In applied mathematics, test functions, known as artificial landscapes, are useful to evaluate characteristics of optimization algorithms, such as:
Here some test functions are presented with the aim of giving an idea about the different situations that optimization algorithms have to face when coping with these kinds of problems. In the first part, some objective functions for single-objective optimization cases are presented. In the second part, test functions with their respective Pareto fronts for multi-objective optimization problems (MOP) are given.
The artificial landscapes presented herein for single-objective optimization problems are taken from Bäck, Haupt et al. and from Rody Oldenhuis software. Given the number of problems (55 in total), just a few are presented here.
The test functions used to evaluate the algorithms for MOP were taken from Deb, Binh et al. and Binh. The software developed by Deb can be downloaded, which implements the NSGA-II procedure with GAs, or the program posted on Internet, which implements the NSGA-II procedure with ES.
Just a general form of the equation, a plot of the objective function, boundaries of the object variables and the coordinates of global minima are given herein.
Name | Plot | Formula | Global minimum | Search domain |
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Rastrigin function | ![]() |
f
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x
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=
A
n
+
∑
i
=
1
n
{\displaystyle f(\mathbf {x} )=An+\sum _{i=1}^{n}\left}
where: A = 10 {\displaystyle {\text{where: }}A=10} |
f ( 0 , … , 0 ) = 0 {\displaystyle f(0,\dots ,0)=0} | − 5.12 ≤ x i ≤ 5.12 {\displaystyle -5.12\leq x_{i}\leq 5.12} |
Ackley function | ![]() |
f
(
x
,
y
)
=
−
20
exp
{\displaystyle f(x,y)=-20\exp \left}
− exp + e + 20 {\displaystyle -\exp \left+e+20} |
f ( 0 , 0 ) = 0 {\displaystyle f(0,0)=0} | − 5 ≤ x , y ≤ 5 {\displaystyle -5\leq x,y\leq 5} |
Sphere function | ![]() |
f ( x ) = ∑ i = 1 n x i 2 {\displaystyle f({\boldsymbol {x}})=\sum _{i=1}^{n}x_{i}^{2}} | f ( x 1 , … , x n ) = f ( 0 , … , 0 ) = 0 {\displaystyle f(x_{1},\dots ,x_{n})=f(0,\dots ,0)=0} | − ∞ ≤ x i ≤ ∞ {\displaystyle -\infty \leq x_{i}\leq \infty } | , 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n}
Rosenbrock function | ![]() |
f ( x ) = ∑ i = 1 n − 1 {\displaystyle f({\boldsymbol {x}})=\sum _{i=1}^{n-1}\left} | Min = { n = 2 → f ( 1 , 1 ) = 0 , n = 3 → f ( 1 , 1 , 1 ) = 0 , n > 3 → f ( 1 , … , 1 ⏟ n times ) = 0 {\displaystyle {\text{Min}}={\begin{cases}n=2&\rightarrow \quad f(1,1)=0,\\n=3&\rightarrow \quad f(1,1,1)=0,\\n>3&\rightarrow \quad f(\underbrace {1,\dots ,1} _{n{\text{ times}}})=0\\\end{cases}}} | − ∞ ≤ x i ≤ ∞ {\displaystyle -\infty \leq x_{i}\leq \infty } | , 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n}
Beale function | ![]() |
f
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x
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y
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=
(
1.5
−
x
+
x
y
)
2
+
(
2.25
−
