Test functions (known as artificial landscapes) are useful to evaluate characteristics of optimization algorithms, such as:
Velocity of convergence.
Precision.
Robustness.
General performance. Here some test functions are presented with the aim of giving an idea about the different situations that optimization algorithms have to face when coping with these kind of problems. In the first part, some objective functions for single-objective optimization cases are presented. In the second part, test functions with their respective Pareto fronts for multi-objective optimization problems (MOP) are given.
The artificial landscapes presented herein for single-objective optimization problems are taken from Bäck[ 1] [ 2] [ 3] [ 4]
The test functions used to evaluate the algorithms for MOP were taken from Deb[ 5] [ 6] [ 7] [ 8] [ 9]
Just a general form of the equation, a plot of the objective function, boundaries of the object variables and the coordinates of global minima are given herein.
Test functions for single-objective optimization problems
f
(
x
)
=
∑
i
=
1
n
x
i
2
.
{\displaystyle f({\boldsymbol {x}})=\sum _{i=1}^{n}x_{i}^{2}.\quad }
Rosenbrock's function in 3D.
Test function for single-objective optimization problems
Minimum:
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}
f
(
x
)
=
∑
i
=
1
n
−
1
[
100
(
x
i
+
1
−
x
i
2
)
2
+
(
x
i
−
1
)
2
]
.
{\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 }
Minimum
=
{
n
=
2
→
f
(
1
,
1
)
=
0
,
n
=
3
→
f
(
1
,
1
,
1
)
=
0
,
n
>
3
→
f
(
−
1
,
1
,
…
,
1
⏟
(
n
−
1
)
times
)
=
0.
{\displaystyle {\text{Minimum}}={\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(-1,\underbrace {1,\dots ,1} _{(n-1){\text{ times}}}\right)=0.\\\end{cases}}}
for
−
∞
≤
x
i
≤
∞
{\displaystyle -\infty \leq x_{i}\leq \infty }
1
≤
i
≤
n
{\displaystyle 1\leq i\leq n}
f
(
x
,
y
)
=
(
1.5
−
x
+
x
y
)
2
+
(
2.25
−
x
+
x
y
2
)
2
+
(
2.625
−
x
+
x
y
3
)
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 }
Minimum:
f
(
3
,
0.5
)
=
0
{\displaystyle f(3,0.5)=0}
−
4.5
≤
x
,
y
≤
4.5
{\displaystyle -4.5\leq x,y\leq 4.5}
Goldstein Price function:
f
(
x
,
y
)
=
(
1
+
(
x
+
y
+
1
)
2
(
19
−
14
x
+
3
x
2
−
14
y
+
6
x
y
+
3
y
2
)
)
(
30
+
(
2
x
−
3
y
)
2
(
18
−
32
x
+
12
x
2
−
48
y
+
36
x
y
+
27
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)\left(30+\left(2x-3y\right)^{2}\left(18-32x+12x^{2}-48y+36xy+27y^{2}\right)\right).\quad }
Minimum:
f
(
0
,
−
1
)
=
3
{\displaystyle f(0,-1)=3}
−
2
≤
x
,
y
≤
2
{\displaystyle -2\leq x,y\leq 2}
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}.\quad }
Minimum:
f
(
1
,
3
)
=
0
{\displaystyle f(1,3)=0}
−
10
≤
x
,
y
≤
10
{\displaystyle -10\leq x,y\leq 10}
Goldstein Price function.
Test function for single-objective optimization problems
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 }
Minimum:
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}
f
(
x
,
y
)
=
−
20
exp
(
−
0.2
0.5
(
x
2
+
y
2
)
)
−
exp
(
0.5
(
cos
(
2
π
x
)
+
cos
(
2
π
y
)
)
)
+
20
+
e
.
{\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 }
Minimum:
f
(
0
,
0
)
=
0
{\displaystyle f(0,0)=0}
−
5
≤
x
,
y
≤
5
{\displaystyle -5\leq x,y\leq 5}
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.\quad }
Minimum:
f
(
0
,
0
)
=
0
{\displaystyle f(0,0)=0}
−
10
≤
x
,
y
≤
10
{\displaystyle -10\leq x,y\leq 10}
Test function for single-objective optimization problems
f
(
x
,
y
)
=
sin
2
(
3
π
x
)
+
(
x
−
1
)
2
(
1
+
sin
2
(
3
π
y
)
)
+
(
y
−
1
)
2
(
1
+
sin
2
(
2
π
y
)
)
.
