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2013-11-21T19:01:58Z

Dating maintenance tags: {{Intro too long}}

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In applied mathematics, '''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,<ref>{{cite book|last=Bäck|first=Thomas|title=Evolutionary algorithms in theory and practice : evolution strategies, evolutionary programming, genetic algorithms|year=1995|publisher=Oxford University Press|location=Oxford|isbn=0-19-509971-0|pages=328}}</ref> Haupt et. al.<ref>{{cite book|last=Haupt|first=Randy L. Haupt, Sue Ellen|title=Practical genetic algorithms with DC-Rom|year=2004|publisher=J. Wiley|location=New York|isbn=0-471-45565-2|edition=2nd ed.}}</ref> and from Rody Oldenhuis software.<ref>{{cite web|last=Oldenhuis|first=Rody|title=Many test functions for global optimizers|url=http://www.mathworks.com/matlabcentral/fileexchange/23147-many-testfunctions-for-global-optimizers|publisher=Mathworks|accessdate=1 November 2012}}</ref> Given the amount of problems (55 in total), just a few are presented here. The complete list of test functions is found on the Mathworks website.<ref>{{cite web|last=Ortiz|first=Gilberto A.|title=Evolution Strategies (ES)|url=http://www.mathworks.com/matlabcentral/fileexchange/35801-evolution-strategies-es|publisher=Mathworks|accessdate=1 November 2012}}</ref>

The test functions used to evaluate the algorithms for MOP were taken from Deb,<ref name="Deb:2002">Deb, Kalyanmoy (2002) Multiobjective optimization using evolutionary algorithms (Repr. ed.). Chichester [u.a.]: Wiley. ISBN 0-471-87339-X.</ref> Binh et. al.<ref name="Binh97">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</ref> and Binh.<ref name="Binh99">Binh T. (1999) A multiobjective evolutionary algorithm. The study cases. Technical report. Institute for Automation and Communication. Barleben, Germany</ref> You can download the software developed by Deb,<ref name="Deb_nsga">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</ref> which implements the NSGA-II procedure with GAs, or the program posted on Internet,<ref>{{cite web|last=Ortiz|first=Gilberto A.|title=Multi-objective optimization using ES as Evolutionary Algorithm.|url=http://www.mathworks.com/matlabcentral/fileexchange/35824-multi-objective-optimization-using-evolution-strategies-es-as-evolutionary-algorithm-ea|publisher=Mathworks|accessdate=1 November 2012}}</ref> which implements the NSGA-II procedure with ES.

