MOP系列
MOP1:
{ f 1 ( x ) = ( 1 + g ( x ) ) x 1 f 2 ( x ) = ( 1 + g ( x ) ) ( 1 − x 1 ) \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. { f1(x)=(1+g(x))x1f2(x)=(1+g(x))(1−x1)
where
g ( x ) = 2 sin ( π x 1 ) ∑ i = 2 n ( − 0.9 t i 2 + ∣ t i ∣ 0.6 ) 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) g(x)=2sin(πx1)∑i=2n(−0.9ti2+∣ti∣0.6);
t i = x i − sin ( 0.5 π x 1 ) t_{i}=x_{i}-\sin \left(0.5 \pi x_{1}\right) ti=xi−sin(0.5πx1);
its PF is f 2 = 1 − f 1 , 0 ≤ f 1 ≤ 1 f_{2}=1-\sqrt{f_{1}}, 0 \leq f_{1} \leq 1 f2=1−f1,0≤f1≤1;
its PS is { ( x 1 , ⋯ , x n ) ∣ 0 < x 1 < 1 , x j = sin ( 0.5 π x 1 ) \left\{\left(x_{1}, \cdots, x_{n}\right) \mid 0<x_{1}<1, x_{j}=\sin \left(0.5 \pi x_{1}\right)\right. {
(x1,⋯,xn)∣0<x1<1,xj=sin(0.5πx1); j = 2 , … , n ; or x 1 = 0 , 1. } \left.j=2, \ldots, n ; \text { or } x_{1}=0,1 .\right\} j=2,…,n; or x1=0,1.}.
Domain: [ 0 , 1 ] n [0,1]^{n} [0,1]n; Number of Variables = 10
MOP2:
{ f 1 ( x ) = ( 1 + g ( x ) ) x 1 f 2 ( x ) = ( 1 + g ( x ) ) ( 1 − x 1 2 ) \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. { f1(x)=(1+g(x))x1f2(x)=(1+g(x))(1−x12)
where
g ( x ) = 10 sin ( π x 1 ) ∑ i = 2 n ∣ t i ∣ 1 + e 5 ∣ t i ∣ 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|}} g(x)=10sin(πx1)∑i=2n1+e5∣ti∣∣ti∣;
t i = x i − sin ( 0.5 π x 1 ) t_{i}=x_{i}-\sin \left(0.5 \pi x_{1}\right) ti=xi−sin(0.5πx1);
its PF is f 2 = 1 − f 1 2 , 0 ≤ f 1 ≤ 1 f_{2}=1-f_{1}^{2}, 0 \leq f_{1} \leq 1 f2=1−f12,0≤f1≤1;
its PS is { ( x 1 , ⋯ , x n ) ∣ 0 < x 1 < 1 , x j = sin ( 0.5 π x 1 ) \left\{ \left( {
{x}_{1}},\cdots ,{
{x}_{n}} \right)\mid 0<{
{x}_{1}}<1,{
{x}_{j}}=\sin \left( 0.5\pi {
{x}_{1}} \right) \right. {
(x1,⋯,xn)∣0<x1<1,xj=sin(0.5πx1); j = 2 , … , n ; or x 1 = 0 , 1. } \left.j=2, \ldots, n ; \text { or } x_{1}=0,1 .\right\} j=2,…,n; or x1=0,1.}.
Domain: [ 0 , 1 ] n [0,1]^{n} [0,1]n; Number of Variables = 10
MOP3:
where
g ( x ) = 10 sin ( π x 1 2 ) ∑ i = 2 n ∣ t i ∣ 1 + e 5 ∣ t i ∣ 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|}} g(x)=10sin(2πx1)∑i=2n1+e5∣ti∣∣ti∣;
t i = x i − sin ( 0.5 π x 1 ) t_{i}=x_{i}-\sin \left(0.5 \pi x_{1}\right) ti=xi−sin(0.5πx1);
its PF is f 2 = 1 − f 1 2 , 0 ≤ f 1 ≤ 1 f_{2}=\sqrt{1-f_{1}^{2}}, 0 \leq f_{1} \leq 1 f2=1−f12,0≤f1≤1;
its PS is { ( x 1 , ⋯ , x n ) ∣ 0 < x 1 ≤ 1 , x j = sin ( 0.5 π x 1 ) \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. {
(x1,⋯,xn)∣0<x1≤1,xj=sin(0.5πx1); j = 2 , … , n ; or x 1 = 0 , 1. } \left.j=2, \ldots, n ; \text { or } x_{1}=0,1 .\right\} j=2,…,n; or x1=0,1.}.
