Table 1 Benchmark functions.

Sphere

\({f_1}({\text{x}})=\sum\limits_{{i=1}}^{n} {x_{i}^{2}}\)

0

Schwefel 2.22

\({f_2}({\text{x}})=\sum\limits_{{i=1}}^{D} {\left| {{x_i}} \right|+\prod\limits_{{i=1}}^{D} {\left| {{x_i}} \right|} }\)

0

Schwefel 1.2

\({f_3}(x)=\sum\limits_{{i=1}}^{d} {(\sum\limits_{{j=1}}^{i} {{{\text{x}}_j}} } {)^2}\)

0

Schwefel 2.21

\({f_4}(x)=\hbox{max} \left| {{x_i}} \right|\)

0

Zakharov

\({f_5}({\text{x}})=\sum\limits_{{i=1}}^{D} {{x_i}^{2}} +{(\sum\limits_{{i=1}}^{D} {\frac{i}{2}{x_i}} )^2}+{(\sum\limits_{{i=1}}^{D} {\frac{i}{2}{x_i}} )^4}\)

0

Step

\({f_6}(x)=\sum\limits_{{i=1}}^{n} {{{(\left\lfloor {{x_i}+0.5} \right\rfloor )}^2}}\)

0

Quartic

\({f_7}(x)=\sum\limits_{{i=1}}^{n} {ix_{i}^{4}} +random\left[ {0,1} \right]\)

0

Rosenbrock

\({f_8}(x)=\sum\limits_{{{\text{i=1}}}}^{{n - 1}} {[100{{(x_{i}^{2} - {x_{{\text{i}}+1}})}^2}+{{({x_i} - 1)}^2}]}\)

0

Rastrigin

\({f_9}(x)=\sum\limits_{{{\text{i=1}}}}^{n} {(x_{i}^{2} - 10\cos (2\pi {{\text{x}}_i})+10)}\)

0

Ackley

\({f_{10}}(x)= - 20\exp ( - 0.2\sqrt {\frac{1}{n}\sum\limits_{{i=1}}^{n} {x_{i}^{2}} } ) - \exp (\frac{1}{n}\sum\limits_{{i=1}}^{n} {\cos (2\pi {x_i})} )+20+e\)

0

Griewank

\({f_{11}}(x)=\frac{1}{{4000}}\sum\limits_{{i=1}}^{d} {x_{i}^{2}} - \prod\limits_{{i=1}}^{d} {\cos (\frac{{{x_i}}}{{\sqrt i }}} )+1\)

0

Penalized 1

\(\begin{aligned} f_{{12}} (x) & = \frac{\pi }{D}\left\{ {10\sin ^{2} \left( {\pi y_{1} } \right) + \sum\limits_{{i = 1}}^{{D - 1}} {\left( {y_{i} - 1} \right)^{2} } \left[ {1 + 10\sin ^{2} \left( {\pi y_{{i + 1}} } \right)} \right] + \left( {y_{D} - 1} \right)^{2} } \right\} \\ & {\kern 1pt} {\kern 1pt} + {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{{i = 1}}^{D} {\mu (x_{i} ,10,100,4)} ,y_{i} = 1 + \frac{1}{4}\left( {x_{i} + 1} \right),\mu \left( {x_{i} ,10,100,4} \right) = \left\{ {\begin{array}{*{20}c} {k(x_{i} - a)^{m} ,x_{i} > a} \\ {0, - a \le x_{i} \le a} \\ {k( - x_{i} - a)^{m} ,x_{i} < - a} \\ \end{array} } \right. \\ \end{aligned}\)

0

Penalized 2

\(\begin{gathered} {f_{13}}(x)=0.1\{ {\sin ^2}(3\pi {x_1})+\sum\limits_{{i=1}}^{D} {{{({x_i} - 1)}^2}[1+{{\sin }^2}(3\pi {x_i}+1)]} +{\kern 1pt} {\kern 1pt} {({x_D} - 1)^2}[1 \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} +{\sin ^2}(2\pi {x_D})]\} +\sum\limits_{{i=1}}^{D} {\mu ({x_i},5,100,4)} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mu ({x_i},a,k,m)=\left\{ {\begin{array}{*{20}{c}} {k{{({x_i} - m)}^m},{x_i}>a} \\ {0, - a \leqslant {x_i} \leqslant a} \\ {k{{( - {x_i} - m)}^m},{x_i}< - a} \end{array}} \right. \hfill \\ \end{gathered}\)

