Table 2 Multimodal functions \(F_{{8}} (x) - F_{{{13}}} (x)\) with dimension \(n\).

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

30

[− 500, 500]

\(- 418.9829 \times {\text{Dim}}\)

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

30

[− 5.12, 15.12]

0

\(\begin{aligned} F_{{{10}}} (x) & = - 20\exp \left( { - 0.2\sqrt {\tfrac{1}{n}\sum\limits_{i = 1}^{n} {x_{i}^{2} } } } \right){\kern 1pt} - \exp \left( {\tfrac{1}{n}\sum\limits_{i = 1}^{n} {\cos (2\pi x_{i} )} } \right) \\ & \quad + 20 + \exp \\ \end{aligned}\)

30

[− 32, 32]

0

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

30

[− 600, 600]

0

\(\begin{aligned} & F_{{{12}}} (x) = \tfrac{\pi }{n}\left\{ {10\sin \left( {\pi y_{1} } \right) + \sum\limits_{i = 1}^{n - 1} {\left( {y_{i} - 1} \right)^{2} } [1 + 10\sin^{2} \left( {\pi y_{i + 1} } \right)]} \right. \\ & \quad \left. { + \left( {y_{n}^{2} - 1} \right)^{2} } \right\} + \sum\limits_{i = 1}^{n} {u(x_{i} ,10,100,4)} \\ & u(x_{i} ,a,k,m) = \left\{ {\begin{array}{*{20}l} {k(x_{i} - a)^{m} } \hfill & {x_{i} > a} \hfill \\ 0 \hfill & { - a < x_{i} < a} \hfill \\ {k( - x_{i} - a)^{m} } \hfill & {x < - a} \hfill \\ \end{array} } \right. \\ \end{aligned}\)

30

[− 50, 50]

0

\(\begin{aligned} F_{{{13}}} (x) & = {0}{\text{.1}}\left\{ {{\text{sin}}^{{2}} (3\pi x_{1} ) + \sum\limits_{i = 1}^{n} {(x_{i} - 1)^{2} [1 + \sin^{2} (3\pi x_{i} + 1)] + (x_{n} - 1)^{2} [1 + \sin^{2} (2\pi x_{n} )]} } \right\} \\ & \quad + \sum\limits_{i = 1}^{n} {u(x_{i} ,5,100,4)} \\ \end{aligned}\)

30

[− 50, 50]

0