Тестові функції для оптимізації

У прикладній математиці тестові функції для оптимізації (штучні ландшафти) — нелінійні функції, які використовують для оцінки характеристик алгоритмів оптимізації, таких як: швидкість збіжності; точність; грубість; загальні характеристики.

Нижче наведені деякі функції тестування оптимізаційних алгоритмів, що дозволяють отримати уявлення про різні характерні ситуації, з якими стикаються алгоритми оптимізації при вирішенні задач такого роду. У першій частині наведені функції для тестування алгоритмів пошуку глобального мінімуму (максимуму). У другій частині функції з відповідними фронтами для алгоритмів багатокритеріальної оптимізації.

Штучні ландшафти, наведені для тестування оптимізаційних алгоритмів, взяті з декількох джерел (див. Посилання).

Загальний вигляд рівняння, графік цільової функції, межі змінних об'єкта і координати глобального мінімуму наведені в таблиці.

Функції для алгоритмів пошуку глобального мінімуму

Назва / Рисунок	Формула	Мінімум	Область пошуку
Ackley's function	${\displaystyle f(x,y)=-20\exp \left(-0.2{\sqrt {0.5\left(x^{2}+y^{2}\right)}}\right)}$ ${\displaystyle -\exp \left(0.5\left(\cos \left(2\pi x\right)+\cos \left(2\pi y\right)\right)\right)+e+20}$	${\displaystyle f(0,0)=0}$	${\displaystyle -5\leq x,y\leq 5}$
Sphere function	${\displaystyle f({\boldsymbol {x}})=\sum _{i=1}^{n}x_{i}^{2}}$	${\displaystyle f(0,\dots ,0)=0}$	${\displaystyle -\infty \leq x_{i}\leq \infty }$, ${\displaystyle 1\leq i\leq n}$
Функція Розенброка	${\displaystyle f({\boldsymbol {x}})=\sum _{i=1}^{n-1}\left[100\left(x_{i+1}-x_{i}^{2}\right)^{2}+\left(x_{i}-1\right)^{2}\right]}$	${\displaystyle {\text{Min}}={\begin{cases}n=2&\rightarrow \quad f(1,1)=0,\\n=3&\rightarrow \quad f(1,1,1)=0,\\n>3&\rightarrow \quad f\left(\underbrace {1,\dots ,1} _{(n){\text{ times}}}\right)=0\\\end{cases}}}$	${\displaystyle -\infty \leq x_{i}\leq \infty }$, ${\displaystyle 1\leq i\leq n}$
Beale's function	${\displaystyle f(x,y)=\left(1.5-x+xy\right)^{2}+\left(2.25-x+xy^{2}\right)^{2}}$ ${\displaystyle +\left(2.625-x+xy^{3}\right)^{2}}$	${\displaystyle f(3,0.5)=0}$	${\displaystyle -4.5\leq x,y\leq 4.5}$
Goldstein–Price function	${\displaystyle f(x,y)=}$ ${\displaystyle \left(1+\left(x+y+1\right)^{2}\left(19-14x+3x^{2}-14y+6xy+3y^{2}\right)\right)}$ ${\displaystyle \left(30+\left(2x-3y\right)^{2}\left(18-32x+12x^{2}+48y-36xy+27y^{2}\right)\right)}$	${\displaystyle f(0,-1)=3}$	${\displaystyle -2\leq x,y\leq 2}$
Booth's function	${\displaystyle f(x,y)=\left(x+2y-7\right)^{2}+\left(2x+y-5\right)^{2}}$	${\displaystyle f(1,3)=0}$	${\displaystyle -10\leq x,y\leq 10}$
Bukin function N.6	${\displaystyle f(x,y)=100{\sqrt {\left|y-0.01x^{2}\right|}}+0.01\left|x+10\right|.\quad }$	${\displaystyle f(-10,1)=0}$	${\displaystyle -15\leq x\leq -5}$, ${\displaystyle -3\leq y\leq 3}$
Matyas function	${\displaystyle f(x,y)=0.26\left(x^{2}+y^{2}\right)-0.48xy}$	${\displaystyle f(0,0)=0}$	${\displaystyle -10\leq x,y\leq 10}$
