Table 4.

Multimodal benchmark functions.

Function	Dim	Range	fmin
$f_8(x)={\sum }_{i=1^n-x_i\sin (\sqrt{|x_i|})}$	50	[−500,500]	−2094.9145
$f_9(x)={\sum }_{i=1^n[x_i^2-10\cos (2\pi x_i)+10]}$	50	[−5.12, 5.12]	0
$f_{10}(x)=-20\exp \left(-0.2\sqrt{\frac{1}{n}{\sum }_{i=1^n}{x}_i^2}\right)-\exp \left(\frac{1}{n}{\sum }_{i=1^n}\cos \left(2\pi x_i\right)\right)+20+e$	50	[−32,32]	0
$f_{11}(x)=\frac{1}{4000}{\sum }_{i=1^n{x}_i^2-\prod _{i=1}^n\cos \left(\frac{x_i}{\sqrt{i}}\right)}+1$	50	[−600,600]	0
$\begin{array}{c}{f}_{12}(x)=\frac{\pi }{n}\left\{10\sin \left(\pi y_1\right)\right)+{\sum }_{i=1}^{n-1}{(y_i-1)}^2\left[1+10{\sin }^2\left(\pi y_{i+1}\right)\right]+{(y_n-1)}^2+{\sum }_{i=1}^{n}u\left(x_i,10,100,4\right)\hfill \\ y_i=1+\frac{x_i+1}{4}u\left(x_i,a,k,m\right)\left\{\begin{array}{c}k\left(x_i-a\right)x_i>a\hfill \\ 0-a<x_i<a\hfill \\ k\left(x_i-a\right)x_i<-a\hfill \end{array}\right)\hfill \end{array}$	50	[−50,50]	0
$\begin{array}{c}{f}_{13}(x)=0.1\left\{{\sin }^2\left(3\pi x_1\right)\right)+{\sum }_{\mathrm{i}=1^{\mathrm{n}}{(x_{\mathrm{i}}-1])}^2}\left[1+{\sin }^2\left(3\pi x_{\mathrm{i}}+1\right)\right]+\hfill \\ \left({\left(x_{\mathrm{n}}-1\right)}^2\left[1+{\sin }^2\left(2\pi x_{\mathrm{n}}\right)\right]\right\}+{\sum }_{\mathrm{i}=1^{\mathrm{n}}u(x_{\mathrm{i}},5,100,4)}\hfill \end{array}$	50	[−50,50]	0