Optimum:} \> $f(x) = 0.0$ at $x = (1.0, 3.0)$ \\ \textbf{Operator:}       \> BoothEvaluator \\ \textbf{Charts:}         \> \\ \end{tabbing} \begin{figure}[ht] \includegraphics[width=\textwidth]{Images/Booth} \caption{Booth function [-10.0, 10.0].} \end{figure} \newpage \section*{Griewank Function} \begin{equation*} f(x) = 1 + \sum_{i=1}^n \frac{x_i^2}{4000} - \prod_{i=1}^n \cos(\frac{x_i}{\sqrt i}) \end{equation*} \textbf{Domain:}         \> $-600.0 \leq x_i \leq 600.0$ \\ \textbf{Global Optimum:} \> $f(x) = 0.0$ at $x = (0.0, 0.0, \dots, 0.0)$ \\ \textbf{Operator:}       \> GriewangkEvaluator \\ \textbf{Charts:}         \> \\ \end{tabbing} \begin{center} \includegraphics[width=0.45\textwidth]{Images/Griewangk_large} \hfill \includegraphics[width=0.45\textwidth]{Images/Griewangk_small} \end{center} \textbf{Operator:}       \> GriewankEvaluator \\ \textbf{Charts:}         \> \\ \end{tabbing} \begin{figure}[ht] \begin{subfigure}{0.49\textwidth} \includegraphics[width=\linewidth]{Images/Griewank_large} \caption{[-600.0, 600.0]} \end{subfigure} \begin{subfigure}{0.49\textwidth} \includegraphics[width=\linewidth]{Images/Griewank_small} \caption{[-10.0, 10.0]} \end{subfigure} \caption{Griewank function plots.} \end{figure} \newpage \section*{Levy Function} \begin{equation*} f(x)=\sin^2(\pi w_1)+\sum\limits_{i=1}^{n-1}(w_i-1)^2[1+10\sin^2(\pi w_i+1)]+(w_n-1)^2[1+\sin^2(2\pi w_n)] \end{equation*} \begin{equation*} w_i=1+\frac{x_i - 1}{4}, i=1,\dots,n \end{equation*} \begin{tabbing} \hspace{5cm}\=\kill \textbf{Dimensions:}     \> $n$ \\ \textbf{Domain:}         \> $-10.0 \leq x_i \leq 10.0$ \\ \textbf{Global Optimum:} \> $f(x) = 0.0$ at $x = (1.0, 1.0)$ \\ \textbf{Operator:}       \> LevyEvaluator \\ \textbf{Charts:}         \> \\ \end{tabbing} \begin{figure}[ht] \includegraphics[width=\textwidth]{Images/Levy} \caption{Levy function [-10.0, 10.0].} \end{figure} \newpage \section*{Matyas Function} \begin{equation*} f(x)=0.26(x_1^2+x_2^2)-0.48x_1x_2 \end{equation*} \begin{tabbing} \hspace{5cm}\=\kill \textbf{Dimensions:}     \> $2$ \\ \textbf{Domain:}         \> $-10.0 \leq x_i \leq 10.0$ \\ \textbf{Global Optimum:} \> $f(x) = 0.0$ at $x = (0.0, 0.0)$ \\ \textbf{Operator:}       \> MatyasEvaluator \\ \textbf{Charts:}         \> \\ \end{tabbing} \begin{figure}[ht] \includegraphics[width=\textwidth]{Images/Matyas} \caption{Matyas function [-10.0, 10.0].} \end{figure} \newpage \section*{Rastrigin Function} \begin{equation*} f(x)=10n+\sum\limits_{i=1}^n[x_i^2-10\cos(2\pi x_i)] \end{equation*} \begin{tabbing} \hspace{5cm}\=\kill \textbf{Dimensions:}     \> $n$ \\ \textbf{Domain:}         \> $-5.12 \leq x_i \leq 5.12$ \\ \textbf{Global Optimum:} \> $f(x) = 0.0$ at $x = (0.0, 0.0, \dots, 0.0)$ \\ \textbf{Operator:}       \> RastriginEvaluator \\ \textbf{Charts:}         \> \\ \end{tabbing} \begin{figure}[ht] \begin{subfigure}{0.49\textwidth} \includegraphics[width=\linewidth]{Images/Rastrigin_large} \caption{[-5.12, 5.12]} \end{subfigure} \begin{subfigure}{0.49\textwidth} \includegraphics[width=\linewidth]{Images/Rastrigin_small} \caption{[-2.0, 2.0]} \end{subfigure} \caption{Rastrigin function plots.