x
+
x
y
2
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2
{\displaystyle f(x,y)=\left(1.5-x+xy\right)^{2}+\left(2.25-x+xy^{2}\right)^{2}}
+ ( 2.625 − x + x y 3 ) 2 {\displaystyle +\left(2.625-x+xy^{3}\right)^{2}} |
f ( 3 , 0.5 ) = 0 {\displaystyle f(3,0.5)=0} | − 4.5 ≤ x , y ≤ 4.5 {\displaystyle -4.5\leq x,y\leq 4.5} |
Goldstein–Price function | ![]() |
f
(
x
,
y
)
=
{\displaystyle f(x,y)=\left}
{\displaystyle \left} |
f ( 0 , − 1 ) = 3 {\displaystyle f(0,-1)=3} | − 2 ≤ x , y ≤ 2 {\displaystyle -2\leq x,y\leq 2} |
Booth function | ![]() |
f ( x , y ) = ( x + 2 y − 7 ) 2 + ( 2 x + y − 5 ) 2 {\displaystyle f(x,y)=\left(x+2y-7\right)^{2}+\left(2x+y-5\right)^{2}} | f ( 1 , 3 ) = 0 {\displaystyle f(1,3)=0} | − 10 ≤ x , y ≤ 10 {\displaystyle -10\leq x,y\leq 10} |
Bukin function N.6 | ![]() |
f ( x , y ) = 100 | y − 0.01 x 2 | + 0.01 | x + 10 | . {\displaystyle f(x,y)=100{\sqrt {\left|y-0.01x^{2}\right|}}+0.01\left|x+10\right|.\quad } | f ( − 10 , 1 ) = 0 {\displaystyle f(-10,1)=0} | − 15 ≤ x ≤ − 5 {\displaystyle -15\leq x\leq -5} | , − 3 ≤ y ≤ 3 {\displaystyle -3\leq y\leq 3}
Matyas function | ![]() |
f ( x , y ) = 0.26 ( x 2 + y 2 ) − 0.48 x y {\displaystyle f(x,y)=0.26\left(x^{2}+y^{2}\right)-0.48xy} | f ( 0 , 0 ) = 0 {\displaystyle f(0,0)=0} | − 10 ≤ x , y ≤ 10 {\displaystyle -10\leq x,y\leq 10} |
Lévi function N.13 | ![]() |
f
(
x
,
y
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=
sin
2
3
π
x
+
(
x
−
1
)
2
(
1
+
sin
2
3
π
y
)
{\displaystyle f(x,y)=\sin ^{2}3\pi x+\left(x-1\right)^{2}\left(1+\sin ^{2}3\pi y\right)}
+ ( y − 1 ) 2 ( 1 + sin 2 2 π y ) {\displaystyle +\left(y-1\right)^{2}\left(1+\sin ^{2}2\pi y\right)} |
f ( 1 , 1 ) = 0 {\displaystyle f(1,1)=0} | − 10 ≤ x , y ≤ 10 {\displaystyle -10\leq x,y\leq 10} |
Himmelblau's function | ![]() |
f ( x , y ) = ( x 2 + y − 11 ) 2 + ( x + y 2 − 7 ) 2 . {\displaystyle f(x,y)=(x^{2}+y-11)^{2}+(x+y^{2}-7)^{2}.\quad } | Min = { f ( 3.0 , 2.0 ) = 0.0 f ( − 2.805118 , 3.131312 ) = 0.0 f ( − 3.779310 , − 3.283186 ) = 0.0 f ( 3.584428 , − 1.848126 ) = 0.0 {\displaystyle {\text{Min}}={\begin{cases}f\left(3.0,2.0\right)&=0.0\\f\left(-2.805118,3.131312\right)&=0.0\\f\left(-3.779310,-3.283186\right)&=0.0\\f\left(3.584428,-1.848126\right)&=0.0\\\end{cases}}} | − 5 ≤ x , y ≤ 5 {\displaystyle -5\leq x,y\leq 5} |
Three-hump camel function | ![]() |
f ( x , y ) = 2 x 2 − 1.05 x 4 + x 6 6 + x y + y 2 {\displaystyle f(x,y)=2x^{2}-1.05x^{4}+{\frac {x^{6}}{6}}+xy+y^{2}} | f ( 0 , 0 ) = 0 {\displaystyle f(0,0)=0} | − 5 ≤ x , y ≤ 5 {\displaystyle -5\leq x,y\leq 5} |
Easom function | ![]() |
f ( x , y ) = − cos ( x ) cos ( y ) exp ( − ( ( x − π ) 2 + ( y − π ) 2 ) ) {\displaystyle f(x,y)=-\cos \left(x\right)\cos \left(y\right)\exp \left(-\left(\left(x-\pi \right)^{2}+\left(y-\pi \right)^{2}\right)\right)} | f ( π , π ) = − 1 {\displaystyle f(\pi ,\pi )=-1} | − 100 ≤ x , y ≤ 100 {\displaystyle -100\leq x,y\leq 100} |
Cross-in-tray function | ![]() |
