Table 2.

Multimodal benchmark functions.

Function	Dim	Range	F min
$F_8\left(x\right)={\displaystyle {\sum }_{i=1}^n-x_i\sin \left(\sqrt{\left|x_i\right|}\right)}$	30	[−500, 500]	−418.9829 × Dim
F9(x)=∑i=1n[xi2 − 10  cos(2πxi)+10]	30	[−5.12, 5.12]
$F_{10}\left(x\right)=-20\exp \left(-0.2\sqrt{\left(1/n\right){\displaystyle {\sum }_{i=1}^n{x}_i^2}}\right)-\exp \left(\left(1/n\right){\displaystyle {\sum }_{i=1}^n\cos \left(2\pi x_i\right)}\right)+20+e$	30	[−32, 32]	0
$F_{11}\left(x\right)=\left(1/4000\right){\displaystyle {\sum }_{i=1}^n{x}_i^2}-{\displaystyle {\prod }_{i=1}^n\cos \left(x_i/\sqrt{i}\right)}+1$	30	[−600, 600]	0
$\begin{array}{c}{F}_{12}\left(x\right)=\left(\pi /n\right)\left\{10\sin \left(\pi y_1\right)+{{\displaystyle {\sum }_{i=1}^{n-1}\left(y_i-1\right)}}^2\left[1+10{\sin }^2\left(\pi y_{i+1}\right)\right]+{\left(y_n-1\right)}^2\right\}\\ +{\displaystyle {\sum }_{i=1}^{n}u\left(x_i,\mathrm{10,100,4}\right)},\text{where\hspace{0.17em}}{y}_i=1+\left(x_i+1\right)/4,\\ u\left(x_i,a,k,m\right)=\left\{\begin{array}{cc}k{\left(x_i-a\right)}^m,& x_i>a,\\ 0,& -a<x_i<a,\\ k{\left(-x_i-a\right)}^m,& x_i<-a\end{array}\right.\end{array}$	30	[−50, 50]	0
$\begin{array}{c}{F}_{13}\left(x\right)=0.1\left({\sin }^2\left(3\pi x_1\right)+{\displaystyle {\sum }_{i=1}^n{\left(x_i-1\right)}^2\left[1+{\sin }^2\left(3\pi x_i+1\right)\right]}\right)\\ +{\left(x_n-1\right)}^2\left[1+{\sin }^2\left(2\pi x_n\right)\right]\left(\right)+{\displaystyle {\sum }_{i=1}^{n}u\left(x_i,\mathrm{5,100,4}\right)}\end{array}$	30	[−50, 50]	0