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Test functions for optimization

From Wikipedia, the free encyclopedia
Functions used to evaluate optimization algorithms

In applied mathematics, test functions, known as artificial landscapes, are useful to evaluate characteristics of optimization algorithms, such as convergence rate, precision, robustness and general performance.

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,[1] Haupt et al.[2] and from Rody Oldenhuis software.[3] 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,[4] Binh et al.[5] and Binh.[6] The software developed by Deb can be downloaded,[7] which implements the NSGA-II procedure with GAs, or the program posted on Internet,[8] 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.

Test functions for single-objective optimization

[edit ]
Name Plot Formula Global minimum Search domain
Rastrigin function Rastrigin function for n=2 f ( x ) = A n + ∑ i = 1 n [ x i 2 − A cos ⁡ ( 2 π x i ) ] {\displaystyle f(\mathbf {x} )=An+\sum _{i=1}^{n}\left[x_{i}^{2}-A\cos(2\pi x_{i})\right]} {\displaystyle f(\mathbf {x} )=An+\sum _{i=1}^{n}\left[x_{i}^{2}-A\cos(2\pi x_{i})\right]}

where:  A = 10 {\displaystyle {\text{where: }}A=10} {\displaystyle {\text{where: }}A=10}

f ( 0 , … , 0 ) = 0 {\displaystyle f(0,\dots ,0)=0} {\displaystyle f(0,\dots ,0)=0} − 5.12 ≤ x i ≤ 5.12 {\displaystyle -5.12\leq x_{i}\leq 5.12} {\displaystyle -5.12\leq x_{i}\leq 5.12}
Ackley function Ackley's function for n=2 f ( x , y ) = − 20 exp ⁡ [ − 0.2 0.5 ( x 2 + y 2 ) ] {\displaystyle f(x,y)=-20\exp \left[-0.2{\sqrt {0.5\left(x^{2}+y^{2}\right)}}\right]} {\displaystyle f(x,y)=-20\exp \left[-0.2{\sqrt {0.5\left(x^{2}+y^{2}\right)}}\right]}

− exp ⁡ [ 0.5 ( cos ⁡ 2 π x + cos ⁡ 2 π y ) ] + e + 20 {\displaystyle -\exp \left[0.5\left(\cos 2\pi x+\cos 2\pi y\right)\right]+e+20} {\displaystyle -\exp \left[0.5\left(\cos 2\pi x+\cos 2\pi y\right)\right]+e+20}

f ( 0 , 0 ) = 0 {\displaystyle f(0,0)=0} {\displaystyle f(0,0)=0} − 5 ≤ x , y ≤ 5 {\displaystyle -5\leq x,y\leq 5} {\displaystyle -5\leq x,y\leq 5}
Sphere function Sphere function for n=2 f ( x ) = ∑ i = 1 n x i 2 {\displaystyle f({\boldsymbol {x}})=\sum _{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} {\displaystyle f(x_{1},\dots ,x_{n})=f(0,\dots ,0)=0} − ∞ ≤ x i ≤ ∞ {\displaystyle -\infty \leq x_{i}\leq \infty } {\displaystyle -\infty \leq x_{i}\leq \infty }, 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n} {\displaystyle 1\leq i\leq n}
Rosenbrock function Rosenbrock's function for n=2 f ( x ) = ∑ i = 1 n − 1 [ 100 ( x i + 1 − x i 2 ) 2 + ( 1 − x i ) 2 ] {\displaystyle f({\boldsymbol {x}})=\sum _{i=1}^{n-1}\left[100\left(x_{i+1}-x_{i}^{2}\right)^{2}+\left(1-x_{i}\right)^{2}\right]} {\displaystyle f({\boldsymbol {x}})=\sum _{i=1}^{n-1}\left[100\left(x_{i+1}-x_{i}^{2}\right)^{2}+\left(1-x_{i}\right)^{2}\right]} 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}}} {\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 } {\displaystyle -\infty \leq x_{i}\leq \infty }, 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n} {\displaystyle 1\leq i\leq n}
Beale function Beale's function f ( x , y ) = ( 1.5 − x + x y ) 2 + ( 2.25 − x + x y 2 ) 2 {\displaystyle f(x,y)=\left(1.5-x+xy\right)^{2}+\left(2.25-x+xy^{2}\right)^{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}} {\displaystyle +\left(2.625-x+xy^{3}\right)^{2}}

