test_functions 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.

Bartels-Conn function

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.

Beale function

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.

Bird function

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.

Bohachevsky function

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]

Branin 1 function

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.

Branin 2 function

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.

Brent function

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.

Brown function

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]

Bukin 2 function

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]

Bukin 4 function

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]

Bukin 6 function

Two-dimensional Bukin 6 function

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

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