N-D Test Functions B¶

class BartelsConn(dimensions=2)¶

Bartels-Conn objective function.

The BartelsConn global optimization problem is a multimodal minimization problem defined as follows:

$f_{\text{BartelsConn}}(x) = \lvert {x_1^2 + x_2^2 + x_1x_2} \rvert + \lvert {\sin(x_1)} \rvert + \lvert {\cos(x_2)} \rvert$

with $x_i \in [-500, 500]$ for $i = 1, 2$ .

Two-dimensional BartelsConn function

Global optimum: $f(x) = 1$ 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 Beale(dimensions=2)¶

Beale objective function.

The Beale global optimization problem is a multimodal minimization problem defined as follows:

$f_{\text{Beale}}(x) = \left(x_1 x_2 - x_1 + 1.5\right)^{2} + \left(x_1 x_2^{2} - x_1 + 2.25\right)^{2} + \left(x_1 x_2^{3} - x_1 + 2.625\right)^{2}$

with $x_i \in [-4.5, 4.5]$ for $i = 1, 2$ .

Two-dimensional Beale function

Global optimum: $f(x) = 0$ for $x=[3, 0.5]$

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 BiggsExp02(dimensions=2)¶

BiggsExp02 objective function.

The BiggsExp02 global optimization problem is a multimodal minimization problem defined as follows

$\begin{matrix} \\ f_{\text{BiggsExp02}}(x) = \sum_{i=1}^{10} (e^{-t_i x_1} - 5 e^{-t_i x_2} - y_i)^2 \\ t_i = 0.1 i \\ y_i = e^{-t_i} - 5 e^{-10t_i} \\ \end{matrix}$

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

Two-dimensional BiggsExp02 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 BiggsExp03(dimensions=3)¶

BiggsExp03 objective function.

The BiggsExp03 global optimization problem is a multimodal minimization problem defined as follows

$\begin{matrix} f_{\text{BiggsExp03}}(x) = \sum_{i=1}^{10} (e^{-t_i x_1} - x_3e^{-t_i x_2} - y_i)^2 \\ t_i = 0.1i \\ y_i = e^{-t_i} - 5e^{-10 t_i} \\ \end{matrix}$

with $x_i \in [0, 20]$ for $i = 1, 2, 3$ .

Global optimum: $f(x) = 0$ for $x = [1, 10, 5]$

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 BiggsExp04(dimensions=4)¶

BiggsExp04 objective function.

The BiggsExp04 global optimization problem is a multimodal minimization problem defined as follows

$\begin{matrix} \\ f_{\text{BiggsExp04}}(x) = \sum_{i=1}^{10} (x_3 e^{-t_i x_1} - x_4 e^{-t_i x_2} - y_i)^2 \\ t_i = 0.1i \\ y_i = e^{-t_i} - 5 e^{-10 t_i} \\ \end{matrix}$

with $x_i \in [0, 20]$ for $i = 1, ..., 4$ .

Global optimum: $f(x) = 0$ for $x = [1, 10, 1, 5]$

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 BiggsExp05(dimensions=5)¶

BiggsExp05 objective function.

The BiggsExp05 global optimization problem is a multimodal minimization problem defined as follows

$\begin{matrix} \\ f_{\text{BiggsExp05}}(x) = \sum_{i=1}^{11}(x_3 e^{-t_i x_1} - x_4 e^{-t_i x_2} + 3 e^{-t_i x_5} - y_i)^2 \\ t_i = 0.1i \\ y_i = e^{-t_i} - 5e^{-10 t_i} + 3e^{-4 t_i} \\ \end{matrix}$

with $x_i \in [0, 20]$ for $i=1, ..., 5$ .

Global optimum: $f(x) = 0$ for $x = [1, 10, 1, 5, 4]$

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 BiggsExp06(dimensions=6)¶: BiggsExp06 objective function.

class Bird(dimensions=2)¶

Bird objective function.

The Bird global optimization problem is a multimodal minimization problem defined as follows

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

with $x_i \in [-2\pi, 2\pi]$

Two-dimensional Bird function

Global optimum: $f(x) = -106.7645367198034$ for $x = [4.701055751981055, 3.152946019601391]$ or $x = [-1.582142172055011, -3.130246799635430]$

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 Bohachevsky01(dimensions=2)¶

Bohachevsky 1 objective function.

The Bohachevsky 1 global optimization problem is a multimodal minimization problem defined as follows

$f_{\text{Bohachevsky01}}(x) = \sum_{i=1}^{n-1}\left[x_i^2 + 2x_{i+1}^2 - 0.3 \cos(3 \pi x_i) - 0.4 \cos(4 \pi x_{i+1}) + 0.7 \right]$

Here, $n$ represents the number of dimensions and $x_i \in [-15, 15]$ for $i = 1, ..., n$ .

Two-dimensional Bohachevsky01 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.

Todo

equation needs to be fixed up in the docstring. see Jamil#17

class Bohachevsky02(dimensions=2)¶

Bohachevsky 2 objective function.

The Bohachevsky 2 global optimization problem is a multimodal minimization problem defined as follows

$f_{\text{Bohachevsky02}}(x) = \sum_{i=1}^{n-1}\left[x_i^2 + 2 x_{i+1}^2 - 0.3 \cos(3 \pi x_i) - 0.4 \cos(4 \pi x_{i + 1}) + 0.7 \right]$

Here, $n$ represents the number of dimensions and $x_i \in [-15, 15]$ for $i = 1, ..., n$ .

Two-dimensional Bohachevsky02 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.