f ( 3 , 0.5 ) = 0 {\displaystyle 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} {\displaystyle -4.5\leq x,y\leq 4.5}
Goldstein–Price function Goldstein–Price function f ( x , y ) = [ 1 + ( x + y + 1 ) 2 ( 19 − 14 x + 3 x 2 − 14 y + 6 x y + 3 y 2 ) ] {\displaystyle f(x,y)=\left[1+\left(x+y+1\right)^{2}\left(19-14x+3x^{2}-14y+6xy+3y^{2}\right)\right]} {\displaystyle f(x,y)=\left[1+\left(x+y+1\right)^{2}\left(19-14x+3x^{2}-14y+6xy+3y^{2}\right)\right]}

[ 30 + ( 2 x − 3 y ) 2 ( 18 − 32 x + 12 x 2 + 48 y − 36 x y + 27 y 2 ) ] {\displaystyle \left[30+\left(2x-3y\right)^{2}\left(18-32x+12x^{2}+48y-36xy+27y^{2}\right)\right]} {\displaystyle \left[30+\left(2x-3y\right)^{2}\left(18-32x+12x^{2}+48y-36xy+27y^{2}\right)\right]}

f ( 0 , − 1 ) = 3 {\displaystyle f(0,-1)=3} {\displaystyle f(0,-1)=3} − 2 ≤ x , y ≤ 2 {\displaystyle -2\leq x,y\leq 2} {\displaystyle -2\leq x,y\leq 2}
Booth function Booth's 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}} {\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} {\displaystyle f(1,3)=0} − 10 ≤ x , y ≤ 10 {\displaystyle -10\leq x,y\leq 10} {\displaystyle -10\leq x,y\leq 10}
Bukin function N.6 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 } {\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} {\displaystyle f(-10,1)=0} − 15 ≤ x ≤ − 5 {\displaystyle -15\leq x\leq -5} {\displaystyle -15\leq x\leq -5}, − 3 ≤ y ≤ 3 {\displaystyle -3\leq y\leq 3} {\displaystyle -3\leq y\leq 3}
Matyas function 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} {\displaystyle f(x,y)=0.26\left(x^{2}+y^{2}\right)-0.48xy} f ( 0 , 0 ) = 0 {\displaystyle f(0,0)=0} {\displaystyle f(0,0)=0} − 10 ≤ x , y ≤ 10 {\displaystyle -10\leq x,y\leq 10} {\displaystyle -10\leq x,y\leq 10}
Lévi function N.13 Lévi function N.13 f ( x , y ) = 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)} {\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)} {\displaystyle +\left(y-1\right)^{2}\left(1+\sin ^{2}2\pi y\right)}

