N-D Test Functions M¶

class Matyas(dimensions=2)¶

Matyas objective function.

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

$f_{\text{Matyas}}(x) = 0.26(x_1^2 + x_2^2) - 0.48 x_1 x_2$

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

Two-dimensional Matyas 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 McCormick(dimensions=2)¶

McCormick objective function.

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

$f_{\text{McCormick}}(x) = - x_{1} + 2 x_{2} + \left(x_{1} - x_{2}\right)^{2} + \sin\left(x_{1} + x_{2}\right) + 1$

with $x_1\in [-1.5, 4]$ , $x_2\in [-3, 4]$ .

Two-dimensional McCormick function

Global optimum: $f(x) = -1.913222954981037$ for $x = [-0.5471975602214493, -1.547197559268372]$

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 Meyer(dimensions=3)¶

Meyer objective function.

http://www.itl.nist.gov/div898/strd/nls/data/mgh10.shtml

Todo

NIST regression standard

class MeyerRoth(dimensions=3)¶: MeyerRoth objective function.

class Michalewicz(dimensions=2)¶

Michalewicz objective function.

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

$f_{\text{Michalewicz}}(x) = - \sum_{i=1}^{2} \sin\left(x_i\right) \sin^{2 m}\left(\frac{i x_i^{2}}{\pi}\right)$

Where, in this exercise, $m = 10$ .

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

Two-dimensional Michalewicz function

Global optimum: $f(x_i) = -1.8013$ for $x = [0, 0]$

Adorio, E. MVF - “Multivariate Test Functions Library in C for Unconstrained Global Optimization”, 2005

Todo

could change dimensionality, but global minimum might change.

class MieleCantrell(dimensions=4)¶

Miele-Cantrell objective function.

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

$f_{\text{MieleCantrell}}({x}) = (e^{-x_1} - x_2)^4 + 100(x_2 - x_3)^6 + \tan^4(x_3 - x_4) + x_1^8$

with $x_i \in [-1, 1]$ for $i = 1, ..., 4$ .

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

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 Mishra01(dimensions=2)¶

Mishra 1 objective function.

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

$f_{\text{Mishra01}}(x) = (1 + x_n)^{x_n}$

where

$x_n = n - \sum_{i=1}^{n-1} x_i$

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

Two-dimensional Mishra01 function

Global optimum: $f(x) = 2$ for $x_i = 1$ 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 Mishra02(dimensions=2)¶

Mishra 2 objective function.

This class defines the Mishra 2 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Mishra02}}({x}) = (1 + x_n)^{x_n}$

with

$x_n = n - \sum_{i=1}^{n-1} \frac{(x_i + x_{i+1})}{2}$

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

Two-dimensional Mishra02 function

Global optimum: $f(x) = 2$ for $x_i = 1$ 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 Mishra03(dimensions=2)¶

Mishra 3 objective function.

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

$f_{\text{Mishra03}}(x) = \sqrt{\lvert \cos{\sqrt{\lvert x_1^2 + x_2^2 \rvert}} \rvert} + 0.01(x_1 + x_2)$

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

Two-dimensional Mishra03 function

Global optimum: $f(x) = -0.1999$ for $x = [-9.99378322, -9.99918927]$

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

I think that Jamil#76 has the wrong global minimum, a smaller one is possible

class Mishra04(dimensions=2)¶

Mishra 4 objective function.

This class defines the Mishra 4 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Mishra04}}({x}) = \sqrt{\lvert \sin{\sqrt{\lvert x_1^2 + x_2^2 \rvert}} \rvert} + 0.01(x_1 + x_2)$

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

Two-dimensional Mishra04 function

Global optimum: $f(x) = -0.17767$ for $x = [-8.71499636, -9.0533148]$

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

I think that Jamil#77 has the wrong minimum, not possible

class Mishra05(dimensions=2)¶

Mishra 5 objective function.

