N-D Test Functions L¶

class go_benchmark.Langermann(dimensions=2)¶

Langermann test objective function.

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

$f_{\text{Langermann}}(\mathbf{x}) = - \sum_{i=1}^{5} \frac{c_i \cos\left\{\pi \left[\left(x_{1}- a_i\right)^{2} + \left(x_{2} - b_i \right)^{2}\right]\right\}}{e^{\frac{\left( x_{1} - a_i\right)^{2} + \left( x_{2} - b_i\right)^{2}}{\pi}}}$

Here, $n$ represents the number of dimensions and $x_i \in [0, 10]$ for $i=1,2$ .

Two-dimensional Langermann function

Global optimum: $f(x_i) = -5.1621259$ for $\mathbf{x} = [2.00299219, 1.006096]$

class go_benchmark.LennardJones(dimensions=6)¶

LennardJones test objective function.

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

$f_{\text{LennardJones}}(\mathbf{x}) = \sum_{i=0}^{n-2}\sum_{j>1}^{n-1}\frac{1}{r_{ij}^{12}} - \frac{1}{r_{ij}^{6}}$

Where, in this exercise:

$r_{ij} = \sqrt{(x_{3i}-x_{3j})^2 + (x_{3i+1}-x_{3j+1})^2) + (x_{3i+2}-x_{3j+2})^2}$

Valid for any dimension, $n = 3*k, k=2,3,4,...,20$ . $k$ is the number of atoms in 3-D space constraints: unconstrained type: multi-modal with one global minimum; non-separable

Value-to-reach: $minima[k-2] + 0.0001$ . See array of minima below; additional minima available at the Cambridge cluster database:

http://www-wales.ch.cam.ac.uk/~jon/structures/LJ/tables.150.html

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

Global optimum:

$minima = [-1.,-3.,-6.,-9.103852,-12.712062,-16.505384,-19.821489,-24.113360, \\ -28.422532,-32.765970,-37.967600,-44.326801,-47.845157,-52.322627, \\ -56.815742,-61.317995, -66.530949,-72.659782,-77.1777043]$

class go_benchmark.Leon(dimensions=2)¶

Leon test objective function.

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

$f_{\text{Leon}}(\mathbf{x}) = \left(1 - x_{1}\right)^{2} + 100 \left(x_{2} - x_{1}^{2} \right)^{2}$

Here, $n$ represents the number of dimensions and $x_i \in [-1.2, 1.2]$ for $i=1,2$ .

Two-dimensional Leon function

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

class go_benchmark.Levy03(dimensions=2)¶

Levy 3 test objective function.

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

$f_{\text{Levy03}}(\mathbf{x}) = \sin^2(\pi y_1)+\sum_{i=1}^{n-1}(y_i-1)^2[1+10\sin^2(\pi y_{i+1})]+(y_n-1)^2$

Where, in this exercise:

$y_i=1+\frac{x_i-1}{4}$

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

Two-dimensional Levy 3 function

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

class go_benchmark.Levy05(dimensions=2)¶

Levy 5 test objective function.

This class defines the Levy 5 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Levy05}}(\mathbf{x}) = \sum_{i=1}^{5} i \cos \left[(i-1)x_1 + i \right] \times \sum_{j=1}^{5} j \cos \left[(j+1)x_2 + j \right] + (x_1 + 1.42513)^2 + (x_2 + 0.80032)^2$

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

Two-dimensional Levy 5 function

Global optimum: $f(x_i) = -176.1375$ for $\mathbf{x} = [-1.3068, -1.4248]$ .

class go_benchmark.Levy13(dimensions=2)¶

Levy13 test objective function.

This class defines the Levy13 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Levy13}}(\mathbf{x}) = \left(x_{1} -1\right)^{2} \left[\sin^{2}\left(3 \pi x_{2}\right) + 1\right] + \left(x_{2} -1\right)^{2} \left[\sin^{2}\left(2 \pi x_{2}\right) + 1\right] + \sin^{2}\left(3 \pi x_{1}\right)$

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

Two-dimensional Levy13 function

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

N-D Test Functions L¶

Previous topic

Next topic

This Page

Navigation

N-D Test Functions L¶

Previous topic

Next topic

This Page

Quick search

Navigation