Funcția de testare (optimizare)

Funcțiile de testare sunt funcții concepute și utilizate pentru a testa funcționarea și eficiența algoritmilor de optimizare. Aspectele algoritmului care sunt de obicei interesate de testare sunt viteza de convergență , precizia rezultatului și robustețea algoritmului. Funcțiile de testare sunt adesea probleme artificiale care testează algoritmi în situații deosebit de incomode, de exemplu în căutarea minimelor în funcții deosebit de plane (cum ar fi un punct minim al unei funcții continue în care se anulează multe derivate succesive), funcții al căror comportament global se apropie de a unei funcții unimodale, dar care are de fapt alte extreme locale, funcții cu un număr mare de puncte optime locale semnificative sau funcții a căror tendință globală nu oferă informații semnificative cu privire la poziția punctelor optime. ^[1]

Mai jos sunt câteva dintre cele mai cunoscute funcții de testare cu o expresie în formă generală și caracteristicile lor principale.

Funcțiile principale de testare

Funcțiile de testare prezentate mai jos sunt raportate în Bäck, ^[2] Haupt et. la. ^[3] și din biblioteca software a lui Rody Oldenhuis. ^[4]

Nume	Expresie	Minim	Căutare domeniu
Funcția Ackley	$f(x,y)=-20\exp \left(-0.2{\sqrt {0.5\left(x^{2}+y^{2}\right)}}\right)$ ${\ displaystyle f (x, y) = - 20 \ exp \ left (-0.2 {\ sqrt {0.5 \ left (x ^ {2} + y ^ {2} \ right)}} \ right)}$ ${\ displaystyle f (x, y) = - 20 \ exp \ left (-0.2 {\ sqrt {0.5 \ left (x ^ {2} + y ^ {2} \ right)}} \ right)}$ $-\exp \left(0.5\left(\cos \left(2\pi x\right)+\cos \left(2\pi y\right)\right)\right)+20+e.\quad$ ${\ displaystyle - \ exp \ left (0,5 \ left (\ cos \ left (2 \ pi x \ right) + \ cos \ left (2 \ pi y \ right) \ right) \ right) + 20 + e. \ Quad}$ ${\ displaystyle - \ exp \ left (0,5 \ left (\ cos \ left (2 \ pi x \ right) + \ cos \ left (2 \ pi y \ right) \ right) \ right) + 20 + e. \ Quad}$	$f(0,0)=0$ ${\ displaystyle f (0,0) = 0}$ ${\ displaystyle f (0,0) = 0}$	$-5\leq x,y\leq 5$ ${\ displaystyle -5 \ leq x, y \ leq 5}$ ${\ displaystyle -5 \ leq x, y \ leq 5}$
Funcția sferică	$f({\boldsymbol {x}})=\sum _{i=1}^{n}x_{i}^{2}.\quad$ ${\ displaystyle f ({\ boldsymbol {x}}) = \ sum _ {i = 1} ^ {n} x_ {i} ^ {2}. \ quad}$ ${\ displaystyle f ({\ boldsymbol {x}}) = \ sum _ {i = 1} ^ {n} x_ {i} ^ {2}. \ quad}$	$f(x_{1},\dots ,x_{n})=f(0,\dots ,0)=0$ ${\ displaystyle f (x_ {1}, \ dots, x_ {n}) = f (0, \ dots, 0) = 0}$ ${\ displaystyle f (x_ {1}, \ dots, x_ {n}) = f (0, \ dots, 0) = 0}$	$-\infty \leq x_{i}\leq \infty$ ${\ displaystyle - \ infty \ leq x_ {i} \ leq \ infty}$ ${\ displaystyle - \ infty \ leq x_ {i} \ leq \ infty}$ , $1\leq i\leq n$ ${\ displaystyle 1 \ leq i \ leq n}$ ${\ displaystyle 1 \ leq i \ leq n}$
Funcția Rosenbrock	$f({\boldsymbol {x}})=\sum _{i=1}^{n-1}\left[100\left(x_{i+1}-x_{i}^{2}\right)^{2}+\left(x_{i}-1\right)^{2}\right].\quad$ ${\ displaystyle f ({\ boldsymbol {x}}) = \ sum _ {i = 1} ^ {n-1} \ left [100 \ left (x_ {i + 1} -x_ {i} ^ {2} \ right) ^ {2} + \ left (x_ {i} -1 \ right) ^ {2} \ right]. \ quad}$ ${\ displaystyle f ({\ boldsymbol {x}}) = \ sum _ {i = 1} ^ {n-1} \ left [100 \ left (x_ {i + 1} -x_ {i} ^ {2} \ right) ^ {2} + \ left (x_ {i} -1 \ right) ^ {2} \ right]. \ quad}$	${\text{Min}}={\begin{cases}n=2&\rightarrow \quad f(1,1)=0,\\n=3&\rightarrow \quad f(1,1,1)=0,\\n>3&\rightarrow \quad f\left(\underbrace {1,\dots ,1} _{(n){\text{ times}}}\right)=0.\\\end{cases}}$ ${\ displaystyle {\ text {Min}} = {\ begin {cases} n = 2 & \ rightarrow \ quad f (1,1) = 0, \\ n = 3 & \ rightarrow \ quad f (1,1, 1) = 0, \\ n> 3 & \ rightarrow \ quad f \ left (\ underbrace {1, \ dots, 1} _ {(n) {\ text {times}}} \ right) = 0. \\ \ end {cases}}}$ ${\ displaystyle {\ text {Min}} = {\ begin {cases} n = 2 & \ rightarrow \ quad f (1,1) = 0, \\ n = 3 & \ rightarrow \ quad f (1,1, 1) = 0, \\ n> 3 & \ rightarrow \ quad f \ left (\ underbrace {1, \ dots, 1} _ {(n) {\ text {times}}} \ right) = 0. \\ \ end {cases}}}$	$-\infty \leq x_{i}\leq \infty$ ${\ displaystyle - \ infty \ leq x_ {i} \ leq \ infty}$ ${\ displaystyle - \ infty \ leq x_ {i} \ leq \ infty}$ , $1\leq i\leq n$ ${\ displaystyle 1 \ leq i \ leq n}$ ${\ displaystyle 1 \ leq i \ leq n}$
Funcția Powell ^[5]	$f({\boldsymbol {x}})=\sum _{i=1}^{n/4}\left[(x_{4i-3}+10x_{4i-2})^{2}+5(x_{4i-1}-x_{4i})^{2}\right.$ ${\ displaystyle f ({\ boldsymbol {x}}) = \ sum _ {i = 1} ^ {n / 4} \ left [(x_ {4i-3} + 10x_ {4i-2}) ^ {2} +5 (x_ {4i-1} -x_ {4i}) ^ {2} \ right.}$ ${\ displaystyle f ({\ boldsymbol {x}}) = \ sum _ {i = 1} ^ {n / 4} \ left [(x_ {4i-3} + 10x_ {4i-2}) ^ {2} +5 (x_ {4i-1} -x_ {4i}) ^ {2} \ right.}$ $\left.+(x_{4i-2}-2x_{4i-1})^{2}+(10x_{4i-3}-x_{4i})^{4}\right].\quad$ ${\ displaystyle \ left. + (x_ {4i-2} -2x_ {4i-1}) ^ {2} + (10x_ {4i-3} -x_ {4i}) ^ {4} \ right]. \ quad }$ ${\ displaystyle \ left. + (x_ {4i-2} -2x_ {4i-1}) ^ {2} + (10x_ {4i-3} -x_ {4i}) ^ {4} \ right]. \ quad }$	$f(x_{1},\dots ,x_{n})=f(0,\dots ,0)=0$ ${\ displaystyle f (x_ {1}, \ dots, x_ {n}) = f (0, \ dots, 0) = 0}$ ${\ displaystyle f (x_ {1}, \ dots, x_ {n}) = f (0, \ dots, 0) = 0}$	$x_{i}\in [-4,5]$ ${\ displaystyle x_ {i} \ în [-4.5]}$ ${\ displaystyle x_ {i} \ în [-4.5]}$ $\forall i=1,\,...,\,n$ ${\ displaystyle \ forall i = 1, \, ..., \, n}$ ${\ displaystyle \ forall i = 1, \, ..., \, n}$
Funcția Beale	$f(x,y)=\left(1.5-x+xy\right)^{2}+\left(2.25-x+xy^{2}\right)^{2}$ ${\ displaystyle f (x, y) = \ left (1,5-x + xy \ right) ^ {2} + \ left (2,25-x + xy ^ {2} \ right) ^ {2}}$ ${\ displaystyle f (x, y) = \ left (1,5-x + xy \ right) ^ {2} + \ left (2,25-x + xy ^ {2} \ right) ^ {2}}$ $+\left(2.625-x+xy^{3}\right)^{2}.\quad$ ${\ displaystyle + \ left (2.625-x + xy ^ {3} \ right) ^ {2}. \ quad}$ ${\ displaystyle + \ left (2.625-x + xy ^ {3} \ right) ^ {2}. \ quad}$	$f(3,0.5)=0$ ${\ displaystyle f (3,0.5) = 0}$ ${\ displaystyle f (3,0.5) = 0}$	$-4.5\leq x,y\leq 4.5$ ${\ displaystyle -4,5 \ leq x, y \ leq 4.5}$ ${\ displaystyle -4,5 \ leq x, y \ leq 4.5}$
