View source for Test functions for optimization

{{Short description|Functions used to evaluate optimization algorithms}}
In applied mathematics, '''test functions''', known as '''artificial landscapes''', are useful to evaluate characteristics of optimization algorithms, such as:

* Convergence rate.
* 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 kinds 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 [[Engineering:Pareto front|Pareto fronts]] for [[Multi-objective optimization|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=978-0-19-509971-3|page=328}}</ref> Haupt et al.<ref>{{cite book|last=Haupt|first=Randy L. Haupt, Sue Ellen|title=Practical genetic algorithms with CD-Rom|year=2004|publisher=J. Wiley|location=New York|isbn=978-0-471-45565-3|edition=2nd}}</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|access-date=1 November 2012}}</ref> Given the number of problems (55 in total), just a few are presented here.

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) [https://web.archive.org/web/20190801183649/https://pdfs.semanticscholar.org/cf68/41a6848ca2023342519b0e0e536b88bdea1d.pdf 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) [https://www.researchgate.net/profile/Thanh_Binh_To/publication/2446107_A_Multiobjective_Evolutionary_Algorithm_The_Study_Cases/links/53eb422f0cf28f342f45251d.pdf A multiobjective evolutionary algorithm. The study cases.] Technical report. Institute for Automation and Communication. Barleben, Germany</ref> The software developed by Deb can be downloaded,<ref name="Deb_nsga">Deb K. (2011) Software for multi-objective NSGA-II code in C. Available at URL: https://www.iitk.ac.in/kangal/codes.shtml</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|access-date=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==