Domain: [ 0 , 1 ] n [0,1]^{n} [0,1]n; Number of Variables = 10
MOP4:
where
g ( x ) = 10 sin ( π x 1 ) ∑ i = 2 n ∣ t i ∣ 1 + e 5 ∣ t i ∣ 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} g(x)=10sin(πx1)∑i=2n1+e5∣ti∣∣ti∣;
t i = x i − sin ( 0.5 π x 1 ) t_{i}=x_{i}-\sin \left(0.5 \pi x_{1}\right) ti=xi−sin(0.5πx1);
its PF is discontinuous;
its PS is { ( x 1 , ⋯ , x n ) ∣ 0 < x 1 < 1 , x j = sin ( 0.5 π x 1 ) \left\{\left(x_{1}, \cdots, x_{n}\right) \mid 0<x_{1}<1, x_{j}=\sin \left(0.5 \pi x_{1}\right)\right. {
(x1,⋯,xn)∣0<x1<1,xj=sin(0.5πx1); j = 2 , … , n ; or x 1 = 0 , 1. } \left.j=2, \ldots, n ; \text { or } x_{1}=0,1 .\right\} j=2,…,n; or x1=0,1.}.
Domain: [ 0 , 1 ] n [0,1]^{n} [0,1]n; Number of Variables = 10
MOP5:
{ f 1 ( x ) = ( 1 + g ( x ) ) x 1 f 2 ( x ) = ( 1 + g ( x ) ) ( 1 − x 1 ) \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. { f1(x)=(1+g(x))x1f2(x)=(1+g(x))(1−x1)
where
g ( x ) = 2 ∣ cos ( π x 1 ) ∣ ∑ i = 2 n ( − 0.9 t i 2 + ∣ t i ∣ 0.6 ) 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) g(x)=2∣cos(πx1)∣∑i=2n(−0.9ti2+∣ti∣0.6);
t i = x i − sin ( 0.5 π x 1 ) t_{i}=x_{i}-\sin \left(0.5 \pi x_{1}\right) ti=xi−sin(0.5πx1);
its PF is f 2 = 1 − f 1 , 0 ≤ f 1 ≤ 1 f_{2}=1-\sqrt{f_{1}}, 0 \leq f_{1} \leq 1 f2=1−f1,0≤f1≤1;
its PS is { ( x 1 , ⋯ , x n ) ∣ 0 ≤ x 1 ≤ 1 , x j = sin ( 0.5 π x 1 ) \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. {
(x1,⋯,xn)∣0≤x1≤1,xj=sin(0.5πx1); j = 2 , … , n ; or x 1 = 0.5 } \left.j=2, \ldots, n ; \text { or } x_{1}=0.5\right\} j=2,…,n; or x1=0.5}.
