Fonction de test pour l'optimisation

En mathématiques appliquées, les fonctions de test sont des fonctions d'évaluation des caractéristiques des algorithmes d'optimisation, telles que taux de convergence ; précision ; robustesse ; performances générales.

Cette page représente que les fonctions de test les plus classiques:

  • à objectif unique.
  • multi-objectifs (MOP), avec leur fronts de Pareto.

Les représentations graphiques sont tirés de Bäck[1], Haupt et al.[2] et du logiciel Rody Oldenhuis[3]. Compte tenu du nombre de problèmes (55 au total), seuls quelques-uns sont présentés ici.

Les fonctions de test utilisées pour évaluer les algorithmes de MOP sont tirées de Deb[4], Binh et al.[5] et Binh[6]. On peut télécharger le logiciel développé par Deb[7], qui implémente la procédure NSGA-II avec GAs, ou le programme mis en ligne sur Internet[8], qui implémente la procédure NSGA-II avec ES.

Dans les présentations suivantes, seront juste données la forme générale de l'équation, un tracé de la fonction objectif, les limites des variables d'objet et les coordonnées des minima globaux.

Optimisations à objectif unique

Nom Représentation Formule Minimum absolu Domaine d'application
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]}

où:  A = 10 {\displaystyle {\text{où: }}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}
Fonction d'Ackley 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]}

− 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}
Sphère Sphere function for n=2 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 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]} Min = { n = 2 → f ( 1 , 1 ) = 0 , n = 3 → f ( 1 , 1 , 1 ) = 0 , n > 3 → f ( 1 , … , 1 ⏟ n  fois ) = 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{ fois}}})=0\\\end{cases}}} − ∞ ≤ x i ≤ ∞ {\displaystyle -\infty \leq x_{i}\leq \infty } , 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n}
Fonction de Beale 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}}

+ ( 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 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 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}} f ( 1 , 3 ) = 0 {\displaystyle f(1,3)=0} − 10 ≤ x , y ≤ 10 {\displaystyle -10\leq x,y\leq 10}
Bukin 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 } 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}
Fonction de Matyas 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}
Fonction de Lévi 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)}

+ ( 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 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 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 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 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[9] 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}
Table de Hölder 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|} 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 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 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}}}} f ( 0 , 0 ) = 0 {\displaystyle f(0,0)=0} − 100 ≤ x , y ≤ 100 {\displaystyle -100\leq x,y\leq 100}
Schaffer 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}}}} 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 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  fois ) < − 39.16616 n {\displaystyle -39.16617n<f(\underbrace {-2.903534,\ldots ,-2.903534} _{n{\text{ fois}}})<-39.16616n} − 5 ≤ x i ≤ 5 {\displaystyle -5\leq x_{i}\leq 5} , 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n} ..

Optimisations contraintes

Name Plot Formula Global minimum Search domain
Rosenbrock function constrained with a cubic and a line[10] 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[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}} ,

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[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}} ,

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)[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)} ,

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}} 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)[15] Gomez and Levy Function f ( x , y ) = 4 x 2 − 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[16] Simionescu function f ( x , y ) = 0.1 x y {\displaystyle f(x,y)=0.1xy} ,

subjected to: x 2 + y 2 ≤ [ r T + r S cos ⁡ ( n arctan ⁡ x y ) ] 2 {\displaystyle x^{2}+y^{2}\leq \left[r_{T}+r_{S}\cos \left(n\arctan {\frac {x}{y}}\right)\right]^{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}

Optimisations multi-objectifs

Name Plot Functions Constraints Search domain
Binh and Korn function: 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[17] : 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}}} 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[18] : 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}}} − 4 ≤ x i ≤ 4 {\displaystyle -4\leq x_{i}\leq 4} , 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n}
Test function 4: 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}}} 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: 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}}} − 5 ≤ x i ≤ 5 {\displaystyle -5\leq x_{i}\leq 5} , 1 ≤ i ≤ 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}}} − 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: 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: 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}}}

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[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}}} 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[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}}} 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[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}}} 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[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}}} 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[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}}} 0 ≤ x i ≤ 1 {\displaystyle 0\leq x_{i}\leq 1} , 1 ≤ i ≤ 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}}} 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)[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}}} 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[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}}} 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: 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} .