{| class="sortable wikitable" 
! Name 
! Plot 
! Formula 
! Global minimum 
! Search domain
|-
| [[Rastrigin function]]
| [[File:Rastrigin contour plot.svg|200px|Rastrigin function for n=2]]
|<math>f(\mathbf{x}) = A n + \sum_{i=1}^n \left[x_i^2 - A\cos(2 \pi x_i)\right]</math>
<math>\text{where: }  A=10</math>
|<math>f(0, \dots, 0) = 0</math>
|<math>-5.12\le x_{i} \le 5.12 </math>
|-
| [[Ackley function]]
| [[File:Ackley contour function.svg|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 2\pi x + \cos 2\pi y \right)\right] + e + 20</math>
|<math>f(0,0) = 0</math>
|<math>-5\le x,y \le 5</math>
|-
| Sphere function 
| [[File:Sphere contour.svg|200px|Sphere function for n=2]]
| <math>f(\boldsymbol{x}) = \sum_{i=1}^{n} x_{i}^{2}</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 contour.svg|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(1 - x_{i}\right)^{2}\right]</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(\underbrace{1,\dots,1}_{n \text{ times}}) = 0 \\
\end{cases}
</math>
| <math>-\infty \le x_{i} \le \infty</math>, <math>1 \le i \le n</math> 
|-
| Beale function
| [[File:Beale contour.svg|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}</math>
| <math>f(3, 0.5) = 0</math>
| <math>-4.5 \le x,y \le 4.5</math> 
|-
| Goldstein–Price function
| [[File:Goldstein-Price contour.svg|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]</math>
| <math>f(0, -1) = 3</math>
| <math>-2 \le x,y \le 2</math>
|-
| Booth function
| [[File:Booth contour.svg|200px|Booth's function]]
|<math>f(x,y) = \left( x + 2y -7\right)^{2} + \left(2x +y - 5\right)^{2}</math>
|<math>f(1,3) = 0</math>
|<math>-10 \le x,y \le 10</math>
|-
| Bukin function N.6
| [[File:Bukin 6 contour.svg|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 contour.svg|200px|Matyas function]]
| <math>f(x,y) = 0.26 \left( x^{2} + y^{2}\right) - 0.48 xy</math>
| <math>f(0,0) = 0</math>
| <math>-10\le x,y \le 10</math>
|-
| Lévi function N.13
|[[File:Levi13 contour.svg|200px|Lévi function N.13]]
| <math>f(x,y) = \sin^{2} 3\pi x + \left(x-1\right)^{2}\left(1+\sin^{2} 3\pi y\right)</math>
<math>+\left(y-1\right)^{2}\left(1+\sin^{2} 2\pi y\right)</math>
| <math>f(1,1) = 0</math>
| <math>-10\le x,y \le 10</math>
|-
| [[Himmelblau's function]]
|[[File:Himmelblau contour plot.svg|200px|Himmelblau's function]]
| <math>f(x, y) = (x^2+y-11)^2 + (x+y^2-7)^2.\quad</math>
| <math>\text{Min} =
\begin{cases}
      f\left(3.0,  2.0\right) & = 0.0 \\
      f\left(-2.805118, 3.131312\right) & = 0.0 \\
      f\left(-3.779310, -3.283186\right) & = 0.0 \\
      f\left(3.584428, -1.848126\right) & = 0.0 \\
\end{cases}
</math>
| <math>-5\le x,y \le 5</math>
|-
| Three-hump camel function
| [[File:Three-hump-camel contour.svg|200px|Three Hump Camel function]]
| <math>f(x,y) = 2x^{2} - 1.05x^{4} + \frac{x^{6}}{6} + xy + y^{2}</math>
| <math>f(0,0) = 0</math>
| <math>-5\le x,y \le 5</math>
|-
| Easom function
| [[File:Easom contour.svg|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)</math>
| <math>f(\pi , \pi) = -1</math>
| <math>-100\le x,y \le 100</math>
|-
| Cross-in-tray function
| [[File:Cross-in-tray contour.svg|200px|Cross-in-tray function]]
| <math>f(x,y) = -0.0001 \left[ \left| \sin x \sin y \exp \left(\left|100 - \frac{\sqrt{x^{2} + y^{2}}}{\pi} \right|\right)\right| + 1 \right]^{0.1}</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<ref name="Whitley Rana Dzubera Mathias 1996 pp. 245–276">{{cite journal | last1=Whitley | first1=Darrell | last2=Rana | first2=Soraya | last3=Dzubera | first3=John | last4=Mathias | first4=Keith E. | title=Evaluating evolutionary algorithms | journal=Artificial Intelligence | publisher=Elsevier BV | volume=85 | issue=1–2 | year=1996 | issn=0004-3702 | doi=10.1016/0004-3702(95)00124-7 | pages=264| doi-access=free }}</ref><ref name="vanaret2015hybridation">Vanaret C. (2015) [https://www.researchgate.net/publication/337947149_Hybridization_of_interval_methods_and_evolutionary_algorithms_for_solving_difficult_optimization_problems Hybridization of interval methods and evolutionary algorithms for solving difficult optimization problems.] PhD thesis. Ecole Nationale de l'Aviation Civile. Institut National Polytechnique de Toulouse, France.</ref>
| [[File:Eggholder contour.svg|200px|Eggholder function]]
| <math>f(x,y) = - \left(y+47\right) \sin \sqrt{\left|\frac{x}{2}+\left(y+47\right)\right|} - x \sin \sqrt{\left|x - \left(y + 47 \right)\right|}</math>
| <math>f(512, 404.2319) = -959.6407</math>
| <math>-512\le x,y \le 512</math>
|-
| Hölder table function
| [[File:Hoelder table contour.svg|200px|Holder table function]]
| <math>f(x,y) = - \left|\sin x \cos y \exp \left(\left|1 - \frac{\sqrt{x^{2} + y^{2}}}{\pi} \right|\right)\right|</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 contour.svg|200px|McCormick function]]
| <math>f(x,y) = \sin \left(x+y\right) + \left(x-y\right)^{2} - 1.5x + 2.5y + 1</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:Schaffer2 contour.svg|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}}</math>
| <math>f(0, 0) = 0</math>
| <math>-100\le x,y \le 100</math>
|-
| Schaffer function N. 4
| [[File:Schaffer4 contour.svg|200px|Schaffer function N.4]]
| <math>f(x,y) = 0.5 + \frac{\cos^{2}\left[\sin \left( \left|x^{2} - y^{2}\right|\right)\right] - 0.5}{\left[1 + 0.001\left(x^{2} + y^{2}\right) \right]^{2}}</math>
| <math>\text{Min} =
\begin{cases}
    f\left(0,1.25313\right) & = 0.292579 \\
    f\left(0,-1.25313\right) & = 0.292579 \\
    f\left(1.25313,0\right) & = 0.292579 \\
    f\left(-1.25313,0\right) & = 0.292579
\end{cases}
</math>
| <math>-100\le x,y \le 100</math>
|-
| Styblinski–Tang function
| [[File:Styblinski-Tang contour.svg|200px|Styblinski-Tang function]]
| <math>f(\boldsymbol{x}) = \frac{\sum_{i=1}^{n} x_{i}^{4} - 16x_{i}^{2} + 5x_{i}}{2}</math>
| <math>-39.16617n < f(\underbrace{-2.903534, \ldots, -2.903534}_{n \text{ times}} ) < -39.16616n</math>
| <math>-5\le x_{i} \le 5</math>, <math>1\le i \le n</math>..
|-
| [[Shekel function]]
| [[Image:Shekel_2D.jpg|200px|A Shekel function in 2 dimensions and with 10 maxima]]
| <math>
f(\vec{x}) = \sum_{i = 1}^{m} \; \left( c_{i} + \sum\limits_{j = 1}^{n} (x_{j} - a_{ji})^2 \right)^{-1}
</math>
or, similarly,
<math>
f(x_1,x_2,...,x_{n-1},x_n) = \sum_{i = 1}^{m} \; \left( c_{i} + \sum\limits_{j = 1}^{n} (x_{j} - a_{ij})^2 \right)^{-1}
</math>
|
| <math>-\infty \le x_{i} \le \infty</math>, <math>1 \le i \le n</math> 
|}