Goldstein - Funcția de preț	$f(x,y)=\left(1+\left(x+y+1\right)^{2}\left(19-14x+3x^{2}-14y+6xy+3y^{2}\right)\right)$ ${\ displaystyle f (x, y) = \ left (1+ \ left (x + y + 1 \ right) ^ {2} \ left (19-14x + 3x ^ {2} -14y + 6xy + 3y ^ { 2} \ right) \ right)}$ ${\ displaystyle f (x, y) = \ left (1+ \ left (x + y + 1 \ right) ^ {2} \ left (19-14x + 3x ^ {2} -14y + 6xy + 3y ^ { 2} \ right) \ right)}$ $\left(30+\left(2x-3y\right)^{2}\left(18-32x+12x^{2}+48y-36xy+27y^{2}\right)\right).\quad$ ${\ displaystyle \ left (30+ \ left (2x-3y \ right) ^ {2} \ left (18-32x + 12x ^ {2} + 48y-36xy + 27y ^ {2} \ right) \ right). \ Quad}$ ${\ displaystyle \ left (30+ \ left (2x-3y \ right) ^ {2} \ left (18-32x + 12x ^ {2} + 48y-36xy + 27y ^ {2} \ right) \ right). \ Quad}$	$f(0,-1)=3$ ${\ displaystyle f (0, -1) = 3}$ ${\ displaystyle f (0, -1) = 3}$	$-2\leq x,y\leq 2$ ${\ displaystyle -2 \ leq x, y \ leq 2}$ ${\ displaystyle -2 \ leq x, y \ leq 2}$
Funcția stand	$f(x,y)=\left(x+2y-7\right)^{2}+\left(2x+y-5\right)^{2}.\quad$ ${\ displaystyle f (x, y) = \ left (x + 2y-7 \ right) ^ {2} + \ left (2x + y-5 \ right) ^ {2}. \ quad}$ ${\ displaystyle f (x, y) = \ left (x + 2y-7 \ right) ^ {2} + \ left (2x + y-5 \ right) ^ {2}. \ quad}$	$f(1,3)=0$ ${\ displaystyle f (1,3) = 0}$ ${\ displaystyle f (1,3) = 0}$	$-10\leq x,y\leq 10$ ${\ displaystyle -10 \ leq x, y \ leq 10}$ ${\ displaystyle -10 \ leq x, y \ leq 10}$ .
Funcția Bukin # 6	$f(x,y)=100{\sqrt {\left\|y-0.01x^{2}\right\|}}+0.01\left\|x+10\right\|.\quad$ ${\ displaystyle f (x, y) = 100 {\ sqrt {\ left \| y-0.01x ^ {2} \ right \|}} + 0.01 \ left \| x + 10 \ right \|. \ quad}$ ${\ displaystyle f (x, y) = 100 {\ sqrt {\ left \| y-0.01x ^ {2} \ right \|}} + 0.01 \ left \| x + 10 \ right \|. \ quad}$	$f(-10,1)=0$ ${\ displaystyle f (-10.1) = 0}$ ${\ displaystyle f (-10.1) = 0}$	$-15\leq x\leq -5$ ${\ displaystyle -15 \ leq x \ leq -5}$ ${\ displaystyle -15 \ leq x \ leq -5}$ , $-3\leq y\leq 3$ ${\ displaystyle -3 \ leq y \ leq 3}$ ${\ displaystyle -3 \ leq y \ leq 3}$
Funcția Matyas	$f(x,y)=0.26\left(x^{2}+y^{2}\right)-0.48xy.\quad$ ${\ displaystyle f (x, y) = 0.26 \ left (x ^ {2} + y ^ {2} \ right) -0.48xy. \ quad}$ ${\ displaystyle f (x, y) = 0.26 \ left (x ^ {2} + y ^ {2} \ right) -0.48xy. \ quad}$	$f(0,0)=0$ ${\ displaystyle f (0,0) = 0}$ ${\ displaystyle f (0,0) = 0}$	$-10\leq x,y\leq 10$ ${\ displaystyle -10 \ leq x, y \ leq 10}$ ${\ displaystyle -10 \ leq x, y \ leq 10}$
Funcția lui Levi n.13	$f(x,y)=\sin ^{2}\left(3\pi x\right)+\left(x-1\right)^{2}\left(1+\sin ^{2}\left(3\pi y\right)\right)$ ${\ displaystyle f (x, y) = \ sin ^ {2} \ left (3 \ pi x \ right) + \ left (x-1 \ right) ^ {2} \ left (1+ \ sin ^ {2 } \ left (3 \ pi y \ right) \ right)}$ ${\ displaystyle f (x, y) = \ sin ^ {2} \ left (3 \ pi x \ right) + \ left (x-1 \ right) ^ {2} \ left (1+ \ sin ^ {2 } \ left (3 \ pi y \ right) \ right)}$ $+\left(y-1\right)^{2}\left(1+\sin ^{2}\left(2\pi y\right)\right).\quad$ ${\ displaystyle + \ left (y-1 \ right) ^ {2} \ left (1+ \ sin ^ {2} \ left (2 \ pi y \ right) \ right). \ quad}$ ${\ displaystyle + \ left (y-1 \ right) ^ {2} \ left (1+ \ sin ^ {2} \ left (2 \ pi y \ right) \ right). \ quad}$	$f(1,1)=0$ ${\ displaystyle f (1,1) = 0}$ ${\ displaystyle f (1,1) = 0}$	$-10\leq x,y\leq 10$ ${\ displaystyle -10 \ leq x, y \ leq 10}$ ${\ displaystyle -10 \ leq x, y \ leq 10}$
Funcție de cămilă cu trei cocoașe	$f(x,y)=2x^{2}-1.05x^{4}+{\frac {x^{6}}{6}}+xy+y^{2}.\quad$ ${\ displaystyle f (x, y) = 2x ^ {2} -1,05x ^ {4} + {\ frac {x ^ {6}} {6}} + xy + y ^ {2}. \ quad}$ ${\ displaystyle f (x, y) = 2x ^ {2} -1,05x ^ {4} + {\ frac {x ^ {6}} {6}} + xy + y ^ {2}. \ quad}$	$f(0,0)=0$ ${\ displaystyle f (0,0) = 0}$ ${\ displaystyle f (0,0) = 0}$	$-5\leq x,y\leq 5$ ${\ displaystyle -5 \ leq x, y \ leq 5}$ ${\ displaystyle -5 \ leq x, y \ leq 5}$
Funcția Easom	$f(x,y)=-\cos \left(x\right)\cos \left(y\right)\exp \left(-\left(\left(x-\pi \right)^{2}+\left(y-\pi \right)^{2}\right)\right).\quad$ ${\ displaystyle f (x, y) = - \ cos \ left (x \ right) \ cos \ left (y \ right) \ exp \ left (- \ left (\ left (x- \ pi \ right) ^ { 2} + \ left (y- \ pi \ right) ^ {2} \ right) \ right). \ Quad}$ ${\ displaystyle f (x, y) = - \ cos \ left (x \ right) \ cos \ left (y \ right) \ exp \ left (- \ left (\ left (x- \ pi \ right) ^ { 2} + \ left (y- \ pi \ right) ^ {2} \ right) \ right). \ Quad}$	$f(\pi ,\pi )=-1$ ${\ displaystyle f (\ pi, \ pi) = - 1}$ ${\ displaystyle f (\ pi, \ pi) = - 1}$	$-100\leq x,y\leq 100$ ${\ displaystyle -100 \ leq x, y \ leq 100}$ ${\ displaystyle -100 \ leq x, y \ leq 100}$
Funcție încrucișată	$f(x,y)=-0.0001\left(\left\|\sin \left(x\right)\sin \left(y\right)\exp \left(\left\|100-{\frac {\sqrt {x^{2}+y^{2}}}{\pi }}\right\|\right)\right\|+1\right)^{0.1}.\quad$ ${\ displaystyle f (x, y) = - 0.0001 \ left (\ left \| \ sin \ left (x \ right) \ sin \ left (y \ right) \ exp \ left (\ left \| 100 - {\ frac { \ sqrt {x ^ {2} + y ^ {2}}} {\ pi}} \ right \| \ right) \ right \| +1 \ right) ^ {0.1}. \ quad}$ ${\ displaystyle f (x, y) = - 0.0001 \ left (\ left \| \ sin \ left (x \ right) \ sin \ left (y \ right) \ exp \ left (\ left \| 100 - {\ frac { \ sqrt {x ^ {2} + y ^ {2}}} {\ pi}} \ right \| \ right) \ right \| +1 \ right) ^ {0.1}. \ quad}$	${\text{Min}}={\begin{cases}f\left(1.34941,-1.34941\right)&=-2.06261\\f\left(1.34941,1.34941\right)&=-2.06261\\f\left(-1.34941,1.34941\right)&=-2.06261\\f\left(-1.34941,-1.34941\right)&=-2.06261\\\end{cases}}$ ${\ displaystyle {\ text {Min}} = {\ begin {cases} f \ left (1.34941, -1.34941 \ right) & = - 2.06261 \\ f \ left (1.34941,1.34941 \ right) & = - 2.06261 \\ f \ left (-1.34941,1.34941 \ right) & = - 2.06261 \\ f \ left (-1.34941, -1.34941 \ right) & = - 2.06261 \\\ end {cases}}}$ ${\ displaystyle {\ text {Min}} = {\ begin {cases} f \ left (1.34941, -1.34941 \ right) & = - 2.06261 \\ f \ left (1.34941,1.34941 \ right) & = - 2.06261 \\ f \ left (-1.34941,1.34941 \ right) & = - 2.06261 \\ f \ left (-1.34941, -1.34941 \ right) & = - 2.06261 \\\ end {cases}}}$	$-10\leq x,y\leq 10$ ${\ displaystyle -10 \ leq x, y \ leq 10}$ ${\ displaystyle -10 \ leq x, y \ leq 10}$
