N-D Test Functions X¶

class XinSheYang01(dimensions=2)¶

Xin-She Yang 1 objective function.

This class defines the Xin-She Yang 1 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{XinSheYang01}}(x) = \sum_{i=1}^{n} \epsilon_i \lvert x_i \rvert^i$

The variable $\epsilon_i, (i = 1, ..., n)$ is a random variable uniformly distributed in $[0, 1]$ .

Here, $n$ represents the number of dimensions and $x_i \in [-5, 5]$ for $i = 1, ..., n$ .

Two-dimensional XinSheYang01 function

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

class XinSheYang02(dimensions=2)¶

Xin-She Yang 2 objective function.

This class defines the Xin-She Yang 2 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{XinSheYang02}}(x) = \frac{\sum_{i=1}^{n} \lvert{x_{i}}\rvert} {e^{\sum_{i=1}^{n} \sin\left(x_{i}^{2.0} \right)}}$

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

Two-dimensional XinSheYang02 function

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

class XinSheYang03(dimensions=2)¶

Xin-She Yang 3 objective function.

This class defines the Xin-She Yang 3 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{XinSheYang03}}(x) = e^{-\sum_{i=1}^{n} (x_i/\beta)^{2m}} - 2e^{-\sum_{i=1}^{n} x_i^2} \prod_{i=1}^{n} \cos^2(x_i)$

Where, in this exercise, $\beta = 15$ and $m = 3$ .

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

Two-dimensional XinSheYang03 function

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

class XinSheYang04(dimensions=2)¶

Xin-She Yang 4 objective function.

This class defines the Xin-She Yang 4 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{XinSheYang04}}(x) = \left[ \sum_{i=1}^{n} \sin^2(x_i) - e^{-\sum_{i=1}^{n} x_i^2} \right ] e^{-\sum_{i=1}^{n} \sin^2 \sqrt{ \lvert x_i \rvert }}$

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

Two-dimensional XinSheYang04 function

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

class Xor(dimensions=9)¶

Xor objective function.

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

$f_{\text{Xor}}(x) = \left[ 1 + \exp \left( - \frac{x_7}{1 + \exp(-x_1 - x_2 - x_5)} - \frac{x_8}{1 + \exp(-x_3 - x_4 - x_6)} - x_9 \right ) \right ]^{-2} \\ + \left [ 1 + \exp \left( -\frac{x_7}{1 + \exp(-x_5)} - \frac{x_8}{1 + \exp(-x_6)} - x_9 \right ) \right] ^{-2} \\ + \left [1 - \left\{1 + \exp \left(-\frac{x_7}{1 + \exp(-x_1 - x_5)} - \frac{x_8}{1 + \exp(-x_3 - x_6)} - x_9 \right ) \right\}^{-1} \right ]^2 \\ + \left [1 - \left\{1 + \exp \left(-\frac{x_7}{1 + \exp(-x_2 - x_5)} - \frac{x_8}{1 + \exp(-x_4 - x_6)} - x_9 \right ) \right\}^{-1} \right ]^2$