==Test functions for constrained optimization==

{| class="wikitable" style="text-align:center"
|-
! Name !! Plot           !! Formula !! Global minimum !! Search domain
|-
| Rosenbrock function constrained with a cubic and a line<ref>{{cite conference |author1=Simionescu, P.A. |author2=Beale, D. |title=New Concepts in Graphic Visualization of Objective Functions |conference=ASME 2002 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference |location=Montreal, Canada |date=September 29 – October 2, 2002|pages=891–897 |url=http://faculty.tamucc.edu/psimionescu/PDFs/DETC02-DAC-34129.pdf |access-date=7 January 2017 }}</ref>
|| [[File:Rosenbrock cubic constraint.svg|200px|Rosenbrock function constrained with a cubic and a line]]
|| <math>f(x,y) = (1-x)^2 + 100(y-x^2)^2</math>,

subjected to: <math> (x-1)^3 - y + 1 \le 0 \text{  and  } x + y - 2 \le 0 </math>
|| <math>f(1.0,1.0) = 0</math>
|| <math>-1.5\le x \le 1.5</math>, <math>-0.5\le y \le 2.5</math>
|-
| Rosenbrock function constrained to a disk<ref>{{Cite web|url=https://www.mathworks.com/help/optim/ug/example-nonlinear-constrained-minimization.html?requestedDomain=www.mathworks.com|title=Solve a Constrained Nonlinear Problem - MATLAB & Simulink|website=www.mathworks.com|access-date=2017-08-29}}</ref>
|| [[File:Rosenbrock circle constraint.svg|200px|Rosenbrock function constrained to a disk]]
|| <math>f(x,y) = (1-x)^2 + 100(y-x^2)^2</math>,