Lévi function N.13	${\displaystyle f(x,y)=\sin ^{2}\left(3\pi x\right)+\left(x-1\right)^{2}\left(1+\sin ^{2}\left(3\pi y\right)\right)}$ ${\displaystyle +\left(y-1\right)^{2}\left(1+\sin ^{2}\left(2\pi y\right)\right)}$	${\displaystyle f(1,1)=0}$	${\displaystyle -10\leq x,y\leq 10}$
Three-hump camel function	${\displaystyle f(x,y)=2x^{2}-1.05x^{4}+{\frac {x^{6}}{6}}+xy+y^{2}}$	${\displaystyle f(0,0)=0}$	${\displaystyle -5\leq x,y\leq 5}$
Easom function	${\displaystyle f(x,y)=-\cos \left(x\right)\cos \left(y\right)}$ ${\displaystyle \exp \left(-\left(\left(x-\pi \right)^{2}+\left(y-\pi \right)^{2}\right)\right)}$	${\displaystyle f(\pi ,\pi )=-1}$	${\displaystyle -100\leq x,y\leq 100}$
Cross-in-tray function	${\displaystyle f(x,y)=-0.0001}$ ${\displaystyle \left(\left|\sin \left(x\right)\sin \left(y\right)\exp \left(\left|100-{\frac {\sqrt {x^{2}+y^{2}}}{\pi }}\right|\right)\right|+1\right)^{0.1}}$	${\displaystyle {\text{Min}}={\begin{cases}f\left(1.34941,-1.34941\right)&=-2.06261\\f\left(1.34941,1.34941\right)&=-2.06261\\f\left(-1.34941,1.34941\right)&=-2.06261\\f\left(-1.34941,-1.34941\right)&=-2.06261\\\end{cases}}}$	${\displaystyle -10\leq x,y\leq 10}$
Eggholder function	${\displaystyle f(x,y)=-\left(y+47\right)\sin \left({\sqrt {\left|{\frac {x}{2}}+\left(y+47\right)\right|}}\right)}$ ${\displaystyle -x\sin \left({\sqrt {\left|x-\left(y+47\right)\right|}}\right)}$	${\displaystyle f(512,404.2319)=-959.6407}$	${\displaystyle -512\leq x,y\leq 512}$
Hölder table function	${\displaystyle f(x,y)=-\left|\sin \left(x\right)\cos \left(y\right)\exp \left(\left|1-{\frac {\sqrt {x^{2}+y^{2}}}{\pi }}\right|\right)\right|}$	${\displaystyle {\text{Min}}={\begin{cases}f\left(8.05502,9.66459\right)&=-19.2085\\f\left(-8.05502,9.66459\right)&=-19.2085\\f\left(8.05502,-9.66459\right)&=-19.2085\\f\left(-8.05502,-9.66459\right)&=-19.2085\end{cases}}}$	${\displaystyle -10\leq x,y\leq 10}$
McCormick function	${\displaystyle f(x,y)=\sin \left(x+y\right)+\left(x-y\right)^{2}-1.5x+2.5y+1}$	${\displaystyle f(-0.54719,-1.54719)=-1.9133}$	${\displaystyle -1.5\leq x\leq 4}$, ${\displaystyle -3\leq y\leq 4}$
Schaffer function N. 2	${\displaystyle f(x,y)=0.5+{\frac {\sin ^{2}\left(x^{2}-y^{2}\right)-0.5}{\left(1+0.001\left(x^{2}+y^{2}\right)\right)^{2}}}}$	${\displaystyle f(0,0)=0}$	${\displaystyle -100\leq x,y\leq 100}$
Schaffer function N. 4	${\displaystyle f(x,y)=0.5+{\frac {\cos ^{2}\left(\sin \left(\left|x^{2}-y^{2}\right|\right)\right)-0.5}{\left(1+0.001\left(x^{2}+y^{2}\right)\right)^{2}}}}$	${\displaystyle f(0,1.25313)=0.292579}$	${\displaystyle -100\leq x,y\leq 100}$
Styblinski–Tang function	${\displaystyle f({\boldsymbol {x}})={\frac {\sum _{i=1}^{n}x_{i}^{4}-16x_{i}^{2}+5x_{i}}{2}}}$	${\displaystyle X=\underbrace {-2.903534,\ldots ,-2.903534} _{(n){\text{ times}}}}$ ${\displaystyle -39.16617n<f(X)<-39.16616n}$	${\displaystyle -5\leq x_{i}\leq 5}$, ${\displaystyle 1\leq i\leq n}$.
