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\title{The Generalized Rastrigin Function}
\date{}

\maketitle

The \emph{Generalized Rastrigin Function} (Equation \ref{eq}) is a
typical example of non-linear multimodal function. This function
was first proposed by Rastrigin as a 2-dimensional function
\cite{TZ89} and has been generalized by M\"{u}hlenbein et al in
\cite{MSB91}. This function is a fairly difficult problem due to
its large search space and its large number of local minima.

\begin{eqnarray}
\label{eq} F(\vec{x}) = A\cdot{}n + \sum_{i=1}^{n}{x_{i}^{2} -
A\cdot{}\cos (\omega{}\cdot{}x_{i})} \\ \nonumber A = 10 \ ; \
\omega = 2\cdot{}\pi{} \ ; \ x_{i} \in [-5.12,5.12]
\end{eqnarray}


The Rastrigin function has a complexity of $\Theta (n\cdot{}\ln
(n))$, where $n$ is the dimension of the problem. The surface of
the function is determined by the external variables $A$ and
$\omega{}$, which control the amplitude and frequency modulation
respectively.


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