N-D Test Functions D¶

class Damavandi(dimensions=2)¶

Damavandi objective function.

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

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

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

Two-dimensional Damavandi function

Global optimum: $f(x) = 0.0$ for $x_i = 2$ 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 Deb01(dimensions=2)¶

Deb 1 objective function.

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

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

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

Two-dimensional Deb01 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.

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 Deb03(dimensions=2)¶

Deb 3 objective function.

This class defines the Deb 3 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Deb03}}(x) = - \frac{1}{N} \sum_{i=1}^n \sin^6 \left[ 5 \pi \left ( x_i^{3/4} - 0.05 \right) \right ]$

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

Two-dimensional Deb03 function

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

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 Decanomial(dimensions=2)¶

Decanomial objective function.

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

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

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

Two-dimensional Decanomial function

Global optimum: $f(x) = 0$ for $x = [2, -3]$

Gavana, A. Global Optimization Benchmarks and AMPGO

class Deceptive(dimensions=2)¶

Deceptive objective function.

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

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

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

Gavana, A. Global Optimization Benchmarks and AMPGO

Todo

this function was taken from the Gavana website. The following code is based on his code. His code and the website don’t match, the equations are wrong.

class DeckkersAarts(dimensions=2)¶

Deckkers-Aarts objective function.

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

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

with $x_i \in [-20, 20]$ for $i = 1, 2$ .

Two-dimensional DeckkersAarts function

Global optimum: $f(x) = -24776.518242168$ for $x = [0, \pm 14.9451209]$

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 solution and global minimum are slightly wrong.

class DeflectedCorrugatedSpring(dimensions=2)¶

DeflectedCorrugatedSpring objective function.

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

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

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 DeflectedCorrugatedSpring function

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

Gavana, A. Global Optimization Benchmarks and AMPGO

Todo

website has a different equation to the gavana codebase. The function below is different to the equation above. Also, the global minimum is wrong.

class DeJong5(dimensions=2)¶: De Jong Function No. 5

Two-dimensional DeJong5 function

class DeVilliersGlasser01(dimensions=4)¶

DeVilliers-Glasser 1 objective function.

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

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

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 $i = 1, ..., 4$ .

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

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 DeVilliersGlasser02(dimensions=5)¶

DeVilliers-Glasser 2 objective function.

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

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

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

with $x_i \in [0.5, 60]$ for $i = 1, ..., 5$ .

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

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 DixonPrice(dimensions=2)¶

Dixon and Price objective function.

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

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

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

Two-dimensional DixonPrice function

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

Gavana code not correct. i array should start from 2.

class Dolan(dimensions=5)¶

Dolan objective function.

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

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

with $x_i \in [-100, 100]$ for $i = 1, ..., 5$ .

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

Gavana, A. Global Optimization Benchmarks and AMPGO

class DropWave(dimensions=2)¶

DropWave objective function.

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

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

with $x_i \in [-5.12, 5.12]$ for $i = 1, 2$ .

Two-dimensional DropWave function

Global optimum: $f(x) = -1$ for $x = [0, 0]$

Gavana, A. Global Optimization Benchmarks and AMPGO

N-D Test Functions D¶

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