Funcția Eggholder	$f(x,y)=-\left(y+47\right)\sin \left({\sqrt {\left\|y+{\frac {x}{2}}+47\right\|}}\right)-x\sin \left({\sqrt {\left\|x-\left(y+47\right)\right\|}}\right).\quad$ ${\ displaystyle f (x, y) = - \ left (y + 47 \ right) \ sin \ left ({\ sqrt {\ left \| y + {\ frac {x} {2}} + 47 \ right \|} } \ right) -x \ sin \ left ({\ sqrt {\ left \| x- \ left (y + 47 \ right) \ right \|}} \ right). \ quad}$ ${\ displaystyle f (x, y) = - \ left (y + 47 \ right) \ sin \ left ({\ sqrt {\ left \| y + {\ frac {x} {2}} + 47 \ right \|} } \ right) -x \ sin \ left ({\ sqrt {\ left \| x- \ left (y + 47 \ right) \ right \|}} \ right). \ quad}$	$f(512,404.2319)=-959.6407$ ${\ displaystyle f (512.404.2319) = - 959.6407}$ ${\ displaystyle f (512.404.2319) = - 959.6407}$	$-512\leq x,y\leq 512$ ${\ displaystyle -512 \ leq x, y \ leq 512}$ ${\ displaystyle -512 \ leq x, y \ leq 512}$
Funcția Hölder	$f(x,y)=-\left\|\sin \left(x\right)\cos \left(y\right)\exp \left(\left\|1-{\frac {\sqrt {x^{2}+y^{2}}}{\pi }}\right\|\right)\right\|.\quad$ ${\ displaystyle f (x, y) = - \ left \| \ sin \ left (x \ right) \ cos \ left (y \ right) \ exp \ left (\ left \| 1 - {\ frac {\ sqrt {x ^ {2} + y ^ {2}}} {\ pi}} \ right \| \ right) \ right \|. \ Quad}$ ${\ displaystyle f (x, y) = - \ left \| \ sin \ left (x \ right) \ cos \ left (y \ right) \ exp \ left (\ left \| 1 - {\ frac {\ sqrt {x ^ {2} + y ^ {2}}} {\ pi}} \ right \| \ right) \ right \|. \ Quad}$	${\text{Min}}={\begin{cases}f\left(8.05502,9.66459\right)&=-19.2085\\f\left(-8.05502,9.66459\right)&=-19.2085\\f\left(8.05502,-9.66459\right)&=-19.2085\\f\left(-8.05502,-9.66459\right)&=-19.2085\end{cases}}$ ${\ displaystyle {\ text {Min}} = {\ begin {cases} f \ left (8.05502,9.66459 \ right) & = - 19.2085 \\ f \ left (-8.05502,9.66459 \ right) & = - 19.2085 \\ f \ left (8.05502, -9.66459 \ right) & = - 19.2085 \\ f \ left (-8.05502, -9.66459 \ right) & = - 19.2085 \ end {cases}}}$ ${\ displaystyle {\ text {Min}} = {\ begin {cases} f \ left (8.05502,9.66459 \ right) & = - 19.2085 \\ f \ left (-8.05502,9.66459 \ right) & = - 19.2085 \\ f \ left (8.05502, -9.66459 \ right) & = - 19.2085 \\ f \ left (-8.05502, -9.66459 \ right) & = - 19.2085 \ end {cases}}}$	$-10\leq x,y\leq 10$ ${\ displaystyle -10 \ leq x, y \ leq 10}$ ${\ displaystyle -10 \ leq x, y \ leq 10}$
Funcția McCormick	$f(x,y)=\sin \left(x+y\right)+\left(x-y\right)^{2}-1.5x+2.5y+1.\quad$ ${\ displaystyle f (x, y) = \ sin \ left (x + y \ right) + \ left (xy \ right) ^ {2} -1,5x + 2,5y + 1. \ quad}$ ${\ displaystyle f (x, y) = \ sin \ left (x + y \ right) + \ left (x-y \ right) ^ {2} -1,5x + 2,5y + 1. \ quad}$	$f(-0.54719,-1.54719)=-1.9133$ ${\ displaystyle f (-0.54719, -1.54719) = - 1.9133}$ ${\ displaystyle f (-0.54719, -1.54719) = - 1.9133}$	$-1.5\leq x\leq 4$ ${\ displaystyle -1,5 \ leq x \ leq 4}$ ${\ displaystyle -1,5 \ leq x \ leq 4}$ , $-3\leq y\leq 4$ ${\ displaystyle -3 \ leq y \ leq 4}$ ${\ displaystyle -3 \ leq y \ leq 4}$
Funcția Schaffer nr. 2	$f(x,y)=0.5+{\frac {\sin ^{2}\left(x^{2}-y^{2}\right)-0.5}{\left(1+0.001\left(x^{2}+y^{2}\right)\right)^{2}}}.\quad$ ${\ displaystyle f (x, y) = 0,5 + {\ frac {\ sin ^ {2} \ left (x ^ {2} -y ^ {2} \ right) -0,5} {\ left (1 + 0,001 \ stânga (x ^ {2} + y ^ {2} \ dreapta) \ dreapta) ^ {2}}}. \ quad}$ ${\ displaystyle f (x, y) = 0,5 + {\ frac {\ sin ^ {2} \ left (x ^ {2} -y ^ {2} \ right) -0,5} {\ left (1 + 0,001 \ stânga (x ^ {2} + y ^ {2} \ dreapta) \ dreapta) ^ {2}}}. \ quad}$	$f(0,0)=0$ ${\ displaystyle f (0,0) = 0}$ ${\ displaystyle f (0,0) = 0}$	$-100\leq x,y\leq 100$ ${\ displaystyle -100 \ leq x, y \ leq 100}$ ${\ displaystyle -100 \ leq x, y \ leq 100}$
Funcția Schaffer nr. 4	$f(x,y)=0.5+{\frac {\cos \left(\sin \left(\left\|x^{2}-y^{2}\right\|\right)\right)-0.5}{\left(1+0.001\left(x^{2}+y^{2}\right)\right)^{2}}}.\quad$ ${\ displaystyle f (x, y) = 0,5 + {\ frac {\ cos \ left (\ sin \ left (\ left \| x ^ {2} -y ^ {2} \ right \| \ right) \ right) - 0,5} {\ left (1 + 0,001 \ left (x ^ {2} + y ^ {2} \ right) \ right) ^ {2}}}. \ Quad}$ ${\ displaystyle f (x, y) = 0,5 + {\ frac {\ cos \ left (\ sin \ left (\ left \| x ^ {2} -y ^ {2} \ right \| \ right) \ right) - 0,5} {\ left (1 + 0,001 \ left (x ^ {2} + y ^ {2} \ right) \ right) ^ {2}}}. \ Quad}$	$f(0,1.25313)=0.292579$ ${\ displaystyle f (0,1.25313) = 0,292579}$ ${\ displaystyle f (0,1.25313) = 0,292579}$	$-100\leq x,y\leq 100$ ${\ displaystyle -100 \ leq x, y \ leq 100}$ ${\ displaystyle -100 \ leq x, y \ leq 100}$
Styblinski - Funcția Tang	$f({\boldsymbol {x}})={\frac {\sum _{i=1}^{n}x_{i}^{4}-16x_{i}^{2}+5x_{i}}{2}}.\quad$ ${\ displaystyle f ({\ boldsymbol {x}}) = {\ frac {\ sum _ {i = 1} ^ {n} x_ {i} ^ {4} -16x_ {i} ^ {2} + 5x_ { i}} {2}}. \ quad}$ ${\ displaystyle f ({\ boldsymbol {x}}) = {\ frac {\ sum _ {i = 1} ^ {n} x_ {i} ^ {4} -16x_ {i} ^ {2} + 5x_ { i}} {2}}. \ quad}$	$f\left(\underbrace {-2.903534,\ldots ,-2.903534} _{(n){\text{ volte}}}\right)=-39.16599n$ ${\ displaystyle f \ left (\ underbrace {-2.903534, \ ldots, -2.903534} _ {(n) {\ text {times}}} \ right) = - 39.16599n}$ ${\ displaystyle f \ left (\ underbrace {-2.903534, \ ldots, -2.903534} _ {(n) {\ text {times}}} \ right) = - 39.16599n}$	$-5\leq x_{i}\leq 5$ ${\ displaystyle -5 \ leq x_ {i} \ leq 5}$ ${\ displaystyle -5 \ leq x_ {i} \ leq 5}$ , $1\leq i\leq n$ ${\ displaystyle 1 \ leq i \ leq n}$ ${\ displaystyle 1 \ leq i \ leq n}$ .
Funcția Simionescu ^[6]	$f(x,y)=0.1xy$ ${\ displaystyle f (x, y) = 0.1xy}$ ${\ displaystyle f (x, y) = 0.1xy}$ , ${\text{subjected to: }}x^{2}+y^{2}\leq \left(r_{T}+r_{S}\cos \left(n\arctan {\frac {x}{y}}\right)\right)^{2}$ ${\ displaystyle {\ text {supus:}} x ^ {2} + y ^ {2} \ leq \ left (r_ {T} + r_ {S} \ cos \ left (n \ arctan {\ frac {x } {y}} \ right) \ right) ^ {2}}$ ${\ displaystyle {\ text {supus:}} x ^ {2} + y ^ {2} \ leq \ left (r_ {T} + r_ {S} \ cos \ left (n \ arctan {\ frac {x } {y}} \ right) \ right) ^ {2}}$ ${\text{where: }}r_{T}=1,r_{S}=0.2{\text{ and }}n=8$ ${\ displaystyle {\ text {unde:}} r_ {T} = 1, r_ {S} = 0.2 {\ text {și}} n = 8}$ ${\ displaystyle {\ text {unde:}} r_ {T} = 1, r_ {S} = 0.2 {\ text {și}} n = 8}$	$f(\pm 0.85586214,\mp 0.85586214)=-0.072625$ ${\ displaystyle f (\ pm 0.85586214, \ mp 0.85586214) = - 0.072625}$ ${\ displaystyle f (\ pm 0.85586214, \ mp 0.85586214) = - 0.072625}$	$-1.25\leq x,y\leq 1.25$ ${\ displaystyle -1.25 \ leq x, y \ leq 1.25}$ ${\ displaystyle -1.25 \ leq x, y \ leq 1.25}$

