Table 2 Details of selected benchmark functions

Unimodal

\({F}_{1}(x)={\sum }_{i=1}^{D}{{x}_{i}}^{2}\)

[− 100,100]

\({F}_{2}\left(x\right)={\left({x}_{1}-1\right)}^{2}+{\sum }_{i=2}^{D}i{\left(2{{x}_{i}}^{2}-{x}_{i-1}\right)}^{2}\)

[− 10,10]

\({F}_{3}(x)=\sum_{i=1}^{D}{\left(\sum_{j=1}^{i}{x}_{j}\right)}^{2}\)

[− 100,100]

Multimodal

\({F}_{4}(x)=\sum_{i=1}^{D-1}{\left({w}_{i}-1\right)}^{2}\left[1+10{\mathrm{sin}}^{2}\left(\pi {w}_{i}+1\right)\right]+{\left({w}_{D}-1\right)}^{2}\left[1+10{\mathrm{sin}}^{2}\left(2\pi {w}_{D}+\right)\right]\)

[− 10,10]

\({F}_{5}\left(x\right)=\frac{1}{4000}\sum_{i=1}^{D}{{x}_{i}}^{2}-\prod_{i=1}^{n}\mathrm{cos}\left(\frac{{x}_{i}}{{i}^{0.5}}\right)+1\)

[− 600,600]

\({F}_{6}\left(x\right)=0.1\left\{{\mathrm{sin}}^{2}\left(3\pi {x}_{1}\right)+\sum_{i=1}^{D}{\left({x}_{i}-1\right)}^{2}\left[1+{\mathrm{sin}}^{2}\left(3\pi {x}_{1}\right)\right]+{\left({x}_{n}-1\right)}^{2}\left[1+{\mathrm{sin}}^{2}\left(2\pi {x}_{n}\right)\right]\right\}+\sum_{i=1}^{D}g\left({x}_{i},\mathrm{5,100,4}\right)\)

\(g\left( {x_{i} ,5,100,4} \right) = \left\{ {\begin{array}{*{20}l} {b\left( {x_{i} - a} \right)^{c} } \hfill & {x_{i} > a} \hfill \\ 0 \hfill & { - a < x_{i} > a} \hfill \\ {b\left( { - x_{i} - a} \right)^{c} } \hfill & {x_{i} < - a} \hfill \\ \end{array} } \right.\)

[− 50,50]

\({F}_{7}\left(x\right)=\sum_{i=1}^{D}{x}_{i}\mathrm{sin}\left({x}_{i}\right)+0.1{x}_{i}\)

[− 10,10]

\({F}_{8}\left(x\right)=\sum_{i=1}^{D}1-\mathrm{cos}\left(2\pi \sqrt{\sum_{i=1}^{D}{{x}_{i}}^{2}}\right)+0.1\sqrt{\sum_{i=1}^{D}{{x}_{i}}^{2}}\)

[− 100,100]

\({F}_{9}\left(x\right)={\left({{x}_{1}}^{6}+{{x}_{2}}^{4}-17\right)}^{2}+{\left(2{x}_{1}+{x}_{2}-4\right)}^{2}\)

[− 500,500]

\({F}_{10}\left(x\right)=-\frac{1}{30}\mathrm{exp}\left(1-\frac{\sqrt{{{x}_{1}}^{2}+{{x}_{2}}^{2}}}{\pi }\right) {\mathrm{cos}}^{2}\left({x}_{1}\right){\mathrm{ cos}}^{2}\left({x}_{2}\right)\)

[− 10,10]