Table 5.

Multimodal benchmark functions.

Function	Dim	Range	f min
$f_{\mathrm{8}}\left(x\right)={\displaystyle \sum }_{i=\mathrm{1}}^n-x_i\sin \left(\sqrt{\left|x_i\right|}\right)$	30	[-500,500]	-418.9829∗5
$f_{\mathrm{9}}\left(x\right)={\displaystyle \sum }_{i=\mathrm{1}}^n\left[{x_i}^{\mathrm{2}}-\mathrm{10}\cos \left(\mathrm{2}\pi x_i\right)+\mathrm{10}\right]$	30	[-5.12,5.12]	0
$f_{\mathrm{10}}\left(x\right)=-\mathrm{20}\exp \left(-\mathrm{0.2}\sqrt{\frac{\mathrm{1}}{n}{\displaystyle \sum }_{i=\mathrm{1}}^n{x_i}^{\mathrm{2}}}\right)-\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{\mathrm{1}}{n}{\displaystyle \sum }_{i=\mathrm{1}}^n\cos \left(\mathrm{2}\pi x_i\right)\right)+\mathrm{20}+e$	30	[-32,32]	0
$f_{\mathrm{11}}\left(x\right)=\frac{\mathrm{1}}{\mathrm{4000}}{\displaystyle \sum }_{i=\mathrm{1}}^n{x_i}^{\mathrm{2}}-{\displaystyle \prod }_{i=\mathrm{1}}^n\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{x_i}{\sqrt{i}}\right)+\mathrm{1}$	30	[-600,600]	0
$f_{\mathrm{12}}\left(x\right)=\frac{\pi }{n}\left\{\mathrm{10}\sin \left(\pi y_{\mathrm{1}}\right)+{\displaystyle \sum }_{i=\mathrm{1}}^{n-\mathrm{1}}{\left(y_i-\mathrm{1}\right)}^{\mathrm{2}}\left[\mathrm{1}+\mathrm{10}\hspace{0.17em}\mathrm{s}\mathrm{i}{\mathrm{n}}^{\mathrm{2}}\left(\pi y_{i+\mathrm{1}}\right)\right]+(y_n-\mathrm{1}{)}^{\mathrm{2}}\right\}$	30	[-50,50]	0
$+{\displaystyle \sum }_{i=\mathrm{1}}^{n}u\left(x_i,\mathrm{10,100,4}\right)$
$y_i=\mathrm{1}+\frac{x_i+\mathrm{1}}{\mathrm{4}}u\left(x_i,a,k,m\right)\left\{\begin{array}{cc}k\left(x_i-a\right)\hfill & x_i>a\hfill \\ \mathrm{0}\hfill & -a<x_i<a\hfill \\ k\left(x_i-a\right)\hfill & x_i<-a\hfill \end{array}\right.$
$f_{\mathrm{13}}\left(x\right)=\mathrm{0.1}\left\{\mathrm{s}\mathrm{i}{\mathrm{n}}^{\mathrm{2}}\left(\mathrm{3}\pi x_{\mathrm{1}}\right)+{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}{(x_{\mathrm{i}}-\mathrm{1}])}^{\mathrm{2}}\left[\mathrm{1}+\mathrm{s}\mathrm{i}{\mathrm{n}}^{\mathrm{2}}\left(\mathrm{3}\pi x_{\mathrm{i}}+\mathrm{1}\right)\right]+{\left(x_{\mathrm{n}}-\mathrm{1}\right)}^{\mathrm{2}}\left[\mathrm{1}+\mathrm{s}\mathrm{i}{\mathrm{n}}^{\mathrm{2}}\left(\mathrm{2}\pi x_{\mathrm{n}}\right)\right]\right\}$	30	[-50,50]	0
$+{\displaystyle \sum }_{\mathrm{i}=\mathrm{1}}^{\mathrm{n}}u(x_{\mathrm{i}},\mathrm{5,100,4})$