Table 2 Mathematical equation, range and the best value for the studied benchmark functions.

Sphere

\(\:f\left(z\right)={\sum\:}_{i=1}^{N}{z}_{i}^{2}\)

−100

100

Ellipsoid

\(\:f\left(z\right)={\sum\:}_{i=1}^{N}{{(10}^{6})}^{\frac{i-1}{N-1}}{\:z}_{i}^{2}\)

−100

100

Bent Cigar

\(\:f\left(z\right)={z}_{1}^{2}+{10}^{6}\text{*}{\sum\:}_{i=2}^{N}{z}_{i}^{2}\)

−100

100

Discus

\(\:f\left(z\right)={10}^{6}\text{*}{z}_{1}^{2}+{\sum\:}_{i=2}^{N}{z}_{i}^{2}\)

−100

100

Different Powers

\(\:f\left(z\right)={\sum\:}_{i=2}^{N}{\left|{\text{x}}_{\text{i}}\right|}^{2}+\frac{4\text{i}}{\text{N}}\)

−100

100

Rosenbrock

\(\:f\left(z\right)={\sum\:}_{i=1}^{N-1}100{\left({{z}_{i+1}-z}_{i}^{2}\right)}^{2}+{({z}_{i}-1)}^{2}\)

−100

100

Rosenbrock Rotated

\(\:f\left(z\right)={\sum\:}_{i=1}^{N-1}100{\left({{u}_{i+1}-u}_{i}^{2}\right)}^{2}+{({u}_{i}-1)}^{2}\)

where \(\:u=M\left(z-o\right)\)

with [-π,π] rotation matrix M

and shifted global optimum vector o

−100

100

Elliptic

\(\:f\left(z\right)={\sum\:}_{i=1}^{N}{{(10}^{6})}^{\frac{i-1}{N-1}}{\:z}_{i}^{2}\)

−100

100

Rastrigin

\(\:f\left(z\right)=10N+{\sum\:}_{i=1}^{N}({z}_{i}^{2}-10\text{c}\text{o}\text{s}(2\pi\:{z}_{i}\left)\right)\)

−5.12

5.12

Rastrigin Non-separable

\(\:f\left(z\right)={\sum\:}_{i=1}^{N}({z}_{i}^{2}-10\text{c}\text{o}\text{s}(2\pi\:{z}_{i}\left)\right)\)

−5.12

5.12

Ackley

\(\:f\left(z\right)=-20exp\left(-0.2\sqrt{\frac{1}{N}{\sum\:}_{i=1}^{N}{z}_{i}^{2}}\right)-exp\left(\frac{1}{N}{\sum\:}_{i=1}^{N}\text{cos}\left(2\pi\:{z}_{i}\right)\right)+20+e\)

−32

32

Ackley Rotated

\(\:f\left(z\right)=-20exp\left(-0.2\sqrt{\frac{1}{N}{\sum\:}_{i=1}^{N}{u}_{i}^{2}}\right)-exp\left(\frac{1}{N}{\sum\:}_{i=1}^{N}\text{cos}\left(2\pi\:{u}_{i}\right)\right)+20+e\)

where \(\:u=M\left(z-o\right)\) with [-π,π] rotation matrix M

and shifted global optimum vector o

−32

32