Table 4 Multimodal benchmark functions

\( {f}_8(x)={\sum}_{i=1}^n-{x}_i\sin \left(\sqrt{\mid {x}_i\mid}\right) \)

[− 500,500]

\( {f}_9(x)={\sum}_{i=1}^n\left[{x_i}^2-10\cos \left(2\pi {x}_i\right)+10\right] \)

[− 5.12,5.12]

\( {f}_{10}(x)=-20\exp \left(-0.2\sqrt{\frac{1}{n}{\sum}_{i=1}^n{x_i}^2}\right)-\exp \left(\frac{1}{n}{\sum}_{i=1}^n\cos \left(2\pi {x}_i\right)\right)+20+e\kern0.50em \)

[− 32,32]

\( {f}_{11}(x)=\frac{1}{4000}{\sum}_{i=1}^n{x_i}^2-{\prod}_{i=1}^n\cos \left(\frac{x_i}{\sqrt{i}}\right)+1 \)

[− 600,600]

\( {\displaystyle \begin{array}{l}{f}_{12}(x)=\frac{\pi }{n}\left\{10\sin \left(\pi {y}_1\right)+{\sum}_{i=1}^{n-1}{\left({y}_i-1\right)}^2\left[1+10{\sin}^2\left(\pi {y}_{i+1}\right)\right]+{\left({y}_n-1\right)}^2\right\}+{\sum}_{i=1}^nu\left({x}_i,10,100,4\right)\\ {}{y}_i=1+\frac{x_i+1}{4}\\ {}u\left({x}_i,a,k,m\right)\left\{\begin{array}{l}k{\left({x}_i-a\right)}^m\kern0.75em {x}_i>a\\ {}0\kern3.75em -a<{x}_i<a\\ {}k{\left(-{x}_i-a\right)}^m\kern0.75em {x}_i<-a\end{array}\right.\\ {}\end{array}} \)

[− 50,50]

\( {\displaystyle \begin{array}{l}{f}_{13}(x)=0.1\left\{{\sin}^2\left(3\uppi {x}_1\right)+{\sum}_{\mathrm{i}=1}^{\mathrm{n}}\left({x}_{\mathrm{i}}\hbox{-} 1\right]\Big){}^2\left[1+{\sin}^2\left(3\uppi {x}_{\mathrm{i}}+1\right)\right]+{\left({x}_{\mathrm{n}}-1\right)}^2\left[1+{\sin}^2\left(2\uppi {x}_{\mathrm{n}}\right)\right]\right\}\\ {}+{\sum}_{\mathrm{i}=1}^{\mathrm{n}}u\left({x}_{\mathrm{i}},5,100,4\right)\end{array}} \)

[− 50,50]