} \end{figure} \newpage \section*{Rosenbrock Function} \begin{equation*} f(x)=\sum\limits_{i=1}^{n-1}[100(x_i^2-x_{i+1})^2+(x_i-1)^2] \end{equation*} \begin{tabbing} \hspace{5cm}\=\kill \textbf{Dimensions:}     \> $n$ \\ \textbf{Domain:}         \> $-2.048 \leq x_i \leq 2.048$ \\ \textbf{Global Optimum:} \> $f(x) = 0.0$ at $x = (1.0, 1.0, \dots, 1.0)$ \\ \textbf{Operator:}       \> RosenbrockEvaluator \\ \textbf{Charts:}         \> \\ \end{tabbing} \begin{figure}[ht] \includegraphics[width=\textwidth]{Images/Rosenbrock} \caption{Rosenbrock function [-2.048, 2.048].} \end{figure} \newpage \section*{Schwefel Function} \begin{equation*} f(x)=418.982887272433n - \sum\limits_{i=1}^n x_i\sin(\sqrt{|x_i|}) \end{equation*} \begin{tabbing} \hspace{5cm}\=\kill \textbf{Dimensions:}     \> $n$ \\ \textbf{Domain:}         \> $-500.0 \leq x_i \leq 500.0$ \\ \textbf{Global Optimum:} \> $f(x) \approx 0.0$ at $x = (420.9687, 420.9687, \dots, 420.9687)$ \\ \textbf{Operator:}       \> SchwefelEvaluator \\ \textbf{Charts:}         \> \\ \end{tabbing} \begin{figure}[ht] \includegraphics[width=\textwidth]{Images/Schwefel} \caption{Schwefel function [-500.0, 500.0].} \end{figure} \newpage \section*{Sphere Function} \begin{equation*} f(x)=\sum\limits_{i=1}^n x_i^2 \end{equation*} \begin{tabbing} \hspace{5cm}\=\kill \textbf{Dimensions:}     \> $n$ \\ \textbf{Domain:}         \> $-5.12 \leq x_i \leq 5.12$ \\ \textbf{Global Optimum:} \> $f(x) = 0.0$ at $x = (0.0, 0.0, \dots, 0.0)$ \\ \textbf{Operator:}       \> SphereEvaluator \\ \textbf{Charts:}         \> \\ \end{tabbing} \begin{figure}[ht] \includegraphics[width=\textwidth]{Images/Sphere} \caption{Sphere function [-5.12, 5.12].} \end{figure} \newpage \section*{Sum Squares Function} \begin{equation*} f(x)=\sum\limits_{i=1}^n ix_i^2 \end{equation*} \begin{tabbing} \hspace{5cm}\=\kill \textbf{Dimensions:}     \> $n$ \\ \textbf{Domain:}         \> $-10.0 \leq x_i \leq 10.0$ \\ \textbf{Global Optimum:} \> $f(x) = 0.0$ at $x = (0.0, 0.0, \dots, 0.0)$ \\ \textbf{Operator:}       \> SumSquaresEvaluator \\ \textbf{Charts:}         \> \\ \end{tabbing} \begin{figure}[ht] \includegraphics[width=\textwidth]{Images/SumSquares} \caption{Sum squares function [-10.0, 10.0].} \end{figure} \newpage \section*{Zakharov Function} \begin{equation*} f(x)=\sum\limits_{i=1}^n x_i^2+\left(\sum\limits_{i=1}^n 0.5ix_i\right)^2+\left(\sum\limits_{i=1}^n 0.5ix_i\right)^4 \end{equation*} \begin{tabbing} \hspace{5cm}\=\kill \textbf{Dimensions:}     \> $n$ \\ \textbf{Domain:}         \> $-5.0 \leq x_i \leq 10.0$ \\ \textbf{Global Optimum:} \> $f(x) = 0.0$ at $x = (0.0, 0.0, \dots, 0.0)$ \\ \textbf{Operator:}       \> ZakharovEvaluator \\ \textbf{Charts:}         \> \\ \end{tabbing} \begin{figure}[ht] \includegraphics[width=\textwidth]{Images/Zakharov} \caption{Zakharov function [-5.0, 10.0].} \end{figure} \end{document}