Testați funcțiile pentru probleme MOP

Următoarele funcții de testare pentru algoritmi de optimizare multiobiectivă provin de la Deb, ^[7] Binh et. la. ^[8] și Binh. ^[9] ^[10] ^[11]

Nume	Funcții	Constrângeri	Căutare domeniu
Funcția Binh și Korn	${\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)&=4x^{2}+4y^{2}\\f_{2}\left(x,y\right)&=\left(x-5\right)^{2}+\left(y-5\right)^{2}\\\end{cases}}$ ${\ displaystyle {\ text {Minimize}} = {\ begin {cases} f_ {1} \ left (x, y \ right) & = 4x ^ {2} + 4y ^ {2} \\ f_ {2} \ left (x, y \ right) & = \ left (x-5 \ right) ^ {2} + \ left (y-5 \ right) ^ {2} \\\ end {cases}}}$ ${\ displaystyle {\ text {Minimize}} = {\ begin {cases} f_ {1} \ left (x, y \ right) & = 4x ^ {2} + 4y ^ {2} \\ f_ {2} \ left (x, y \ right) & = \ left (x-5 \ right) ^ {2} + \ left (y-5 \ right) ^ {2} \\\ end {cases}}}$	${\text{s.t.}}={\begin{cases}g_{1}\left(x,y\right)&=\left(x-5\right)^{2}+y^{2}\leq 25\\g_{2}\left(x,y\right)&=\left(x-8\right)^{2}+\left(y+3\right)^{2}\geq 7.7\\\end{cases}}$ ${\ displaystyle {\ text {st}} = {\ begin {cases} g_ {1} \ left (x, y \ right) & = \ left (x-5 \ right) ^ {2} + y ^ {2 } \ leq 25 \\ g_ {2} \ left (x, y \ right) & = \ left (x-8 \ right) ^ {2} + \ left (y + 3 \ right) ^ {2} \ geq 7.7 \\\ end {cases}}}$ ${\ displaystyle {\ text {st}} = {\ begin {cases} g_ {1} \ left (x, y \ right) & = \ left (x-5 \ right) ^ {2} + y ^ {2 } \ leq 25 \\ g_ {2} \ left (x, y \ right) & = \ left (x-8 \ right) ^ {2} + \ left (y + 3 \ right) ^ {2} \ geq 7.7 \\\ end {cases}}}$	$0\leq x\leq 5$ ${\ displaystyle 0 \ leq x \ leq 5}$ ${\ displaystyle 0 \ leq x \ leq 5}$ , $0\leq y\leq 3$ ${\ displaystyle 0 \ leq y \ leq 3}$ ${\ displaystyle 0 \ leq y \ leq 3}$
Funcția Chakong și Haimes	${\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)&=2+\left(x-2\right)^{2}+\left(y-1\right)^{2}\\f_{2}\left(x,y\right)&=9x+\left(y-1\right)^{2}\\\end{cases}}$ ${\ displaystyle {\ text {Minimize}} = {\ begin {cases} f_ {1} \ left (x, y \ right) & = 2+ \ left (x-2 \ right) ^ {2} + \ left (y-1 \ right) ^ {2} \\ f_ {2} \ left (x, y \ right) & = 9x + \ left (y-1 \ right) ^ {2} \\\ end {cases} }}$ ${\ displaystyle {\ text {Minimize}} = {\ begin {cases} f_ {1} \ left (x, y \ right) & = 2+ \ left (x-2 \ right) ^ {2} + \ left (y-1 \ right) ^ {2} \\ f_ {2} \ left (x, y \ right) & = 9x + \ left (y-1 \ right) ^ {2} \\\ end {cases} }}$	${\text{s.t.}}={\begin{cases}g_{1}\left(x,y\right)&=x^{2}+y^{2}\leq 225\\g_{2}\left(x,y\right)&=x-3y+10\leq 0\\\end{cases}}$ ${\ displaystyle {\ text {st}} = {\ begin {cases} g_ {1} \ left (x, y \ right) & = x ^ {2} + y ^ {2} \ leq 225 \\ g_ { 2} \ left (x, y \ right) & = x-3y + 10 \ leq 0 \\\ end {cases}}}$ ${\ displaystyle {\ text {st}} = {\ begin {cases} g_ {1} \ left (x, y \ right) & = x ^ {2} + y ^ {2} \ leq 225 \\ g_ { 2} \ left (x, y \ right) & = x-3y + 10 \ leq 0 \\\ end {cases}}}$	$-20\leq x,y\leq 20$ ${\ displaystyle -20 \ leq x, y \ leq 20}$ ${\ displaystyle -20 \ leq x, y \ leq 20}$
Funcția Fonseca și Fleming	${\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)&=1-\exp \left(-\sum _{i=1}^{n}\left(x_{i}-{\frac {1}{\sqrt {n}}}\right)^{2}\right)\\f_{2}\left({\boldsymbol {x}}\right)&=1-\exp \left(-\sum _{i=1}^{n}\left(x_{i}+{\frac {1}{\sqrt {n}}}\right)^{2}\right)\\\end{cases}}$ ${\ displaystyle {\ text {Minimize}} = {\ begin {cases} f_ {1} \ left ({\ boldsymbol {x}} \ right) & = 1- \ exp \ left (- \ sum _ {i = 1} ^ {n} \ left (x_ {i} - {\ frac {1} {\ sqrt {n}}} \ right) ^ {2} \ right) \\ f_ {2} \ left ({\ boldsymbol {x}} \ right) & = 1- \ exp \ left (- \ sum _ {i = 1} ^ {n} \ left (x_ {i} + {\ frac {1} {\ sqrt {n}} } \ right) ^ {2} \ right) \\\ end {cases}}}$ ${\ displaystyle {\ text {Minimize}} = {\ begin {cases} f_ {1} \ left ({\ boldsymbol {x}} \ right) & = 1- \ exp \ left (- \ sum _ {i = 1} ^ {n} \ left (x_ {i} - {\ frac {1} {\ sqrt {n}}} \ right) ^ {2} \ right) \\ f_ {2} \ left ({\ boldsymbol {x}} \ right) & = 1- \ exp \ left (- \ sum _ {i = 1} ^ {n} \ left (x_ {i} + {\ frac {1} {\ sqrt {n}} } \ right) ^ {2} \ right) \\\ end {cases}}}$		$-4\leq x_{i}\leq 4$ ${\ displaystyle -4 \ leq x_ {i} \ leq 4}$ ${\ displaystyle -4 \ leq x_ {i} \ leq 4}$ , $1\leq i\leq n$ ${\ displaystyle 1 \ leq i \ leq n}$ ${\ displaystyle 1 \ leq i \ leq n}$
Funcția de testare n. 4 ^[9]	${\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)&=x^{2}-y\\f_{2}\left(x,y\right)&=-0.5x-y-1\\\end{cases}}$ ${\ displaystyle {\ text {Minimize}} = {\ begin {cases} f_ {1} \ left (x, y \ right) & = x ^ {2} -y \\ f_ {2} \ left (x, y \ right) & = - 0,5xy-1 \\\ end {cases}}}$ ${\ displaystyle {\ text {Minimize}} = {\ begin {cases} f_ {1} \ left (x, y \ right) & = x ^ {2} -y \\ f_ {2} \ left (x, y \ right) & = - 0,5xy-1 \\\ end {cases}}}$	${\text{s.t.}}={\begin{cases}g_{1}\left(x,y\right)&=6.5-{\frac {x}{6}}-y\geq 0\\g_{2}\left(x,y\right)&=7.5-0.5x-y\geq 0\\g_{3}\left(x,y\right)&=30-5x-y\geq 0\\\end{cases}}$ ${\ displaystyle {\ text {st}} = {\ begin {cases} g_ {1} \ left (x, y \ right) & = 6.5 - {\ frac {x} {6}} - y \ geq 0 \ \ g_ {2} \ left (x, y \ right) & = 7.5-0.5xy \ geq 0 \\ g_ {3} \ left (x, y \ right) & = 30-5x-y \ geq 0 \\ \ end {cases}}}$ ${\ displaystyle {\ text {st}} = {\ begin {cases} g_ {1} \ left (x, y \ right) & = 6.5 - {\ frac {x} {6}} - y \ geq 0 \ \ g_ {2} \ left (x, y \ right) & = 7.5-0.5xy \ geq 0 \\ g_ {3} \ left (x, y \ right) & = 30-5x-y \ geq 0 \\ \ end {cases}}}$	$-7\leq x,y\leq 4$ ${\ displaystyle -7 \ leq x, y \ leq 4}$ ${\ displaystyle -7 \ leq x, y \ leq 4}$
Funcția Kursawe	${\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)&=\sum _{i=1}^{2}\left[-10\exp \left(-0.2{\sqrt {x_{i}^{2}+x_{i+1}^{2}}}\right)\right]\\&\\f_{2}\left({\boldsymbol {x}}\right)&=\sum _{i=1}^{3}\left[\left\|x_{i}\right\|^{0.8}+5\sin \left(x_{i}^{3}\right)\right]\\\end{cases}}$ ${\ displaystyle {\ text {Minimize}} = {\ begin {cases} f_ {1} \ left ({\ boldsymbol {x}} \ right) & = \ sum _ {i = 1} ^ {2} \ left [-10 \ exp \ left (-0.2 {\ sqrt {x_ {i} ^ {2} + x_ {i + 1} ^ {2}}} \ right) \ right] \\ & \\ f_ {2} \ left ({\ boldsymbol {x}} \ right) & = \ sum _ {i = 1} ^ {3} \ left [\ left \| x_ {i} \ right \| ^ {0.8} +5 \ sin \ left (x_ {i} ^ {3} \ right) \ right] \\\ end {cases}}}$ ${\ displaystyle {\ text {Minimize}} = {\ begin {cases} f_ {1} \ left ({\ boldsymbol {x}} \ right) & = \ sum _ {i = 1} ^ {2} \ left [-10 \ exp \ left (-0.2 {\ sqrt {x_ {i} ^ {2} + x_ {i + 1} ^ {2}}} \ right) \ right] \\ & \\ f_ {2} \ left ({\ boldsymbol {x}} \ right) & = \ sum _ {i = 1} ^ {3} \ left [\ left \| x_ {i} \ right \| ^ {0.8} +5 \ sin \ left (x_ {i} ^ {3} \ right) \ right] \\\ end {cases}}}$		$-5\leq x_{i}\leq 5$ ${\ displaystyle -5 \ leq x_ {i} \ leq 5}$ ${\ displaystyle -5 \ leq x_ {i} \ leq 5}$ , $1\leq i\leq 3$ ${\ displaystyle 1 \ leq i \ leq 3}$ ${\ displaystyle 1 \ leq i \ leq 3}$ .