0

Schwefel 2.26

\({f_{14}}(x)=\sum\limits_{{i=1}}^{d} { - {x_i}\sin (\sqrt {|{x_i}|} )}\)

−418.98d

Salomon

\({f_{15}}(x)=1{\text{-}}\cos (2\pi \sqrt {\sum\limits_{{i=1}}^{d} {x_{i}^{2}} } )+0.1\sqrt {\sum\limits_{{i=1}}^{d} {x_{i}^{2}} }\)

0

Alpine

\({f_{16}}(x)=\sum\limits_{{i=1}}^{d} {\left| {{x_i}\sin ({x_i})+0.1{x_i}} \right|}\)

0

Schaffer 7

\({f_{17}}(x)=\frac{1}{{N - 1}}\sum\limits_{{i=1}}^{{N - 1}} {[{{(x_{i}^{2}+x_{{i+1}}^{2})}^{0.25}}+} {(x_{i}^{2}+x_{{i+1}}^{2})^{0.25}}{\sin ^2}(50{(x_{i}^{2}+x_{{i+1}}^{2})^{0.1}})]\)

0

Expansion 10

\(\begin{gathered} {f_{18}}(x)={f_{18}}({x_1},{x_2})+{f_{18}}({x_2},{x_3})+...+{f_{18}}({x_{i - 1}},{x_i})+{f_{18}}({x_d},{x_1}) \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {f_{18}}(x,y)={({x^2}+{y^2})^{0.25}}[{\sin ^2}(50{({x^2}+{y^2})^{0.1}})+1] \hfill \\ \end{gathered}\)

0

Lévy

\(\begin{gathered} {f_{19}}({\text{x}}){\text{=}}\sum\limits_{{{\text{i=1}}}}^{{d - 1}} {{{({\omega _i} - 1)}^2}[1+10{{\sin }^2}(\pi {\omega _i}+1)]} +{({\omega _d} - 1)^2}[1+{\sin ^2}(2\pi {\omega _d})]+ \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\sin ^2}(\pi {\omega _1}),{\omega _i}=1+\frac{{{x_i} - 1}}{4} \hfill \\ \end{gathered}\)

0

Powell

\(\begin{gathered} {f_{20}}(x)=\sum\limits_{{i=1}}^{{D/4}} {[{{({x_{4i - 3}}+10{x_{4i - 2}})}^2}+5{{({x_{4i - 1}} - {x_{4i}})}^2}+{{({x_{4i - 2}} - 2{x_{4i - 1}})}^4}} + \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 10{({x_{4i - 3}} - {x_{4i}})^2}] \hfill \\ \end{gathered}\)

0

Dixon–Price

\({f_{21}}(x)={({x_1} - 1)^2}+\sum\limits_{{i=2}}^{D} {i{{(2x_{i}^{2} - {x_i} - 1)}^2}}\)

0

Weierstrass

\(\begin{gathered} {f_{22}}(x)=\sum\limits_{{i=1}}^{D} {((\sum\limits_{{k=0}}^{{{k_{\hbox{max} }}}} {[{a^k}\cos (2\pi {b^k}({x_i}+0.5)]} } ) - {\text{D}}\sum\limits_{{k=0}}^{{{k_{\hbox{max} }}}} {[{a^k}\cos (2\pi {b^k}0.5)]} \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} a=0.5,b=3,{k_{\hbox{max} }}=20 \hfill \\ \end{gathered}\)

0

Schaffer

\({f_{23}}(x)=0.5+\frac{{{{(\sin \sqrt {\sum\nolimits_{{i=1}}^{{\text{D}}} {x_{i}^{2}} } )}^2} - 0.5}}{{{{(1+0.001\sqrt {\sum\nolimits_{{i=1}}^{{\text{D}}} {x_{i}^{2}} } )}^2}}}\)

0

Masters

\(\begin{gathered} {f_{24}}(x)= - \sum\limits_{{i=1}}^{{D - 1}} {(\exp (\frac{{ - (x_{i}^{2}+x_{{i+1}}^{2}+0.5{x_i}{x_{i+1}})}}{8})} \cos (4\sqrt {x_{i}^{2}+x_{{i+1}}^{2}+0.5{x_i}{x_{i+1}}} )) \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} +D - 1 \hfill \\ \end{gathered}\)

0