Domain: [ 0 , 1 ] n [0,1]^{n} [0,1]n; Number of Variables = 10
MOP6:
{ f 1 ( x ) = ( 1 + g ( x ) ) x 1 x 2 f 2 ( x ) = ( 1 + g ( x ) ) x 1 ( 1 − x 2 ) f 3 ( x ) = ( 1 + g ( x ) ) ( 1 − x 1 ) \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. ⎩⎨⎧f1(x)=(1+g(x))x1x2f2(x)=(1+g(x))x1(1−x2)f3(x)=(1+g(x))(1−x1)
where
g ( x ) = 2 sin ( π x 1 ) ∑ i = 3 n ( − 0.9 t i 2 + ∣ t i ∣ 0.6 ) 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)} g(x)=2sin(πx1)i=3∑n(−0.9ti2+∣ti∣0.6);
t i = x i − x 1 x 2 {
{t}_{i}}={
{x}_{i}}-{
{x}_{1}}{
{x}_{2}} ti=xi−x1x2;
its PF is f 1 + f 2 + f 3 = 1 , 0 ≤ f 1 , f 2 , f 3 ≤ 1 {
{f}_{1}}+{
{f}_{2}}+{
{f}_{3}}=1,0\le {
{f}_{1}},{
{f}_{2}},{
{f}_{3}}\le 1 f1+f2+f3=1,0≤f1,f2,f3≤1;
its PS is { ( x 1 , ⋯ , x n ) ∣ 0 < x 1 < 1 , 0 ≤ x 2 ≤ 1 \left\{ \left( {
{x}_{1}},\cdots ,{
{x}_{n}} \right)\mid 0<{
{x}_{1}}<1,0\le {
{x}_{2}}\le 1 \right. {
(x1,⋯,xn)∣0<x1<1,0≤x2≤1; x j = x 1 x 2 , j = 3 , … , n ; or x 1 = 0 , 1. } \left. {
{x}_{j}}={
{x}_{1}}{
{x}_{2}},j=3,\ldots ,n;\text{ or }{
{x}_{1}}=0,1. \right\} xj=x1x2,j=3,…,n; or x1=0,1.}
Domain: [ 0 , 1 ] n [0,1]^{n} [0,1]n; Number of Variables = 10
MOP7:
where
g ( x ) = 2 sin ( π x 1 ) ∑ i = 3 n ( − 0.9 t i 2 + ∣ t i ∣ 0.6 ) 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)} g(x)=2sin(πx1)i=3∑n(−0.9ti2+∣ti∣0.6);
t i = x i − x 1 x 2 {
{t}_{i}}={
{x}_{i}}-{
{x}_{1}}{
{x}_{2}} ti=xi−x1x2;
its PF is f 1 2 + f 2 2 + f 3 2 = 1 , 0 ≤ f 1 , f 2 , f 3 ≤ 1 f_{1}^{2}+f_{2}^{2}+f_{3}^{2}=1,0\le {
{f}_{1}},{
{f}_{2}},{
{f}_{3}}\le 1 f12+f22+f32=1,0≤f1,f2,f3≤1;
its PS is { ( x 1 , ⋯ , x n ) ∣ 0 < x 1 < 1 , 0 ≤ x 2 ≤ 1 \left\{ \left( {
{x}_{1}},\cdots ,{
{x}_{n}} \right)\mid 0<{
{x}_{1}}<1,0\le {
{x}_{2}}\le 1 \right. {
(x1,⋯,xn)∣0<x1<1,0≤x2≤1; x j = x 1 x 2 , j = 3 , … , n ; or x 1 = 0 , 1. } \left. {
{x}_{j}}={
{x}_{1}}{
{x}_{2}},j=3,\ldots ,n;\text{ or }{
{x}_{1}}=0,1. \right\} xj=x1x2,j=3,…,n; or x1=0,1.}.
Domain: [ 0 , 1 ] n [0,1]^{n} [0,1]n; Number of Variables = 10
F系列
F1:
f 1 ( x ) = ( 1 + g ( x ) ) x 1 f_{1}(x)=(1+g(x)) x_{1} f1(x)=(1+g(x))x1
f 2 ( x ) = ( 1 + g ( x ) ) ( 1 − x 1 ) 5 f_{2}(x)=(1+g(x))(1-\sqrt{x_{1}})^{5} f2(x)=(1+g(x))(1−x1)5
g ( x ) = 2 sin ( 0.5 π x 1 ) ( n − 1 + ∑ i = 2 n ( y i 2 − cos ( 2 π y i ) ) ) 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) g(x)=2sin(0.5πx1)(n−1+∑i=2n(yi2−cos(2πyi)))
