Table 1.

Details of 23 benchmark functions.

No.	Function	Dimension	Range	f min
F1	f 1(x)=∑i=1nxi2f2(x)=∑i=1n|xi|+∏i=1n|xi|	30	[−100,100]	0
F2	f 2(x)=∑fminn|xi|+∏i=1n|xi|	30	[−10,10]	0
F3	f 3(x)=∑i=1n(∑j−1ixj)2	30	[−100,100]	0
F4	$f_4\left(x\right)=\underset{i}{\min }\left\{\left|x_i\right|,1\le i\le n\right\}$	30	[−100,100]	0
F5	f 5(x)=∑i=1n−1100(xi+1 − xi2)2+(xi − 1)2	30	[−30,30]	0
F6	f 6(x)=∑i=1n([xi+0.5])2	30	[−100,100]	0
F7	f 7(x)=∑i=1nixi4+random[0,1)	30	[−1.28,1.28]	0
F8	$f_8\left(x\right)={\displaystyle {\sum }_{i=1}^n-x_i\sin \left(\sqrt{\left|x_i\right|}\right)}$	30	[−500,500]	$\begin{array}{c}-418.9829\\ \times \dim \end{array}$
F9	f 9(x)=∑i=1n[xi2 − 10  cos(2πxi)+10]	30	[−5.12,5.12]	0
F10	$f_{10}\left(x\right)=-20\exp \left(-0.2\sqrt{\frac{1}{n}{\displaystyle {\sum }_{i=1}^n{x}_i^2}}\right)-\exp \left(\left(1/n\right){\displaystyle {\sum }_{i=1}^n\cos \left(2\pi x_i\right)}\right)+20+e$	30	[−32,32]	0
F11	$f_{11}\left(x\right)=1/4000{\displaystyle {\sum }_{i=1}^n}{\displaystyle \sum x_i^2-{\displaystyle {\prod }_{i=1}^n\cos \left(x_i/\sqrt{i}\right)+1}}$	30	[−600,600]	0
F12	$\begin{array}{c}{f}_{12}\left(x\right)=\pi /n\left\{{\displaystyle {\sum }_{i=1}^{n-1}{\left(y_i-1\right)}^2\left[1+10{\sin }^2\left(\pi y_{i+1}\right)\right]+{\left(y_n-1\right)}^2}\right\}\\ +{\displaystyle {\sum }_{i=1}^{n}u\left(x_i,\mathrm{10,100,4}\right)}+\pi /n10\sin \left(\pi y_1\right)\\ y_i=1+x_i+\left(1/4\right)u\left(x_i,a,k,m\right)=\left\{\begin{array}{cc}k{\left(x_i-a\right)}^m\hfill & x_i>a\hfill \\ 0\hfill & -a<x_i<a\hfill \\ k{\left(-x_i-a\right)}^m\hfill & x_i<-a\hfill \end{array}\right.\end{array}$	30	[−50,50]	0
F13	$\begin{array}{c}{f}_{13}\left(x\right)=0.1\left\{{\displaystyle {\sum }_{i=1}^n{\left(x_i-1\right)}^2\left[1+{\sin }^2\left(3\pi x_i+1\right)\right]+{\left(x_n-1\right)}^2\left[1+{\sin }^2\left(2\pi x_n\right)\right]}\right\}\\ +0.1{\sin }^2\left(3\pi x_1\right)+{\displaystyle {\sum }_{i=1}^{n}u\left(x_i,\mathrm{5,100,4}\right)}\end{array}$	30	[−50,50]	0
F14	f 14(x)=((1/500)+∑j=1251/j+∑i=12(xi − aij)6)−1	2	[−65,65]	1
F15	f 15(x)=∑i=111[ai − x1(bi2+bix2)/bi2+bix3+x4]2	4	[−5,5]	0.00030
F16	f 16(x)=4x12 − 2.1x14+1/3x16+x1x2 − 4x22+4x24	2	[−5,5]	−1.0316
F17	f 17(x)=(x2 − 5.1/4π2x12+5/πx1 − 6)2+10(1 − (1/8π))cos  x1+10	2	[−5,5]	0.398
F18	$\begin{array}{c}{f}_{18}\left(x\right)=\left[1+{\left(x_1+x_2+1\right)}^2\left(19-14x_1+3x_1^2-14x_2+6x_1{x}_2+3x_2^2\right)\right]\times \\ \left[30+{\left(2x_1-3x_2\right)}^2\times \left(18-32x_1+12x_1^2+48x_2-36x_1{x}_2+27x_2^2\right)\right]\end{array}$	2	[−2,2]	3
F19	f 19(x)=−∑i=14ciexp[−∑j=13aij(xj − pij)2]	3	[1, 3]	−3.86
F20	f 20(x)=−∑i=14ciexp[−∑j=16aij(xj − pij)2]	6	[0,1]	−3.32
F21	f 21(x)=−∑i=15[(X − ai)(X − ai)T+ci]−1	4	[0,10]	−10.1532
F22	f 22(x)=−∑i=17[(X − ai)(X − ai)T+ci]−1	4	[0,10]	−10.4028
F23	f 23(x)=−∑i=110[(X − ai)(X − ai)T+ci]−1	4	[0,10]	−10.5363