{\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 }
Minimum:
f
(
1
,
1
)
=
0
{\displaystyle f(1,1)=0}
−
10
≤
x
,
y
≤
10
{\displaystyle -10\leq x,y\leq 10}
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}.\quad }
Minimum:
f
(
0
,
0
)
=
0
{\displaystyle f(0,0)=0}
−
5
≤
x
,
y
≤
5
{\displaystyle -5\leq x,y\leq 5}
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).\quad }
Minimum:
f
(
π
,
π
)
=
−
1
{\displaystyle f(\pi ,\pi )=-1}
−
100
≤
x
,
y
≤
100
{\displaystyle -100\leq x,y\leq 100}
Three Hump Camel function.
Test function for single-objective optimization problems
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 \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 }
Test function for single-objective optimization problems
Minima
=
{
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{Minima}}={\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}}}
for
−
10
≤
x
,
y
≤
10
{\displaystyle -10\leq x,y\leq 10}
f
(
x
,
y
)
=
−
(
y
+
47
)
sin
(
|
y
+
x
2
+
47
|
)
−
x
sin
(
|
x
−
(
y
+
47
)
|
)
.
{\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 }
Minimum:
f
(
512
,
404.2319
)
=
−
959.6407
{\displaystyle f(512,404.2319)=-959.6407}
−
512
≤
x
,
y
≤
512
{\displaystyle -512\leq x,y\leq 512}
Test function for single-objective optimization problems
f
(
x
,
y
)
=
−
|
sin
(
x
)
cos
(
y
)
exp
(
|
1
−
x
2
+
y
2
π
|
)
|
.
{\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 }
Minima
=
{
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{Minima}}={\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}}}
for
−
10
≤
x
,
y
≤
10
{\displaystyle -10\leq x,y\leq 10}
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.\quad }
Minimum:
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}
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}}}.\quad }
Minimum:
f
(
0
,
0
)
=
0
{\displaystyle f(0,0)=0}
−
100
≤
x
,
y
≤
100
{\displaystyle -100\leq x,y\leq 100}
f
(
x
,
y
)
=
0.5
+
cos
(
sin
(
|
x
2
−
y
2
|
)
)
−
0.5
(
1
+
0.001
(
x
2
+
y
2
)
)
2
.
{\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 }
Minimum:
f
(
0
,
1.25313
)
=
0.292579
{\displaystyle f(0,1.25313)=0.292579}
−
100
≤
x
,
y
≤
100
{\displaystyle -100\leq x,y\leq 100}
Styblinski-Tang function:
f
(
x
,
y
)
=
∑
i
=
1
n
x
i
4
−
16
x
i
2
+
5
x
2
.
{\displaystyle f(x,y)={\frac {\sum _{i=1}^{n}x_{i}^{4}-16x_{i}^{2}+5x}{2}}.\quad }
Minimum:
f
(
−
2.903534
,
…
,
−
2.903534
⏟
(
n
)
times
)
=
−
39.16599
n
{\displaystyle f\left(\underbrace {-2.903534,\ldots ,-2.903534} _{(n){\text{ times}}}\right)=-39.16599n}
−
5
≤
x
i
≤
5
{\displaystyle -5\leq x_{i}\leq 5}
1
≤
i
≤
n
{\displaystyle 1\leq i\leq n}
Styblinski-Tang function.
Test function for single-objective optimization problems
Test functions for multi-objective optimization problems
Chakong and Haimes function.
Test function for multi-objective optimization problems
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}}}
for
0
≤
x
≤
5
{\displaystyle 0\leq x\leq 5}
0
≤
y
≤
3
{\displaystyle 0\leq y\leq 3}
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}}}
for
−
20
≤
x
,
y
≤
20
{\displaystyle -20\leq x,y\leq 20}
Fonseca and Fleming function: Fonseca and Fleming function.