f ( x , y ) = − 0.0001 0.1 {\displaystyle f(x,y)=-0.0001\left^{0.1}} | Min = { f ( 1.34941 , − 1.34941 ) = − 2.06261 f ( 1.34941 , 1.34941 ) = − 2.06261 f ( − 1.34941 , 1.34941 ) = − 2.06261 f ( − 1.34941 , − 1.34941 ) = − 2.06261 {\displaystyle {\text{Min}}={\begin{cases}f\left(1.34941,-1.34941\right)&=-2.06261\\f\left(1.34941,1.34941\right)&=-2.06261\\f\left(-1.34941,1.34941\right)&=-2.06261\\f\left(-1.34941,-1.34941\right)&=-2.06261\\\end{cases}}} | − 10 ≤ x , y ≤ 10 {\displaystyle -10\leq x,y\leq 10} |
Eggholder function | ![]() |
f ( x , y ) = − ( y + 47 ) sin | x 2 + ( y + 47 ) | − x sin | x − ( y + 47 ) | {\displaystyle f(x,y)=-\left(y+47\right)\sin {\sqrt {\left|{\frac {x}{2}}+\left(y+47\right)\right|}}-x\sin {\sqrt {\left|x-\left(y+47\right)\right|}}} | f ( 512 , 404.2319 ) = − 959.6407 {\displaystyle f(512,404.2319)=-959.6407} | − 512 ≤ x , y ≤ 512 {\displaystyle -512\leq x,y\leq 512} |
Hölder table function | ![]() |
f ( x , y ) = − | sin x cos y exp ( | 1 − x 2 + y 2 π | ) | {\displaystyle f(x,y)=-\left|\sin x\cos y\exp \left(\left|1-{\frac {\sqrt {x^{2}+y^{2}}}{\pi }}\right|\right)\right|} | Min = { f ( 8.05502 , 9.66459 ) = − 19.2085 f ( − 8.05502 , 9.66459 ) = − 19.2085 f ( 8.05502 , − 9.66459 ) = − 19.2085 f ( − 8.05502 , − 9.66459 ) = − 19.2085 {\displaystyle {\text{Min}}={\begin{cases}f\left(8.05502,9.66459\right)&=-19.2085\\f\left(-8.05502,9.66459\right)&=-19.2085\\f\left(8.05502,-9.66459\right)&=-19.2085\\f\left(-8.05502,-9.66459\right)&=-19.2085\end{cases}}} | − 10 ≤ x , y ≤ 10 {\displaystyle -10\leq x,y\leq 10} |
McCormick function | ![]() |
f ( x , y ) = sin ( x + y ) + ( x − y ) 2 − 1.5 x + 2.5 y + 1 {\displaystyle f(x,y)=\sin \left(x+y\right)+\left(x-y\right)^{2}-1.5x+2.5y+1} | f ( − 0.54719 , − 1.54719 ) = − 1.9133 {\displaystyle f(-0.54719,-1.54719)=-1.9133} | − 1.5 ≤ x ≤ 4 {\displaystyle -1.5\leq x\leq 4} | , − 3 ≤ y ≤ 4 {\displaystyle -3\leq y\leq 4}
Schaffer function N. 2 | ![]() |
f ( x , y ) = 0.5 + sin 2 ( x 2 − y 2 ) − 0.5 2 {\displaystyle f(x,y)=0.5+{\frac {\sin ^{2}\left(x^{2}-y^{2}\right)-0.5}{\left^{2}}}} | f ( 0 , 0 ) = 0 {\displaystyle f(0,0)=0} | − 100 ≤ x , y ≤ 100 {\displaystyle -100\leq x,y\leq 100} |
Schaffer function N. 4 | ![]() |
f ( x , y ) = 0.5 + cos 2 − 0.5 2 {\displaystyle f(x,y)=0.5+{\frac {\cos ^{2}\left-0.5}{\left^{2}}}} | Min = { f ( 0 , 1.25313 ) = 0.292579 f ( 0 , − 1.25313 ) = 0.292579 f ( 1.25313 , 0 ) = 0.292579 f ( − 1.25313 , 0 ) = 0.292579 {\displaystyle {\text{Min}}={\begin{cases}f\left(0,1.25313\right)&=0.292579\\f\left(0,-1.25313\right)&=0.292579\\f\left(1.25313,0\right)&=0.292579\\f\left(-1.25313,0\right)&=0.292579\end{cases}}} | − 100 ≤ x , y ≤ 100 {\displaystyle -100\leq x,y\leq 100} |
Styblinski–Tang function | ![]() |
f ( x ) = ∑ i = 1 n x i 4 − 16 x i 2 + 5 x i 2 {\displaystyle f({\boldsymbol {x}})={\frac {\sum _{i=1}^{n}x_{i}^{4}-16x_{i}^{2}+5x_{i}}{2}}} | − 39.16617 n < f ( − 2.903534 , … , − 2.903534 ⏟ n times ) < − 39.16616 n {\displaystyle -39.16617n<f(\underbrace {-2.903534,\ldots ,-2.903534} _{n{\text{ times}}})<-39.16616n} | − 5 ≤ x i ≤ 5 {\displaystyle -5\leq x_{i}\leq 5} | , 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n} ..
