Table 4 Description of multimodal benchmark functions.

_F8

\(f(x)={\sum }_{i=1}^{n}-{x}_{i}\mathit{sin}(\sqrt{\left|{x}_{i}\right|})\)

[− 500, 500]

_F9

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

[− 5.12, 5.12]

_F10

\(f(x)=-20exp(-0.2\sqrt{\frac{1}{n}\sum_{i=0}^{n}{x}_{i}^{2})}- exp(\frac{1}{n}\sum_{i=1}^{n}\mathit{cos}(2\pi {x}_{i}))+20+e\)

[− 30, 30]

_F11

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

[− 600, 600]

_F12

\(f(x)=\frac{\pi }{n}\left\{10\mathit{sin}(\pi {y}_{1})+\sum_{i=1}^{n-1}{({y}_{i}-1)}^{2}[1+10{sin}^{2}(\pi {y}_{i+1})]{({y}_{n}-1)}^{2}\right\}+\sum_{i=1}^{n}u\left({x}_{i}.\text{10.100.4}\right). where {y}_{i}=1+\frac{{x}_{i}+1}{4}\)

\(u({x}_{i}.a.k.m)=\left\{\begin{array}{c}k{({x}_{i}-a)}^{m}{x}_{i}>a\\ 0 -a<{x}_{i}<a\\ k{({-x}_{i}-a)}^{m}{x}_{i}<-a\end{array}\right.\)

[− 50, 50]

_F13

\(f(x)=0.1{sin}^{2}(3\pi {x}_{1})+{\sum }_{i=1}^{n}{({x}_{i}-1)}^{2}[1+{sin}^{2}(3\pi {x}_{i}+1)]+{({x}_{n}-1)}^{2}[1+{sin}^{2}(2\pi {x}_{n})]+\sum_{i=1}^{n}u({x}_{i}.\text{5.100.4})\)

[− 50, 50]