Simionescu function[1]	${\displaystyle f(x,y)=0.1xy}$, ${\displaystyle {\text{subjected to: }}x^{2}+y^{2}\leq \left(r_{T}+r_{S}\cos \left(n\arctan {\frac {x}{y}}\right)\right)^{2}}$ ${\displaystyle {\text{where: }}r_{T}=1,r_{S}=0.2{\text{ and }}n=8}$	${\displaystyle f(\pm 0.85586214,\mp 0.85586214)=-0.072625}$	${\displaystyle -1.25\leq x,y\leq 1.25}$

Функції для алгоритмів багатокритеріальної оптимізації

Назва / Рисунок	Формула	Мінімум	Область пошуку
Binh and Korn function	${\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)&=4x^{2}+4y^{2}\\f_{2}\left(x,y\right)&=\left(x-5\right)^{2}+\left(y-5\right)^{2}\\\end{cases}}}$	${\displaystyle {\text{s.t.}}={\begin{cases}g_{1}\left(x,y\right)&=\left(x-5\right)^{2}+y^{2}\leq 25\\g_{2}\left(x,y\right)&=\left(x-8\right)^{2}+\left(y+3\right)^{2}\geq 7.7\\\end{cases}}}$	${\displaystyle 0\leq x\leq 5}$, ${\displaystyle 0\leq y\leq 3}$
Chakong and Haimes function	${\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)&=2+\left(x-2\right)^{2}+\left(y-1\right)^{2}\\f_{2}\left(x,y\right)&=9x-\left(y-1\right)^{2}\\\end{cases}}}$	${\displaystyle {\text{s.t.}}={\begin{cases}g_{1}\left(x,y\right)&=x^{2}+y^{2}\leq 225\\g_{2}\left(x,y\right)&=x-3y+10\leq 0\\\end{cases}}}$	${\displaystyle -20\leq x,y\leq 20}$
Fonseca and Fleming function	${\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)&=1-\exp \left(-\sum _{i=1}^{n}\left(x_{i}-{\frac {1}{\sqrt {n}}}\right)^{2}\right)\\f_{2}\left({\boldsymbol {x}}\right)&=1-\exp \left(-\sum _{i=1}^{n}\left(x_{i}+{\frac {1}{\sqrt {n}}}\right)^{2}\right)\\\end{cases}}}$		${\displaystyle -4\leq x_{i}\leq 4}$, ${\displaystyle 1\leq i\leq n}$
Test function 4	${\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)&=x^{2}-y\\f_{2}\left(x,y\right)&=-0.5x-y-1\\\end{cases}}}$	${\displaystyle {\text{s.t.}}={\begin{cases}g_{1}\left(x,y\right)&=6.5-{\frac {x}{6}}-y\geq 0\\g_{2}\left(x,y\right)&=7.5-0.5x-y\geq 0\\g_{3}\left(x,y\right)&=30-5x-y\geq 0\\\end{cases}}}$	${\displaystyle -7\leq x,y\leq 4}$
Kursawe function	${\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)&=\sum _{i=1}^{2}\left[-10\exp \left(-0.2{\sqrt {x_{i}^{2}+x_{i+1}^{2}}}\right)\right]\\&\\f_{2}\left({\boldsymbol {x}}\right)&=\sum _{i=1}^{3}\left[\left|x_{i}\right|^{0.8}+5\sin \left(x_{i}^{3}\right)\right]\\\end{cases}}}$		${\displaystyle -5\leq x_{i}\leq 5}$, ${\displaystyle 1\leq i\leq 3}$.
Schaffer function N. 1	${\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left(x\right)&=x^{2}\\f_{2}\left(x\right)&=\left(x-2\right)^{2}\\\end{cases}}}$		${\displaystyle -A\leq x\leq A}$. Values of ${\displaystyle A}$ form ${\displaystyle 10}$ to ${\displaystyle 10^{5}}$ have been used successfully. Higher values of ${\displaystyle A}$ increase the difficulty of the problem.