Shekel function | ![]() |
f
(
x
→
)
=
∑
i
=
1
m
(
c
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∑
j
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1
n
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a
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i
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1
{\displaystyle f({\vec {x}})=\sum _{i=1}^{m}\;\left(c_{i}+\sum \limits _{j=1}^{n}(x_{j}-a_{ji})^{2}\right)^{-1}}
or, similarly, f ( x 1 , x 2 , . . . , x n − 1 , x n ) = ∑ i = 1 m ( c i + ∑ j = 1 n ( x j − a i j ) 2 ) − 1 {\displaystyle f(x_{1},x_{2},...,x_{n-1},x_{n})=\sum _{i=1}^{m}\;\left(c_{i}+\sum \limits _{j=1}^{n}(x_{j}-a_{ij})^{2}\right)^{-1}} |
− ∞ ≤ x i ≤ ∞ {\displaystyle -\infty \leq x_{i}\leq \infty } | , 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n}
Name | Plot | Formula | Global minimum | Search domain |
---|---|---|---|---|
Rosenbrock function constrained with a cubic and a line | ![]() |
f
(
x
,
y
)
=
(
1
−
x
)
2
+
100
(
y
−
x
2
)
2
{\displaystyle f(x,y)=(1-x)^{2}+100(y-x^{2})^{2}}
subjected to: ( x − 1 ) 3 − y + 1 ≤ 0 and x + y − 2 ≤ 0 {\displaystyle (x-1)^{3}-y+1\leq 0{\text{ and }}x+y-2\leq 0} |
,
f ( 1.0 , 1.0 ) = 0 {\displaystyle f(1.0,1.0)=0} | − 1.5 ≤ x ≤ 1.5 {\displaystyle -1.5\leq x\leq 1.5} | , − 0.5 ≤ y ≤ 2.5 {\displaystyle -0.5\leq y\leq 2.5}
Rosenbrock function constrained to a disk | ![]() |
f
(
x
,
y
)
=
(
1
−
x
)
2
+
100
(
y
−
x
2
)
2
{\displaystyle f(x,y)=(1-x)^{2}+100(y-x^{2})^{2}}
subjected to: x 2 + y 2 ≤ 2 {\displaystyle x^{2}+y^{2}\leq 2} |
,
f ( 1.0 , 1.0 ) = 0 {\displaystyle f(1.0,1.0)=0} | − 1.5 ≤ x ≤ 1.5 {\displaystyle -1.5\leq x\leq 1.5} | , − 1.5 ≤ y ≤ 1.5 {\displaystyle -1.5\leq y\leq 1.5}
Mishra's Bird function - constrained | ![]() |
f
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x
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y
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=
sin
(
y
)
e
+
cos
(
x
)
e
+
(
x
−
y
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2
{\displaystyle f(x,y)=\sin(y)e^{\left}+\cos(x)e^{\left}+(x-y)^{2}}
subjected to: ( x + 5 ) 2 + ( y + 5 ) 2 < 25 {\displaystyle (x+5)^{2}+(y+5)^{2}<25} |
,
f ( − 3.1302468 , − 1.5821422 ) = − 106.7645367 {\displaystyle f(-3.1302468,-1.5821422)=-106.7645367} | − 10 ≤ x ≤ 0 {\displaystyle -10\leq x\leq 0} | , − 6.5 ≤ y ≤ 0 {\displaystyle -6.5\leq y\leq 0}
Townsend function (modified) | ![]() |
f
(
x
,
y
)
=
−
2
−
x
sin
(
3
x
+
y
)
{\displaystyle f(x,y)=-^{2}-x\sin(3x+y)}
subjected to: x 2 + y 2 < 2 + 2 {\displaystyle x^{2}+y^{2}<\left^{2}+^{2}} where: t = Atan2(x,y) |
,
f ( 2.0052938 , 1.1944509 ) = − 2.0239884 {\displaystyle f(2.0052938,1.1944509)=-2.0239884} | − 2.25 ≤ x ≤ 2.25 {\displaystyle -2.25\leq x\leq 2.25} | , − 2.5 ≤ y ≤ 1.75 {\displaystyle -2.5\leq y\leq 1.75}
Gomez and Levy function (modified) | ![]() |
f
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x
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y
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4
x
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−
2.1
x
4
+
1
3
x
6
+
x
y
−
4
y
2
+
4
y
4
{\displaystyle f(x,y)=4x^{2}-2.1x^{4}+{\frac {1}{3}}x^{6}+xy-4y^{2}+4y^{4}}
