N-D Test Functions C¶

class CarromTable(dimensions=2)¶

CarromTable objective function.

The CarromTable global optimization problem is a multimodal minimization problem defined as follows:

$f_{\text{CarromTable}}(x) = - \frac{1}{30}\left(\cos(x_1)cos(x_2) e^{\left|1 - \frac{\sqrt{x_1^2 + x_2^2}}{\pi}\right|}\right)^2$

with $x_i \in [-10, 10]$ for $i = 1, 2$ .

Two-dimensional CarromTable function

Global optimum: $f(x) = -24.15681551650653$ for $x_i = \pm 9.646157266348881$ for $i = 1, 2$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

class Chichinadze(dimensions=2)¶

Chichinadze objective function.

This class defines the Chichinadze global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Chichinadze}}(x) = x_{1}^{2} - 12 x_{1} + 8 \sin\left(\frac{5}{2} \pi x_{1}\right) + 10 \cos\left(\frac{1}{2} \pi x_{1}\right) + 11 - 0.2 \frac{\sqrt{5}}{e^{\frac{1}{2} \left(x_{2} -0.5\right)^{2}}}$

with $x_i \in [-30, 30]$ for $i = 1, 2$ .

Two-dimensional Chichinadze function

Global optimum: $f(x) = -42.94438701899098$ for $x = [6.189866586965680, 0.5]$

Gavana, A. Global Optimization Benchmarks and AMPGO

Todo

Jamil#33 has a dividing factor of 2 in the sin term. However, f(x) for the given solution does not give the global minimum. i.e. the equation is at odds with the solution. Only by removing the dividing factor of 2, i.e. 8 * sin(5 * pi * x[0]) does the given solution result in the given global minimum. Do we keep the result or equation?

class Cigar(dimensions=2)¶

Cigar objective function.

This class defines the Cigar global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Cigar}}(x) = x_1^2 + 10^6\sum_{i=2}^{n} x_i^2$

Here, $n$ represents the number of dimensions and $x_i \in [-100, 100]$ for $i = 1, ..., n$ .

Two-dimensional Cigar function

Global optimum: $f(x) = 0$ for $x_i = 0$ for $i = 1, ..., n$

Gavana, A. Global Optimization Benchmarks and AMPGO

class Cola(dimensions=17)¶

Cola objective function.

This class defines the Cola global optimization problem. The 17-dimensional function computes indirectly the formula $f(n, u)$ by setting $x_0 = y_0, x_1 = u_0, x_i = u_{2(i2)}, y_i = u_{2(i2)+1}$ :

$f_{\text{Cola}}(x) = \sum_{i<j}^{n} \left (r_{i,j} - d_{i,j} \right )^2$

Where $r_{i, j}$ is given by:

$r_{i, j} = \sqrt{(x_i - x_j)^2 + (y_i - y_j)^2}$

And $d$ is a symmetric matrix given by:

$\mathbf{d} = \left [ d_{ij} \right ] = \begin{pmatrix} \\ 1.27 & & & & & & & & \\ 1.69 & 1.43 & & & & & & & \\ 2.04 & 2.35 & 2.43 & & & & & & \\ 3.09 & 3.18 & 3.26 & 2.85 & & & & & \\ 3.20 & 3.22 & 3.27 & 2.88 & 1.55 & & & & \\ 2.86 & 2.56 & 2.58 & 2.59 & 3.12 & 3.06 & & & \\ 3.17 & 3.18 & 3.18 & 3.12 & 1.31 & 1.64 & 3.00 & \\ 3.21 & 3.18 & 3.18 & 3.17 & 1.70 & 1.36 & 2.95 & 1.32 & \\ 2.38 & 2.31 & 2.42 & 1.94 & 2.85 & 2.81 & 2.56 & 2.91 & 2.97 \end{pmatrix}$

This function has bounds $x_0\in [0, 4]$ and $x_i \in [-4, 4]$ for $i = 1, ..., n-1$ . Global optimum 11.7464.

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

class Colville(dimensions=4)¶

Colville objective function.

This class defines the Colville global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Colville}}(x) = \left(x_{1} -1\right)^{2} + 100 \left(x_{1}^{2} - x_{2}\right)^{2} + 10.1 \left(x_{2} -1\right)^{2} + \left(x_{3} -1\right)^{2} + 90 \left(x_{3}^{2} - x_{4}\right)^{2} + 10.1 \left(x_{4} -1\right)^{2} + 19.8 \frac{x_{4} -1}{x_{2}}$

with $x_i \in [-10, 10]$ for $i = 1, ..., 4$ .

Global optimum: $f(x) = 0$ for $x_i = 1$ for $i = 1, ..., 4$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

Todo

docstring equation is wrong use Jamil#36

class Corana(dimensions=4)¶

Corana test objective function.