Schaffer function N. 2	${\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left(x\right)&={\begin{cases}-x,&{\text{if }}x\leq 1\\x-2,&{\text{if }}1<x\leq 3\\4-x,&{\text{if }}3<x\leq 4\\x-4,&{\text{if }}x>4\\\end{cases}}\\f_{2}\left(x\right)&=\left(x-5\right)^{2}\\\end{cases}}}$		${\displaystyle -5\leq x\leq 10}$.
Poloni's two objective function	${\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)&=\left[1+\left(A_{1}-B_{1}\left(x,y\right)\right)^{2}+\left(A_{2}-B_{2}\left(x,y\right)\right)^{2}\right]\\f_{2}\left(x,y\right)&=\left(x+3\right)^{2}+\left(y+1\right)^{2}\\\end{cases}}}$ ${\displaystyle {\text{where}}={\begin{cases}A_{1}&=0.5\sin \left(1\right)-2\cos \left(1\right)+\sin \left(2\right)-1.5\cos \left(2\right)\\A_{2}&=1.5\sin \left(1\right)-\cos \left(1\right)+2\sin \left(2\right)-0.5\cos \left(2\right)\\B_{1}\left(x,y\right)&=0.5\sin \left(x\right)-2\cos \left(x\right)+\sin \left(y\right)-1.5\cos \left(y\right)\\B_{2}\left(x,y\right)&=1.5\sin \left(x\right)-\cos \left(x\right)+2\sin \left(y\right)-0.5\cos \left(y\right)\end{cases}}}$		${\displaystyle -\pi \leq x,y\leq \pi }$
Zitzler–Deb–Thiele's function N. 1	${\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)&=x_{1}\\f_{2}\left({\boldsymbol {x}}\right)&=g\left({\boldsymbol {x}}\right)h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)\\g\left({\boldsymbol {x}}\right)&=1+{\frac {9}{29}}\sum _{i=2}^{30}x_{i}\\h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)&=1-{\sqrt {\frac {f_{1}\left({\boldsymbol {x}}\right)}{g\left({\boldsymbol {x}}\right)}}}\\\end{cases}}}$		${\displaystyle 0\leq x_{i}\leq 1}$, ${\displaystyle 1\leq i\leq 30}$.
Zitzler–Deb–Thiele's function N. 2	${\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)&=x_{1}\\f_{2}\left({\boldsymbol {x}}\right)&=g\left({\boldsymbol {x}}\right)h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)\\g\left({\boldsymbol {x}}\right)&=1+{\frac {9}{29}}\sum _{i=2}^{30}x_{i}\\h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)&=1-\left({\frac {f_{1}\left({\boldsymbol {x}}\right)}{g\left({\boldsymbol {x}}\right)}}\right)^{2}\\\end{cases}}}$		${\displaystyle 0\leq x_{i}\leq 1}$, ${\displaystyle 1\leq i\leq 30}$.
Zitzler–Deb–Thiele's function N. 3	${\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)&=x_{1}\\f_{2}\left({\boldsymbol {x}}\right)&=g\left({\boldsymbol {x}}\right)h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)\\g\left({\boldsymbol {x}}\right)&=1+{\frac {9}{29}}\sum _{i=2}^{30}x_{i}\\h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)&=1-{\sqrt {\frac {f_{1}\left({\boldsymbol {x}}\right)}{g\left({\boldsymbol {x}}\right)}}}-\left({\frac {f_{1}\left({\boldsymbol {x}}\right)}{g\left({\boldsymbol {x}}\right)}}\right)\sin \left(10\pi f_{1}\left({\boldsymbol {x}}\right)\right)\end{cases}}}$		${\displaystyle 0\leq x_{i}\leq 1}$, ${\displaystyle 1\leq i\leq 30}$.