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==

{| class="wikitable" style="text-align:center"
|-
! Name !! Plot !! Formula !! Minimum !! Search domain
|-
| Ackley's function:
|| [[File:Ackley's function.pdf|200px|Ackley's function for n=2]]
||<math>f(x,y) = -20\exp\left(-0.2\sqrt{0.5\left(x^{2}+y^{2}\right)}\right)</math>
<math>-\exp\left(0.5\left(\cos\left(2\pi x\right)+\cos\left(2\pi y\right)\right)\right) + 20 + e.\quad</math>
||<math>f(0,0) = 0</math>
||<math>-5\le x,y \le 5</math>
|-
| Sphere function
|| [[File:Sphere function in 3D.pdf|200px|Sphere function for n=2]]
|| <math>f(\boldsymbol{x}) = \sum_{i=1}^{n} x_{i}^{2}.\quad</math>
|| <math>f(x_{1}, \dots, x_{n}) = f(0, \dots, 0) = 0</math>
|| <math>-\infty \le x_{i} \le \infty</math>, <math>1 \le i \le n</math>
|-
| [[Rosenbrock function]]
|| [[File:Rosenbrock's function in 3D.pdf|200px|Rosenbrock's function for n=2]]
|| <math>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</math>
|| <math>\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(-1,\underbrace{1,\dots,1}_{(n-1) \text{ times}}\right) = 0. \\
\end{cases}
</math>
|| <math>-\infty \le x_{i} \le \infty</math>, <math>1 \le i \le n</math>
|-
| Beale's function
|| [[File:Beale's function.pdf|200px|Beale's function]]
|| <math>f(x,y) = \left( 1.5 - x + xy \right)^{2} + \left( 2.25 - x + xy^{2}\right)^{2}</math>
<math>+ \left(2.625 - x+ xy^{3}\right)^{2}.\quad</math>
|| <math>f(3, 0.5) = 0</math>
|| <math>-4.5 \le x,y \le 4.5</math>
|-
| Goldstein–Price function:
|| [[File:Goldstein Price function.pdf|200px|Goldstein–Price function]]
|| <math>f(x,y) = \left(1+\left(x+y+1\right)^{2}\left(19-14x+3x^{2}-14y+6xy+3y^{2}\right)\right)</math>
<math>\left(30+\left(2x-3y\right)^{2}\left(18-32x+12x^{2}+48y-36xy+27y^{2}\right)\right).\quad</math>
|| <math>f(0, -1) = 3</math>
|| <math>-2 \le x,y \le 2</math>
|-
| Booth's function:
|| [[File:Booth's function.pdf|200px|Booth's function]]
||<math>f(x,y) = \left( x + 2y -7\right)^{2} + \left(2x +y - 5\right)^{2}.\quad</math>
||<math>f(1,3) = 0</math>
||<math>-10 \le x,y \le 10</math>.
|-
| Bukin function N.6:
|| [[File:Bukin function 6.pdf|200px|Bukin function N.6]]
|| <math>f(x,y) = 100\sqrt{\left|y - 0.01x^{2}\right|} + 0.01 \left|x+10 \right|.\quad</math>
|| <math>f(-10,1) = 0</math>
|| <math>-15\le x \le -5</math>, <math>-3\le y \le 3</math>
|-
| Matyas function:
|| [[File:Matyas function.pdf|200px|Matyas function]]
|| <math>f(x,y) = 0.26 \left( x^{2} + y^{2}\right) - 0.48 xy.\quad</math>
|| <math>f(0,0) = 0</math>
|| <math>-10\le x,y \le 10</math>
|-
| Lévi function N.13:
||[[File:Levi function 13.pdf|200px|Lévi function N.13]]
|| <math>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)</math>
<math>+\left(y-1\right)^{2}\left(1+\sin^{2}\left(2\pi y\right)\right).\quad</math>
|| <math>f(1,1) = 0</math>
|| <math>-10\le x,y \le 10</math>
|-
| Three-hump camel function:
||[[File:Three Hump Camel function.pdf|200px|Three Hump Camel function]]
|| <math>f(x,y) = 2x^{2} - 1.05x^{4} + \frac{x^{6}}{6} + xy + y^{2}.\quad</math>
|| <math>f(0,0) = 0</math>
|| <math>-5\le x,y \le 5</math>
|-
| Easom function:
|| [[File:Easom function.pdf|200px|Easom function]]
|| <math>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</math>
|| <math>f(\pi , \pi) = -1</math>
|| <math>-100\le x,y \le 100</math>
|-
| Cross-in-tray function:
|| [[File:Cross-in-tray function.pdf|200px|Cross-in-tray function]]
|| <math>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</math>
|| <math>\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}
</math>
|| <math>-10\le x,y \le 10</math>
|-
| Eggholder function:
|| [[File:Eggholder function.pdf|200px|Eggholder function]]
|| <math>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</math>
|| <math>f(512, 404.2319) = -959.6407</math>
|| <math>-512\le x,y \le 512</math>
|-
| Hölder table function:
|| [[File:Holder table function.pdf|200px|Holder table function]]
|| <math>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</math>
|| <math>\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}
</math>
|| <math>-10\le x,y \le 10</math>
|-
| McCormick function:
|| [[File:McCormick function.pdf|200px|McCormick function]]
|| <math>f(x,y) = \sin \left(x+y\right) + \left(x-y\right)^{2} - 1.5x + 2.5y + 1.\quad</math>
|| <math>f(-0.54719,-1.54719) = -1.9133</math>
|| <math>-1.5\le x \le 4</math>, <math>-3\le y \le 4</math>
|-
| Schaffer function N. 2:
|| [[File:Schaffer function 2.pdf|200px|Schaffer function N.2]]
|| <math>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</math>
|| <math>f(0, 0) = 0</math>
|| <math>-100\le x,y \le 100</math>
|-
| Schaffer function N. 4:
|| [[File:Schaffer function 4.pdf|200px|Schaffer function N.4]]
|| <math>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</math>
|| <math>f(0,1.25313) = 0.292579</math>
|| <math>-100\le x,y \le 100</math>
|-
| Styblinski–Tang function:
|| [[File:Styblinski-Tang function.pdf|200px|Styblinski-Tang function]]
|| <math>f(\boldsymbol{x}) = \frac{\sum_{i=1}^{n} x_{i}^{4} - 16x_{i}^{2} + 5x_{i}}{2}.\quad</math>
|| <math>f\left(\underbrace{-2.903534, \ldots, -2.903534}_{(n) \text{ times}} \right) = -39.16599n</math>
|| <math>-5\le x_{i} \le 5</math>, <math>1\le i \le n</math>.

