 which optimum is the global optimum
  \item $\sigma_i$: control $G_i(\cdot)$'s coverage range, a small $\sigma_i$ gives a narrow range for $G_i(\cdot)$
  \item $\lambda_i$: control $G_i(\cdot)$'s height
  \item $\omega_i$: weighted value for $G_i(\cdot)$, calculated as follows:
  \begin{equation}\label{eq:weight}
    w_i = \frac{1}{\sqrt{\sum_{j=1}^D(\mathbf{x}_j-\mathbf{x}_j^{opt,i})^2}}\exp(    -\frac{\sum_{j=1}^D(\mathbf{x}_j-\mathbf{x}_j^{opt,i})^2}{2D\sigma_i^2}        )
  \end{equation}
  where $\mathbf{x}^{opt,i}$ represents the optimum position for $G_i(\cdot)$.
  Then normalized $w_i$ to get $\omega_i$:
  $\omega_i = w_i/\sum_{i=1}^Nw_i$.

  When $\mathbf{x} = \mathbf{x}^{opt,i}$, $\omega_j=\left\{\begin{aligned}1\quad& j=i\\0\quad&j \ne i\end{aligned}\right. \quad(j = 1,2,\cdots,N)$,
  such that $F(\mathbf{x})=bias_i+f^{opt,i}$.


\end{itemize}

The constructed functions are multi-modal and non-separable and merge the properties of the sub-functions better and maintains continuity around the global/local optima.
The local optimum which has the smallest bias value is the global optimum.
The optimum of the third basic function is set to the origin as a trip in order to test the algorithms' tendency to converge to the search center.

Note that, the landscape is not only changes along with the selection of basic function, but the optima and $\sigma$ and $\lambda$ can effect it greatly.


\subsection{Composition Function 1}

\begin{itemize}
  \item $N = 5$
  \item $\sigma = [10,20,30,40,50]$
  \item $\lambda = [1e-10, 1e-6,1e-26,1e-6,1e-6]$
  \item $bias=[0,100,200,300,400]$
  \item $G_1$: Rotated Rosenbrock Function
  \item $G_2$: High Conditioned Elliptic Function
  \item $G_3$: Rotated Bent Cigar Function
  \item $G_4$: Rotated Discus Function
  \item $G_5$: High Conditioned Elliptic Function
\end{itemize}

\subsection{Composition Function 2}

\begin{itemize}
  \item $N = 3$
  \item $\sigma = [15, 15, 15]$
  \item $\lambda =  [1,1,1] $
  \item $bias=[0,100,200]$
  \item $G_1$: Expanded Schwefel Function
  \item $G_2$: Rotated Rstrigin Function
  \item $G_3$: Rotated HGBat Function
\end{itemize}



\subsection{Composition Function 3}

\begin{itemize}
  \item $N = 3$
  \item $\sigma = [20,50,40]$
  \item $\lambda = [0.25, 1, 1e-7]$
  \item $bias=[0,100,200]$
  \item $G_1$: Rotated Schwefel Function
  \item $G_2$: Rotated Rastrigin Function
  \item $G_3$: Rotated High Conditioned Elliptic Function
\end{itemize}


\subsection{Composition Function 4}

\begin{itemize}
  \item $N = 5$
  \item $\sigma = [20,15,10,10,40]$
  \item $\lambda = [2.5e-2,0.1,1e-8,0.25,1]$
  \item $bias=[0,100,200,300,400]$
  \item $G_1$: Rotated Schwefel Function
  \item $G_2$: Rotated HappyCat Function
  \item $G_3$: Rotated High Conditioned Elliptic Function
  \item $G_4$: Rotated Weierstrass Function
  \item $G_5$: Rotated Griewank Function
\end{itemize}


\subsection{Composition Function 5}

\begin{itemize}
  \item $N = 5$
  \item $\sigma = [15,15,15,15,15]$
  \item $\lambda = [10,10,2.5,2.5,1e-6]$
  \item $bias=[0,100,200,300,400]$
  \item $G_1$: Rotated HGBat Function
  \item $G_2$: Rotated Rastrigin Function
  \item $G_5$: Rotated Schwefel Function
  \item $G_4$: Rotated Weierstrass Function
  \item $G_3$: Rotated High Conditioned Elliptic Function
\end{itemize}

\subsection{Composition Function 6}

\begin{itemize}
  \item $N = 5$
  \item $\sigma = [10,20,30,40,50]$
  \item $\lambda = [2.5,10,2.5,5e-4,1e-6]$
  \item $bias=[0,100,200,300,400]$
  \item $G_1$: Rotated Expanded Griewank plus Rosenbrock Function
  \item $G_2$: Rotated HappyCat Function
  \item $G_3$: Rotated Schwefel Function
  \item $G_4$: Rotated Expanded Scaffer's F6 Function
  \item $G_5$: High Conditioned Elliptic Function
\end{itemize}


\subsection{Composition Function 7}

\begin{itemize}
  \item $N = 3$
  \item $\sigma = [10,30,50]$
  \item $\lambda = [1, 1, 1]$
  \item $bias=[0,100,200]$
  \item $G_1$: Hybrid Function 1
  \item $G_2$: Hybrid Function 2
  \item $G_3$: Hybrid Function 3
\end{itemize}



\subsection{Composition Function 8}

\begin{itemize}
  \item $N = 3$
  \item $\sigma = [10,30,50]$
  \item $\lambda = [1, 1, 1]$
  \item $bias=[0,100,200]$
  \item $G_1$: Hybrid Function 4
  \item $G_2$: Hybrid Function 5
  \item $G_3$: Hybrid Function 6
\end{itemize}

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