N-D Test Functions D¶

class go_benchmark.Damavandi(dimensions=2)¶

Damavandi test objective function.

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

$f_{\text{Damavandi}}(\mathbf{x}) = \left[ 1 - \lvert{\frac{\sin[\pi(x_1-2)]\sin[\pi(x2-2)]}{\pi^2(x_1-2)(x_2-2)}} \rvert^5 \right] \left[2 + (x_1-7)^2 + 2(x_2-7)^2 \right]$

Here, $n$ represents the number of dimensions and $x_i \in [0, 14]$ for $i=1,...,n$ .

Two-dimensional Damavandi function

Global optimum: $f(x_i) = 0.0$ for $x_i = 2$ for $i=1,...,n$

class go_benchmark.Deb01(dimensions=2)¶

Deb 1 test objective function.

This class defines the Deb 1 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Deb01}}(\mathbf{x}) = - \frac{1}{N} \sum_{i=1}^n \sin^6(5 \pi x_i)$

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

Two-dimensional Deb 1 function

Global optimum: $f(x_i) = 0.0$ . The number of global minima is $5^n$ that are evenly spaced in the function landscape, where $n$ represents the dimension of the problem.

class go_benchmark.Deb02(dimensions=2)¶

Deb 2 test objective function.

This class defines the Deb 2 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Deb02}}(\mathbf{x}) = - \frac{1}{N} \sum_{i=1}^n \sin^6 \left[ 5 \pi \left ( x_i^{3/4} - 0.05 \right) \right ]$

Here, $n$ represents the number of dimensions and $x_i \in [0, 1]$ for $i=1,...,n$ .

Two-dimensional Deb 2 function

Global optimum: $f(x_i) = 0.0$ . The number of global minima is $5^n$ that are evenly spaced in the function landscape, where $n$ represents the dimension of the problem.

class go_benchmark.Decanomial(dimensions=2)¶

Decanomial test objective function.

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

$f_{\text{Decanomial}}(\mathbf{x}) = 0.001 \left(\lvert{x_{2}^{4} + 12 x_{2}^{3} + 54 x_{2}^{2} + 108 x_{2} + 81.0}\rvert + \lvert{x_{1}^{10} - 20 x_{1}^{9} + 180 x_{1}^{8} - 960 x_{1}^{7} + 3360 x_{1}^{6} - 8064 x_{1}^{5} + 13340 x_{1}^{4} - 15360 x_{1}^{3} + 11520 x_{1}^{2} - 5120 x_{1} + 2624.0}\rvert\right)^{2}$

Here, $n$ represents the number of dimensions and $x_i \in [-10, 10]$ for $i=1,2$ .

Two-dimensional Decanomial function

Global optimum: $f(x_i) = 0$ for $\mathbf{x} = [2, -3]$

class go_benchmark.Deceptive(dimensions=2)¶

Deceptive test objective function.

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

$f_{\text{Deceptive}}(\mathbf{x}) = - \left [\frac{1}{n} \sum_{i=1}^{n} g_i(x_i) \right ]^{\beta}$

Where $\beta$ is a fixed non-linearity factor; in this exercise, $\beta = 2$ . The function $g_i(x_i)$ is given by:

$g_i(x_i) = \begin{cases} - \frac{x}{\alpha_i} + \frac{4}{5} & \textrm{if} \hspace{5pt} 0 \leq x_i \leq \frac{4}{5} \alpha_i \\ \frac{5x}{\alpha_i} -4 & \textrm{if} \hspace{5pt} \frac{4}{5} \alpha_i \le x_i \leq \alpha_i \\ \frac{5(x - \alpha_i)}{\alpha_i-1} & \textrm{if} \hspace{5pt} \alpha_i \le x_i \leq \frac{1 + 4\alpha_i}{5} \\ \frac{x - 1}{1 - \alpha_i} & \textrm{if} \hspace{5pt} \frac{1 + 4\alpha_i}{5} \le x_i \leq 1 \end{cases}$

Here, $n$ represents the number of dimensions and $x_i \in [0, 1]$ for $i=1,...,n$ .

Two-dimensional Deceptive function

Global optimum: $f(x_i) = -1$ for $x_i = \alpha_i$ for $i=1,...,n$

class go_benchmark.DeckkersAarts(dimensions=2)¶

Deckkers-Aarts test objective function.

