Table 4 General information of the benchmark functions F1–F12.

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

30

\([-100,100]\)

0

\(F_2(x) = \sum \limits _{i = 1}^{n}\left|x_i\right|+ \prod \limits _{i = 1}^{n} \left|x_i\right|\)

30

\([-10,10]\)

0

\(F_3(x) = \sum \limits _{i = 1}^{n-1} \left[ 100(x_{i+1}- x^2_i)^2 + (x_i - 1)^2 \right]\)

30

\([-30,30]\)

0

\(F_4(x) = \sum \limits _{i = 1}^{n} \left[ x_i^2 - 10\cos (2\pi x_i) + 10\right]\)

30

\(\left[ -5.12,5.12\right]\)

0

\(F_5(x) = -20\exp \left( -0.2\sqrt{\frac{1}{n} \sum \limits _{i = 1}^{n}x_i^2} \right) -\exp \left( \frac{1}{n} \sum \limits _{i = 1}^{n} \cos (2\pi x_i)\right) +20 + e\)

30

\(\left[ -32,32\right]\)

0

\(+ \sum \limits _{i = 1}^{n}u(x_i,10,100,4)\)

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

\(F_6(x) = \frac{\pi }{D} \left\{ 10\sin ^2 (\pi y_1) + \sum \limits _{i = 1}^{n-1} (y_i - 1)^2 \left[ 1 + 10\sin ^2 (\pi y_{i+1})\right] +(y_n - 1)^2 \right\}\)

30

\(\left[ -50,50\right]\)

0

\(F_7(x) = \left( \frac{1}{500} + \sum \limits _{j = 1}^{25}\frac{1}{j+\sum \limits _{i = 1}^{2}(x_i-a_{ij})^6} \right) ^{-1}\)

2

\([-65.536,65.536]\)

1

\(F_8(x) = \left[ 1+(x_1+x_2+1)^2(19-14x_1+3x_1^2-14x_2+6x_1x_2+3x_2^2)\right]\)

2

\({[-30,30]}\)

0.3983

\(\times \left[ 30+(2x_1-3x_2)^2\times (18-32x_1+12x_1^2+48x_2-36x_1x_2+27x_2^2)\right]\)

\(F_9(x) = -\sum \limits _{i = 1}^{4}c_i \exp \left( -\sum \limits _{j = 1}^{3}a_{ij}(x_j-p_{ij})^2\right)\)

3

[0, 1]

− 3.86

\(F_{10}(x) = -\sum \limits _{i = 1}^{4}c_i \exp \left( -\sum \limits _{j = 1}^{6}a_{ij}(x_j-p_{ij})^2\right)\)

6

[0, 1]

− 3.32

\(F_{11}(x) = -\sum \limits _{i = 1}^{4}\left( -\sum \limits _{j = 1}^{5}(x_j-C_{ji})^2 + \beta _i\right) ^{-1}\)

4

[0, 10]

− 10

\(F_{12}(x) = -\sum \limits _{i = 1}^{4}\left( -\sum \limits _{j = 1}^{10}(x_j-C_{ji})^2 + \beta _i\right) ^{-1}\)

4

[0, 10]

− 10