subjected to: − sin ( 4 π x ) + 2 sin 2 ( 2 π y ) ≤ 1.5 {\displaystyle -\sin(4\pi x)+2\sin ^{2}(2\pi y)\leq 1.5} |
,
f ( 0.08984201 , − 0.7126564 ) = − 1.031628453 {\displaystyle f(0.08984201,-0.7126564)=-1.031628453} | − 1 ≤ x ≤ 0.75 {\displaystyle -1\leq x\leq 0.75} | , − 1 ≤ y ≤ 1 {\displaystyle -1\leq y\leq 1}
Simionescu function | ![]() |
f
(
x
,
y
)
=
0.1
x
y
{\displaystyle f(x,y)=0.1xy}
subjected to: x 2 + y 2 ≤ 2 {\displaystyle x^{2}+y^{2}\leq \left^{2}} where: r T = 1 , r S = 0.2 and n = 8 {\displaystyle {\text{where: }}r_{T}=1,r_{S}=0.2{\text{ and }}n=8} |
,
f ( ± 0.84852813 , ∓ 0.84852813 ) = − 0.072 {\displaystyle f(\pm 0.84852813,\mp 0.84852813)=-0.072} | − 1.25 ≤ x , y ≤ 1.25 {\displaystyle -1.25\leq x,y\leq 1.25} |
Name | Plot | Functions | Constraints | Search domain |
---|---|---|---|---|
Binh and Korn function: | ![]() |
Minimize = { f 1 ( x , y ) = 4 x 2 + 4 y 2 f 2 ( x , y ) = ( x − 5 ) 2 + ( y − 5 ) 2 {\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)=4x^{2}+4y^{2}\\f_{2}\left(x,y\right)=\left(x-5\right)^{2}+\left(y-5\right)^{2}\\\end{cases}}} | s.t. = { g 1 ( x , y ) = ( x − 5 ) 2 + y 2 ≤ 25 g 2 ( x , y ) = ( x − 8 ) 2 + ( y + 3 ) 2 ≥ 7.7 {\displaystyle {\text{s.t.}}={\begin{cases}g_{1}\left(x,y\right)=\left(x-5\right)^{2}+y^{2}\leq 25\\g_{2}\left(x,y\right)=\left(x-8\right)^{2}+\left(y+3\right)^{2}\geq 7.7\\\end{cases}}} | 0 ≤ x ≤ 5 {\displaystyle 0\leq x\leq 5} | , 0 ≤ y ≤ 3 {\displaystyle 0\leq y\leq 3}
Chankong and Haimes function: | ![]() |
Minimize = { f 1 ( x , y ) = 2 + ( x − 2 ) 2 + ( y − 1 ) 2 f 2 ( x , y ) = 9 x − ( y − 1 ) 2 {\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)=2+\left(x-2\right)^{2}+\left(y-1\right)^{2}\\f_{2}\left(x,y\right)=9x-\left(y-1\right)^{2}\\\end{cases}}} | s.t. = { g 1 ( x , y ) = x 2 + y 2 ≤ 225 g 2 ( x , y ) = x − 3 y + 10 ≤ 0 {\displaystyle {\text{s.t.}}={\begin{cases}g_{1}\left(x,y\right)=x^{2}+y^{2}\leq 225\\g_{2}\left(x,y\right)=x-3y+10\leq 0\\\end{cases}}} | − 20 ≤ x , y ≤ 20 {\displaystyle -20\leq x,y\leq 20} |
Fonseca–Fleming function: | ![]() |
Minimize = { f 1 ( x ) = 1 − exp f 2 ( x ) = 1 − exp {\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)=1-\exp \left\\f_{2}\left({\boldsymbol {x}}\right)=1-\exp \left\\\end{cases}}} | − 4 ≤ x i ≤ 4 {\displaystyle -4\leq x_{i}\leq 4} | , 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n}|
Test function 4: | ![]() |
Minimize = { f 1 ( x , y ) = x 2 − y f 2 ( x , y ) = − 0.5 x − y − 1 {\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)=x^{2}-y\\f_{2}\left(x,y\right)=-0.5x-y-1\\\end{cases}}} | s.t. = { g 1 ( x , y ) = 6.5 − x 6 − y ≥ 0 g 2 ( x , y ) = 7.5 − 0.5 x − y ≥ 0 g 3 ( x , y ) = 30 − 5 x − y ≥ 0 {\displaystyle {\text{s.t.}}={\begin{cases}g_{1}\left(x,y\right)=6.5-{\frac {x}{6}}-y\geq 0\\g_{2}\left(x,y\right)=7.5-0.5x-y\geq 0\\g_{3}\left(x,y\right)=30-5x-y\geq 0\\\end{cases}}} | − 7 ≤ x , y ≤ 4 {\displaystyle -7\leq x,y\leq 4} |
Kursawe function: | ![]() |