Zitzler–Deb–Thiele's function N. 4	${\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)&=x_{1}\\f_{2}\left({\boldsymbol {x}}\right)&=g\left({\boldsymbol {x}}\right)h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)\\g\left({\boldsymbol {x}}\right)&=91+\sum _{i=2}^{10}\left(x_{i}^{2}-10\cos \left(4\pi x_{i}\right)\right)\\h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)&=1-{\sqrt {\frac {f_{1}\left({\boldsymbol {x}}\right)}{g\left({\boldsymbol {x}}\right)}}}\end{cases}}}$		${\displaystyle 0\leq x_{1}\leq 1}$, ${\displaystyle -5\leq x_{i}\leq 5}$, ${\displaystyle 2\leq i\leq 10}$
Zitzler–Deb–Thiele's function N. 6	${\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)&=1-\exp \left(-4x_{1}\right)\sin ^{6}\left(6\pi x_{1}\right)\\f_{2}\left({\boldsymbol {x}}\right)&=g\left({\boldsymbol {x}}\right)h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)\\g\left({\boldsymbol {x}}\right)&=1+9\left[{\frac {\sum _{i=2}^{10}x_{i}}{9}}\right]^{0.25}\\h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)&=1-\left({\frac {f_{1}\left({\boldsymbol {x}}\right)}{g\left({\boldsymbol {x}}\right)}}\right)^{2}\\\end{cases}}}$		${\displaystyle 0\leq x_{i}\leq 1}$, ${\displaystyle 1\leq i\leq 10}$.
Viennet function	${\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)&=0.5\left(x^{2}+y^{2}\right)+\sin \left(x^{2}+y^{2}\right)\\f_{2}\left(x,y\right)&={\frac {\left(3x-2y+4\right)^{2}}{8}}+{\frac {\left(x-y+1\right)^{2}}{27}}+15\\f_{3}\left(x,y\right)&={\frac {1}{x^{2}+y^{2}+1}}-1.1\exp \left(-\left(x^{2}+y^{2}\right)\right)\\\end{cases}}}$		${\displaystyle -3\leq x,y\leq 3}$.
Osyczka and Kundu function	${\displaystyle F_{1}(x)=-25\left(x_{1}-2\right)^{2}-\left(x_{2}-2\right)^{2}}$ ${\displaystyle -\left(x_{3}-1\right)^{2}-\left(x_{4}-4\right)^{2}-\left(x_{5}-1\right)^{2}}$ ${\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)&=F_{1}(x)\\f_{2}\left({\boldsymbol {x}}\right)&=\sum _{i=1}^{6}x_{i}^{2}\\\end{cases}}}$	${\displaystyle {\text{s.t.}}={\begin{cases}g_{1}\left({\boldsymbol {x}}\right)&=x_{1}+x_{2}-2\geq 0\\g_{2}\left({\boldsymbol {x}}\right)&=6-x_{1}-x_{2}\geq 0\\g_{3}\left({\boldsymbol {x}}\right)&=2-x_{2}+x_{1}\geq 0\\g_{4}\left({\boldsymbol {x}}\right)&=2-x_{1}+3x_{2}\geq 0\\g_{5}\left({\boldsymbol {x}}\right)&=4-\left(x_{3}-3\right)^{2}-x_{4}\geq 0\\g_{6}\left({\boldsymbol {x}}\right)&=\left(x_{5}-3\right)^{2}+x_{6}-4\geq 0\end{cases}}}$	${\displaystyle 0\leq x_{1},x_{2},x_{6}\leq 10}$, ${\displaystyle 1\leq x_{3},x_{5}\leq 5}$, ${\displaystyle 0\leq x_{4}\leq 6}$.
CTP1 function (2 variables)	${\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)&=x\\f_{2}\left(x,y\right)&=\left(1+y\right)\exp \left(-{\frac {x}{1+y}}\right)\end{cases}}}$	${\displaystyle {\text{s.t.}}={\begin{cases}g_{1}\left(x,y\right)&={\frac {f_{2}\left(x,y\right)}{0.858\exp \left(-0.541f_{1}\left(x,y\right)\right)}}\geq 1\\g_{1}\left(x,y\right)&={\frac {f_{2}\left(x,y\right)}{0.728\exp \left(-0.295f_{1}\left(x,y\right)\right)}}\geq 1\end{cases}}}$	${\displaystyle 0\leq x,y\leq 1}$.
Constr-Ex problem	${\displaystyle {\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)&=x\\f_{2}\left(x,y\right)&={\frac {1+y}{x}}\\\end{cases}}}$	${\displaystyle {\text{s.t.}}={\begin{cases}g_{1}\left(x,y\right)&=y+9x\geq 6\\g_{1}\left(x,y\right)&=-y+9x\geq 1\\\end{cases}}}$	${\displaystyle 0.1\leq x\leq 1}$, ${\displaystyle 0\leq y\leq 5}$