N-D Test Functions L¶

class Langermann(dimensions=2)¶

Langermann objective function.

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

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

Where:

$\begin{matrix} a = [3, 5, 2, 1, 7] \\ b = [5, 2, 1, 4, 9]\\ c = [1, 2, 5, 2, 3] \\ \end{matrix}$

Here $x_i \in [0, 10]$ for $i = 1, 2$ .

Two-dimensional Langermann function

Global optimum: $f(x) = -5.1621259$ for $x = [2.00299219, 1.006096]$

Gavana, A. Global Optimization Benchmarks and AMPGO

Todo

Langermann from Gavana is not the same as Jamil #68.

class LennardJones(dimensions=6)¶

LennardJones 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:

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

class Leon(dimensions=2)¶

Leon objective function.

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

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

with $x_i \in [-1.2, 1.2]$ for $i = 1, 2$ .

Two-dimensional Leon function

Global optimum: $f(x) = 0$ 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.

class Levy03(dimensions=2)¶

Levy 3 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 Levy03 function

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

Mishra, S. Global Optimization by Differential Evolution and Particle Swarm Methods: Evaluation on Some Benchmark Functions. Munich Personal RePEc Archive, 2006, 1005

Todo

not clear what the Levy function definition is. Gavana, Mishra, Adorio have different forms. Indeed Levy 3 docstring from Gavana disagrees with the Gavana code! The following code is from the Mishra listing of Levy08.

class Levy05(dimensions=2)¶

Levy 5 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 Levy05 function

Global optimum: $f(x_i) = -176.1375779$ for $\mathbf{x} = [-1.30685, -1.42485]$ .

Mishra, S. Global Optimization by Differential Evolution and Particle Swarm Methods: Evaluation on Some Benchmark Functions. Munich Personal RePEc Archive, 2006, 1005

class Levy13(dimensions=2)¶

Levy13 objective function.

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

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

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

Two-dimensional Levy13 function

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

Mishra, S. Some new test functions for global optimization and performance of repulsive particle swarm method. Munich Personal RePEc Archive, 2006, 2718

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

Two-dimensional LunacekBiRastrigin function

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

Two-dimensional LunacekBiSphere function

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