Funcția Schaffer Nr. 1	${\text{Minimize}}={\begin{cases}f_{1}\left(x\right)&=x^{2}\\f_{2}\left(x\right)&=\left(x-2\right)^{2}\\\end{cases}}$ ${\ displaystyle {\ text {Minimize}} = {\ begin {cases} f_ {1} \ left (x \ right) & = x ^ {2} \\ f_ {2} \ left (x \ right) & = \ left (x-2 \ right) ^ {2} \\\ end {cases}}}$ ${\ displaystyle {\ text {Minimize}} = {\ begin {cases} f_ {1} \ left (x \ right) & = x ^ {2} \\ f_ {2} \ left (x \ right) & = \ left (x-2 \ right) ^ {2} \\\ end {cases}}}$		$-A\leq x\leq A$ ${\ displaystyle -A \ leq x \ leq A}$ ${\ displaystyle -A \ leq x \ leq A}$ . Valorile $LA$ ${\ displaystyle A}$ $LA$ formă $10$ ${\ displaystyle 10}$ $10$ la $10^{5}$ ${\ displaystyle 10 ^ {5}}$ ${\ displaystyle 10 ^ {5}}$ au fost utilizate cu succes. Valori mai mari ale $LA$ ${\ displaystyle A}$ $LA$ crește dificultatea problemei.
Funcția Schaffer Nr. 2	${\text{Minimize}}={\begin{cases}f_{1}\left(x\right)&={\begin{cases}-x,&{\text{if }}x\leq 1\\x-2,&{\text{if }}1<x\leq 3\\4-x,&{\text{if }}3<x\leq 4\\x-4,&{\text{if }}x>4\\\end{cases}}\\f_{2}\left(x\right)&=\left(x-5\right)^{2}\\\end{cases}}$ ${\ displaystyle {\ text {Minimize}} = {\ begin {cases} f_ {1} \ left (x \ right) & = {\ begin {cases} -x, & {\ text {if}} x \ leq 1 \\ x-2, & {\ text {if}} 1 <x \ leq 3 \\ 4-x, & {\ text {if}} 3 <x \ leq 4 \\ x-4, & {\ text {if}} x> 4 \\\ end {cases}} \\ f_ {2} \ left (x \ right) & = \ left (x-5 \ right) ^ {2} \\\ end {cases }}}$ ${\ displaystyle {\ text {Minimize}} = {\ begin {cases} f_ {1} \ left (x \ right) & = {\ begin {cases} -x, & {\ text {if}} x \ leq 1 \\ x-2, & {\ text {if}} 1 <x \ leq 3 \\ 4-x, & {\ text {if}} 3 <x \ leq 4 \\ x-4, & {\ text {if}} x> 4 \\\ end {cases}} \\ f_ {2} \ left (x \ right) & = \ left (x-5 \ right) ^ {2} \\\ end {cases }}}$		$-5\leq x\leq 10$ ${\ displaystyle -5 \ leq x \ leq 10}$ ${\ displaystyle -5 \ leq x \ leq 10}$ .
Funcția Poloni	${\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)&=\left[1+\left(A_{1}-B_{1}\left(x,y\right)\right)^{2}+\left(A_{2}-B_{2}\left(x,y\right)\right)^{2}\right]\\f_{2}\left(x,y\right)&=\left(x+3\right)^{2}+\left(y+1\right)^{2}\\\end{cases}}$ ${\ displaystyle {\ text {Minimize}} = {\ begin {cases} f_ {1} \ left (x, y \ right) & = \ left [1+ \ left (A_ {1} -B_ {1} \ left (x, y \ right) \ right) ^ {2} + \ left (A_ {2} -B_ {2} \ left (x, y \ right) \ right) ^ {2} \ right] \\ f_ {2} \ left (x, y \ right) & = \ left (x + 3 \ right) ^ {2} + \ left (y + 1 \ right) ^ {2} \\\ end {cases}}}$ ${\ displaystyle {\ text {Minimize}} = {\ begin {cases} f_ {1} \ left (x, y \ right) & = \ left [1+ \ left (A_ {1} -B_ {1} \ left (x, y \ right) \ right) ^ {2} + \ left (A_ {2} -B_ {2} \ left (x, y \ right) \ right) ^ {2} \ right] \\ f_ {2} \ left (x, y \ right) & = \ left (x + 3 \ right) ^ {2} + \ left (y + 1 \ right) ^ {2} \\\ end {cases}}}$ ${\text{where}}={\begin{cases}A_{1}&=0.5\sin \left(1\right)-2\cos \left(1\right)+\sin \left(2\right)-1.5\cos \left(2\right)\\A_{2}&=1.5\sin \left(1\right)-\cos \left(1\right)+2\sin \left(2\right)-0.5\cos \left(2\right)\\B_{1}\left(x,y\right)&=0.5\sin \left(x\right)-2\cos \left(x\right)+\sin \left(y\right)-1.5\cos \left(y\right)\\B_{2}\left(x,y\right)&=1.5\sin \left(x\right)-\cos \left(x\right)+2\sin \left(y\right)-0.5\cos \left(y\right)\end{cases}}$ ${\ displaystyle {\ text {where}} = {\ begin {cases} A_ {1} & = 0,5 \ sin \ left (1 \ right) -2 \ cos \ left (1 \ right) + \ sin \ left ( 2 \ dreapta) -1,5 \ cos \ left (2 \ right) \\ A_ {2} & = 1.5 \ sin \ left (1 \ right) - \ cos \ left (1 \ right) +2 \ sin \ left ( 2 \ dreapta) -0,5 \ cos \ left (2 \ right) \\ B_ {1} \ left (x, y \ right) & = 0,5 \ sin \ left (x \ right) -2 \ cos \ left (x \ right) + \ sin \ left (y \ right) -1,5 \ cos \ left (y \ right) \\ B_ {2} \ left (x, y \ right) & = 1.5 \ sin \ left (x \ right ) - \ cos \ left (x \ right) +2 \ sin \ left (y \ right) -0.5 \ cos \ left (y \ right) \ end {cases}}}$ ${\ displaystyle {\ text {where}} = {\ begin {cases} A_ {1} & = 0,5 \ sin \ left (1 \ right) -2 \ cos \ left (1 \ right) + \ sin \ left ( 2 \ dreapta) -1,5 \ cos \ left (2 \ right) \\ A_ {2} & = 1.5 \ sin \ left (1 \ right) - \ cos \ left (1 \ right) +2 \ sin \ left ( 2 \ dreapta) -0,5 \ cos \ left (2 \ right) \\ B_ {1} \ left (x, y \ right) & = 0,5 \ sin \ left (x \ right) -2 \ cos \ left (x \ right) + \ sin \ left (y \ right) -1,5 \ cos \ left (y \ right) \\ B_ {2} \ left (x, y \ right) & = 1.5 \ sin \ left (x \ right ) - \ cos \ left (x \ right) +2 \ sin \ left (y \ right) -0.5 \ cos \ left (y \ right) \ end {cases}}}$		$-\pi \leq x,y\leq \pi$ ${\ displaystyle - \ pi \ leq x, y \ leq \ pi}$ ${\ displaystyle - \ pi \ leq x, y \ leq \ pi}$
Funcția Zitzler - Deb - Thiele n. 1	${\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)&=x_{1}\\f_{2}\left({\boldsymbol {x}}\right)&=g\left({\boldsymbol {x}}\right)h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)\\g\left({\boldsymbol {x}}\right)&=1+{\frac {9}{29}}\sum _{i=2}^{30}x_{i}\\h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)&=1-{\sqrt {\frac {f_{1}\left({\boldsymbol {x}}\right)}{g\left({\boldsymbol {x}}\right)}}}\\\end{cases}}$ ${\ displaystyle {\ text {Minimize}} = {\ begin {cases} f_ {1} \ left ({\ boldsymbol {x}} \ right) & = x_ {1} \\ f_ {2} \ left ({ \ boldsymbol {x}} \ right) & = g \ left ({\ boldsymbol {x}} \ right) h \ left (f_ {1} \ left ({\ boldsymbol {x}} \ right), g \ left ({\ boldsymbol {x}} \ right) \ right) \\ g \ left ({\ boldsymbol {x}} \ right) & = 1 + {\ frac {9} {29}} \ sum _ {i = 2} ^ {30} x_ {i} \\ h \ left (f_ {1} \ left ({\ boldsymbol {x}} \ right), g \ left ({\ boldsymbol {x}} \ right) \ right ) & = 1 - {\ sqrt {\ frac {f_ {1} \ left ({\ boldsymbol {x}} \ right)} {g \ left ({\ boldsymbol {x}} \ right)}}} \\ \ end {cases}}}$ ${\ displaystyle {\ text {Minimize}} = {\ begin {cases} f_ {1} \ left ({\ boldsymbol {x}} \ right) & = x_ {1} \\ f_ {2} \ left ({ \ boldsymbol {x}} \ right) & = g \ left ({\ boldsymbol {x}} \ right) h \ left (f_ {1} \ left ({\ boldsymbol {x}} \ right), g \ left ({\ boldsymbol {x}} \ right) \ right) \\ g \ left ({\ boldsymbol {x}} \ right) & = 1 + {\ frac {9} {29}} \ sum _ {i = 2} ^ {30} x_ {i} \\ h \ left (f_ {1} \ left ({\ boldsymbol {x}} \ right), g \ left ({\ boldsymbol {x}} \ right) \ right ) & = 1 - {\ sqrt {\ frac {f_ {1} \ left ({\ boldsymbol {x}} \ right)} {g \ left ({\ boldsymbol {x}} \ right)}}} \\ \ end {cases}}}$		$0\leq x_{i}\leq 1$ ${\ displaystyle 0 \ leq x_ {i} \ leq 1}$ ${\ displaystyle 0 \ leq x_ {i} \ leq 1}$ , $1\leq i\leq 30$ ${\ displaystyle 1 \ leq i \ leq 30}$ ${\ displaystyle 1 \ leq i \ leq 30}$ .