where
y i = 2 : n = x i − sin ( 0.5 π x i ) y_{i=2: n}=x_{i}-\sin \left(0.5 \pi x_{i}\right) yi=2:n=xi−sin(0.5πxi)
POF: f 2 = ( 1 − f 1 ) 5 f_{2}=(1-\sqrt{f_{1}})^{5} f2=(1−f1)5
POS: x i = sin ( 0.5 π x i ) , i = 2 , … , n x_{i}=\sin \left(0.5 \pi x_{i}\right), i=2, \ldots, n xi=sin(0.5πxi),i=2,…,n
Domain: [ 0 , 1 ] n [0,1]^{n} [0,1]n; Number of Variables = 30
F2:
f 1 ( x ) = ( 1 + g ( x ) ) ( 1 − x 1 ) f_{1}(x)=(1+g(x))\left(1-x_{1}\right) f1(x)=(1+g(x))(1−x1)
f 2 ( x ) = 1 2 ( 1 + g ( x ) ) ( x 1 + x 1 cos 2 ( 4 π x 1 ) ) f_{2}(x)=\frac{1}{2}(1+g(x))\left(x_{1}+\sqrt{x_{1}} \cos ^{2}\left(4 \pi x_{1}\right)\right) f2(x)=21(1+g(x))(x1+x1cos2(4πx1))
g ( x ) = 2 sin ( 0.5 π x 1 ) ( n − 1 + ∑ i = 2 n ( y i 2 − cos ( 2 π y i ) ) ) 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) g(x)=2sin(0.5πx1)(n−1+∑i=2n(yi2−cos(2πyi)))
where
y i = 2 : n = x i − sin ( 0.5 π x i ) y_{i=2: n}=x_{i}-\sin \left(0.5 \pi x_{i}\right) yi=2:n=xi−sin(0.5πxi)
POF: f 2 = 1 2 ( 1 − f 1 + 1 − f 1 cos 2 ( 4 π ( 1 − f 1 ) ) ) 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) f2=21(1−f1+1−f1cos2(4π(1−f1)))
POS: x i = sin ( 0.5 π x i ) , i = 2 , … , n x_{i}=\sin \left(0.5 \pi x_{i}\right), i=2, \ldots, n xi=sin(0.5πxi),i=2,…,n
Domain: [ 0 , 1 ] n [0,1]^{n} [0,1]n; Number of Variables = 30
F3:
f 1 ( x ) = ( 1 + g ( x ) ) x 1 f_{1}(x)=(1+g(x)) x_{1} f1(x)=(1+g(x))x1
f 2 ( x ) = 1 2 ( 1 + g ( x ) ) ( 1 − x 1 0.1 + ( 1 − x 1 ) 2 cos 2 ( 3 π 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) f2(x)=21(1+g(x))(1−x10.1+(1−x1)2cos2(3πx1))
g ( x ) = 2 sin ( 0.5 π x 1 ) ( n − 1 + ∑ i = 2 n ( y i 2 − cos ( 2 π y i ) ) ) 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) g(x)=2sin(0.5πx1)(n−1+∑i=2n(yi2−cos(2πyi)))
where
y i = 2 : n = x i − sin ( 0.5 π x i ) y_{i=2: n}=x_{i}-\sin \left(0.5 \pi x_{i}\right) yi=2:n=xi−sin(0.5πxi)
POF: f 2 = 1 2 ( 1 − f 1 0.1 + ( 1 − f 1 ) 2 cos 2 ( 3 π f 1 ) ) 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) f2=21(1−f10.1+(1−f1)2cos2(3πf1))
POS: x i = sin ( 0.5 π x i ) , ∀ x i ∈ x II x_{i}=\sin \left(0.5 \pi x_{i}\right), \forall x_{i} \in \mathbf{x}_{\text {II }} xi=sin(0.5πxi),∀xi∈xII
Domain: [ 0 , 1 ] n [0,1]^{n} [0,1]n; Number of Variables = 30
F4:
