N-D Test Functions B¶

class go_benchmark.BartelsConn(dimensions=2)¶

Bartels-Conn test objective function.

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

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

Here, $n$ represents the number of dimensions and $x_i \in [-50, 50]$ for $i=1,...,n$ .

Two-dimensional Bartels-Conn function

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

class go_benchmark.Beale(dimensions=2)¶

Beale test objective function.

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

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

Here, $n$ represents the number of dimensions and $x_i \in [-10, 10]$ for $i=1,2$ .

Two-dimensional Beale function

Global optimum: $f(x_i) = 0$ for $\mathbf{x} = [3, 0.5]$

class go_benchmark.Bird(dimensions=2)¶

Bird test objective function.

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

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

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

Two-dimensional Bird function

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

class go_benchmark.Bohachevsky(dimensions=2)¶

Bohachevsky test objective function.

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

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

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

Two-dimensional Bohachevsky function

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

class go_benchmark.BoxBetts(dimensions=3)¶

BoxBetts test objective function.

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

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

Where, in this exercise:

$g(x) = e^{-0.1(i+1)x_1} - e^{-0.1(i+1)x_2} - \left[(e^{-0.1(i+1)}) - e^{-(i+1)}x_3\right]$

And $k = 10$ .

Here, $n$ represents the number of dimensions and $x_1 \in [0.9, 1.2], x_2 \in [9, 11.2], x_3 \in [0.9, 1.2]$ .

Global optimum: $f(x_i) = 0$ for $\mathbf{x} = [1, 10, 1]$

class go_benchmark.Branin01(dimensions=2)¶

Branin 1 test objective function.

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

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

Here, $n$ represents the number of dimensions and $x_1 \in [-5, 10], x_2 \in [0, 15]$

Two-dimensional Branin 1 function

Global optimum: $f(x_i) = 0.39788735772973816$ for $\mathbf{x} = [-\pi, 12.275]$ or $\mathbf{x} = [\pi, 2.275]$ or $\mathbf{x} = [9.42478, 2.475]$

class go_benchmark.Branin02(dimensions=2)¶

Branin 2 test objective function.

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

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

Here, $n$ represents the number of dimensions and $x_i \in [-5, 15]$ for $i=1,2$ .

Two-dimensional Branin 2 function

Global optimum: $f(x_i) = 5.559037$ for $\mathbf{x} = [-3.2, 12.53]$

class go_benchmark.Brent(dimensions=2)¶

Brent test objective function.

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

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

Here, $n$ represents the number of dimensions and $x_i \in [-10, 10]$ for $i=1,2$ .

Two-dimensional Brent function

Global optimum: $f(x_i) = 0$ for $\mathbf{x} = [-10, -10]$

class go_benchmark.Brown(dimensions=2)¶

Brown test objective function.

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

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

Here, $n$ represents the number of dimensions and $x_i \in [-1, 4]$ for $i=1,...,n$ .

Two-dimensional Brown function

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

class go_benchmark.Bukin02(dimensions=2)¶

Bukin 2 test objective function.

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

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

Here, $n$ represents the number of dimensions and $x_1 \in [-15, -5], x_2 \in [-3, 3]$

Two-dimensional Bukin 2 function

Global optimum: $f(x_i) = 0$ for $\mathbf{x} = [-10, 0]$

class go_benchmark.Bukin04(dimensions=2)¶

Bukin 4 test objective function.

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

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

Here, $n$ represents the number of dimensions and $x_1 \in [-15, -5], x_2 \in [-3, 3]$

Two-dimensional Bukin 4 function

Global optimum: $f(x_i) = 0$ for $\mathbf{x} = [-10, 0]$

class go_benchmark.Bukin06(dimensions=2)¶

Bukin 6 test objective function.

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

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

Here, $n$ represents the number of dimensions and $x_1 \in [-15, -5], x_2 \in [-3, 3]$

Two-dimensional Bukin 6 function

Global optimum: $f(x_i) = 0$ for $\mathbf{x} = [-10, 1]$

N-D Test Functions B¶

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