f ( 1 , 1 ) = 0 {\displaystyle f(1,1)=0} {\displaystyle f(1,1)=0} − 10 ≤ x , y ≤ 10 {\displaystyle -10\leq x,y\leq 10} {\displaystyle -10\leq x,y\leq 10}
Griewank function Griewank's function f ( x ) = 1 + 1 4000 ∑ i = 1 n x i 2 − ∏ i = 1 n P i ( x i ) {\displaystyle f({\boldsymbol {x}})=1+{\frac {1}{4000}}\sum _{i=1}^{n}x_{i}^{2}-\prod _{i=1}^{n}P_{i}(x_{i})} {\displaystyle f({\boldsymbol {x}})=1+{\frac {1}{4000}}\sum _{i=1}^{n}x_{i}^{2}-\prod _{i=1}^{n}P_{i}(x_{i})}, where P i ( x i ) = cos ⁡ ( x i i ) {\displaystyle P_{i}(x_{i})=\cos \left({\frac {x_{i}}{\sqrt {i}}}\right)} {\displaystyle P_{i}(x_{i})=\cos \left({\frac {x_{i}}{\sqrt {i}}}\right)} f ( 0 , … , 0 ) = 0 {\displaystyle f(0,\dots ,0)=0} {\displaystyle f(0,\dots ,0)=0} − ∞ ≤ x i ≤ ∞ {\displaystyle -\infty \leq x_{i}\leq \infty } {\displaystyle -\infty \leq x_{i}\leq \infty }, 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n} {\displaystyle 1\leq i\leq n}
Himmelblau's function 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 } {\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}}} {\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} {\displaystyle -5\leq x,y\leq 5}
Three-hump camel function 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}} {\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} {\displaystyle f(0,0)=0} − 5 ≤ x , y ≤ 5 {\displaystyle -5\leq x,y\leq 5} {\displaystyle -5\leq x,y\leq 5}
Easom function 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)} {\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} {\displaystyle f(\pi ,\pi )=-1} − 100 ≤ x , y ≤ 100 {\displaystyle -100\leq x,y\leq 100} {\displaystyle -100\leq x,y\leq 100}
Cross-in-tray function Cross-in-tray function f ( x , y ) = − 0.0001 [ | sin ⁡ x sin ⁡ y exp ⁡ ( | 100 − x 2 + y 2 π | ) | + 1 ] 0.1 {\displaystyle f(x,y)=-0.0001\left[\left|\sin x\sin y\exp \left(\left|100-{\frac {\sqrt {x^{2}+y^{2}}}{\pi }}\right|\right)\right|+1\right]^{0.1}} {\displaystyle f(x,y)=-0.0001\left[\left|\sin x\sin y\exp \left(\left|100-{\frac {\sqrt {x^{2}+y^{2}}}{\pi }}\right|\right)\right|+1\right]^{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}}} {\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} {\displaystyle -10\leq x,y\leq 10}
Eggholder function [9] [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|}}} {\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} {\displaystyle f(512,404.2319)=-959.6407} − 512 ≤ x , y ≤ 512 {\displaystyle -512\leq x,y\leq 512} {\displaystyle -512\leq x,y\leq 512}
Hölder table function Holder 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|} {\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}}} {\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} {\displaystyle -10\leq x,y\leq 10}
McCormick function 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} {\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} {\displaystyle f(-0.54719,-1.54719)=-1.9133} − 1.5 ≤ x ≤ 4 {\displaystyle -1.5\leq x\leq 4} {\displaystyle -1.5\leq x\leq 4}, − 3 ≤ y ≤ 4 {\displaystyle -3\leq y\leq 4} {\displaystyle -3\leq y\leq 4}
Schaffer function N. 2 Schaffer function N.2 f ( x , y ) = 0.5 + sin 2 ⁡ ( x 2 − y 2 ) − 0.5 [ 1 + 0.001 ( x 2 + y 2 ) ] 2 {\displaystyle f(x,y)=0.5+{\frac {\sin ^{2}\left(x^{2}-y^{2}\right)-0.5}{\left[1+0.001\left(x^{2}+y^{2}\right)\right]^{2}}}} {\displaystyle f(x,y)=0.5+{\frac {\sin ^{2}\left(x^{2}-y^{2}\right)-0.5}{\left[1+0.001\left(x^{2}+y^{2}\right)\right]^{2}}}} f ( 0 , 0 ) = 0 {\displaystyle f(0,0)=0} {\displaystyle f(0,0)=0} − 100 ≤ x , y ≤ 100 {\displaystyle -100\leq x,y\leq 100} {\displaystyle -100\leq x,y\leq 100}
Schaffer function N. 4 Schaffer function N.4 f ( x , y ) = 0.5 + cos 2 ⁡ [ sin ⁡ ( | x 2 − y 2 | ) ] − 0.5 [ 1 + 0.001 ( x 2 + y 2 ) ] 2 {\displaystyle f(x,y)=0.5+{\frac {\cos ^{2}\left[\sin \left(\left|x^{2}-y^{2}\right|\right)\right]-0.5}{\left[1+0.001\left(x^{2}+y^{2}\right)\right]^{2}}}} {\displaystyle f(x,y)=0.5+{\frac {\cos ^{2}\left[\sin \left(\left|x^{2}-y^{2}\right|\right)\right]-0.5}{\left[1+0.001\left(x^{2}+y^{2}\right)\right]^{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}}} {\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} {\displaystyle -100\leq x,y\leq 100}
Styblinski–Tang function 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}}} {\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} {\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} {\displaystyle -5\leq x_{i}\leq 5}, 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n} {\displaystyle 1\leq i\leq n}..
Shekel function A Shekel function in 2 dimensions and with 10 maxima f ( x ) = ∑ i = 1 m ( c i + ∑ j = 1 n ( x j − a j i ) 2 ) − 1 {\displaystyle f({\boldsymbol {x}})=\sum _{i=1}^{m}\;\left(c_{i}+\sum \limits _{j=1}^{n}(x_{j}-a_{ji})^{2}\right)^{-1}} {\displaystyle f({\boldsymbol {x}})=\sum _{i=1}^{m}\;\left(c_{i}+\sum \limits _{j=1}^{n}(x_{j}-a_{ji})^{2}\right)^{-1}} − ∞ ≤ x i ≤ ∞ {\displaystyle -\infty \leq x_{i}\leq \infty } {\displaystyle -\infty \leq x_{i}\leq \infty }, 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n} {\displaystyle 1\leq i\leq n}