subjected to: <math> x^2 + y^2 \le 2 </math>
|| <math>f(1.0,1.0) = 0</math>
|| <math>-1.5\le x \le 1.5</math>, <math>-1.5\le y \le 1.5</math>
|-
| Mishra's Bird function - constrained<ref>{{Cite web|url=http://www.phoenix-int.com/software/benchmark_report/bird_constrained.php|title=Bird Problem (Constrained) {{!}} Phoenix Integration|access-date=2017-08-29|url-status=bot: unknown|archive-url=https://web.archive.org/web/20161229032528/http://www.phoenix-int.com/software/benchmark_report/bird_constrained.php|archive-date=2016-12-29}}</ref><ref>{{Cite journal|last=Mishra|first=Sudhanshu|date=2006|title=Some new test functions for global optimization and performance of repulsive particle swarm method|url=https://mpra.ub.uni-muenchen.de/2718/|journal=MPRA Paper}}</ref>
|| [[File:Mishra bird contour.svg|200px|Bird function (constrained)]]
|| <math>f(x,y) = \sin(y) e^{\left [(1-\cos x)^2\right]} + \cos(x) e^{\left [(1-\sin y)^2 \right]} + (x-y)^2</math>,
subjected to: <math> (x+5)^2 + (y+5)^2 < 25 </math>
|| <math>f(-3.1302468,-1.5821422) = -106.7645367</math>
|| <math>-10\le x \le 0</math>, <math>-6.5\le y \le 0</math>
|-
| Townsend function (modified)<ref>{{Cite web|url=http://www.chebfun.org/examples/opt/ConstrainedOptimization.html|title=Constrained optimization in Chebfun|last=Townsend|first=Alex|date=January 2014|website=chebfun.org|access-date=2017-08-29}}</ref>
|| [[File:Townsend contour.svg|200px|Heart constrained multimodal function]]
|| <math>f(x,y) = -[\cos((x-0.1)y)]^2 - x \sin(3x+y)</math>,
subjected to:<math>x^2+y^2 < \left[2\cos t - \frac 1 2 \cos 2t - \frac 1 4 \cos 3t - \frac 1 8 \cos 4t\right]^2 + [2\sin t]^2 </math>
where: {{Math|1=''t'' = Atan2(x,y)}}
|| <math>f(2.0052938,1.1944509) = -2.0239884</math>
|| <math>-2.25\le x \le 2.25</math>, <math>-2.5\le y \le 1.75</math>

|-
| Gomez and Levy function (modified)<ref>{{cite journal |last1=Simionescu |first1=P.A. |date=2020 |title=A collection of bivariate nonlinear optimisation test problems with graphical representations |url= |journal=International Journal of Mathematical Modelling and Numerical Optimisation |volume=10 |issue=4 |pages=365–398 |doi=10.1504/IJMMNO.2020.110704 |access-date=}}</ref>
|| [[File:Gomez-Levi contour.svg|200px|Gomez and Levy Function]]
|| <math>f(x,y) = 4 x^2 - 2.1 x^4 +  \frac 1 3 x^6 + xy - 4y^2 +4 y^4 </math>,
subjected to:<math> -\sin(4 \pi x) + 2\sin^2(2 \pi y) \le 1.5 </math>
||  <math>f(0.08984201,-0.7126564) = -1.031628453</math>           
|| <math>-1\le x \le 0.75</math>, <math>-1\le y \le 1</math>

|-
| Simionescu function<ref>{{cite book|last=Simionescu|first=P.A.|title=Computer Aided Graphing and Simulation Tools for AutoCAD Users|year=2014|publisher=CRC Press|location=Boca Raton, FL|isbn=978-1-4822-5290-3|edition=1st}}</ref>
|| [[File:Simionescu contour.svg|200px|Simionescu function]]
|| <math>f(x,y) = 0.1xy</math>,
subjected to: <math> x^2+y^2\le\left[r_{T}+r_{S}\cos\left(n \arctan \frac{x}{y} \right)\right]^2</math>
<math>\text{where: }  r_{T}=1, r_{S}=0.2 \text{ and } n = 8</math>
|| <math>f(\pm 0.84852813,\mp 0.84852813) = -0.072</math>
|| <math>-1.25\le x,y \le 1.25</math>
|}

