source: trunk/documentation/Test Functions/TestFunctions.tex @ 12075

\documentclass[12pt, a4paper]{article}

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\title{Real Valued Test Functions}
\author{Heuristic and Evolutionary Algorithms Laboratory (HEAL)}
\date{\today}

\begin{document}
  \maketitle

  \section*{Ackley Function}
    \begin{equation*}
      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*}

    \begin{tabbing}
      \hspace{5cm}\=\kill
      \textbf{Dimensions:}     \> $n$ \\
      \textbf{Domain:}         \> $-32.768 \leq x_i \leq 32.768$ \\
      \textbf{Global Optimum:} \> $f(x) = 0.0$ at $x = (0.0, 0.0, \dots, 0.0)$ \\
      \textbf{Operator:}       \> AckleyEvaluator \\
      \textbf{Charts:}         \> \\
    \end{tabbing}

    \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*}

    \begin{tabbing}
      \hspace{5cm}\=\kill
      \textbf{Dimensions:}     \> $n$ \\
      \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:}       \> 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}