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# Table 1 Benchmark functions,DDimension,CCharacteristic,UUnimodal,MMultimodal,SSeparable,NNon-separable

No. Function D C Range Formulation Value
f 1 Step 30 US [−100,100] $f\left(x\right)=\sum _{i=1}^{D}{\left({x}_{i}+0.5\right)}^{2}$ f min =0
f 2 Sphere 30 US [−100,100] $f\left(x\right)=\sum _{i=1}^{D}{x}_{i}^{2}$ f min =0
f 3 SumSquares 30 US [−100,100] $f\left(x\right)=\sum _{i=1}^{D}i{x}_{i}^{2}$ f min =0
f 4 Quartic 30 US [−1.28,1.28] $f\left(x\right)=\sum _{i=1}^{D}i{x}_{i}^{4}+\mathit{random}\left(0,1\right)$ f min =0
f 5 Zakharov 10 UN [−5,10] $f\left(x\right)=\sum _{i=1}^{D}{x}_{i}^{2}+{\left(\sum _{i=1}^{D}0.5i{x}_{i}\right)}^{2}+{\left(\sum _{i=1}^{D}0.5i{x}_{i}\right)}^{4}$ f min =0
f 6 Schwefel 1.2 30 UN [−100,100] $f\left(x\right)=\sum _{i=1}^{D}{\left(\sum _{j=1}^{i}{x}_{j}\right)}^{2}$ f min =0
f 7 Schwefel 2.22 30 UN [−10,10] $f\left(x\right)=\sum _{i=1}^{D}\left|{x}_{i}\right|+\prod _{i=1}^{D}\left|{x}_{i}\right|$ f min =0
f 8 Schwefel 2.21 30   [−100,100] $f\left(x\right)=\begin{array}{c}\hfill \mathit{max}\hfill \\ \hfill i\hfill \end{array}\left\{\left|{x}_{i}\right|,1\le i\le D\right\}$ f min =0
f 9 Bohachevsky1 2 MS [−100,100] $\begin{array}{c}f\left(x\right)={x}_{1}^{2}+2{x}_{2}^{2}-0.3cos\left(3\pi {x}_{1}\right)-0.4\mathrm{cos}\left(4\pi {x}_{2}\right)+0.7\end{array}$ f min =0
f 10 Bohachevsky2 2 MS [−100,100] $\begin{array}{c}f\left(x\right)={x}_{1}^{2}+2{x}_{2}^{2}-0.3cos\left(3\pi {x}_{1}\right)*cos\left(4\pi {x}_{2}\right)+0.3\end{array}$ f min =0
f 11 Bohachevsky3 2 MS [−100,100] $\begin{array}{c}f\left(x\right)={x}_{1}^{2}+2{x}_{2}^{2}-0.3cos\left(\left(3\pi {x}_{1}\right)+\left(4\pi {x}_{2}\right)\right)+0.3\end{array}$ f min =0
f 12 Booth 2 MS [−10,10] f(x) = (x1 + 2x2 − 7)2 + (2x1 + x2 − 5)2 f min =0
f 13 Rastrigin 30 MS [−5.12,5.12] $f\left(x\right)=\sum _{i=1}^{D}\left[{x}_{i}^{2}-10cos\left(2\pi {x}_{i}\right)+10\right]$ f min =0
f 14 Schaffer 2 MN [−100,100] $f\left(x\right)=\frac{\mathit{si}{n}^{2}\left(\sqrt{{x}_{1}^{2}+{x}_{2}^{2}}\right)-0.5}{{\left(1+0.001\left({x}_{1}^{2}+{x}_{2}^{2}\right)\right)}^{2}}$ f min =0
f 15 Six hump camel back 2 MN [−5,5] $f\left(x\right)=4{x}_{1}^{2}-2.1{x}_{1}^{4}+\frac{1}{3}{x}_{1}^{6}+{x}_{1}{x}_{2}-4{x}_{2}^{2}+4{x}_{2}^{4}$ f min = − 1.03163
f 16 Griewank 30 MN [−600,600] $f\left(x\right)=\frac{1}{4000}\sum _{i=1}^{D}{x}_{i}^{2}-\prod _{i=1}^{D}\mathrm{cos}\left(\frac{{x}_{i}}{\sqrt{i}}\right)+1$ f min =0
f 17 Ackley 30 MN [−32,32] $\begin{array}{c}f\left(x\right)=-20exp\left(-0.2\sqrt{\frac{1}{D}}\sum _{i=1}^{D}{x}_{i}^{2}\right)-exp\left(\frac{1}{n}\sum _{i=1}^{D}cos\left(2*\mathit{pi}*{x}_{i}\right)\right)+20+e\end{array}$ f min =0
f 18 Multimod 30 MS [−10,10] $f\left(x\right)=\sum _{i=1}^{`D}\left|{x}_{i}\right|\prod _{i=1}^{D}\left|{x}_{i}\right|$ f min =0
f 19 Noncontinuous rastrigin 30 MS [−5.12,5.12] $\begin{array}{l}f\left(x\right)=\sum _{i=1}^{D}\left[{y}_{i}^{2}-10cos\left(2\pi {y}_{i}\right)+10\right]\phantom{\rule{0.25em}{0ex}}\mathrm{Where}\phantom{\rule{0.25em}{0ex}}{y}_{i}=\left\{\begin{array}{c}\hfill {x}_{i}\phantom{\rule{0.25em}{0ex}}\left|{x}_{i}\right|<0.5\hfill \\ \hfill \frac{\mathit{round}\left(2{x}_{i}\right)}{2}\phantom{\rule{0.25em}{0ex}}\left|{x}_{i}\right|\ge 0.5\hfill \end{array}\phantom{\rule{0.25em}{0ex}}\right\\end{array}$ f min =0
f 20 Weierstrass 30 MS [−0.5, 0.5] $\begin{array}{l}f\left(x\right)=\sum _{i=1}^{D}\left(\sum _{k=0}^{\mathit{\text{kmax}}}\left[{a}^{k}\mathrm{cos}\left(2\pi {b}^{k}\left({x}_{i}+0.5\right)\right)\right]\right)-D\sum _{k=0}^{\mathit{kmax}}\left[{a}^{k}\mathrm{cos}\left(2\pi {b}^{k}\left({x}_{i}+0.5\right)\right)\right],\\ \mathrm{where}\phantom{\rule{0.25em}{0ex}}a=0.5,b=3,\mathit{kmax}=20\end{array}$ f min =0