Test functions for constrained optimization

[edit ]
Name Plot Formula Global minimum Search domain
Rosenbrock function constrained to a disk[11] 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}} {\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} {\displaystyle x^{2}+y^{2}\leq 2}

f ( 1.0 , 1.0 ) = 0 {\displaystyle 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} {\displaystyle -1.5\leq x\leq 1.5}, − 1.5 ≤ y ≤ 1.5 {\displaystyle -1.5\leq y\leq 1.5} {\displaystyle -1.5\leq y\leq 1.5}
Mishra's Bird function - constrained[12] [13] Bird function (constrained) f ( x , y ) = sin ⁡ ( y ) e [ ( 1 − cos ⁡ x ) 2 ] + cos ⁡ ( x ) e [ ( 1 − sin ⁡ y ) 2 ] + ( x − y ) 2 {\displaystyle f(x,y)=\sin(y)e^{\left[(1-\cos x)^{2}\right]}+\cos(x)e^{\left[(1-\sin y)^{2}\right]}+(x-y)^{2}} {\displaystyle f(x,y)=\sin(y)e^{\left[(1-\cos x)^{2}\right]}+\cos(x)e^{\left[(1-\sin y)^{2}\right]}+(x-y)^{2}},

subjected to: ( x + 5 ) 2 + ( y + 5 ) 2 < 25 {\displaystyle (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} {\displaystyle f(-3.1302468,-1.5821422)=-106.7645367} − 10 ≤ x ≤ 0 {\displaystyle -10\leq x\leq 0} {\displaystyle -10\leq x\leq 0}, − 6.5 ≤ y ≤ 0 {\displaystyle -6.5\leq y\leq 0} {\displaystyle -6.5\leq y\leq 0}
Townsend function (modified)[14] Heart constrained multimodal function f ( x , y ) = − [ cos ⁡ ( ( x − 0.1 ) y ) ] 2 − x sin ⁡ ( 3 x + y ) {\displaystyle f(x,y)=-[\cos((x-0.1)y)]^{2}-x\sin(3x+y)} {\displaystyle f(x,y)=-[\cos((x-0.1)y)]^{2}-x\sin(3x+y)},

subjected to: x 2 + y 2 < [ 2 cos ⁡ t − 1 2 cos ⁡ 2 t − 1 4 cos ⁡ 3 t − 1 8 cos ⁡ 4 t ] 2 + [ 2 sin ⁡ t ] 2 {\displaystyle x^{2}+y^{2}<\left[2\cos t-{\frac {1}{2}}\cos 2t-{\frac {1}{4}}\cos 3t-{\frac {1}{8}}\cos 4t\right]^{2}+[2\sin t]^{2}} {\displaystyle x^{2}+y^{2}<\left[2\cos t-{\frac {1}{2}}\cos 2t-{\frac {1}{4}}\cos 3t-{\frac {1}{8}}\cos 4t\right]^{2}+[2\sin t]^{2}} where: t = Atan2(x,y)

f ( 2.0052938 , 1.1944509 ) = − 2.0239884 {\displaystyle 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} {\displaystyle -2.25\leq x\leq 2.25}, − 2.5 ≤ y ≤ 1.75 {\displaystyle -2.5\leq y\leq 1.75} {\displaystyle -2.5\leq y\leq 1.75}
Keane's bump function[15] Keane's bump function f ( x ) = − | [ ∑ i = 1 m cos 4 ⁡ ( x i ) − 2 ∏ i = 1 m cos 2 ⁡ ( x i ) ] ( ∑ i = 1 m i x i 2 ) 0.5 | {\displaystyle f({\boldsymbol {x}})=-\left|{\frac {\left[\sum _{i=1}^{m}\cos ^{4}(x_{i})-2\prod _{i=1}^{m}\cos ^{2}(x_{i})\right]}{{\left(\sum _{i=1}^{m}ix_{i}^{2}\right)}^{0.5}}}\right|} {\displaystyle f({\boldsymbol {x}})=-\left|{\frac {\left[\sum _{i=1}^{m}\cos ^{4}(x_{i})-2\prod _{i=1}^{m}\cos ^{2}(x_{i})\right]}{{\left(\sum _{i=1}^{m}ix_{i}^{2}\right)}^{0.5}}}\right|},