==Test functions for multi-objective optimization==


{| class="wikitable" style="text-align:center"
|-
! Name !! Plot           !! Functions !! Constraints !! Search domain
|-
| Binh and Korn function:<ref name="Binh97"/>
|| [[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>
|-
| Chankong and Haimes function:<ref>{{cite book |last1=Chankong |first1=Vira |last2=Haimes |first2=Yacov Y. |title=Multiobjective decision making. Theory and methodology. |isbn=0-444-00710-5|year=1983 |publisher=North Holland }}</ref>
|| [[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–Fleming function:<ref name="FonzecaFleming:1995">{{cite journal |first1=C. M. |last1=Fonseca |first2=P. J. |last2=Fleming |title=An Overview of Evolutionary Algorithms in Multiobjective Optimization |journal=Evol Comput |volume=3 |issue=1 |pages=1–16 |year=1995 |doi=10.1162/evco.1995.3.1.1 |citeseerx=10.1.1.50.7779 |s2cid=8530790 }}</ref>
|| [[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:<ref name="Kursawe:1991">F. Kursawe, “[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.47.8050&rep=rep1&type=pdf A variant of evolution strategies for vector optimization],” in PPSN I, Vol 496 Lect Notes in Comput Sc. Springer-Verlag, 1991, pp.&nbsp;193–197.</ref>
|| [[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:<ref name="Schaffer:1984">{{cite book |last=Schaffer |first=J. David |date=1984 |chapter=Multiple Objective Optimization with Vector Evaluated Genetic Algorithms |title=Proceedings of the First International Conference on Genetic Algorithms |editor1=G.J.E Grefensette |editor2=J.J. Lawrence Erlbraum |oclc=20004572 }}</ref>
|| [[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> from <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:<ref name="Debetal2002testpr">{{cite book |last1=Deb |first1=Kalyan |last2=Thiele |first2=L. |last3=Laumanns |first3=Marco |last4=Zitzler |first4=Eckart |title=Proceedings of the 2002 Congress on Evolutionary Computation. CEC'02 (Cat. No.02TH8600) |chapter=Scalable multi-objective optimization test problems |date=2002 |volume=1 |pages=825–830 |doi=10.1109/CEC.2002.1007032|isbn=0-7803-7282-4 |s2cid=61001583 }}</ref>
|| [[File:Zitzler-Deb-Thiele's function 1.pdf|200px|Zitzler-Deb-Thiele's function N.1]]
|| <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:<ref name="Debetal2002testpr" />
|| [[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:<ref name="Debetal2002testpr" />
|| [[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:<ref name="Debetal2002testpr" />
|| [[File:Zitzler-Deb-Thiele's function 4.pdf|200px|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:<ref name="Debetal2002testpr" />
|| [[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>.
|-
| Osyczka and Kundu function:<ref name="OsyczkaKundu1995">{{cite journal |last1=Osyczka |first1=A. |last2=Kundu |first2=S. |title=A new method to solve generalized multicriteria optimization problems using the simple genetic algorithm |journal=Structural Optimization |date=1 October 1995 |volume=10 |issue=2 |pages=94–99 |doi=10.1007/BF01743536 |s2cid=123433499 |issn=1615-1488}}</ref>
|| [[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"/><ref name="Jimenezetal2002">{{cite book |last1=Jimenez |first1=F. |last2=Gomez-Skarmeta |first2=A. F. |last3=Sanchez |first3=G. |last4=Deb |first4=K. |title=Proceedings of the 2002 Congress on Evolutionary Computation. CEC'02 (Cat. No.02TH8600) |chapter=An evolutionary algorithm for constrained multi-objective optimization |date=May 2002 |volume=2 |pages=1133–1138 |doi=10.1109/CEC.2002.1004402|isbn=0-7803-7282-4 |s2cid=56563996 }}</ref>
|| [[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_{2}\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_{2}\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>
|-
| 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>.
|}

==References==

<references/>

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

{{Sourceattribution|Test functions for optimization}}