Table 2 Multi-model test functions

1

\(F_{8} (x) = \sum\limits_{i = 1}^{n} { - x_{i} \sin \left( {\sqrt {\left| {x_{i} } \right|} } \right)}\)

30

[− 500,500]

− 418.9829 × Dim

2

\(F_{9} \left( x \right) = - \sum\limits_{I = 1}^{n} {\left[ {x_{i}^{2} - 10\cos \left( {2\pi x_{i} } \right) + 10} \right]}\)

30

[− 5.12,5.12]

0

3

\(F_{10} \left( x \right) = - 20\exp \left( { - 0.2\sqrt {\frac{1}{n}\sum\limits_{i = 1}^{n} {x_{i}^{2} } } } \right) - \exp \left( {\frac{1}{n}\sum\limits_{i = 1}^{n} {\cos \left( {2\pi x_{i} } \right)} } \right) + 20 + e\)

30

[− 32,32]

0

4

\(F_{11} \left( x \right) = \frac{1}{4000}\sum\limits_{i = 1}^{n} {x_{i}^{2} } - \prod\limits_{i = 1}^{n} {\cos \left( {\frac{{x_{i} }}{\sqrt i }} \right) + 1}\)

30

[− 600,600]

0

5

\(\begin{aligned} F_{12} \left( x \right) = & \left| {\frac{\pi }{n}\left\{ {10\sin \left( {\pi y_{1} } \right) + \sum\limits_{i = 1}^{n} {\left( {x_{i} - 1} \right)^{2} \left[ {1 + \sin^{2} \left( {3\pi x_{i} + 1} \right)} \right] + \left( {x_{n} - 1} \right)^{2} \left[ {1 + \sin^{2} \left( {2\pi x_{n} } \right)} \right]} } \right\}} \right. \\ + & \left| { + y_{i} = 1 + \frac{{x_{i} + 1}}{4}u\left( {x_{i} ,a,k,m} \right) = \left\{ {\begin{array}{*{20}c} {\left( {x_{i} - a} \right)^{m} x_{i} } \\ {x_{i} } \\ {\left( { - x_{i} - a} \right)^{m} x_{i} } \\ \end{array} } \right.} \right. \\ \end{aligned}\)

30

[− 50,50]

0

6

\(F_{13} \left( x \right) = 0.1\left\{ {\sin^{2} \left( {3\pi x_{1} } \right) + \sum\limits_{i = 1}^{n} {\left( {x_{i} - 1} \right)^{2} \left[ {1 + \sin^{2} \left( {2\pi x_{i} + 1} \right)} \right]} } \right\} + \sum\limits_{i = 1}^{n} {u\left( {x_{i} ,5,100,4} \right)}\)

30

[− 50,50]

0

7

\(F_{14} \left( x \right) = \left( {\frac{1}{500} + \sum\limits_{j = 1}^{25} {\frac{1}{{j + \sum\nolimits_{i = 1}^{2} {\left( {x_{i} - a_{ij} } \right)^{6} } }}} } \right)^{ - 1}\)

2

[− 65,65]

1

8

\(F_{15} \left( x \right) = \sum\limits_{i = 1}^{11} {\left[ {a_{i} - \frac{{x_{i} \left( {b_{i}^{2} + b_{i} x_{2} } \right)}}{{b_{i}^{2} + b_{i} x_{3} + x_{4} }}} \right]^{2} }\)

4

[− 5,5]

0.00030

9

\(F_{16} \left( x \right) = 4x_{1}^{2} - 2.1x_{1}^{4} + \frac{1}{3}x_{1}^{6} + x_{1} x_{2} - 4x_{2}^{2} + 4x_{2}^{4}\)

2

[− 5,5]

− 1.0316