Table 1 Multimodal benchmark functions

\( {F}_4(X)={\sum}_{i=1}^n{\left({x}_i-1/i\right)}^2 \)

[− 10, 10]

\( {F}_{10}(X)={\sum}_{i=1}^n-{x}_i\sin \left(\sqrt{\left|{x}_i\right|}\right) \)

[− 500,50]

\( {\displaystyle \begin{array}{l}{F}_{12}(X)=\left(\pi /n\right)\Big\{10{\sin}^2\left(\pi {y}_1\right)+{\sum}_{i=1}^{D-1}{\left({y}_i-1\right)}^2\left[1+{\sin}^2\left(\pi {y}_{i+1}\right)\right]+\\ {}\kern3.899998em {\left({y}_n-1\right)}^2\Big\}+{\sum}_{i=1}^Du\left({x}_i,10,100,4\right),\end{array}} \)

[− 50,-50]

\( {F}_{16}(X)=\frac{1}{n}{\sum}_{i=1}^n\left({x}_i^4-16{x}_i^2+5{x}_i\right) \)

[− 5,5]