多目标 benchmark 函数

MOP系列

MOP1:

                                                        \(\left\{\begin{array}{l}f_{1}(x)=(1+g(x)) x_{1} \\ f_{2}(x)=(1+g(x))(1-\sqrt{x_{1}})\end{array}\right.\)

where
        \(g(x)=2 \sin \left(\pi x_{1}\right) \sum_{i=2}^{n}\left(-0.9 t_{i}^{2}+\left|t_{i}\right|^{0.6}\right)\);
        \(t_{i}=x_{i}-\sin \left(0.5 \pi x_{1}\right)\);

its PF is        \(f_{2}=1-\sqrt{f_{1}}, 0 \leq f_{1} \leq 1\);
its PS is        \(\left\{\left(x_{1}, \cdots, x_{n}\right) \mid 0<x_{1}<1, x_{j}=\sin \left(0.5 \pi x_{1}\right)\right.\); \(\left.j=2, \ldots, n ; \text { or } x_{1}=0,1 .\right\}\).

        Domain: \([0,1]^{n}\); Number of Variables = 10
       

MOP2:

                                               \(\left\{\begin{array}{l}f_{1}(x)=(1+g(x)) x_{1} \\ f_{2}(x)=(1+g(x))\left(1-x_{1}^{2}\right)\end{array}\right.\)

where

        \(g(x)=10 \sin \left(\pi x_{1}\right) \sum_{i=2}^{n} \frac{\left|t_{i}\right|}{1+e^{5\left|t_{i}\right|}}\);
        \(t_{i}=x_{i}-\sin \left(0.5 \pi x_{1}\right)\);

its PF is        \(f_{2}=1-f_{1}^{2}, 0 \leq f_{1} \leq 1\);
its PS is         \(\left\{ \left( {{x}_{1}},\cdots ,{{x}_{n}} \right)\mid 0<{{x}_{1}}<1,{{x}_{j}}=\sin \left( 0.5\pi {{x}_{1}} \right) \right.\); \(\left.j=2, \ldots, n ; \text { or } x_{1}=0,1 .\right\}\).

        Domain: \([0,1]^{n}\); Number of Variables = 10
       

MOP3:

where
                        \(g(x)=10 \sin \left(\frac{\pi x_{1}}{2}\right) \sum_{i=2}^{n} \frac{\left|t_{i}\right|}{1+e^{5\left|t_{i}\right|}}\);
                        \(t_{i}=x_{i}-\sin \left(0.5 \pi x_{1}\right)\);
its PF is         \(f_{2}=\sqrt{1-f_{1}^{2}}, 0 \leq f_{1} \leq 1\);
its PS is         \(\left\{\left(x_{1}, \cdots, x_{n}\right) \mid 0<x_{1} \leq 1, x_{j}=\sin \left(0.5 \pi x_{1}\right)\right.\); \(\left.j=2, \ldots, n ; \text { or } x_{1}=0,1 .\right\}\).

        Domain: \([0,1]^{n}\); Number of Variables = 10
       

MOP4:

where
                        \(g(x)=10 \sin \left(\pi x_{1}\right) \sum_{i=2}^{n} \frac{\left|t_{i}\right|}{1+e^{5 \mid t_{i}} \mid}\);
                        \(t_{i}=x_{i}-\sin \left(0.5 \pi x_{1}\right)\);

its PF is         discontinuous;
its PS is         \(\left\{\left(x_{1}, \cdots, x_{n}\right) \mid 0<x_{1}<1, x_{j}=\sin \left(0.5 \pi x_{1}\right)\right.\); \(\left.j=2, \ldots, n ; \text { or } x_{1}=0,1 .\right\}\).

        Domain: \([0,1]^{n}\); Number of Variables = 10
       

MOP5:

                                                        \(\left\{\begin{array}{l}f_{1}(x)=(1+g(x)) x_{1} \\ f_{2}(x)=(1+g(x))(1-\sqrt{x_{1}})\end{array}\right.\)

where

                        \(g(x)=2\left|\cos \left(\pi x_{1}\right)\right| \sum_{i=2}^{n}\left(-0.9 t_{i}^{2}+\left|t_{i}\right|^{0.6}\right)\);
                        \(t_{i}=x_{i}-\sin \left(0.5 \pi x_{1}\right)\);

its PF is         \(f_{2}=1-\sqrt{f_{1}}, 0 \leq f_{1} \leq 1\);
its PS is         \(\left\{\left(x_{1}, \cdots, x_{n}\right) \mid 0 \leq x_{1} \leq 1, x_{j}=\sin \left(0.5 \pi x_{1}\right)\right.\); \(\left.j=2, \ldots, n ; \text { or } x_{1}=0.5\right\}\).

