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 $i=1,2$ .

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 $i=1,2$ .

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 $i=1,2$ .

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 $i=1,...,n$ .

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:

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