Funcția Zitzler - Deb - Thiele n. 2	${\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)&=x_{1}\\f_{2}\left({\boldsymbol {x}}\right)&=g\left({\boldsymbol {x}}\right)h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)\\g\left({\boldsymbol {x}}\right)&=1+{\frac {9}{29}}\sum _{i=2}^{30}x_{i}\\h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)&=1-\left({\frac {f_{1}\left({\boldsymbol {x}}\right)}{g\left({\boldsymbol {x}}\right)}}\right)^{2}\\\end{cases}}$ ${\ displaystyle {\ text {Minimize}} = {\ begin {cases} f_ {1} \ left ({\ boldsymbol {x}} \ right) & = x_ {1} \\ f_ {2} \ left ({ \ boldsymbol {x}} \ right) & = g \ left ({\ boldsymbol {x}} \ right) h \ left (f_ {1} \ left ({\ boldsymbol {x}} \ right), g \ left ({\ boldsymbol {x}} \ right) \ right) \\ g \ left ({\ boldsymbol {x}} \ right) & = 1 + {\ frac {9} {29}} \ sum _ {i = 2} ^ {30} x_ {i} \\ h \ left (f_ {1} \ left ({\ boldsymbol {x}} \ right), g \ left ({\ boldsymbol {x}} \ right) \ right ) & = 1- \ left ({\ frac {f_ {1} \ left ({\ boldsymbol {x}} \ right)} {g \ left ({\ boldsymbol {x}} \ right)}} \ right) ^ {2} \\\ end {cases}}}$ ${\ displaystyle {\ text {Minimize}} = {\ begin {cases} f_ {1} \ left ({\ boldsymbol {x}} \ right) & = x_ {1} \\ f_ {2} \ left ({ \ boldsymbol {x}} \ right) & = g \ left ({\ boldsymbol {x}} \ right) h \ left (f_ {1} \ left ({\ boldsymbol {x}} \ right), g \ left ({\ boldsymbol {x}} \ right) \ right) \\ g \ left ({\ boldsymbol {x}} \ right) & = 1 + {\ frac {9} {29}} \ sum _ {i = 2} ^ {30} x_ {i} \\ h \ left (f_ {1} \ left ({\ boldsymbol {x}} \ right), g \ left ({\ boldsymbol {x}} \ right) \ right ) & = 1- \ left ({\ frac {f_ {1} \ left ({\ boldsymbol {x}} \ right)} {g \ left ({\ boldsymbol {x}} \ right)}} \ right) ^ {2} \\\ end {cases}}}$		$0\leq x_{i}\leq 1$ ${\ displaystyle 0 \ leq x_ {i} \ leq 1}$ ${\ displaystyle 0 \ leq x_ {i} \ leq 1}$ , $1\leq i\leq 30$ ${\ displaystyle 1 \ leq i \ leq 30}$ ${\ displaystyle 1 \ leq i \ leq 30}$ .
Zitzler - Deb - Funcția lui Thiele n. 3	${\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)&=x_{1}\\f_{2}\left({\boldsymbol {x}}\right)&=g\left({\boldsymbol {x}}\right)h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)\\g\left({\boldsymbol {x}}\right)&=1+{\frac {9}{29}}\sum _{i=2}^{30}x_{i}\\h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)&=1-{\sqrt {\frac {f_{1}\left({\boldsymbol {x}}\right)}{g\left({\boldsymbol {x}}\right)}}}-\left({\frac {f_{1}\left({\boldsymbol {x}}\right)}{g\left({\boldsymbol {x}}\right)}}\right)\sin \left(10\pi f_{1}\left({\boldsymbol {x}}\right)\right)\end{cases}}$ ${\ displaystyle {\ text {Minimize}} = {\ begin {cases} f_ {1} \ left ({\ boldsymbol {x}} \ right) & = x_ {1} \\ f_ {2} \ left ({ \ boldsymbol {x}} \ right) & = g \ left ({\ boldsymbol {x}} \ right) h \ left (f_ {1} \ left ({\ boldsymbol {x}} \ right), g \ left ({\ boldsymbol {x}} \ right) \ right) \\ g \ left ({\ boldsymbol {x}} \ right) & = 1 + {\ frac {9} {29}} \ sum _ {i = 2} ^ {30} x_ {i} \\ h \ left (f_ {1} \ left ({\ boldsymbol {x}} \ right), g \ left ({\ boldsymbol {x}} \ right) \ right ) & = 1 - {\ sqrt {\ frac {f_ {1} \ left ({\ boldsymbol {x}} \ right)} {g \ left ({\ boldsymbol {x}} \ right)}}} - \ left ({\ frac {f_ {1} \ left ({\ boldsymbol {x}} \ right)} {g \ left ({\ boldsymbol {x}} \ right)}} \ right) \ sin \ left (10 \ pi f_ {1} \ left ({\ boldsymbol {x}} \ right) \ right) \ end {cases}}}$ ${\ displaystyle {\ text {Minimize}} = {\ begin {cases} f_ {1} \ left ({\ boldsymbol {x}} \ right) & = x_ {1} \\ f_ {2} \ left ({ \ boldsymbol {x}} \ right) & = g \ left ({\ boldsymbol {x}} \ right) h \ left (f_ {1} \ left ({\ boldsymbol {x}} \ right), g \ left ({\ boldsymbol {x}} \ right) \ right) \\ g \ left ({\ boldsymbol {x}} \ right) & = 1 + {\ frac {9} {29}} \ sum _ {i = 2} ^ {30} x_ {i} \\ h \ left (f_ {1} \ left ({\ boldsymbol {x}} \ right), g \ left ({\ boldsymbol {x}} \ right) \ right ) & = 1 - {\ sqrt {\ frac {f_ {1} \ left ({\ boldsymbol {x}} \ right)} {g \ left ({\ boldsymbol {x}} \ right)}}} - \ left ({\ frac {f_ {1} \ left ({\ boldsymbol {x}} \ right)} {g \ left ({\ boldsymbol {x}} \ right)}} \ right) \ sin \ left (10 \ pi f_ {1} \ left ({\ boldsymbol {x}} \ right) \ right) \ end {cases}}}$		$0\leq x_{i}\leq 1$ ${\ displaystyle 0 \ leq x_ {i} \ leq 1}$ ${\ displaystyle 0 \ leq x_ {i} \ leq 1}$ , $1\leq i\leq 30$ ${\ displaystyle 1 \ leq i \ leq 30}$ ${\ displaystyle 1 \ leq i \ leq 30}$ .
Funcția Zitzler - Deb - Thiele n. 4	${\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)&=x_{1}\\f_{2}\left({\boldsymbol {x}}\right)&=g\left({\boldsymbol {x}}\right)h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)\\g\left({\boldsymbol {x}}\right)&=91+\sum _{i=2}^{10}\left(x_{i}^{2}-10\cos \left(4\pi x_{i}\right)\right)\\h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)&=1-{\sqrt {\frac {f_{1}\left({\boldsymbol {x}}\right)}{g\left({\boldsymbol {x}}\right)}}}\end{cases}}$ ${\ displaystyle {\ text {Minimize}} = {\ begin {cases} f_ {1} \ left ({\ boldsymbol {x}} \ right) & = x_ {1} \\ f_ {2} \ left ({ \ boldsymbol {x}} \ right) & = g \ left ({\ boldsymbol {x}} \ right) h \ left (f_ {1} \ left ({\ boldsymbol {x}} \ right), g \ left ({\ boldsymbol {x}} \ right) \ right) \\ g \ left ({\ boldsymbol {x}} \ right) & = 91+ \ sum _ {i = 2} ^ {10} \ left (x_ {i} ^ {2} -10 \ cos \ left (4 \ pi x_ {i} \ right) \ right) \\ h \ left (f_ {1} \ left ({\ boldsymbol {x}} \ right) , g \ left ({\ boldsymbol {x}} \ right) \ right) & = 1 - {\ sqrt {\ frac {f_ {1} \ left ({\ boldsymbol {x}} \ right)} {g \ stânga ({\ boldsymbol {x}} \ right)}}} \ end {cases}}}$ ${\ displaystyle {\ text {Minimize}} = {\ begin {cases} f_ {1} \ left ({\ boldsymbol {x}} \ right) & = x_ {1} \\ f_ {2} \ left ({ \ boldsymbol {x}} \ right) & = g \ left ({\ boldsymbol {x}} \ right) h \ left (f_ {1} \ left ({\ boldsymbol {x}} \ right), g \ left ({\ boldsymbol {x}} \ right) \ right) \\ g \ left ({\ boldsymbol {x}} \ right) & = 91+ \ sum _ {i = 2} ^ {10} \ left (x_ {i} ^ {2} -10 \ cos \ left (4 \ pi x_ {i} \ right) \ right) \\ h \ left (f_ {1} \ left ({\ boldsymbol {x}} \ right) , g \ left ({\ boldsymbol {x}} \ right) \ right) & = 1 - {\ sqrt {\ frac {f_ {1} \ left ({\ boldsymbol {x}} \ right)} {g \ stânga ({\ boldsymbol {x}} \ right)}}} \ end {cases}}}$		$0\leq x_{1}\leq 1$ ${\ displaystyle 0 \ leq x_ {1} \ leq 1}$ ${\ displaystyle 0 \ leq x_ {1} \ leq 1}$ , $-5\leq x_{i}\leq 5$ ${\ displaystyle -5 \ leq x_ {i} \ leq 5}$ ${\ displaystyle -5 \ leq x_ {i} \ leq 5}$ , $2\leq i\leq 10$ ${\ displaystyle 2 \ leq i \ leq 10}$ ${\ displaystyle 2 \ leq i \ leq 10}$