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

$f_{\text{Mishra05}}(x) = \left [ \sin^2 ((\cos(x_1) + \cos(x_2))^2) + \cos^2 ((\sin(x_1) + \sin(x_2))^2) + x_1 \right ]^2 + 0.01(x_1 + x_2)$

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

Two-dimensional Mishra05 function

Global optimum: $f(x) = -0.119829$ for $x = [-1.98682, -10]$

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

Todo

Line 381 in paper

class Mishra06(dimensions=2)¶

Mishra 6 objective function.

This class defines the Mishra 6 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Mishra06}}(x) = -\log{\left [ \sin^2 ((\cos(x_1) + \cos(x_2))^2) - \cos^2 ((\sin(x_1) + \sin(x_2))^2) + x_1 \right ]^2} + 0.01 \left[(x_1 -1)^2 + (x_2 - 1)^2 \right]$

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

Two-dimensional Mishra06 function

Global optimum: $f(x_i) = -2.28395$ for $x = [2.88631, 1.82326]$

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

Todo

line 397

class Mishra07(dimensions=2)¶

Mishra 7 objective function.

This class defines the Mishra 7 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Mishra07}}(x) = \left [\prod_{i=1}^{n} x_i - n! \right]^2$

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

Two-dimensional Mishra07 function

Global optimum: $f(x) = 0$ for $x_i = \sqrt{n}$ 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 Mishra08(dimensions=2)¶

Mishra 8 objective function.

This class defines the Mishra 8 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Mishra08}}(x) = 0.001 \left[\lvert x_1^{10} - 20x_1^9 + 180x_1^8 - 960 x_1^7 + 3360x_1^6 - 8064x_1^5 + 13340x_1^4 - 15360x_1^3 + 11520x_1^2 - 5120x_1 + 2624 \rvert \lvert x_2^4 + 12x_2^3 + 54x_2^2 + 108x_2 + 81 \rvert \right]^2$

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

Two-dimensional Mishra08 function

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

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

Todo

Line 1065

class Mishra09(dimensions=3)¶

Mishra 9 objective function.

This class defines the Mishra 9 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Mishra09}}({x}) = \left[ ab^2c + abc^2 + b^2 + (x_1 + x_2 - x_3)^2 \right]^2$

Where, in this exercise:

$\begin{cases} a = 2x_1^3 + 5x_1x_2 + 4x_3 - 2x_1^2x_3 - 18 \\ b = x_1 + x_2^3 + x_1x_2^2 + x_1x_3^2 - 22 \\ c = 8x_1^2 + 2x_2x_3 + 2x_2^2 + 3x_2^3 - 52 \\ \end{cases}$

with $x_i \in [-10, 10]$ for $i = 1, 2, 3$ .

Global optimum: $f(x) = 0$ for $x = [1, 2, 3]$

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

Todo

Line 1103

class Mishra10(dimensions=2)¶

Mishra 10 objective function.

This class defines the Mishra 10 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Mishra10}}({x}) = \left[ \lfloor x_1 \perp x_2 \rfloor - \lfloor x_1 \rfloor - \lfloor x_2 \rfloor \right]^2$

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

Two-dimensional Mishra10 function

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

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

Todo

int(x) should be used instead of floor(x)!!!!!

class Mishra10b(dimensions=2)¶: Mishra 10b objective function.

Two-dimensional Mishra10b function

class Mishra11(dimensions=2)¶

Mishra 11 objective function.

This class defines the Mishra 11 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Mishra11}}(x) = \left [ \frac{1}{n} \sum_{i=1}^{n} \lvert x_i \rvert - \left(\prod_{i=1}^{n} \lvert x_i \rvert \right )^{\frac{1}{n}} \right]^2$

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

Two-dimensional Mishra11 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 MullerBrown(dimensions=2)¶: MullerBrown objective function.

Two-dimensional MullerBrown function

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

Two-dimensional MultiGaussian function

class MultiModal(dimensions=2)¶

MultiModal objective function.

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

$f_{\text{MultiModal}}(x) = \left( \sum_{i=1}^n \lvert x_i \rvert \right) \left( \prod_{i=1}^n \lvert x_i \rvert \right)$

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

Two-dimensional MultiModal function

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

Gavana, A. Global Optimization Benchmarks and AMPGO

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