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

$f_{\text{DeckkersAarts}}(\mathbf{x}) = 10^5x_1^2 + x_2^2 - (x_1^2 + x_2^2)^2 + 10^{-5}(x_1^2 + x_2^2)^4$

Here, $n$ represents the number of dimensions and $x_i \in [-20, 20]$ for $i=1,2$ .

Two-dimensional Deckkers-Aarts function

Global optimum: $f(x_i) = -24777$ for $\mathbf{x} = [0, \pm 15]$

class go_benchmark.DeflectedCorrugatedSpring(dimensions=2)¶

DeflectedCorrugatedSpring test objective function.

This class defines the Deflected Corrugated Spring function global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{DeflectedCorrugatedSpring}}(\mathbf{x}) = 0.1\sum_{i=1}^n \left[ (x_i - \alpha)^2 - \cos \left( K \sqrt {\sum_{i=1}^n (x_i - \alpha)^2} \right ) \right ]$

Where, in this exercise, $K = 5$ and $\alpha = 5$ .

Here, $n$ represents the number of dimensions and $x_i \in [0, 2\alpha]$ for $i=1,...,n$ .

Two-dimensional Deflected Corrugated Spring function

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

class go_benchmark.DeVilliersGlasser01(dimensions=4)¶

DeVilliers-Glasser 1 test objective function.

This class defines the DeVilliers-Glasser 1 function global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{DeVilliersGlasser01}}(\mathbf{x}) = \sum_{i=1}^{24} \left[ x_1x_2^{t_i} \sin(x_3t_i + x_4) - y_i \right ]^2$

Where, in this exercise, $t_i = 0.1(i-1)$ and $y_i = 60.137(1.371^{t_i}) \sin(3.112t_i + 1.761)$ .

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

Global optimum: $f(x_i) = 0$ for $x_i = 0$ for $\mathbf{x} = [60.137, 1.371, 3.112, 1.761]$ .

class go_benchmark.DeVilliersGlasser02(dimensions=5)¶

DeVilliers-Glasser 2 test objective function.

This class defines the DeVilliers-Glasser 2 function global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{DeVilliersGlasser01}}(\mathbf{x}) = \sum_{i=1}^{24} \left[ x_1x_2^{t_i} \tanh \left [x_3t_i + \sin(x_4t_i) \right] \cos(t_ie^{x_5}) - y_i \right ]^2$

Where, in this exercise, $t_i = 0.1(i-1)$ and $y_i = 53.81(1.27^{t_i}) \tanh (3.012t_i + \sin(2.13t_i)) \cos(e^{0.507}t_i)$ .

Here, $n$ represents the number of dimensions and $x_i \in [1, 60]$ for $i=1,...,n$ .

Global optimum: $f(x_i) = 0$ for $x_i = 0$ for $\mathbf{x} = [53.81, 1.27, 3.012, 2.13, 0.507]$ .

class go_benchmark.DixonPrice(dimensions=2)¶

Dixon and Price test objective function.

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

$f_{\text{DixonPrice}}(\mathbf{x}) = (x_i - 1)^2 + \sum_{i=2}^n i(2x_i^2 - x_{i-1})^2$

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

Two-dimensional Dixon and Price function

Global optimum: $f(x_i) = 0$ for $x_i = 2^{- \frac{(2^i-2)}{2^i}}$ for $i=1,...,n$

class go_benchmark.Dolan(dimensions=5)¶

Dolan test objective function.

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

$f_{\text{Dolan}}(\mathbf{x}) = \lvert (x_1 + 1.7x_2)\sin(x_1) - 1.5x_3 - 0.1x_4\cos(x_5 + x_5 - x_1) + 0.2x_5^2 - x_2 - 1 \rvert$

Here, $n$ represents the number of dimensions and $x_i \in [-100, 100]$ for $i=1,2$ .

Global optimum: $f(x_i) = 10^{-5}$ for $\mathbf{x} = [8.39045925, 4.81424707, 7.34574133, 68.88246895, 3.85470806]$

class go_benchmark.DropWave(dimensions=2)¶

DropWave test objective function.

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

$f_{\text{DropWave}}(\mathbf{x}) = - \frac{1 + \cos\left(12 \sqrt{\sum_{i=1}^{n} x_i^{2}}\right)}{2 + 0.5 \sum_{i=1}^{n} x_i^{2}}$

Here, $n$ represents the number of dimensions and $x_i \in [-5.12, 5.12]$ for $i=1,2$ .

Two-dimensional DropWave function

Global optimum: $f(x_i) = -1$ for $x_i = 0$ for $i=1,2$

N-D Test Functions D¶

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