|}

==Test functions for multi-objective optimization problems==

{| class="wikitable" style="text-align:center"
|-
! Name !! Plot !! Functions !! Constraints !! Search domain
|-
| Binh and Korn function:
|| [[File:Binh and Korn function.pdf|200px|Binh and Korn function]]
|| <math>\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}
</math>
||<math>\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}
</math>
|| <math>0\le x \le 5</math>, <math>0\le y \le 3</math>
|-
| Chakong and Haimes function:
|| [[File:Chakong and Haimes function.pdf|200px|Chakong and Haimes function]]
|| <math>\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}
</math>
|| <math>\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}
</math>
|| <math>-20\le x,y \le 20</math>
|-
| Fonseca and Fleming function:
|| [[File:Fonseca and Fleming function.pdf|200px|Fonseca and Fleming function]]
|| <math>\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}
</math>
||
|| <math>-4\le x_{i} \le 4</math>, <math>1\le i \le n</math>
|-
| Test function 4:<ref name="Binh99"/>
|| [[File:Test function 4 - Binh.pdf|200px|Test function 4.<ref name="Binh99"/>]]
|| <math>\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}
</math>
|| <math>\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}
</math>
|| <math>-7\le x,y \le 4</math>
|-
| Kursawe function:
|| [[File:Kursawe function.pdf|200px|Kursawe function]]
|| <math>\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}
</math>
||
||<math>-5\le x_{i} \le 5</math>, <math>1\le i \le 3</math>.
|-
| Schaffer function N. 1:
|| [[File:Schaffer function 1.pdf|200px|Schaffer function N.1]]
|| <math>\text{Minimize} =
\begin{cases}
f_{1}\left(x\right) & = x^{2} \\
f_{2}\left(x\right) & = \left(x-2\right)^{2} \\
\end{cases}
</math>
||
|| <math>-A\le x \le A</math>. Values of <math>A</math> form <math>10</math> to <math>10^{5}</math> have been used successfully. Higher values of <math>A</math> increase the difficulty of the problem.
|-
| Schaffer function N. 2:
|| [[File:Schaffer function 2 - multi-objective.pdf|200px|Schaffer function N.2]]
|| <math>\text{Minimize} =
\begin{cases}
f_{1}\left(x\right) & = \begin{cases}
-x, & \text{if } x \le 1 \\
x-2, & \text{if } 1 < x \le 3 \\
4-x, & \text{if } 3 < x \le 4 \\
x-4, & \text{if } x > 4 \\
\end{cases} \\
f_{2}\left(x\right) & = \left(x-5\right)^{2} \\
\end{cases}
</math>
||
||<math>-5\le x \le 10</math>.
|-
| Poloni's two objective function:
|| [[File:Poloni's two objective function.pdf|200px|Poloni's two objective function]]
|| <math>\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}
</math>
<math>\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}
</math>
||
||<math>-\pi\le x,y \le \pi</math>
|-
| Zitzler–Deb–Thiele's function N. 1:
|| [[File:Zitzler-Deb-Thiele's function 2.pdf|200px|Zitzler-Deb-Thiele's function N.2]]
|| <math>\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}
</math>
||
||<math>0\le x_{i} \le 1</math>, <math>1\le i \le 30</math>.
|-
| Zitzler–Deb–Thiele's function N. 2:
