Vuxyl

2d Sphere

${\begin{aligned}&f_{1}(x)=\left\|{\vec {x}}-(-1,0,\ldots ,0)\right\|\\&f_{2}(x)=\left\|{\vec {x}}-(1,0,\ldots ,0)\right\|\\\end{aligned}}$

${\begin{aligned}&f_{1}(x)=x\\&f_{2}(x)=\left(2-x^{1/\alpha }\right)^{\alpha }\\\end{aligned}}$

$d(x_{1})=\left(2-x_{1}^{1/\alpha }\right)^{\alpha -1}x_{1}^{1/\alpha -1}$

FOGA

ZDT1

Front

Formulation ${\begin{aligned}&f_{1}=x_{1}\\&f_{2}=g\cdot \left(1.0-{\sqrt {\frac {f_{1}}{g}}}\right)\\&g(x_{2},\ldots ,x_{n})=1.0+{\frac {9}{n-1}}\sum \limits _{i=2}^{n}x_{i}\\&0\leq x_{i}\leq 1,i=1,\ldots ,n\\\end{aligned}}$

Pareto Front $x_{2}=1-{\sqrt {x_{1}}}$

Density

$d(x_{1})={\frac {3}{4x_{1}^{1/4}}}$

Optimal Distribution Helper Functions

$b_{opt}(a,c)=2/9\left(\,a+\,{\sqrt {a(a+3\,c)}}\right)+1/3\,c$

$a_{opt}(r,b)=2/9-4/9\,r+2/9\,{\sqrt {1-2\,r+r^{2}+3\,b}}+2/9\,r^{2}-2/9\,r{\sqrt {1-2\,r+r^{2}+3\,b}}+1/3\,b$

ZDT 3

Formulation

${\begin{aligned}&f_{1}({\vec {x}})=x_{1}\\&f_{2}({\vec {x}})=g({\vec {x}})[1-{\sqrt {x_{1}/g({\vec {x}})}}-x_{1}/g({\vec {x}})\sin(10\pi x_{1})\\&g({\vec {x}})=1+{\frac {9}{n-1}}\left(\sum \limits _{i=2}^{n}{x_{i}}\right)\\&0\leq x_{i}\leq 1,{\text{ }}i=1,\ldots ,n\\\end{aligned}}$

Front

$\displaystyle (x_{1},x_{2})$ with $x_{1}\in F$ and $x_{2}=1-{\sqrt {x_{1}}}-x_{1}\cdot \sin(10\pi x_{1})$ where ${\begin{aligned}F=&[0,{\text{ }}0.0830015349]\cup {\text{ }}\\&(0.1822287280,{\text{ }}0.2577623634]\cup \\&(0.4093136748,{\text{ }}0.4538821041]\cup \\&(0.6183967944,{\text{ }}0.6525117038]\cup \\&(0.8233317983,{\text{ }}0.8518328654]\\\end{aligned}}$

Density

$d(x_{1})=C\cdot \,{\sqrt {\,{\frac {1}{2{\sqrt {x}}}}+\,\sin \left(10\,\pi \,x\right)+10\,x\cos \left(10\,\pi \,x\right)\pi }}$

DTLZ 7

Formulation

${\begin{aligned}f_{1}({\vec {x}})&=x_{1}\\f_{2}({\vec {x}})&=(1+g({\vec {x}}))h(f_{1}({\vec {x}}),g({\vec {x}}))\\g({\vec {x}})&=1+{\frac {9}{\left|{\vec {x}}\right|}}\sum \nolimits _{x_{i}\in {\vec {x}}}{x_{i}}\\h(f_{1}({\vec {x}}),g({\vec {x}}))&=M-{\frac {f_{1}({\vec {x}})}{1+g({\vec {x}})}}(1+\sin(3\pi f_{1}({\vec {x}})))\\&0\leq x_{i}\leq 1,1\leq i\leq n\\\end{aligned}}$

Front

$\displaystyle (x_{1},x_{2})$ with $x_{1}\in F$ and $\displaystyle x_{2}=4-x_{1}(1+\sin(3\pi x_{1}))$ where ${\begin{aligned}F=&[0,{\text{ }}0.2514118360]\cup {\text{ }}\\&(0.6316265307,{\text{ }}0.8594008566]\cup \\&(1.3596178367,{\text{ }}1.5148392681]\cup \\&(2.0518383519,{\text{ }}2.116426807]\\\end{aligned}}$

