Table 2 Multimodal benchmark functions.

Schwefel

\(f_{5} ({\varvec{x}}) = \sum\nolimits_{i = 1}^{n} { - x_{i} } \sin (\sqrt {\left| {x_{i} } \right|} )\)

\([ - 500,500]\)

− 418.9829*n

Rastrigin

\(f_{6} ({\varvec{x}}) = \sum\nolimits_{i = 1}^{n} {\left[ {x_{i} - 10\cos (2\pi x_{i} ) + 10} \right]}\)

\([ - 5.12,5.12]\)

0

Ackley

\(f_{7} ({\varvec{x}}) = \sum\nolimits_{(i = 1)}^{n} { - 20exp} ( - 0.2\sqrt {\frac{1}{n}\sum\nolimits_{i = 1}^{n} {x_{i}^{2} } } ) - exp\left( {\frac{1}{n} \, \sum\nolimits_{i = 1}^{n} {cos} \left( {{2}\pi x_{i} } \right)} \right) + 20 + e\)

\([ - 32,32]\)

0

Griewank

\(f_{8} ({\varvec{x}}) = \frac{1}{4000}\sum\nolimits_{i = 1}^{n} {x_{i}^{2} - \prod\nolimits_{i = 1}^{n} {\cos (\frac{{x_{i} }}{\sqrt i })} } + 1\)

\([ - 600,600]\)

0