Test function for multi-objective optimization problems
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}}}
for
−
4
≤
x
i
≤
4
{\displaystyle -4\leq x_{i}\leq 4}
1
≤
i
≤
n
{\displaystyle 1\leq i\leq n}
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}}}
for
−
7
≤
x
,
y
≤
4
{\displaystyle -7\leq x,y\leq 4}
Test function for multi-objective optimization problems
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}}}
for
−
5
≤
x
i
≤
5
{\displaystyle -5\leq x_{i}\leq 5}
1
≤
i
≤
3
{\displaystyle 1\leq i\leq 3}
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}}}
for
−
A
≤
x
≤
A
{\displaystyle -A\leq x\leq A}
A
{\displaystyle A}
10
{\displaystyle 10}
10
5
{\displaystyle 10^{5}}
A
{\displaystyle A}
Poloni's two objective function.
Test function for multi-objective optimization problems
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}}}
for
−
5
≤
x
≤
10
{\displaystyle -5\leq x\leq 10}
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}}}
for
−
π
≤
x
,
y
≤
π
{\displaystyle -\pi \leq x,y\leq \pi }
Zitzler-Deb-Thiele's function N. 1: Zitzler-Deb-Thiele's function N. 1.
Zitzler-Deb-Thiele's function N. 2.
Test function for multi-objective optimization problems
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}}}
for
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:
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}}}
for
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:
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}}}
for
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:
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}}}
for
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:
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}}}
for
0
≤
x
i
≤
1
{\displaystyle 0\leq x_{i}\leq 1}
1
≤
i
≤
10
{\displaystyle 1\leq i\leq 10}
Zitzler-Deb-Thiele's function N. 3.
Zitzler-Deb-Thiele's function N. 4.
Zitzler-Deb-Thiele's function N. 6.
Test function for multi-objective optimization problems
Osyczka and Kundu function.
Test function for multi-objective optimization problems
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}}}
for
−
3
≤
x
,
y
≤
3
{\displaystyle -3\leq x,y\leq 3}
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}}}
for
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)[ 5] CTP1 function (2 variables)
[ 5] .
Test function for multi-objective optimization problems
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
1
(
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_{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}}}
for
0
≤
x
,
y
≤
1
{\displaystyle 0\leq x,y\leq 1}
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
1
(
x
,
y
)
=
−
y
+
9
x
≥
1
{\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}}}
for
0.1
≤
x
≤
1
{\displaystyle 0.1\leq x\leq 1}
0
≤
y
≤
5
{\displaystyle 0\leq y\leq 5}
References
^ Bäck, Thomas (1995). Evolutionary algorithms in theory and practice : evolution strategies, evolutionary programming, genetic algorithms . Oxford: Oxford University Press. p. 328. ISBN 0-19-509971-0 ^ Haupt, Randy L. Haupt, Sue Ellen (2004). Practical genetic algorithms with DC-Rom (2nd ed. ed.). New York: J. Wiley. ISBN 0-471-45565-2 CS1 maint: multiple names: authors list (link ) ^ Oldenhuis, Rody. "Many test functions for global optimizers" . Mathworks. Retrieved 1 November 2012 . ^ Ortiz, Gilberto A. "Evolution Strategies (ES)" . Mathworks. Retrieved 1 November 2012 . ^ a b c d e Deb, Kalyanmoy (2002) Multiobjective optimization using evolutionary algorithms (Repr. ed.). Chichester [u.a.]: 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
^ Deb K. (2011) Software for multi-objective NSGA-II code in C. Available at URL:http://www.iitk.ac.in/kangal/codes.shtml . Revision 1.1.6
^ Ortiz, Gilberto A. "Multi-objective optimization using ES as Evolutionary Algorithm" . Mathworks. Retrieved 1 November 2012 . 
![{\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 }](/media/api/rest_v1/media/math/render/svg/41f02e77049ab851fe048db982cf1e71246365e6)
![{\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}}}](/media/api/rest_v1/media/math/render/svg/a545df8c08a178973284eae8924aab67ce46077a)
![{\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}}}](/media/api/rest_v1/media/math/render/svg/e6b4a1275cbe615b285fbbcdb840a9a488360cc5)
![{\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}}}](/media/api/rest_v1/media/math/render/svg/73a519e6f7ca429b41031cd0873fce161941b009)
