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Test functions for optimization: Difference between revisions

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{| class="sortable wikitable" style="text-align:center"
|-
! 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, \\
Line 52:
\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 \\
Line 104:
\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 \\
Line 129:
\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 \\
Line 148:
\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 \\
Line 173:
\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>
Line 191:
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>
|}
 
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