subjected to: 0.75 − ∏ i = 1 m x i < 0 {\displaystyle 0.75-\prod _{i=1}^{m}x_{i}<0} {\displaystyle 0.75-\prod _{i=1}^{m}x_{i}<0}, and ∑ i = 1 m x i − 7.5 m < 0 {\displaystyle \sum _{i=1}^{m}x_{i}-7.5m<0} {\displaystyle \sum _{i=1}^{m}x_{i}-7.5m<0}

f ( ( 1.60025376 , 0.468675907 ) ) = − 0.364979746 {\displaystyle f((1.60025376,0.468675907))=-0.364979746} {\displaystyle f((1.60025376,0.468675907))=-0.364979746} 0 < x i < 10 {\displaystyle 0<x_{i}<10} {\displaystyle 0<x_{i}<10}

Test functions for multi-objective optimization

[edit ]

[further explanation needed ]

Name Plot Functions Constraints Search domain
Binh and Korn function:[5] 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}}} {\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}}} {\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} {\displaystyle 0\leq x\leq 5}, 0 ≤ y ≤ 3 {\displaystyle 0\leq y\leq 3} {\displaystyle 0\leq y\leq 3}
Chankong and Haimes function:[16] Chakong 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}}} {\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}}} {\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} {\displaystyle -20\leq x,y\leq 20}
Fonseca–Fleming function:[17] Fonseca and Fleming function Minimize = { f 1 ( x ) = 1 − exp ⁡ [ − ∑ i = 1 n ( x i − 1 n ) 2 ] f 2 ( x ) = 1 − exp ⁡ [ − ∑ i = 1 n ( x i + 1 n ) 2 ] {\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)=1-\exp \left[-\sum _{i=1}^{n}\left(x_{i}-{\frac {1}{\sqrt {n}}}\right)^{2}\right]\\f_{2}\left({\boldsymbol {x}}\right)=1-\exp \left[-\sum _{i=1}^{n}\left(x_{i}+{\frac {1}{\sqrt {n}}}\right)^{2}\right]\\\end{cases}}} {\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)=1-\exp \left[-\sum _{i=1}^{n}\left(x_{i}-{\frac {1}{\sqrt {n}}}\right)^{2}\right]\\f_{2}\left({\boldsymbol {x}}\right)=1-\exp \left[-\sum _{i=1}^{n}\left(x_{i}+{\frac {1}{\sqrt {n}}}\right)^{2}\right]\\\end{cases}}} − 4 ≤ x i ≤ 4 {\displaystyle -4\leq x_{i}\leq 4} {\displaystyle -4\leq x_{i}\leq 4}, 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n} {\displaystyle 1\leq i\leq n}
Test function 4:[6] Test function 4.[6] 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}}} {\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}}} {\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} {\displaystyle -7\leq x,y\leq 4}
Kursawe function:[18] Kursawe function Minimize = { f 1 ( x ) = ∑ i = 1 2 [ − 10 exp ⁡ ( − 0.2 x i 2 + x i + 1 2 ) ] f 2 ( x ) = ∑ i = 1 3 [ | x i | 0.8 + 5 sin ⁡ ( x i 3 ) ] {\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)=\sum _{i=1}^{2}\left[-10\exp \left(-0.2{\sqrt {x_{i}^{2}+x_{i+1}^{2}}}\right)\right]\\&\\f_{2}\left({\boldsymbol {x}}\right)=\sum _{i=1}^{3}\left[\left|x_{i}\right|^{0.8}+5\sin \left(x_{i}^{3}\right)\right]\\\end{cases}}} {\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)=\sum _{i=1}^{2}\left[-10\exp \left(-0.2{\sqrt {x_{i}^{2}+x_{i+1}^{2}}}\right)\right]\\&\\f_{2}\left({\boldsymbol {x}}\right)=\sum _{i=1}^{3}\left[\left|x_{i}\right|^{0.8}+5\sin \left(x_{i}^{3}\right)\right]\\\end{cases}}} − 5 ≤ x i ≤ 5 {\displaystyle -5\leq x_{i}\leq 5} {\displaystyle -5\leq x_{i}\leq 5}, 1 ≤ i ≤ 3 {\displaystyle 1\leq i\leq 3} {\displaystyle 1\leq i\leq 3}.
Schaffer function N. 1:[19] 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}}} {\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} {\displaystyle -A\leq x\leq A}. Values of A {\displaystyle A} {\displaystyle A} from 10 {\displaystyle 10} {\displaystyle 10} to 10 5 {\displaystyle 10^{5}} {\displaystyle 10^{5}} have been used successfully. Higher values of A {\displaystyle A} {\displaystyle A} increase the difficulty of the problem.
