N-D Test Functions E¶

class Easom(dimensions=2)¶

Easom objective function.

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

$f_{\text{Easom}}({x}) = a - \frac{a}{e^{b \sqrt{\frac{\sum_{i=1}^{n} x_i^{2}}{n}}}} + e - e^{\frac{\sum_{i=1}^{n} \cos\left(c x_i\right)} {n}}$

Where, in this exercise, $a = 20, b = 0.2$ and $c = 2 \pi$ .

Here, $x_i \in [-100, 100]$ for $i = 1, 2$ .

Two-dimensional Easom function

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

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 Eckerle4(dimensions=3)¶

Eckerle4 objective function. Eckerle, K., NIST (1979). Circular Interference Transmittance Study.

http://www.itl.nist.gov/div898/strd/nls/data/eckerle4.shtml

class EggCrate(dimensions=2)¶

Egg Crate objective function.

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

$f_{\text{EggCrate}}(x) = x_1^2 + x_2^2 + 25 \left[ \sin^2(x_1) + \sin^2(x_2) \right]$

with $x_i \in [-5, 5]$ for $i = 1, 2$ .

Two-dimensional EggCrate function

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

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 EggHolder(dimensions=2)¶

Egg Holder objective function.

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

$f_{\text{EggHolder}}=\sum_{1}^{n - 1}\left[-\left(x_{i + 1} + 47 \right ) \sin\sqrt{\lvert x_{i+1} + x_i/2 + 47 \rvert} - x_i \sin\sqrt{\lvert x_i - (x_{i + 1} + 47)\rvert}\right ]$

Here, $n$ represents the number of dimensions and $x_i \in [-512, 512]$ for $i = 1, ..., n$ .

Two-dimensional EggHolder function

Global optimum: $f(x) = -959.640662711$ for ${x} = [512, 404.2319]$

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 is missing a minus sign on the fglob value

class ElAttarVidyasagarDutta(dimensions=2)¶

El-Attar-Vidyasagar-Dutta objective function.

This class defines the El-Attar-Vidyasagar-Dutta function global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{ElAttarVidyasagarDutta}}(x) = (x_1^2 + x_2 - 10)^2 + (x_1 + x_2^2 - 7)^2 + (x_1^2 + x_2^3 - 1)^2$

with $x_i \in [-100, 100]$ for $i = 1, 2$ .

Two-dimensional ElAttarVidyasagarDutta function

Global optimum: $f(x) = 1.712780354$ for $x= [3.40918683, -2.17143304]$

Gavana, A. Global Optimization Benchmarks and AMPGO

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

Two-dimensional Engvall function

class Exp2(dimensions=2)¶

Exp2 objective function.

This class defines the Exp2 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Exp2}}(x) = \sum_{i=0}^9 \left ( e^{-ix_1/10} - 5e^{-ix_2/10} - e^{-i/10} + 5e^{-i} \right )^2$

with $x_i \in [0, 20]$ for $i = 1, 2$ .

Two-dimensional Exp2 function

Global optimum: $f(x) = 0$ for $x = [1, 10.]$

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 Exponential(dimensions=2)¶

Exponential [1] objective function.

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

$f_{\text{Exponential}}(x) = -e^{-0.5 \sum_{i=1}^n x_i^2}$

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

Two-dimensional Exponential function

Global optimum: $f(x_i) = -1$ 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.

Todo

Jamil are missing a minus sign on fglob

N-D Test Functions E¶

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