Table 2.

Multimodal benchmark functions.

Function	Dim	Range	fmin
$F_8(x)={\displaystyle {\sum }_{i=1}^n-x_i\sin (\sqrt{\left|x_i\right|})}$	30	[−500,500]	−12,569.487
$F_9(x)={\displaystyle {\sum }_{i=1}^n[x_i^2-10\cos (2\pi x_i)+10]}$	30	[−5.12,5.12]	0
$F_{10}(x)=-20\exp (-0.2\sqrt{\frac{1}{n}{\displaystyle {\sum }_{i=1}^n{x}_i^2}})-\exp (\frac{1}{n}{\displaystyle {\sum }_{i=1}^n\cos (2\pi x_i)})+20+e$	30	[−32,32]	0
$F_{11}(x)=\frac{1}{4000}{\displaystyle {\sum }_{i=1}^n{x}_i^2}-{\displaystyle {\prod }_{i=1}^n\cos (\frac{x_i}{\sqrt{i}})}+1$	30	[−600,600]	0
$\begin{array}{l}{F}_{12}(x)=\frac{\pi }{n}\{10\sin (\pi y_1)+{{\displaystyle {\sum }_{i=1}^{n-1}(y_i-1)}}^2[1+10{\sin }^2(\pi y_{i+1})]+{(y_n-1)}^2\}\\ +{\displaystyle {\sum }_{i=1}^{n}u(x_i,10,100,4)},\begin{array}{c}\end{array}\mathrm{where} y_i=1+\frac{x_i+1}{4},\\ u(x_i,a,k,m)=\left\{\begin{array}{cc}k{(x_i-a)}^m& x_i>a\\ 0& -a<x_i<a\\ k{(-x_i-a)}^m& x_i<-a\end{array}\right.\end{array}$	30	[−50,50]	0
$\begin{array}{l}{F}_{13}(x)=0.1({\sin }^2(3\pi x_1)+{\displaystyle {\sum }_{i=1}^n{(x_i-1)}^2}[1+{\sin }^2(3\pi x_i+1)]\\ +{(x_n-1)}^2[1+{\sin }^2(2\pi x_n)])+{\displaystyle {\sum }_{i=1}^{n}u(x_i,5,100,4)}\end{array}$	30	[−50,50]	0