Minimize = { f 1 ( x ) = ∑ i = 1 2 f 2 ( x ) = ∑ i = 1 3 {\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)=\sum _{i=1}^{2}\left\\&\\f_{2}\left({\boldsymbol {x}}\right)=\sum _{i=1}^{3}\left\\\end{cases}}} | − 5 ≤ x i ≤ 5 {\displaystyle -5\leq x_{i}\leq 5} | , 1 ≤ i ≤ 3 {\displaystyle 1\leq i\leq 3} .|
Schaffer function N. 1: | ![]() |
Minimize = { f 1 ( x ) = x 2 f 2 ( x ) = ( x − 2 ) 2 {\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left(x\right)=x^{2}\\f_{2}\left(x\right)=\left(x-2\right)^{2}\\\end{cases}}} | − A ≤ x ≤ A {\displaystyle -A\leq x\leq A} | . Values of A {\displaystyle A} from 10 {\displaystyle 10} to 10 5 {\displaystyle 10^{5}} have been used successfully. Higher values of A {\displaystyle A} increase the difficulty of the problem.|
Schaffer function N. 2: | ![]() |
Minimize = { f 1 ( x ) = { − x , if x ≤ 1 x − 2 , if 1 < x ≤ 3 4 − x , if 3 < x ≤ 4 x − 4 , if x > 4 f 2 ( x ) = ( x − 5 ) 2 {\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left(x\right)={\begin{cases}-x,&{\text{if }}x\leq 1\\x-2,&{\text{if }}1<x\leq 3\\4-x,&{\text{if }}3<x\leq 4\\x-4,&{\text{if }}x>4\\\end{cases}}\\f_{2}\left(x\right)=\left(x-5\right)^{2}\\\end{cases}}} | − 5 ≤ x ≤ 10 {\displaystyle -5\leq x\leq 10} | .|
Poloni's two objective function: | ![]() |
Minimize
=
{
f
1
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x
,
y
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=
f
2
(
x
,
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=
(
x
+
3
)
2
+
(
y
+
1
)
2
{\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)=\left\\f_{2}\left(x,y\right)=\left(x+3\right)^{2}+\left(y+1\right)^{2}\\\end{cases}}}
where = { A 1 = 0.5 sin ( 1 ) − 2 cos ( 1 ) + sin ( 2 ) − 1.5 cos ( 2 ) A 2 = 1.5 sin ( 1 ) − cos ( 1 ) + 2 sin ( 2 ) − 0.5 cos ( 2 ) B 1 ( x , y ) = 0.5 sin ( x ) − 2 cos ( x ) + sin ( y ) − 1.5 cos ( y ) B 2 ( x , y ) = 1.5 sin ( x ) − cos ( x ) + 2 sin ( y ) − 0.5 cos ( y ) {\displaystyle {\text{where}}={\begin{cases}A_{1}=0.5\sin \left(1\right)-2\cos \left(1\right)+\sin \left(2\right)-1.5\cos \left(2\right)\\A_{2}=1.5\sin \left(1\right)-\cos \left(1\right)+2\sin \left(2\right)-0.5\cos \left(2\right)\\B_{1}\left(x,y\right)=0.5\sin \left(x\right)-2\cos \left(x\right)+\sin \left(y\right)-1.5\cos \left(y\right)\\B_{2}\left(x,y\right)=1.5\sin \left(x\right)-\cos \left(x\right)+2\sin \left(y\right)-0.5\cos \left(y\right)\end{cases}}} |
− π ≤ x , y ≤ π {\displaystyle -\pi \leq x,y\leq \pi } | |
Zitzler–Deb–Thiele's function N. 1: | ![]() |
Minimize = { f 1 ( x ) = x 1 f 2 ( x ) = g ( x ) h ( f 1 ( x ) , g ( x ) ) g ( x ) = 1 + 9 29 ∑ i = 2 30 x i h ( f 1 ( x ) , g ( x ) ) = 1 − f 1 ( x ) g ( x ) {\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)=x_{1}\\f_{2}\left({\boldsymbol {x}}\right)=g\left({\boldsymbol {x}}\right)h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)\\g\left({\boldsymbol {x}}\right)=1+{\frac {9}{29}}\sum _{i=2}^{30}x_{i}\\h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)=1-{\sqrt {\frac {f_{1}\left({\boldsymbol {x}}\right)}{g\left({\boldsymbol {x}}\right)}}}\\\end{cases}}} | 0 ≤ x i ≤ 1 {\displaystyle 0\leq x_{i}\leq 1} | , 1 ≤ i ≤ 30 {\displaystyle 1\leq i\leq 30} .|
Zitzler–Deb–Thiele's function N. 2: | ![]() |
