Table 1.

The 18 classical benchmark test functions

Function equation	Domain	fmin
$f_1(x)={\sum }_{i=1}^D{x}_i^2$	[− 100, 100]D	0
$f_2(x)={\sum }_{i=1}^D|x_i|+{\prod }_{i=1}^D|x_i|$	[− 10, 10]D	0
$f_3(x)={\text{max}}_i\left\{|x_i|,1\le x_i\le D\right\}$	[− 100, 100]D	0
$f_4(x)={\sum }_{i=1}^D\left[100{(x_{i+1}-x_i^2)}^2+{(x_i-1)}^2\right]$	[− 30, 30]D	0
$f_5(x)={\sum }_{i=1}^{D}i{x}_i^4+\text{random}[0,1)$	[− 1.28, 1.28]D	0
$f_6(x)={\sum }_{i=1}^{D}i{x}_i^2$	[− 10, 10]D	0
$f_7(x)={\sum }_{i=1}^D{\left|x_i\right|}^{(i+1)}$	[− 1, 1]D	0
$f_8(x)={\sum }_{i=1}^D{\left({10}^6\right)}^{(i-1)/(D-1)}{x}_i^2$	[− 100, 100]D	0
$f_9(x)={\sum }_{i=1}^D\left[x_i^2-10cos(2\pi x_i)+10\right]$	[− 5.12, 5.12]D	0
$f_{10}(x)=-20exp\left(-0.2\sqrt{\frac{1}{D}{\sum }_{i=1}^D{x}_i^2}\right)-exp\left(\frac{1}{D}{\sum }_{i=1}^{D}cos(2\pi x_i)\right)+20+e$	[− 32, 32]D	0
$f_{11}(x)=\frac{1}{4000}{\sum }_{i=1}^D{x}_i^2-{\prod }_{i=1}^{D}cos\left(\frac{x_i}{\sqrt{i}}\right)+1$	[− 600, 600]D	0
$f_{12}(x)={\sum }_{i=1}^D\left|x_i·sin(x_i)+0.1·x_i\right|$	[− 10, 10]D	0
$f_{13}(x)={\text{sin}}^2(\pi x_1)+{\sum }_{i=1}^{D-1}\left[x_i^2·\left(1+10{sin}^2(\pi x_1)\right)+{(x_i-1)}^2·{sin}^2\left(2\pi x_i\right)\right]$	[− 10, 10]D	0
$f_{14}(x)=1-cos\left(2\pi \sqrt{{\sum }_{i=1}^D{x}_i^2}\right)+0.1\sqrt{{\sum }_{i=1}^D{x}_i^2}$	[− 100, 100]D	0
$f_{15}(x)=0.1\left({sin}^2(3\pi x_1)+{\sum }_{i=1}^{D-1}{(x_i-1)}^2\left(1+{sin}^2\left(3\pi x_{i+1}\right)\right)+{(x_D-1)}^2\left(1+{sin}^2\left(2\pi x_D\right)\right)\right)$	[− 5, 5]D	0
$f_{16}(x)={\sum }_{i=1}^D\left(0.2x_i^2+0.1x_i^2·sin\left(2x_i\right)\right)$	[− 10, 10]D	0
$f_{17}(x)={\sum }_{i=1}^{D-1}{\left(x_i^2+2x_{i+1}^2\right)}^{0.25}·\left({\left(sin50{(x_i^2+x_{i+1}^2)}^{0.1}\right)}^2+1\right)$	[− 10, 10]D	0
$f_{18}(x)={\sum }_{i=1}^D{x}_i^6·\left(2+sin\frac{1}{x_i}\right)$	[− 1, 1]D	0