This class defines the Corana global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Corana}}(x) = \begin{cases} \sum_{i=1}^n 0.15 d_i [z_i - 0.05\textrm{sgn}(z_i)]^2 & \textrm{if }|x_i-z_i| < 0.05 \\ d_ix_i^2 & \textrm{otherwise}\end{cases}$

Where, in this exercise:

$z_i = 0.2 \lfloor |x_i/s_i|+0.49999\rfloor\textrm{sgn}(x_i), d_i=(1,1000,10,100, ...)$

Here, $n$ represents the number of dimensions and $x_i \in [-5, 5]$ for $i=1,...,4$ .

Two-dimensional Corana function

Global optimum: $f(x_i) = 0$ for $x_i = 0$ for $i=1,...,4$

Gavana, A. Global Optimization Benchmarks and AMPGO

class CosineMixture(dimensions=2)¶

Cosine Mixture objective function.

This class defines the Cosine Mixture global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{CosineMixture}}(x) = -0.1 \sum_{i=1}^n \cos(5 \pi x_i) - \sum_{i=1}^n x_i^2$

Here, $n$ represents the number of dimensions and $x_i \in [-1, 1]$ for $i = 1, ..., N$ .

Two-dimensional CosineMixture function

Global optimum: $f(x) = -0.1N$ for $x_i = -1$ for $i = 1, ..., N$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

Todo

Jamil #38 has wrong minimum and wrong fglob. I plotted it. -(x**2) term is always negative if x is negative. cos(5 * pi * x) is equal to -1 for x=-1.

class Crescent(dimensions=2)¶: Crescent objective function.

Two-dimensional Crescent function

class CrossInTray(dimensions=2)¶

Cross-in-Tray objective function.

This class defines the Cross-in-Tray global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{CrossInTray}}(x) = - 0.0001 \left(\left|{e^{\left|{100 - \frac{\sqrt{x_{1}^{2} + x_{2}^{2}}}{\pi}}\right|} \sin\left(x_{1}\right) \sin\left(x_{2}\right)}\right| + 1\right)^{0.1}$

with $x_i \in [-15, 15]$ for $i = 1, 2$ .

Two-dimensional CrossInTray function

Global optimum: $f(x) = -2.062611870822739$ for $x_i = \pm 1.349406608602084$ for $i = 1, 2$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

class CrossLegTable(dimensions=2)¶

Cross-Leg-Table objective function.

This class defines the Cross-Leg-Table global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{CrossLegTable}}(x) = - \frac{1}{\left(\left|{e^{\left|{100 - \frac{\sqrt{x_{1}^{2} + x_{2}^{2}}}{\pi}}\right|} \sin\left(x_{1}\right) \sin\left(x_{2}\right)}\right| + 1\right)^{0.1}}$

with $x_i \in [-10, 10]$ for $i = 1, 2$ .

Two-dimensional CrossLegTable function

Global optimum: $f(x) = -1$ . The global minimum is found on the planes $x_1 = 0$ and $x_2 = 0$

Mishra, S. Global Optimization by Differential Evolution and Particle Swarm Methods: Evaluation on Some Benchmark Functions Munich University, 2006

class CrownedCross(dimensions=2)¶

Crowned Cross objective function.

This class defines the Crowned Cross global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{CrownedCross}}(x) = 0.0001 \left(\left|{e^{\left|{100 - \frac{\sqrt{x_{1}^{2} + x_{2}^{2}}}{\pi}}\right|} \sin\left(x_{1}\right) \sin\left(x_{2}\right)}\right| + 1\right)^{0.1}$

with $x_i \in [-10, 10]$ for $i = 1, 2$ .

Two-dimensional CrownedCross function

Global optimum: $f(x_i) = 0.0001$ . The global minimum is found on the planes $x_1 = 0$ and $x_2 = 0$

Mishra, S. Global Optimization by Differential Evolution and Particle Swarm Methods: Evaluation on Some Benchmark Functions Munich University, 2006

class Csendes(dimensions=2)¶

Csendes objective function.

This class defines the Csendes global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Csendes}}(x) = \sum_{i=1}^n x_i^6 \left[ 2 + \sin \left( \frac{1}{x_i} \right ) \right]$

Here, $n$ represents the number of dimensions and $x_i \in [-1, 1]$ for $i = 1, ..., N$ .

Two-dimensional Csendes function

Global optimum: $f(x) = 0.0$ for $x_i = 0$ for $i = 1, ..., N$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

class Cube(dimensions=2)¶

Cube objective function.

This class defines the Cube global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Cube}}(x) = 100(x_2 - x_1^3)^2 + (1 - x1)^2$

Here, $n$ represents the number of dimensions and $x_i \in [-10, 10]$ for $i=1,...,N$ .

Two-dimensional Cube function

Global optimum: $f(x_i) = 0.0$ for $x = [1, 1]$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

Todo

jamil#41 has the wrong solution.

N-D Test Functions C¶

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