Minimize = { f 1 ( x ) = x 1 f 2 ( x ) = g ( x ) h ( f 1 ( x ) , g ( x ) ) g ( x ) = 1 + 9 29 ∑ i = 2 30 x i h ( f 1 ( x ) , g ( x ) ) = 1 − ( f 1 ( x ) g ( x ) ) 2 {\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)=x_{1}\\f_{2}\left({\boldsymbol {x}}\right)=g\left({\boldsymbol {x}}\right)h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)\\g\left({\boldsymbol {x}}\right)=1+{\frac {9}{29}}\sum _{i=2}^{30}x_{i}\\h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)=1-\left({\frac {f_{1}\left({\boldsymbol {x}}\right)}{g\left({\boldsymbol {x}}\right)}}\right)^{2}\\\end{cases}}} | 0 ≤ x i ≤ 1 {\displaystyle 0\leq x_{i}\leq 1} | , 1 ≤ i ≤ 30 {\displaystyle 1\leq i\leq 30} .|
Zitzler–Deb–Thiele's function N. 3: | ![]() |
Minimize = { f 1 ( x ) = x 1 f 2 ( x ) = g ( x ) h ( f 1 ( x ) , g ( x ) ) g ( x ) = 1 + 9 29 ∑ i = 2 30 x i h ( f 1 ( x ) , g ( x ) ) = 1 − f 1 ( x ) g ( x ) − ( f 1 ( x ) g ( x ) ) sin ( 10 π f 1 ( x ) ) {\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)=x_{1}\\f_{2}\left({\boldsymbol {x}}\right)=g\left({\boldsymbol {x}}\right)h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)\\g\left({\boldsymbol {x}}\right)=1+{\frac {9}{29}}\sum _{i=2}^{30}x_{i}\\h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)=1-{\sqrt {\frac {f_{1}\left({\boldsymbol {x}}\right)}{g\left({\boldsymbol {x}}\right)}}}-\left({\frac {f_{1}\left({\boldsymbol {x}}\right)}{g\left({\boldsymbol {x}}\right)}}\right)\sin \left(10\pi f_{1}\left({\boldsymbol {x}}\right)\right)\end{cases}}} | 0 ≤ x i ≤ 1 {\displaystyle 0\leq x_{i}\leq 1} | , 1 ≤ i ≤ 30 {\displaystyle 1\leq i\leq 30} .|
Zitzler–Deb–Thiele's function N. 4: | ![]() |
Minimize = { f 1 ( x ) = x 1 f 2 ( x ) = g ( x ) h ( f 1 ( x ) , g ( x ) ) g ( x ) = 91 + ∑ i = 2 10 ( x i 2 − 10 cos ( 4 π x i ) ) h ( f 1 ( x ) , g ( x ) ) = 1 − f 1 ( x ) g ( x ) {\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)=x_{1}\\f_{2}\left({\boldsymbol {x}}\right)=g\left({\boldsymbol {x}}\right)h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)\\g\left({\boldsymbol {x}}\right)=91+\sum _{i=2}^{10}\left(x_{i}^{2}-10\cos \left(4\pi x_{i}\right)\right)\\h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)=1-{\sqrt {\frac {f_{1}\left({\boldsymbol {x}}\right)}{g\left({\boldsymbol {x}}\right)}}}\end{cases}}} | 0 ≤ x 1 ≤ 1 {\displaystyle 0\leq x_{1}\leq 1} | , − 5 ≤ x i ≤ 5 {\displaystyle -5\leq x_{i}\leq 5} , 2 ≤ i ≤ 10 {\displaystyle 2\leq i\leq 10}|
Zitzler–Deb–Thiele's function N. 6: | ![]() |
Minimize = { f 1 ( x ) = 1 − exp ( − 4 x 1 ) sin 6 ( 6 π x 1 ) f 2 ( x ) = g ( x ) h ( f 1 ( x ) , g ( x ) ) g ( x ) = 1 + 9 0.25 h ( f 1 ( x ) , g ( x ) ) = 1 − ( f 1 ( x ) g ( x ) ) 2 {\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)=1-\exp \left(-4x_{1}\right)\sin ^{6}\left(6\pi x_{1}\right)\\f_{2}\left({\boldsymbol {x}}\right)=g\left({\boldsymbol {x}}\right)h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)\\g\left({\boldsymbol {x}}\right)=1+9\left^{0.25}\\h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)=1-\left({\frac {f_{1}\left({\boldsymbol {x}}\right)}{g\left({\boldsymbol {x}}\right)}}\right)^{2}\\\end{cases}}} | 0 ≤ x i ≤ 1 {\displaystyle 0\leq x_{i}\leq 1} | , 1 ≤ i ≤ 10 {\displaystyle 1\leq i\leq 10} .|