Funcția Zitzler - Deb - Thiele n. 6	${\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)&=1-\exp \left(-4x_{1}\right)\sin ^{6}\left(6\pi x_{1}\right)\\f_{2}\left({\boldsymbol {x}}\right)&=g\left({\boldsymbol {x}}\right)h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)\\g\left({\boldsymbol {x}}\right)&=1+9\left[{\frac {\sum _{i=2}^{10}x_{i}}{9}}\right]^{0.25}\\h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)&=1-\left({\frac {f_{1}\left({\boldsymbol {x}}\right)}{g\left({\boldsymbol {x}}\right)}}\right)^{2}\\\end{cases}}$ ${\ displaystyle {\ text {Minimize}} = {\ begin {cases} f_ {1} \ left ({\ boldsymbol {x}} \ right) & = 1- \ exp \ left (-4x_ {1} \ right ) \ sin ^ {6} \ left (6 \ pi x_ {1} \ right) \\ f_ {2} \ left ({\ boldsymbol {x}} \ right) & = g \ left ({\ boldsymbol {x }} \ right) h \ left (f_ {1} \ left ({\ boldsymbol {x}} \ right), g \ left ({\ boldsymbol {x}} \ right) \ right) \\ g \ left ( {\ boldsymbol {x}} \ right) & = 1 + 9 \ left [{\ frac {\ sum _ {i = 2} ^ {10} x_ {i}} {9}} \ right] ^ {0.25} \\ h \ left (f_ {1} \ left ({\ boldsymbol {x}} \ right), g \ left ({\ boldsymbol {x}} \ right) \ right) & = 1- \ left ({\ frac {f_ {1} \ left ({\ boldsymbol {x}} \ right)} {g \ left ({\ boldsymbol {x}} \ right)}} \ right) ^ {2} \\\ end {cases }}}$ ${\ displaystyle {\ text {Minimize}} = {\ begin {cases} f_ {1} \ left ({\ boldsymbol {x}} \ right) & = 1- \ exp \ left (-4x_ {1} \ right ) \ sin ^ {6} \ left (6 \ pi x_ {1} \ right) \\ f_ {2} \ left ({\ boldsymbol {x}} \ right) & = g \ left ({\ boldsymbol {x }} \ right) h \ left (f_ {1} \ left ({\ boldsymbol {x}} \ right), g \ left ({\ boldsymbol {x}} \ right) \ right) \\ g \ left ( {\ boldsymbol {x}} \ right) & = 1 + 9 \ left [{\ frac {\ sum _ {i = 2} ^ {10} x_ {i}} {9}} \ right] ^ {0.25} \\ h \ left (f_ {1} \ left ({\ boldsymbol {x}} \ right), g \ left ({\ boldsymbol {x}} \ right) \ right) & = 1- \ left ({\ frac {f_ {1} \ left ({\ boldsymbol {x}} \ right)} {g \ left ({\ boldsymbol {x}} \ right)}} \ right) ^ {2} \\\ end {cases }}}$		$0\leq x_{i}\leq 1$ ${\ displaystyle 0 \ leq x_ {i} \ leq 1}$ ${\ displaystyle 0 \ leq x_ {i} \ leq 1}$ , $1\leq i\leq 10$ ${\ displaystyle 1 \ leq i \ leq 10}$ ${\ displaystyle 1 \ leq i \ leq 10}$ .
Funcția Viennet	${\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)&=0.5\left(x^{2}+y^{2}\right)+\sin \left(x^{2}+y^{2}\right)\\f_{2}\left(x,y\right)&={\frac {\left(3x-2y+4\right)^{2}}{8}}+{\frac {\left(x-y+1\right)^{2}}{27}}+15\\f_{3}\left(x,y\right)&={\frac {1}{x^{2}+y^{2}+1}}-1.1\exp \left(-\left(x^{2}+y^{2}\right)\right)\\\end{cases}}$ ${\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)&=0.5\left(x^{2}+y^{2}\right)+\sin \left(x^{2}+y^{2}\right)\\f_{2}\left(x,y\right)&={\frac {\left(3x-2y+4\right)^{2}}{8}}+{\frac {\left(x-y+1\right)^{2}}{27}}+15\\f_{3}\left(x,y\right)&={\frac {1}{x^{2}+y^{2}+1}}-1.1\exp \left(-\left(x^{2}+y^{2}\right)\right)\\\end{cases}}$ ${\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)&=0.5\left(x^{2}+y^{2}\right)+\sin \left(x^{2}+y^{2}\right)\\f_{2}\left(x,y\right)&={\frac {\left(3x-2y+4\right)^{2}}{8}}+{\frac {\left(x-y+1\right)^{2}}{27}}+15\\f_{3}\left(x,y\right)&={\frac {1}{x^{2}+y^{2}+1}}-1.1\exp \left(-\left(x^{2}+y^{2}\right)\right)\\\end{cases}}$		$-3\leq x,y\leq 3$ $-3\leq x,y\leq 3$ $-3\leq x,y\leq 3$ .
Funzione di Osyczka e Kundu	${\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)&=-25\left(x_{1}-2\right)^{2}-\left(x_{2}-2\right)^{2}-\left(x_{3}-1\right)^{2}-\left(x_{4}-4\right)^{2}-\left(x_{5}-1\right)^{2}\\f_{2}\left({\boldsymbol {x}}\right)&=\sum _{i=1}^{6}x_{i}^{2}\\\end{cases}}$ ${\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)&=-25\left(x_{1}-2\right)^{2}-\left(x_{2}-2\right)^{2}-\left(x_{3}-1\right)^{2}-\left(x_{4}-4\right)^{2}-\left(x_{5}-1\right)^{2}\\f_{2}\left({\boldsymbol {x}}\right)&=\sum _{i=1}^{6}x_{i}^{2}\\\end{cases}}$ ${\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)&=-25\left(x_{1}-2\right)^{2}-\left(x_{2}-2\right)^{2}-\left(x_{3}-1\right)^{2}-\left(x_{4}-4\right)^{2}-\left(x_{5}-1\right)^{2}\\f_{2}\left({\boldsymbol {x}}\right)&=\sum _{i=1}^{6}x_{i}^{2}\\\end{cases}}$	${\text{s.t.}}={\begin{cases}g_{1}\left({\boldsymbol {x}}\right)&=x_{1}+x_{2}-2\geq 0\\g_{2}\left({\boldsymbol {x}}\right)&=6-x_{1}-x_{2}\geq 0\\g_{3}\left({\boldsymbol {x}}\right)&=2-x_{2}+x_{1}\geq 0\\g_{4}\left({\boldsymbol {x}}\right)&=2-x_{1}+3x_{2}\geq 0\\g_{5}\left({\boldsymbol {x}}\right)&=4-\left(x_{3}-3\right)^{2}-x_{4}\geq 0\\g_{6}\left({\boldsymbol {x}}\right)&=\left(x_{5}-3\right)^{2}+x_{6}-4\geq 0\end{cases}}$ ${\text{st}}={\begin{cases}g_{1}\left({\boldsymbol {x}}\right)&=x_{1}+x_{2}-2\geq 0\\g_{2}\left({\boldsymbol {x}}\right)&=6-x_{1}-x_{2}\geq 0\\g_{3}\left({\boldsymbol {x}}\right)&=2-x_{2}+x_{1}\geq 0\\g_{4}\left({\boldsymbol {x}}\right)&=2-x_{1}+3x_{2}\geq 0\\g_{5}\left({\boldsymbol {x}}\right)&=4-\left(x_{3}-3\right)^{2}-x_{4}\geq 0\\g_{6}\left({\boldsymbol {x}}\right)&=\left(x_{5}-3\right)^{2}+x_{6}-4\geq 0\end{cases}}$ ${\text{s.t.}}={\begin{cases}g_{1}\left({\boldsymbol {x}}\right)&=x_{1}+x_{2}-2\geq 0\\g_{2}\left({\boldsymbol {x}}\right)&=6-x_{1}-x_{2}\geq 0\\g_{3}\left({\boldsymbol {x}}\right)&=2-x_{2}+x_{1}\geq 0\\g_{4}\left({\boldsymbol {x}}\right)&=2-x_{1}+3x_{2}\geq 0\\g_{5}\left({\boldsymbol {x}}\right)&=4-\left(x_{3}-3\right)^{2}-x_{4}\geq 0\\g_{6}\left({\boldsymbol {x}}\right)&=\left(x_{5}-3\right)^{2}+x_{6}-4\geq 0\end{cases}}$	$0\leq x_{1},x_{2},x_{6}\leq 10$ $0\leq x_{1},x_{2},x_{6}\leq 10$ $0\leq x_{1},x_{2},x_{6}\leq 10$ , $1\leq x_{3},x_{5}\leq 5$ $1\leq x_{3},x_{5}\leq 5$ $1\leq x_{3},x_{5}\leq 5$ , $0\leq x_{4}\leq 6$ $0\leq x_{4}\leq 6$ $0\leq x_{4}\leq 6$ .
Funzione CTP1 ^[7]	${\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)&=x\\f_{2}\left(x,y\right)&=\left(1+y\right)\exp \left(-{\frac {x}{1+y}}\right)\end{cases}}$ ${\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)&=x\\f_{2}\left(x,y\right)&=\left(1+y\right)\exp \left(-{\frac {x}{1+y}}\right)\end{cases}}$ ${\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)&=x\\f_{2}\left(x,y\right)&=\left(1+y\right)\exp \left(-{\frac {x}{1+y}}\right)\end{cases}}$	${\text{s.t.}}={\begin{cases}g_{1}\left(x,y\right)&={\frac {f_{2}\left(x,y\right)}{0.858\exp \left(-0.541f_{1}\left(x,y\right)\right)}}\geq 1\\g_{1}\left(x,y\right)&={\frac {f_{2}\left(x,y\right)}{0.728\exp \left(-0.295f_{1}\left(x,y\right)\right)}}\geq 1\end{cases}}$ ${\text{st}}={\begin{cases}g_{1}\left(x,y\right)&={\frac {f_{2}\left(x,y\right)}{0.858\exp \left(-0.541f_{1}\left(x,y\right)\right)}}\geq 1\\g_{1}\left(x,y\right)&={\frac {f_{2}\left(x,y\right)}{0.728\exp \left(-0.295f_{1}\left(x,y\right)\right)}}\geq 1\end{cases}}$ ${\text{s.t.}}={\begin{cases}g_{1}\left(x,y\right)&={\frac {f_{2}\left(x,y\right)}{0.858\exp \left(-0.541f_{1}\left(x,y\right)\right)}}\geq 1\\g_{1}\left(x,y\right)&={\frac {f_{2}\left(x,y\right)}{0.728\exp \left(-0.295f_{1}\left(x,y\right)\right)}}\geq 1\end{cases}}$	$0\leq x,y\leq 1$ $0\leq x,y\leq 1$ $0\leq x,y\leq 1$ .
Problema Constr-Ex ^[7]	${\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)&=x\\f_{2}\left(x,y\right)&={\frac {1+y}{x}}\\\end{cases}}$ ${\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)&=x\\f_{2}\left(x,y\right)&={\frac {1+y}{x}}\\\end{cases}}$ ${\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)&=x\\f_{2}\left(x,y\right)&={\frac {1+y}{x}}\\\end{cases}}$	${\text{s.t.}}={\begin{cases}g_{1}\left(x,y\right)&=y+9x\geq 6\\g_{1}\left(x,y\right)&=-y+9x\geq 1\\\end{cases}}$ ${\text{st}}={\begin{cases}g_{1}\left(x,y\right)&=y+9x\geq 6\\g_{1}\left(x,y\right)&=-y+9x\geq 1\\\end{cases}}$ ${\text{s.t.}}={\begin{cases}g_{1}\left(x,y\right)&=y+9x\geq 6\\g_{1}\left(x,y\right)&=-y+9x\geq 1\\\end{cases}}$	$0.1\leq x\leq 1$ $0.1\leq x\leq 1$ $0.1\leq x\leq 1$ , $0\leq y\leq 5$ $0\leq y\leq 5$ $0\leq y\leq 5$