Schaffer function N. 2: 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円-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}}} {\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円-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} {\displaystyle -5\leq x\leq 10}.
Poloni's two objective function: Poloni's two objective function Minimize = { f 1 ( x , y ) = [ 1 + ( A 1 − B 1 ( x , y ) ) 2 + ( A 2 − B 2 ( x , y ) ) 2 ] f 2 ( x , y ) = ( x + 3 ) 2 + ( y + 1 ) 2 {\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)=\left[1+\left(A_{1}-B_{1}\left(x,y\right)\right)^{2}+\left(A_{2}-B_{2}\left(x,y\right)\right)^{2}\right]\\f_{2}\left(x,y\right)=\left(x+3\right)^{2}+\left(y+1\right)^{2}\\\end{cases}}} {\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)=\left[1+\left(A_{1}-B_{1}\left(x,y\right)\right)^{2}+\left(A_{2}-B_{2}\left(x,y\right)\right)^{2}\right]\\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}}} {\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 } {\displaystyle -\pi \leq x,y\leq \pi }
Zitzler–Deb–Thiele's function N. 1:[20] 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}}} {\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} {\displaystyle 0\leq x_{i}\leq 1}, 1 ≤ i ≤ 30 {\displaystyle 1\leq i\leq 30} {\displaystyle 1\leq i\leq 30}.
Zitzler–Deb–Thiele's function N. 2:[20] 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}}} {\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} {\displaystyle 0\leq x_{i}\leq 1}, 1 ≤ i ≤ 30 {\displaystyle 1\leq i\leq 30} {\displaystyle 1\leq i\leq 30}.
Zitzler–Deb–Thiele's function N. 3:[20] 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}}} {\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} {\displaystyle 0\leq x_{i}\leq 1}, 1 ≤ i ≤ 30 {\displaystyle 1\leq i\leq 30} {\displaystyle 1\leq i\leq 30}.
Zitzler–Deb–Thiele's function N. 4:[20] 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}}} {\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} {\displaystyle 0\leq x_{1}\leq 1}, − 5 ≤ x i ≤ 5 {\displaystyle -5\leq x_{i}\leq 5} {\displaystyle -5\leq x_{i}\leq 5}, 2 ≤ i ≤ 10 {\displaystyle 2\leq i\leq 10} {\displaystyle 2\leq i\leq 10}
Zitzler–Deb–Thiele's function N. 6:[20] 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 [ ∑ i = 2 10 x i 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[{\frac {\sum _{i=2}^{10}x_{i}}{9}}\right]^{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}}} {\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[{\frac {\sum _{i=2}^{10}x_{i}}{9}}\right]^{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} {\displaystyle 0\leq x_{i}\leq 1}, 1 ≤ i ≤ 10 {\displaystyle 1\leq i\leq 10} {\displaystyle 1\leq i\leq 10}.
Osyczka and Kundu function:[21] 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}}} {\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}}} {\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} {\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} {\displaystyle 1\leq x_{3},x_{5}\leq 5}, 0 ≤ x 4 ≤ 6 {\displaystyle 0\leq x_{4}\leq 6} {\displaystyle 0\leq x_{4}\leq 6}.
CTP1 function (2 variables):[4] [22] CTP1 function (2 variables).[4] 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}}} {\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}}} {\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} {\displaystyle 0\leq x,y\leq 1}.
Constr-Ex problem:[4] Constr-Ex problem.[4] 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}}} {\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}}} {\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} {\displaystyle 0.1\leq x\leq 1}, 0 ≤ y ≤ 5 {\displaystyle 0\leq y\leq 5} {\displaystyle 0\leq y\leq 5}
Viennet function: 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}}} {\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} {\displaystyle -3\leq x,y\leq 3}.

References

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