Osyczka and Kundu function: | ![]() |
Minimize = { f 1 ( x ) = − 25 ( x 1 − 2 ) 2 − ( x 2 − 2 ) 2 − ( x 3 − 1 ) 2 − ( x 4 − 4 ) 2 − ( x 5 − 1 ) 2 f 2 ( x ) = ∑ i = 1 6 x i 2 {\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)=-25\left(x_{1}-2\right)^{2}-\left(x_{2}-2\right)^{2}-\left(x_{3}-1\right)^{2}-\left(x_{4}-4\right)^{2}-\left(x_{5}-1\right)^{2}\\f_{2}\left({\boldsymbol {x}}\right)=\sum _{i=1}^{6}x_{i}^{2}\\\end{cases}}} | s.t. = { g 1 ( x ) = x 1 + x 2 − 2 ≥ 0 g 2 ( x ) = 6 − x 1 − x 2 ≥ 0 g 3 ( x ) = 2 − x 2 + x 1 ≥ 0 g 4 ( x ) = 2 − x 1 + 3 x 2 ≥ 0 g 5 ( x ) = 4 − ( x 3 − 3 ) 2 − x 4 ≥ 0 g 6 ( x ) = ( x 5 − 3 ) 2 + x 6 − 4 ≥ 0 {\displaystyle {\text{s.t.}}={\begin{cases}g_{1}\left({\boldsymbol {x}}\right)=x_{1}+x_{2}-2\geq 0\\g_{2}\left({\boldsymbol {x}}\right)=6-x_{1}-x_{2}\geq 0\\g_{3}\left({\boldsymbol {x}}\right)=2-x_{2}+x_{1}\geq 0\\g_{4}\left({\boldsymbol {x}}\right)=2-x_{1}+3x_{2}\geq 0\\g_{5}\left({\boldsymbol {x}}\right)=4-\left(x_{3}-3\right)^{2}-x_{4}\geq 0\\g_{6}\left({\boldsymbol {x}}\right)=\left(x_{5}-3\right)^{2}+x_{6}-4\geq 0\end{cases}}} | 0 ≤ x 1 , x 2 , x 6 ≤ 10 {\displaystyle 0\leq x_{1},x_{2},x_{6}\leq 10} | , 1 ≤ x 3 , x 5 ≤ 5 {\displaystyle 1\leq x_{3},x_{5}\leq 5} , 0 ≤ x 4 ≤ 6 {\displaystyle 0\leq x_{4}\leq 6} .
CTP1 function (2 variables): | ![]() |
Minimize = { f 1 ( x , y ) = x f 2 ( x , y ) = ( 1 + y ) exp ( − x 1 + y ) {\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)=x\\f_{2}\left(x,y\right)=\left(1+y\right)\exp \left(-{\frac {x}{1+y}}\right)\end{cases}}} | s.t. = { g 1 ( x , y ) = f 2 ( x , y ) 0.858 exp ( − 0.541 f 1 ( x , y ) ) ≥ 1 g 2 ( x , y ) = f 2 ( x , y ) 0.728 exp ( − 0.295 f 1 ( x , y ) ) ≥ 1 {\displaystyle {\text{s.t.}}={\begin{cases}g_{1}\left(x,y\right)={\frac {f_{2}\left(x,y\right)}{0.858\exp \left(-0.541f_{1}\left(x,y\right)\right)}}\geq 1\\g_{2}\left(x,y\right)={\frac {f_{2}\left(x,y\right)}{0.728\exp \left(-0.295f_{1}\left(x,y\right)\right)}}\geq 1\end{cases}}} | 0 ≤ x , y ≤ 1 {\displaystyle 0\leq x,y\leq 1} | .
Constr-Ex problem: | ![]() |
Minimize = { f 1 ( x , y ) = x f 2 ( x , y ) = 1 + y x {\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)=x\\f_{2}\left(x,y\right)={\frac {1+y}{x}}\\\end{cases}}} | s.t. = { g 1 ( x , y ) = y + 9 x ≥ 6 g 2 ( x , y ) = − y + 9 x ≥ 1 {\displaystyle {\text{s.t.}}={\begin{cases}g_{1}\left(x,y\right)=y+9x\geq 6\\g_{2}\left(x,y\right)=-y+9x\geq 1\\\end{cases}}} | 0.1 ≤ x ≤ 1 {\displaystyle 0.1\leq x\leq 1} | , 0 ≤ y ≤ 5 {\displaystyle 0\leq y\leq 5}
Viennet function: | ![]() |
Minimize = { f 1 ( x , y ) = 0.5 ( x 2 + y 2 ) + sin ( x 2 + y 2 ) f 2 ( x , y ) = ( 3 x − 2 y + 4 ) 2 8 + ( x − y + 1 ) 2 27 + 15 f 3 ( x , y ) = 1 x 2 + y 2 + 1 − 1.1 exp ( − ( x 2 + y 2 ) ) {\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)=0.5\left(x^{2}+y^{2}\right)+\sin \left(x^{2}+y^{2}\right)\\f_{2}\left(x,y\right)={\frac {\left(3x-2y+4\right)^{2}}{8}}+{\frac {\left(x-y+1\right)^{2}}{27}}+15\\f_{3}\left(x,y\right)={\frac {1}{x^{2}+y^{2}+1}}-1.1\exp \left(-\left(x^{2}+y^{2}\right)\right)\\\end{cases}}} | − 3 ≤ x , y ≤ 3 {\displaystyle -3\leq x,y\leq 3} | .