|| [[File:Zitzler-Deb-Thiele's function 2.pdf|200px|Zitzler-Deb-Thiele's function N.2]]
|| <math>\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}
</math>
||
|| <math>0\le x_{i} \le 1</math>, <math>1\le i \le 30</math>.
|-
| Zitzler–Deb–Thiele's function N. 3:
|| [[File:Zitzler-Deb-Thiele's function 3.pdf|200px|Zitzler-Deb-Thiele's function N.3]]
||<math>\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}
</math>
||
||<math>0\le x_{i} \le 1</math>, <math>1\le i \le 30</math>.
|-
| Zitzler–Deb–Thiele's function N. 4:
|| [[File:Zitzler-Deb-Thiele's function 4.pdf| caption2 = Zitzler-Deb-Thiele's function N.4]]
|| <math>\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}
</math>
||
||<math>0\le x_{1} \le 1</math>, <math>-5\le x_{i} \le 5</math>, <math>2\le i \le 10</math>
|-
| Zitzler–Deb–Thiele's function N. 6:
|| [[File:Zitzler-Deb-Thiele's function 6.pdf|200px|Zitzler-Deb-Thiele's function N.6]]
||<math>\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}
</math>
||
||<math>0\le x_{i} \le 1</math>, <math>1\le i \le 10</math>.
|-
| Viennet function:
|| [[File:Viennet function.pdf|200px|Viennet function]]
|| <math>\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}
</math>
||
||<math>-3\le x,y \le 3</math>.
|-
| Osyczka and Kundu function:
|| [[File:Osyczka and Kundu function.pdf|200px|Osyczka and Kundu function]]
||<math>\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}
</math>
||<math>\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}
</math>
|| <math>0\le x_{1},x_{2},x_{6} \le 10</math>, <math>1\le x_{3},x_{5} \le 5</math>, <math>0\le x_{4} \le 6</math>.
|-
| CTP1 function (2 variables):<ref name="Deb:2002"/>
|| [[File:CTP1 function (2 variables).pdf|200px|CTP1 function (2 variables).<ref name="Deb:2002"/>]]
|| <math>\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}
</math>
||<math>\text{s.t.} =
\begin{cases}
g_{1}\left(x,y\right) & = \frac{f_{2}\left(x,y\right)}{0.858 \exp \left(-0.541 f_{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(-0.295 f_{1}\left(x,y\right)\right)} \geq 1
\end{cases}
</math>
|| <math>0\le x,y \le 1</math>.
|-
| Constr-Ex problem:<ref name="Deb:2002"/>
|| [[File:Constr-Ex problem.pdf|200px|Constr-Ex problem.<ref name="Deb:2002"/>]]
|| <math>\text{Minimize} =
\begin{cases}
f_{1}\left(x,y\right) & = x \\
f_{2}\left(x,y\right) & = \frac{1 + y}{x} \\
\end{cases}
</math>
|| <math>\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}
</math>
|| <math>0.1\le x \le 1</math>, <math>0\le y \le 5</math>

|}

== See also ==
* [[Himmelblau's function]]
* [[Rosenbrock function]]
* [[Rastrigin function]]
* [[Shekel function]]
* [[MOEA Framework]], an open source Java library for multiobjective optimization

==References==

<references/>

{{DEFAULTSORT:Test functions for optimization}}
[[Category:Mathematical optimization]]
[[Category:Constraint programming]]
[[Category:Convex optimization]]
[[Category:Types of functions]]

Photon Structure Function - Revision history

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