Table 3.

Multimodal benchmark functions.

F	Function	Dim	Range	f min
F8	$f_8(x)={\displaystyle {\displaystyle \sum }_{i=1}^n}-x_i\sin \left(\sqrt{\left|x_i\right|}\right)$	30	[−500, 500]	−418.9829×n
F9	$f_9(x)={\displaystyle {\displaystyle \sum }_{i=1}^n}{x}_i^2-10\cos \left(2\pi x_i\right)+10$	30	[−5.12, 5.12]	0
F10	$\begin{array}{l}{f}_{10}(x)=-20\exp \left(-0.2\sqrt{\frac{1}{n}{\displaystyle {\displaystyle \sum }_{i=1}^n}{x}_i^2}\right)-\\ \exp \left(\frac{1}{n}{\displaystyle {\displaystyle \sum }_i^n}\cos \left(2\pi x_i\right)\right)+20+e\end{array}$	30	[−32, 32]	0
F11	$f_{11}(x)={\displaystyle {\displaystyle \sum }_{i=1}^n}\frac{x_i^2}{4000}-{\displaystyle {\displaystyle \prod }_{i=1}^n}\cos \left(\frac{x_i}{\sqrt{i}}\right)-1$	30	[−600, 600]	0
F12	$\begin{array}{l}{f}_{12}(x)=\frac{\pi }{\mathrm{n}}\left\{\begin{array}{l}10{\sin }^2\left(\pi y_1\right)+{\displaystyle {\displaystyle \sum }_{i=1}^{n-1}}{\left(y_i-1\right)}^2\\ \left[1+10{\sin }^2\left(\pi y_{i+1}\right)\right]+{\left(y_n-1\right)}^2\end{array}\right\}\\ +{\displaystyle {\displaystyle \sum }_{i=1}^n}u\left(x_i,10,100,4\right)\\ y_i=1+\frac{x_i+1}{4},\\ u\left(x_i,a,k,m\right)=\left\{\begin{array}{c}k{\left(x_i-a\right)}^m{x}_i>a\\ 0-a<x_i<a\\ k{\left(-x_i-a\right)}^m{x}_i<-a\end{array}\right\}\end{array}$	30	[−50, 50]	0
F13	$\begin{array}{l}{f}_{13}(x)=0.1\left\{\begin{array}{l}\begin{array}{l}{\sin }^2\left(3\pi x_i\right)+{\displaystyle {\displaystyle \sum }_{i=1}^n}{\left(x_i-1\right)}^2\\ \left[1+{\sin }^2\left(3\pi x_i+1\right)\right]\end{array}\hfill \\ +{\left(x_n-1\right)}^2\left[1+{\sin }^2\left(2\pi x_n\right)\right]\hfill \end{array}\right\}\\ +{\displaystyle {\displaystyle \sum }_{i=1}^n}u\left(x_i,5,100,4\right)\end{array}$	30	[−50, 50]	0