Table 1 12 benchmark functions

\(F_1(x)=\sum _{i=1}^{n}x_i^2\)

30

[\(-100,100\)]

0

\(F_2(x)=\sum _{i=1}^{n}\mid x_i\mid +\prod _{i=1}^{n}\mid x_i\mid\)

30

[\(-10,10\)]

0

\(F_3(x)=(\frac{1}{500}+\sum _{j=1}^{25}\frac{1}{j+\sum _{i=1}^{2}(x_i-a_{ij})^6})^{-1}\)

2

[\(-65,65\)]

1

\(F_4(x)=max_i\{\mid x_i\mid ,1\le i\le n\}\)

30

[\(-100,100\)]

0

\(F_5(x)=\sum _{i=1}^{n-1}[100(x_{i+1}-x_i^2)^2+(x_i-1)^2]\)

30

[\(-30,30\)]

0

\(F_6(x)=\sum _{i=1}^{n}([x_i+0.5])^2\)

30

[\(-100,100\)]

0

\(F_7(x)=\sum _{i=1}^{n}ix_i^4+\mathrm{{random}}[0,1)\)

30

[\(-1.28,1.28\)]

0

\(F_8(x)=\sum _{i=1}^{n}[x_i^2-10\cos (2\pi x_i)+10]\)

30

[\(-5.12,5.12\)]

0

\(F_9(x)=\frac{1}{4000}\sum _{i=1}^{n}x_i^2-\prod _{i=1}^{n}\cos (\frac{x_i}{\sqrt{i}})+1\)  

30

[\(-600,600\)]

0

\(F_{10}(x)=\frac{\pi }{n}\{10\sin (\pi y_1)+\sum _{i=1}^{n-1}(y_i-1)^2[1+10\sin _2(\pi y_{i+1})]+(y_n-1)^2\}+\sum _{i=1}^{n}u(x_i,10,100,4)\)

\(y_i=1+\frac{x_i+1}{4}u(x_i,a,k,m)= {\left\{ \begin{array}{ll} k(x_i-a)^m \qquad \qquad x_i>a \\ 0 \qquad \qquad \qquad -a<x_i<a \\ k(-x_i-a)^m \qquad \quad x_i<-a \end{array}\right. }\)

30

[\(-50,50\)]

0

\(F_{11}(x)=(x_2-\frac{5.1}{4\pi ^2}x_1^2+\frac{5}{\pi }x_1-6)^2+10(1-\frac{1}{8\pi })\cos x_1+10\)

2

[\(-5,5\)]

0.398

\(F_{12}(x)=-\sum _{i=1}^{7}[(X-a_i)(X-a_i)^T+c_i]^{-1}\)

4

[0,10]

-10.4028