Density

$d(x_{1})=C\cdot {\sqrt {1+\sin(3\pi x_{1})+3\pi x_{1}\cos(3\pi x_{1})}}$

ZDT2

Optimal Distribution Helper Functions

$b_{opt}(a,c)=1/3\,c+1/3\,{\sqrt {c^{2}+3\,a^{2}}}$

Density

$d(x_{2})={\frac {3}{2}}{\sqrt {x_{1}}}$

ZDT6

Formulation

${\begin{aligned}&f_{1}({\vec {x}})=1-e^{-4x_{1}}\cdot \sin ^{6}(6\pi x_{1})\\&f_{2}({\vec {x}})=g({\vec {x}})\cdot \left[{}^{\left(\sum \limits _{i=2}^{n}{x_{i}}\right)}\!\!\diagup \!\!{}_{(n-1)}\;\right]^{0.25}\\&0\leq x_{i}\leq 1,i=1,\ldots ,n\\\end{aligned}}$

Front

$\displaystyle x_{2}=1-x_{1}^{2}$ with $x_{1}\in [{\frac {\arctan {9\pi }}{6\pi }}\approx 0.0815,1]$

Density

$\delta (x_{1})={\frac {3}{2}}\left(1-{\frac {\arctan(9\pi )}{6\pi }}^{3/2}\right)^{-1}{\sqrt {x_{1}}}$

DTLZ 1

Formulation

${\begin{aligned}&f_{1}({\vec {x}})={\frac {1}{2}}x_{1}(1+g({\vec {x}}))\\&f_{2}({\vec {x}})={\frac {1}{2}}(1-x_{1})(1+g({\vec {x}}))\\&g({\vec {x}})=100\left[\left|{\vec {x}}\right|+\sum \nolimits _{x_{i}\in {\vec {x}}}{(x_{1}-0.5)^{2}}-\cos(20\cdot \pi (x_{i}-0.5))\right]\\&0\leq x_{i}\leq 1,i=1,\ldots ,n\\\end{aligned}}$

Front

$x_{2}={\frac {1}{2}}-x_{1}$

Density

$\displaystyle d(x_{1})=2$

DTLZ 2

Front

$x_{2}={\sqrt {1-x_{1}^{2}}}$

Formulation

${\begin{aligned}f_{1}({\vec {x}})&=(1+g({\vec {x}}))\cos \left(x_{1}{\frac {\pi }{2}}\right)\\f_{2}({\vec {x}})&=(1+g({\vec {x}}))\sin \left(x_{1}{\frac {\pi }{2}}\right)\\g({\vec {x}})&=\sum \nolimits _{x_{i}\in {\vec {x}}}{(x_{i}-0.5)^{2}}\\&0\leq x_{i}\leq 1,i=1,\ldots ,n\\\end{aligned}}$

DTLZ 3

Formulation

${\begin{aligned}&f_{1}({\vec {x}})=(1+g({\vec {x}}))\cos \left(x_{1}{\frac {\pi }{2}}\right)\\&f_{2}({\vec {x}})=(1+g({\vec {x}}))\sin \left(x_{1}{\frac {\pi }{2}}\right)\\&g({\vec {x}})=100\cdot [\left|{\vec {x}}\right|+\sum \nolimits _{x_{i}\in {\vec {x}}}{(x_{i}-0.5)^{2}}-\cos(20\pi (x_{i}-0.5))]\\&0\leq x_{i}\leq 1,i=1,\ldots ,n\\\end{aligned}}$

Front

$x_{2}={\sqrt {1-x_{1}^{2}}}$

DTLZ 4

Formulation

${\begin{aligned}&f_{1}({\vec {x}})=(1+g({\vec {x}}))\cos \left(x_{1}^{\alpha }{\frac {\pi }{2}}\right)\\&f_{2}({\vec {x}})=(1+g({\vec {x}}))\sin \left(x_{1}^{\alpha }{\frac {\pi }{2}}\right)\\&g({\vec {x}})=\sum \nolimits _{x_{i}\in {\vec {x}}}{(x_{i}-0.5)^{2}}\\&\alpha =100\\&0\leq x_{i}\leq 1,i=1,\ldots ,n\\\end{aligned}}$