        Domain: \([0,1]^{n}\); Number of Variables = 10

MOP6:

                                        \(\left\{\begin{array}{l}f_{1}(x)=(1+g(x)) x_{1} x_{2} \\ f_{2}(x)=(1+g(x)) x_{1}\left(1-x_{2}\right) \\ f_{3}(x)=(1+g(x))\left(1-x_{1}\right)\end{array}\right.\)

where

                        \(g(x)=2\sin \left( \pi {{x}_{1}} \right)\sum\limits_{i=3}^{n}{\left( -0.9t_{i}^{2}+{{\left| {{t}_{i}} \right|}^{0.6}} \right)}\);
                        \({{t}_{i}}={{x}_{i}}-{{x}_{1}}{{x}_{2}}\);

its PF is         \({{f}_{1}}+{{f}_{2}}+{{f}_{3}}=1,0\le {{f}_{1}},{{f}_{2}},{{f}_{3}}\le 1\);
its PS is         \(\left\{ \left( {{x}_{1}},\cdots ,{{x}_{n}} \right)\mid 0<{{x}_{1}}<1,0\le {{x}_{2}}\le 1 \right.\); \(\left. {{x}_{j}}={{x}_{1}}{{x}_{2}},j=3,\ldots ,n;\text{ or }{{x}_{1}}=0,1. \right\}\)

        Domain: \([0,1]^{n}\); Number of Variables = 10
       

MOP7:

where

                     \(g(x)=2\sin \left( \pi {{x}_{1}} \right)\sum\limits_{i=3}^{n}{\left( -0.9t_{i}^{2}+{{\left| {{t}_{i}} \right|}^{0.6}} \right)}\);
                     \({{t}_{i}}={{x}_{i}}-{{x}_{1}}{{x}_{2}}\);

its PF is        \(f_{1}^{2}+f_{2}^{2}+f_{3}^{2}=1,0\le {{f}_{1}},{{f}_{2}},{{f}_{3}}\le 1\);
its PS is        \(\left\{ \left( {{x}_{1}},\cdots ,{{x}_{n}} \right)\mid 0<{{x}_{1}}<1,0\le {{x}_{2}}\le 1 \right.\); \(\left. {{x}_{j}}={{x}_{1}}{{x}_{2}},j=3,\ldots ,n;\text{ or }{{x}_{1}}=0,1. \right\}\).

        Domain: \([0,1]^{n}\); Number of Variables = 10
       
       

F系列

       

F1:

                     \(f_{1}(x)=(1+g(x)) x_{1}\)
                     \(f_{2}(x)=(1+g(x))(1-\sqrt{x_{1}})^{5}\)
                     \(g(x)=2 \sin \left(0.5 \pi x_{1}\right)\left(n-1+\sum_{i=2}^{n}\left(y_{i}^{2}-\cos \left(2 \pi y_{i}\right)\right)\right)\)

where

                      \(y_{i=2: n}=x_{i}-\sin \left(0.5 \pi x_{i}\right)\)
       

POF:         \(f_{2}=(1-\sqrt{f_{1}})^{5}\)
POS:         \(x_{i}=\sin \left(0.5 \pi x_{i}\right), i=2, \ldots, n\)

       Domain: \([0,1]^{n}\); Number of Variables = 30
       

F2:

                     \(f_{1}(x)=(1+g(x))\left(1-x_{1}\right)\)
                     \(f_{2}(x)=\frac{1}{2}(1+g(x))\left(x_{1}+\sqrt{x_{1}} \cos ^{2}\left(4 \pi x_{1}\right)\right)\)
                     \(g(x)=2 \sin \left(0.5 \pi x_{1}\right)\left(n-1+\sum_{i=2}^{n}\left(y_{i}^{2}-\cos \left(2 \pi y_{i}\right)\right)\right)\)

where

                      \(y_{i=2: n}=x_{i}-\sin \left(0.5 \pi x_{i}\right)\)
       

