Name Plot Formula Global minimum Search domain Rastrigin function 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]} 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 [ − 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]} − 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}
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 [ 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]} 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 ( 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}} + ( 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 ) = [ 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]} [ 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]}
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 ) = 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 [ | 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}} 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[9] [10] 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 [ 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}}}} 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 [ 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}}}} 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 i + ∑ j = 1 n ( x j − a j i ) 2 ) − 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} 
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