Table 3 Classic multimodal functions F9–F18.

\(F9 = \sum\limits_{i = 1}^{n} {\left( {x_{i}^{2} - 10\cos \left( {2\pi x_{i} } \right) + 10} \right)}\)

30

[-5.12,5.12]

0

\(F10 = \sum\limits_{i = 1}^{n} {\frac{{x_{i}^{2} }}{4000}} - \prod\limits_{i = 1}^{D} {\cos \left( {\frac{{x_{i} }}{\sqrt i }} \right)} + 1\)

30

[-600,600]

0

\(F11 = \sum\limits_{i = 1}^{n} {\left| {x_{i} \sin (x_{i} ) + 0.1x_{i} } \right|}\)

30

[-10,10]

0

\(\begin{gathered} F12 = \sin^{2} (3\pi x_{1} ) + (x_{1} - 1)^{2} [1 + \sin^{2} (3\pi x_{2} )] \hfill \\ \, + (x_{2} - 1)^{2} [1 + \sin^{2} (2\pi x_{2} )] \hfill \\ \end{gathered}\)

2

[-10,10]

0

\(\begin{gathered} F13 = - (x_{2} + 47)\sin (\sqrt {\left| {x_{2} + \frac{{x_{1} }}{2} + 47} \right|} ) \hfill \\ \, - x_{1} \sin (\sqrt {\left| {x_{1} - (x_{2} + 47)} \right|} ) \hfill \\ \end{gathered}\)

2

[-512,512]

−959.6407

\(\begin{gathered} F14 = \sin^{2} (\pi w_{1} ) + \sum\limits_{i = 1}^{n - 1} {(w_{i} - 1)^{2} } [1 + 10\sin^{2} (\pi w_{i} + 1)] \hfill \\ \, + (w_{n} - 1)^{2} [1 + \sin^{2} (2\pi w_{n} )], \hfill \\ \, w_{i} = 1 + \frac{{x_{i} - 1}}{4},i = 1,2,...n \hfill \\ \end{gathered}\)

30

[-10,10]

0

\(F15 = \sum\nolimits_{i = 1}^{n} {\left( {\left\lfloor {x_{i} + 0.5} \right\rfloor } \right)^{2} }\)

30

[-100,100]

0

\(F16 = \sum\nolimits_{i = 1}^{n} {\left( {x_{i} - 1} \right)^{2} } + \sum\nolimits_{i = 2}^{n} {x_{i} x_{i - 1} }\)

6

[-36,36]

−50

\(F17 = \sum\nolimits_{i = 1}^{n} {\left( {x_{i} - 1} \right)^{2} } + \sum\nolimits_{i = 2}^{n} {x_{i} x_{i - 1} }\)

10

[-100,100]

−210

\(\begin{gathered} F18 = x_{1}^{2} + 2x_{2}^{2} - 0.3\cos \left( {3\pi x_{1} } \right) \hfill \\ \, - 0.4\cos \left( {4\pi x_{2} } \right) + 0.7 \hfill \\ \end{gathered}\)

2

[-100,100]

0