POF:         \(f_{2}=\frac{1}{2}\left(1-f_{1}+\sqrt{1-f_{1}} \cos ^{2}\left(4 \pi\left(1-f_{1}\right)\right)\right)\)
POS:         \(x_{i}=\sin \left(0.5 \pi x_{i}\right), i=2, \ldots, n\)

       Domain: \([0,1]^{n}\); Number of Variables = 30
       

F3:

                     \(f_{1}(x)=(1+g(x)) x_{1}\)
                     \(f_{2}(x)=\frac{1}{2}(1+g(x))\left(1-x_{1}^{0.1}+(1-\sqrt{x_{1}})^{2} \cos ^{2}\left(3 \pi x_{1}\right)\right)\)
                     \(g(x)=2 \sin \left(0.5 \pi x_{1}\right)\left(n-1+\sum_{i=2}^{n}\left(y_{i}^{2}-\cos \left(2 \pi y_{i}\right)\right)\right)\)

where
                      \(y_{i=2: n}=x_{i}-\sin \left(0.5 \pi x_{i}\right)\)
       

POF:         \(f_{2}=\frac{1}{2}\left(1-f_{1}^{0.1}+(1-\sqrt{f_{1}})^{2} \cos ^{2}\left(3 \pi f_{1}\right)\right)\)
POS:         \(x_{i}=\sin \left(0.5 \pi x_{i}\right), \forall x_{i} \in \mathbf{x}_{\text {II }}\)

       Domain: \([0,1]^{n}\); Number of Variables = 30
       

F4:

                     \(f_{1}(x)=(1+g(x))\left(\frac{x_{1}}{\sqrt{x_{2} x_{3}}}\right)\)
                     \(f_{2}(x)=(1+g(x))\left(\frac{x_{2}}{\sqrt{x_{1} x_{3}}}\right)\)
                     \(f_{3}(x)=(1+g(x))\left(\frac{x_{3}}{\sqrt{x_{1} x_{2}}}\right)\)

where
                      \(g(x)=\sum_{i=4}^{n}\left(x_{i}-2\right)^{2}\)
       

POF:        \(f_{1} f_{2} f_{3}=1\)
POS:         \(x_{i}=2, i=3, \ldots, n\)

       Domain: \({{[1,4]}^{n}}\); Number of Variables = 30
       

F5:

                     \(f_{1}(x)=(1+g(x))\left(\left(1-x_{1}\right) x_{2}\right)\)
                     \(f_{2}(x)=(1+g(x))\left(x_{1}\left(1-x_{2}\right)\right)\)
                     \(f_{3}(x)=(1+g(x))\left(1-x_{1}-x_{2}+2 x_{1} x_{2}\right)^{6}\)

where
                      \(g(x)=\sum_{i=3}^{n}\left(x_{i}-0.5\right)^{2}\)
       

POF:         \(f_{3}=\left(1-f_{1}-f_{2}\right)^{6}\)
POS:        \(x_{i}=0.5, i=3, \ldots, n\)

       Domain: \([0,1]^{n}\); Number of Variables = 30
       

F6:

                     \(f_{1}(x)=\cos ^{4}\left(0.5 \pi x_{1}\right) \cos ^{4}\left(0.5 \pi x_{2}\right)\)
                     \(f_{2}(x)=\cos ^{4}\left(0.5 \pi x_{1}\right) \sin ^{4}\left(0.5 \pi x_{2}\right)\)
                     \(f_{3}(x)=\left(\frac{1+g(x)}{1+\cos ^{2}\left(0.5 \pi x_{1}\right)}\right)^{\frac{1}{1+g(x)}}\)

where
                      \(\left.g(x)=\frac{1}{10} \sum_{i=3}^{n}\left(1+x_{i}^{2}-\cos \left(2 \pi x_{i}\right)\right)\right)\)
       

POF:         \(f_{3}(1+\sqrt{f_{1}}+\sqrt{f_{2}})=1\)
POS:        \(x_{i}=0, i=3, \ldots, n\)

       Domain: \([0,1]^{n}\); Number of Variables = 30