Note

^ Neculai Andrei, An Unconstrained Optimization Test Functions Collection , in Advanced Modeling and Optimization , vol. 10, n. 1, 2008.
^ Thomas Bäck, Evolutionary algorithms in theory and practice: evolution strategies, evolutionary programming, genetic algorithms , Oxford, Oxford University Press, 1995, p. 328, ISBN 0-19-509971-0 .
^ Randy L. Haupt e Sue Ellen, Practical genetic algorithms with DC-Rom , 2ª ed., New York, J. Wiley, 2004, ISBN 0-471-45565-2 .
^ Rody Oldenhuis, Many test functions for global optimizers , su mathworks.com , Mathworks. URL consultato il 1º novembre 2012 .
^ Sonja Surjanovich e Derek Bingham, Powell Function , su sfu.ca , Simon Fraser University. URL consultato il 21 maggio 2014 ( archiviato il 21 maggio 2014) .
^ PA Simionescu, Computer Aided Graphing and Simulation Tools for AutoCAD Users , 1st, Boca Raton, FL, CRC Press, 2014, ISBN 978-1-4822-5290-3 .
^ ^a ^b ^c ^d ^e Deb, Kalyanmoy (2002) Multiobjective optimization using evolutionary algorithms (Repr. ed.). Chichester [ua]: Wiley. ISBN 0-471-87339-X .
^ 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
^ ^a ^b ^c Binh T. (1999) A multiobjective evolutionary algorithm. The study cases. Technical report. Institute for Automation and Communication. Barleben, Germany
^ Il software sviluppato da K. Deb è disponibile presso http://www.iitk.ac.in/kangal/codes.shtml
^ Gilberto A. Ortiz, Multi-objective optimization using ES as Evolutionary Algorithm. , su mathworks.com , Mathworks. URL consultato il 1º novembre 2012 .

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[1] Neculai Andrei, An Unconstrained Optimization Test Functions Collection , in Advanced Modeling and Optimization , vol. 10, n. 1, 2008.

[2] Thomas Bäck, Evolutionary algorithms in theory and practice: evolution strategies, evolutionary programming, genetic algorithms , Oxford, Oxford University Press, 1995, p. 328, ISBN 0-19-509971-0 .

[3] Randy L. Haupt e Sue Ellen, Practical genetic algorithms with DC-Rom , 2ª ed., New York, J. Wiley, 2004, ISBN 0-471-45565-2 .

[4] Rody Oldenhuis, Many test functions for global optimizers , su mathworks.com , Mathworks. URL consultato il 1º novembre 2012 .

[5] Sonja Surjanovich e Derek Bingham, Powell Function , su sfu.ca , Simon Fraser University. URL consultato il 21 maggio 2014 ( archiviato il 21 maggio 2014) .

[6] PA Simionescu, Computer Aided Graphing and Simulation Tools for AutoCAD Users , 1st, Boca Raton, FL, CRC Press, 2014, ISBN 978-1-4822-5290-3 .

[Deb:2002-7] Deb, Kalyanmoy (2002) Multiobjective optimization using evolutionary algorithms (Repr. ed.). Chichester [ua]: Wiley. ISBN 0-471-87339-X .

[Binh97-8] 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

[Binh99-9] Binh T. (1999) A multiobjective evolutionary algorithm. The study cases. Technical report. Institute for Automation and Communication. Barleben, Germany

[10] Il software sviluppato da K. Deb è disponibile presso http://www.iitk.ac.in/kangal/codes.shtml

[11] Gilberto A. Ortiz, Multi-objective optimization using ES as Evolutionary Algorithm. , su mathworks.com , Mathworks. URL consultato il 1º novembre 2012 .

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