f 1 ( x ) = ( 1 + g ( x ) ) ( x 1 x 2 x 3 ) f_{1}(x)=(1+g(x))\left(\frac{x_{1}}{\sqrt{x_{2} x_{3}}}\right) f1(x)=(1+g(x))(x2x3x1)
f 2 ( x ) = ( 1 + g ( x ) ) ( x 2 x 1 x 3 ) f_{2}(x)=(1+g(x))\left(\frac{x_{2}}{\sqrt{x_{1} x_{3}}}\right) f2(x)=(1+g(x))(x1x3x2)
f 3 ( x ) = ( 1 + g ( x ) ) ( x 3 x 1 x 2 ) f_{3}(x)=(1+g(x))\left(\frac{x_{3}}{\sqrt{x_{1} x_{2}}}\right) f3(x)=(1+g(x))(x1x2x3)
where
g ( x ) = ∑ i = 4 n ( x i − 2 ) 2 g(x)=\sum_{i=4}^{n}\left(x_{i}-2\right)^{2} g(x)=∑i=4n(xi−2)2
POF: f 1 f 2 f 3 = 1 f_{1} f_{2} f_{3}=1 f1f2f3=1
POS: x i = 2 , i = 3 , … , n x_{i}=2, i=3, \ldots, n xi=2,i=3,…,n
Domain: [ 1 , 4 ] n {
{[1,4]}^{n}} [1,4]n; Number of Variables = 30
F5:
f 1 ( x ) = ( 1 + g ( x ) ) ( ( 1 − x 1 ) x 2 ) f_{1}(x)=(1+g(x))\left(\left(1-x_{1}\right) x_{2}\right) f1(x)=(1+g(x))((1−x1)x2)
f 2 ( x ) = ( 1 + g ( x ) ) ( x 1 ( 1 − x 2 ) ) f_{2}(x)=(1+g(x))\left(x_{1}\left(1-x_{2}\right)\right) f2(x)=(1+g(x))(x1(1−x2))
f 3 ( x ) = ( 1 + g ( x ) ) ( 1 − x 1 − x 2 + 2 x 1 x 2 ) 6 f_{3}(x)=(1+g(x))\left(1-x_{1}-x_{2}+2 x_{1} x_{2}\right)^{6} f3(x)=(1+g(x))(1−x1−x2+2x1x2)6
where
g ( x ) = ∑ i = 3 n ( x i − 0.5 ) 2 g(x)=\sum_{i=3}^{n}\left(x_{i}-0.5\right)^{2} g(x)=∑i=3n(xi−0.5)2
POF: f 3 = ( 1 − f 1 − f 2 ) 6 f_{3}=\left(1-f_{1}-f_{2}\right)^{6} f3=(1−f1−f2)6
POS: x i = 0.5 , i = 3 , … , n x_{i}=0.5, i=3, \ldots, n xi=0.5,i=3,…,n
Domain: [ 0 , 1 ] n [0,1]^{n} [0,1]n; Number of Variables = 30
F6:
f 1 ( x ) = cos 4 ( 0.5 π x 1 ) cos 4 ( 0.5 π x 2 ) f_{1}(x)=\cos ^{4}\left(0.5 \pi x_{1}\right) \cos ^{4}\left(0.5 \pi x_{2}\right) f1(x)=cos4(0.5πx1)cos4(0.5πx2)
f 2 ( x ) = cos 4 ( 0.5 π x 1 ) sin 4 ( 0.5 π x 2 ) f_{2}(x)=\cos ^{4}\left(0.5 \pi x_{1}\right) \sin ^{4}\left(0.5 \pi x_{2}\right) f2(x)=cos4(0.5πx1)sin4(0.5πx2)
f 3 ( x ) = ( 1 + g ( x ) 1 + cos 2 ( 0.5 π x 1 ) ) 1 1 + g ( x ) f_{3}(x)=\left(\frac{1+g(x)}{1+\cos ^{2}\left(0.5 \pi x_{1}\right)}\right)^{\frac{1}{1+g(x)}} f3(x)=(1+cos2(0.5πx1)1+g(x))1+g(x)1
where
g ( x ) = 1 10 ∑ i = 3 n ( 1 + x i 2 − cos ( 2 π x i ) ) ) \left.g(x)=\frac{1}{10} \sum_{i=3}^{n}\left(1+x_{i}^{2}-\cos \left(2 \pi x_{i}\right)\right)\right) g(x)=101∑i=3n(1+xi2−cos(2πxi)))
POF: f 3 ( 1 + f 1 + f 2 ) = 1 f_{3}(1+\sqrt{f_{1}}+\sqrt{f_{2}})=1 f3(1+f1+f2)=1
POS: x i = 0 , i = 3 , … , n x_{i}=0, i=3, \ldots, n xi=0,i=3,…,n
Domain: [ 0 , 1 ] n [0,1]^{n} [0,1]n; Number of Variables = 30