Table 1 Benchmark functions, D Dimension, C Characteristic, U Unimodal, M Multimodal, S Separable, N Non-separable

Step

30

US

[−100,100]

$f\left(x\right)={\displaystyle \sum }_{i=1}^D{\left(x_i+0.5\right)}^2$

f _min =0

Sphere

30

US

[−100,100]

$f\left(x\right)={\displaystyle \sum }_{i=1}^D{x}_i^2$

f _min =0

SumSquares

30

US

[−100,100]

$f\left(x\right)={\displaystyle \sum }_{i=1}^{D}i{x}_i^2$

f _min =0

Quartic

30

US

[−1.28,1.28]

$f\left(x\right)={\displaystyle \sum }_{i=1}^{D}i{x}_i^4+\mathit{random}\left(0,1\right)$

f _min =0

Zakharov

10

UN

[−5,10]

$f\left(x\right)={\displaystyle \sum }_{i=1}^D{x}_i^2+{\left({\displaystyle \sum }_{i=1}^{D}0.5i{x}_i\right)}^2+{\left({\displaystyle \sum }_{i=1}^{D}0.5i{x}_i\right)}^4$

f _min =0

Schwefel 1.2

30

UN

[−100,100]

$f\left(x\right)={\displaystyle \sum }_{i=1}^D{\left({\displaystyle \sum }_{j=1}^i{x}_j\right)}^2$

f _min =0

Schwefel 2.22

30

UN

[−10,10]

$f\left(x\right)={\displaystyle \sum }_{i=1}^D\left|x_i\right|+{\displaystyle \prod }_{i=1}^D\left|x_i\right|$

f _min =0

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

Bohachevsky1

2

MS

[−100,100]

$\begin{array}{c}f\left(x\right)=x_1^2+2x_2^2-0.3cos\left(3\pi x_1\right)-0.4\cos \left(4\pi x_2\right)+0.7\end{array}$

f _min =0

Bohachevsky2

2

MS

[−100,100]

$\begin{array}{c}f\left(x\right)=x_1^2+2x_2^2-0.3cos\left(3\pi x_1\right)*cos\left(4\pi x_2\right)+0.3\end{array}$

f _min =0

Bohachevsky3

2

MS

[−100,100]

$\begin{array}{c}f\left(x\right)=x_1^2+2x_2^2-0.3cos(\left(3\pi x_1\right)+\left(4\pi x_2\right))+0.3\end{array}$

f _min =0

Booth

2

MS

[−10,10]

f(x) = (x₁ + 2x₂ − 7)² + (2x₁ + x₂ − 5)²

f _min =0

Rastrigin

30

MS

[−5.12,5.12]

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

f _min =0

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

Six hump camel back

2

MN

[−5,5]

$f\left(x\right)=4x_1^2-2.1x_1^4+\frac{1}{3}{x}_1^6+x_1{x}_2-4x_2^2+4x_2^4$

f _min = − 1.03163

Griewank

30

MN

[−600,600]

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

f _min =0

Ackley

30

MN

[−32,32]

$\begin{array}{c}f\left(x\right)=-20exp\left(-0.2\sqrt{\frac{1}{D}}{\displaystyle \sum }_{i=1}^D{x}_i^2\right)-exp\left(\frac{1}{n}{\displaystyle \sum }_{i=1}^{D}cos\left(2*\mathit{pi}*x_i\right)\right)+20+e\end{array}$

f _min =0

Multimod

30

MS

[−10,10]

$f\left(x\right)={\displaystyle \sum }_{i=1}^{`D}\left|x_i\right|{\displaystyle \prod }_{i=1}^D\left|x_i\right|$

f _min =0

Noncontinuous rastrigin

30

MS

[−5.12,5.12]

$\begin{array}{l}f\left(x\right)={\displaystyle \sum }_{i=1}^D\left[y_i^2-10cos\left(2\pi y_i\right)+10\right]\mathrm{Where}{y}_i=\left\{\begin{array}{c}\hfill x_i\left|x_i\right|<0.5\hfill \\ \hfill \frac{\mathit{round}\left(2x_i\right)}{2}\left|x_i\right|\ge 0.5\hfill \end{array}\right.\end{array}$

f _min =0

Weierstrass

30

MS

[−0.5, 0.5]

$\begin{array}{l}f\left(x\right)={\displaystyle \sum }_{i=1}^D\left({\displaystyle \sum }_{k=0}^{\mathit{\text{kmax}}}\left[a^k\cos \left(2\pi b^k\left(x_i+0.5\right)\right)\right]\right)-D{\displaystyle \sum }_{k=0}^{\mathit{kmax}}\left[a^k\cos \left(2\pi b^k\left(x_i+0.5\right)\right)\right],\\ \mathrm{where}a=0.5,b=3,\mathit{kmax}=20\end{array}$

f _min =0