test_functions N-D Test Functions W

class go_benchmark.Watson(dimensions=6)

Watson test objective function.

This class defines the Watson global optimization problem. This is a unimodal minimization problem defined as follows:

f_{\text{Watson}}(\mathbf{x}) = \sum_{i=0}^{29} \left\{ \sum_{j=0}^4 ((j - 1)a_i^j x_{j+1}) - \left[ \sum_{j=0}^5 a_i^j x_{j+1} \right ]^2 - 1 \right\}^2 + x_1^2

Where, in this exercise, a_i = i/29.

Here, n represents the number of dimensions and x_i \in [-5, 5] for i=1,...,6.

Global optimum: f(x_i) = 0.002288 for \mathbf{x} = [-0.0158, 1.012, -0.2329, 1.260, -1.513, 0.9928]


class go_benchmark.Wavy(dimensions=2)

W / Wavy test objective function.

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

f_{\text{Wavy}}(\mathbf{x}) = 1 - \frac{1}{n} \sum_{i=1}^{n} \cos(kx_i)e^{-\frac{x_i^2}{2}}

Where, in this exercise, k = 10. The number of local minima is kn and (k + 1)n for odd and even k respectively.

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

W / Wavy function

Two-dimensional W / Wavy function

Global optimum: f(x_i) = 0 for x_i = 0 for i=1,2


class go_benchmark.WayburnSeader01(dimensions=2)

Wayburn and Seader 1 test objective function.

This class defines the Wayburn and Seader 1 global optimization problem. This is a unimodal minimization problem defined as follows:

f_{\text{WayburnSeader01}}(\mathbf{x}) = (x_1^6 + x_2^4 - 17)^2 + (2x_1 + x_2 - 4)^2

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

Wayburn and Seader 1 function

Two-dimensional Wayburn and Seader 1 function

Global optimum: f(x_i) = 0 for \mathbf{x} = [1, 2]


class go_benchmark.WayburnSeader02(dimensions=2)

Wayburn and Seader 2 test objective function.

This class defines the Wayburn and Seader 2 global optimization problem. This is a unimodal minimization problem defined as follows:

f_{\text{WayburnSeader02}}(\mathbf{x}) = \left[ 1.613 - 4(x_1 - 0.3125)^2 - 4(x_2 - 1.625)^2 \right]^2 + (x_2 - 1)^2

Here, n represents the number of dimensions and x_i \in [-500, 500] for i=1,2.

Wayburn and Seader 2 function

Two-dimensional Wayburn and Seader 2 function

Global optimum: f(x_i) = 0 for \mathbf{x} = [0.2, 1]


class go_benchmark.Weierstrass(dimensions=2)

Weierstrass test objective function.

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

f_{\text{Weierstrass}}(\mathbf{x}) = \sum_{i=1}^{n} \left [ \sum_{k=0}^{kmax} a^k \cos \left( 2 \pi b^k (x_i + 0.5) \right) - n \sum_{k=0}^{kmax} a^k \cos(\pi b^k) \right ]

Where, in this exercise, kmax = 20, a = 0.5 and b = 3.

Here, n represents the number of dimensions and x_i \in [-0.5, 0.5] for i=1,...,n.

Weierstrass function

Two-dimensional Weierstrass function

Global optimum: f(x_i) = 4 for x_i = 0 for i=1,...,n


class go_benchmark.Whitley(dimensions=2)

Whitley test objective function.

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

f_{\text{Whitley}}(\mathbf{x}) = \sum_{i=1}^n \sum_{j=1}^n \left[\frac{(100(x_i^2-x_j)^2 + (1-x_j)^2)^2}{4000} - \cos(100(x_i^2-x_j)^2 + (1-x_j)^2)+1 \right]

Here, n represents the number of dimensions and x_i \in [-10.24, 10.24] for i=1,...,n.

Whitley function

Two-dimensional Whitley function

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


class go_benchmark.Wolfe(dimensions=3)

Wolfe test objective function.

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

f_{\text{Wolfe}}(\mathbf{x}) = \frac{4}{3}(x_1^2 + x_2^2 - x_1x_2)^{0.75} + x_3

Here, n represents the number of dimensions and x_i \in [0, 2] for i=1,2,3.

Global optimum: f(x_i) = 0 for x_i = 0 for i=1,2,3

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