Todo

equation needs to be fixed up in the docstring. Jamil is also wrong. There should be no 0.4 factor in front of the cos term

class Bohachevsky03(dimensions=2)¶

Bohachevsky 3 objective function.

The Bohachevsky 3 global optimization problem is a multimodal minimization problem defined as follows

$f_{\text{Bohachevsky02}}(x) = \sum_{i=1}^{n-1}\left[x_i^2 + 2 x_{i+1}^2 - 0.3 \cos(3 \pi x_i) - 0.4 \cos(4 \pi x_{i + 1}) + 0.7 \right]$

Here, $n$ represents the number of dimensions and $x_i \in [-15, 15]$ for $i = 1, ..., n$ .

Two-dimensional Bohachevsky03 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.

Todo

equation needs to be fixed up in the docstring. Jamil#19

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

Two-dimensional Booth function

class BoxBetts(dimensions=3)¶

BoxBetts objective function.

The BoxBetts global optimization problem is a multimodal minimization problem defined as follows

$f_{\text{BoxBetts}}(x) = \sum_{i=1}^k g(x_i)^2$

Where, in this exercise:

$g(x) = e^{-0.1i x_1} - e^{-0.1i x_2} - x_3\left[e^{-0.1i} - e^{-i}\right]$

And $k = 10$ .

Here, $x_1\in [0.9, 1.2], x_2\in [9, 11.2], x_3\in [0.9, 1.2]$ .

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

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 Brad(dimensions=3)¶: Brad objective function.

class Branin01(dimensions=2)¶

Branin01 objective function.

The Branin01 global optimization problem is a multimodal minimization problem defined as follows

$f_{\text{Branin01}}(x) = \left(- 1.275 \frac{x_1^{2}}{\pi^{2}} + 5 \frac{x_1}{\pi} + x_2 -6\right)^{2} + \left(10 -\frac{5}{4 \pi} \right) \cos\left(x_1\right) + 10$

with $x_1\in [-5, 10], x_2\in [0, 15]$

Two-dimensional Branin01 function

Global optimum: $f(x) = 0.39788735772973816$ for $x = [-\pi, 12.275]$ or $x = [\pi, 2.275]$ or $x = [3\pi, 2.475]$

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#22, one of the solutions is different

class Branin02(dimensions=2)¶

Branin02 objective function.

The Branin02 global optimization problem is a multimodal minimization problem defined as follows

$f_{\text{Branin02}}(x) = \left(- 1.275 \frac{x_1^{2}}{\pi^{2}} + 5 \frac{x_1}{\pi} + x_2 - 6 \right)^{2} + \left(10 - \frac{5}{4 \pi} \right) \cos\left(x_1\right) \cos\left(x_2\right) + \log(x_1^2+x_2^2 + 1) + 10$

with $x_i \in [-5, 15]$ for $i = 1, 2$ .

Two-dimensional Branin02 function

Global optimum: $f(x) = 5.559037$ for $x = [-3.2, 12.53]$

Gavana, A. Global Optimization Benchmarks and AMPGO

class Brent(dimensions=2)¶

Brent objective function.

The Brent global optimization problem is a multimodal minimization problem defined as follows:

$f_{\text{Brent}}(x) = (x_1 + 10)^2 + (x_2 + 10)^2 + e^{(-x_1^2 -x_2^2)}$

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

Two-dimensional Brent function

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

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.

Todo

solution is different to Jamil#24

class Brown(dimensions=2)¶

Brown objective function.

The Brown global optimization problem is a multimodal minimization problem defined as follows:

$f_{\text{Brown}}(x) = \sum_{i=1}^{n-1}\left[ \left(x_i^2\right)^{x_{i + 1}^2 + 1} + \left(x_{i + 1}^2\right)^{x_i^2 + 1}\right]$

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

Two-dimensional Brown function

Global optimum: $f(x_i) = 0$ 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 BuecheRastrigin(dimensions=2)¶: BuecheRastrigin objective function.

Two-dimensional BuecheRastrigin function

class Bukin02(dimensions=2)¶

Bukin02 objective function.

The Bukin02 global optimization problem is a multimodal minimization problem defined as follows:

$f_{\text{Bukin02}}(x) = 100 (x_2^2 - 0.01x_1^2 + 1) + 0.01(x_1 + 10)^2$

with $x_1\in [-15, -5], x_2\in [-3, 3]$

Two-dimensional Bukin02 function

Global optimum: $f(x) = -124.75$ for $x = [-15, 0]$

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.

Todo

I think that Gavana and Jamil are wrong on this function. In both sources the x[1] term is not squared. As such there will be a minimum at the smallest value of x[1].

class Bukin04(dimensions=2)¶

Bukin04 objective function.

The Bukin04 global optimization problem is a multimodal minimization problem defined as follows:

$f_{\text{Bukin04}}(x) = 100 x_2^{2} + 0.01 \lvert{x_1 + 10} \rvert$

with $x_1\in [-15, -5], x_2\in [-3, 3]$

Two-dimensional Bukin04 function

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

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 Bukin06(dimensions=2)¶

Bukin06 objective function.

The Bukin06 global optimization problem is a multimodal minimization problem defined as follows:

$f_{\text{Bukin06}}(x) = 100 \sqrt{ \lvert{x_2 - 0.01 x_1^{2}} \rvert} + 0.01 \lvert{x_1 + 10} \rvert$

with $x_1\in [-15, -5], x_2\in [-3, 3]$

Two-dimensional Bukin06 function

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

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.

N-D Test Functions B¶

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