Table 2 Definition of the benchmark problems and their features

Sphere

\(f_{Sph}(x)=\sum_{i=1}^{p}x_{i}^{2}\)

no

yes

n/a

\(x_{i}\in [-5.12,5.12]\), \(x^{*}=(0,0,\ldots,0)\), \(f_{Sph}(x^{*})=0\).

Schwefel’s

\(f_{SchDS}(x)=\sum_{i=1}^{p}(\sum_{j=1}^{i}x_{j})^{2}\)

no

no

n/a

\(x_{i}\in [-65.536,65.536]\), \(x^{*}=(0,0,\ldots,0)\), \(f_{SchDS}(x^{*})=0\).

Griewank

\(f_{Gri}(x)=1+\sum_{i=1}^{p}\frac{x_{i}^{2}}{4{,}000}-\prod_{i=1}^{p}\cos \frac{x_{i}}{i}\)

yes

no

yes

\(x_{i}\in [-600,600]\), \(x^{*}=(0,0,\ldots,0)\), \(f_{Gri}(x^{*})=0\).

Rosenbrock

\(f_{Ros}(x)=\sum_{i=1}^{p}[100(x_{i+1}-x_{i}^{2})^{2}+(x_{i}-1)^{2}]\)

no

no

n/a

\(x_{i}\in [-2.048,2.048]\), \(x^{*}=(1,1,\ldots,1)\), \(f_{Ros}(x^{*})=0\).

Ackley

\(f_{Ack}(x)=20+e-20 e^{-0.2\sqrt{\frac{1}{p}\sum _{i=1}^{p}x_{i}^{2}}}-e^{\frac{1}{p}\sum _{i=1}^{p}\cos (2\pi x_{i})}\)

yes

no

yes

\(x_{i}\in [-30,30]\), \(x^{*}=(0,0,\ldots,0)\), \(f_{Ack}(x^{*})=0\).