Table 1.

Benchmark functions for single-objective problem optimization.

Function	Dimension	Range
F1(x)=∑i=1nxi2	10	[−100, 100]
F2(x)=∑i=1n|xi|+∏i=1n|xi|	10	[−10, 10]
$F3\left(x\right)={\sum }_{i=1}^n{\left({\sum }_{J-1}^{\dot{I}}{X}_J\right)}^2$	10	[−100, 100]
F4(x)=maxi{|xi|, 1 ≤ i ≤ n}	10	[−100, 100]
F5(x)=∑i=1n−1[100(xi+1 − xi2)2+(xi − 1)2]	10	[−30, 30]
F6(x)=∑i=1n([xi+0.5])2	10	[−100, 100]
$F7\left(x\right)={\sum }_{\dot{I}=1}^{n}i{x}_i^4+\text{random}\left[\mathrm{0,1}\right)$	10	[−1.28, 1.28]
$F8\left(x\right)={\sum }_{i=1}^n-x_i\sin \left(\sqrt{\left|x_i\right|}\right)$	10	[−500, 500]
F9(x)=∑i=0n[xi2 − 10 cos(2πxi)+10]	10	[−5.12, 5.12]
$F10\left(x\right)=-20\hspace{0.17em}\hspace{0.17em}\exp \left(-0.2\sqrt{\left(1/n\right){\sum }_{i=1}^n{x}_i^2}\right)-\left(\exp \left(1/n\right){\sum }_{i=1}^n\cos \left(2\pi x_i\right)\right)+20+e$	10	[−32, 32]
$F11\left(x\right)=\left(1/4000\right){\sum }_{i=1}^n{x}_i^2-{\prod }_{i=1}^n\cos \left(\left(x_i/\sqrt{i}\right)+1\right)$	10	[−600, 600]
$\begin{array}{c}F12\left(x\right)=\left(\pi /n\right)\left\{10\sin \left(\pi y_1\right)+{\sum }_{i=1}^{n-1}{\left(y_i-1\right)}^2\left[1+10{\sin }^2\left(\pi y_{i+1}\right)\right]+{\left(y_n-1\right)}^2\right\}+{\sum }_{i=1}^{n}u\left(x_i,\mathrm{10,100,4}\right)\\ y_i=\left(\left(x_i+1\right)/4\right)\\ u\left(x_i,a,k,m\right)=\hspace{0.17em}\hspace{0.17em}\left\{\begin{array}{cc}k{\left(x_i-a\right)}^m& x_i>a\\ 0& -a<x_i<a\\ k{\left(-x_i-a\right)}^m& x_i<-a\end{array}\right\}\end{array}$	10	[−50, 50]
F13(x)=0.1{sin2(3πx1)+∑i=1n(xi − 1)2[1+sin2(3πxi+1)]+(xn − 1)2[1+sin2(2πxn)]}+∑i=1nu(xi, 5,100,4)	10	[−50, 50]
F14(x)=∑i=1nxi2	10	[−5, 5]
$F15\left(x\right)={\sum }_{i=1}^n{\left(\left(x_i^2\right)/{\left(4000\right)}_i\right)}^2-{\prod }_{i=1}^n\cos \left(\left(x_i/\sqrt{i}\right)+1\right)$	10	[−5, 5]