Voir aussi

  • Fonction d'Ackley
  • Fonction de Himmelblau
  • Fonction Rastrigin
  • Fonction de Rosenbrock
  • Fonction shekel
  • Fonction binh

Références

  1. ↑ Thomas Bäck, Evolutionary algorithms in theory and practice : evolution strategies, evolutionary programming, genetic algorithms, Oxford, Oxford University Press, (ISBN 978-0-19-509971-3), p. 328
  2. ↑ Randy L. Haupt, Sue Ellen Haupt, Practical genetic algorithms with CD-Rom, New York, 2nd, (ISBN 978-0-471-45565-3)
  3. ↑ Oldenhuis, « Many test functions for global optimizers », Mathworks (consulté le )
  4. ↑ a b c d et e Deb, Kalyanmoy (2002) Multiobjective optimization using evolutionary algorithms (Repr. ed.). Chichester [u.a.]: Wiley. (ISBN 0-471-87339-X).
  5. ↑ Binh T. and Korn U. (1997) MOBES: A Multiobjective Evolution Strategy for Constrained Optimization Problems. In: Proceedings of the Third International Conference on Genetic Algorithms. Czech Republic. pp. 176–182
  6. ↑ a et b Binh T. (1999) A multiobjective evolutionary algorithm. The study cases. Technical report. Institute for Automation and Communication. Barleben, Germany
  7. ↑ Deb K. (2011) Software for multi-objective NSGA-II code in C. Available at URL: https://www.iitk.ac.in/kangal/codes.shtml
  8. ↑ Ortiz, « Multi-objective optimization using ES as Evolutionary Algorithm. », Mathworks (consulté le )
  9. ↑ Whitley, Rana, Dzubera et Mathias, « Evaluating evolutionary algorithms », Artificial Intelligence, Elsevier BV, vol. 85, nos 1-2,‎ , p. 264 (ISSN 0004-3702, DOI 10.1016/0004-3702(95)00124-7)
  10. ↑ Simionescu, P.A. et Beale, D. « New Concepts in Graphic Visualization of Objective Functions » (September 29 – October 2, 2002) (lire en ligne, consulté le )
    —ASME 2002 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
  11. ↑ « Solve a Constrained Nonlinear Problem - MATLAB & Simulink », www.mathworks.com (consulté le )
  12. ↑ « Bird Problem (Constrained) | Phoenix Integration » [archive du ] (consulté le )
  13. ↑ Mishra, « Some new test functions for global optimization and performance of repulsive particle swarm method », MPRA Paper,‎ (lire en ligne)
  14. ↑ Townsend, « Constrained optimization in Chebfun », chebfun.org, (consulté le )
  15. ↑ Simionescu, « A collection of bivariate nonlinear optimisation test problems with graphical representations », International Journal of Mathematical Modelling and Numerical Optimisation, vol. 10, no 4,‎ , p. 365–398 (DOI 10.1504/IJMMNO.2020.110704)
  16. ↑ P.A. Simionescu, Computer Aided Graphing and Simulation Tools for AutoCAD Users, Boca Raton, FL, 1st, (ISBN 978-1-4822-5290-3)
  17. ↑ Vira Chankong et Yacov Y. Haimes, Multiobjective decision making. Theory and methodology., (ISBN 0-444-00710-5)
  18. ↑ Fonseca et Fleming, « An Overview of Evolutionary Algorithms in Multiobjective Optimization », Evol Comput, vol. 3, no 1,‎ , p. 1–16 (DOI 10.1162/evco.1995.3.1.1, S2CID 8530790, CiteSeerx 10.1.1.50.7779)
  19. ↑ J. David Schaffer, Proceedings of the First International Conference on Genetic Algorithms, (OCLC 20004572), « Multiple Objective Optimization with Vector Evaluated Genetic Algorithms »
  20. ↑ a b c d et e Kalyan Deb, L. Thiele, Marco Laumanns et Eckart Zitzler, Proceedings of the 2002 IEEE Congress on Evolutionary Computation, vol. 1, , 825–830 p. (ISBN 0-7803-7282-4, DOI 10.1109/CEC.2002.1007032, S2CID 61001583), « Scalable multi-objective optimization test problems »
  21. ↑ Osyczka et Kundu, « A new method to solve generalized multicriteria optimization problems using the simple genetic algorithm », Structural Optimization, vol. 10, no 2,‎ , p. 94–99 (ISSN 1615-1488, DOI 10.1007/BF01743536, S2CID 123433499)
  22. ↑ Jimenez, Gomez-Skarmeta, Sanchez et Deb, « An evolutionary algorithm for constrained multi-objective optimization », Proceedings of the 2002 Congress on Evolutionary Computation. CEC'02 (Cat. No.02TH8600), vol. 2,‎ , p. 1133–1138 (ISBN 0-7803-7282-4, DOI 10.1109/CEC.2002.1004402, S2CID 56563996)
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