Test functions for optimization: Difference between revisions

Revision as of 22:43, 17 March 2018

Revision as of 00:39, 19 March 2018

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==Test functions for single-objective optimization==

{| class="wikitable" style="text-align:center"

|-

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||<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}</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(1 - x_{i}\right)^{2}\right]</math>

|| <math>\text{Min} =

\begin{cases}

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\end{cases}

</math>

|| <math>-\infty \le x_{i} \le \infty</math>, <math>1 \le i \le n</math>

|-

| [[Beale 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}</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]</math>

|| <math>f(0, -1) = 3</math>

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| [[Hölder table function]]

|| [[File:Holder table function.pdf|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}

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f\left(-8.05502,-9.66459\right) & = -19.2085

\end{cases}

</math>

|| <math>-10\le x,y \le 10</math>

|-

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|| [[File:Schaffer function 4.pdf|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>f(0,1.25313) = 0.292579</math>

|| <math>-100\le x,y \le 100</math>

|-

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==Test functions for constrained optimization==

{| class="wikitable" style="text-align:center"

|-

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|| <math>-2.25\le x \le 2.5</math>, <math>-2.5\le y \le 1.75</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, FloridaFL|isbn=978-1-4822-5290-3|edition=1st}}</ref>

|| [[File:Simionescu's function.PNG|200px|Simionescu function]]

|| <math>f(x,y) = 0.1xy</math>,

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==Test functions for multi-objective optimization==

{{explain|reason=What does it mean to minimize two objective functions?|date=September 2016}}

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==References==

<references/>