Table 3.

Multi-modal test functions.

Funtion	Dim	Range	fmin
$f8\left(x\right)={\sum }_{i=1}^n-x_i\sin (\sqrt{|x_i|})$	10	[− 500, 500]	–2820.8
$f9\left(x\right)={\sum }_{i=1}^n[x_i^2-10\cos \left(2,\pi ,x_i\right)+10]$	10	[− 5.12, 5.12]	0
$f10\left(x\right)=-20\exp \left(-,0.2,\sqrt{\frac{1}{n}{\sum }_{i=1}^n{x}_i^2}\right)-exp\left(\frac{1}{n},{\sum }_{i=1}^n,\cos ,\left(2,\pi ,x_i\right)\right)+20+e$	10	[− 32, 32]	0
$f11\left(x\right)=\frac{1}{4000}{\sum }_{i=1}^n{x}_i^2-{\prod }_{i=1}^n\cos \left(\frac{x_i}{\sqrt{i}}\right)+1$	10	[− 600, 600]	0
$f12\left(x\right)=\frac{\pi }{n}\left\{10,{\sin }^2,(\pi y_1),+,{\sum }_{i=1}^n,{(y_i-1)}^2,\left[1,+,10,{\sin }^2,(\pi y_{i+1})\right],+,{(y_n-1)}^2,+,{\sum }_{i=1}^n,u,(x_i,10,100,4)\right\}$
$y_i=1+\frac{x_i+1}{4}$
$u\left(x_i,,,a,,,k,,,m\right)=\left\{\begin{array}{c}k{\left(x_i,-,a\right)}^m{x}_i>a\\ 0-a<x_i<a\\ k{\left({-x}_i,-,a\right)}^m{x}_i<-a\end{array}\right)$	10	[− 50, 50]	0
$f13\left(x\right)=0.1\left\{{\sin }^2,\left(3,\pi ,x_1\right),+,{\sum }_{i=1}^n,{\left(x_i,-,1\right)}^2,\left[1,+,{\sin }^2,\left(3,\pi ,x_i,+,1\right)\right],+,{\left(x_n,-,1\right)}^2,[1+{\sin }^2(2\pi x_n)]\right\}+{\sum }_{i=1}^{n}u(x_i,5,100,4)$	10	[− 50, 50]	0