N-D Test Functions K¶

class go_benchmark.Katsuura(dimensions=2)¶

Katsuura test objective function.

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

$f_{\text{Katsuura}}(\mathbf{x}) = \prod_{i=0}^{n-1} \left [ 1 + (i+1) \sum_{k=1}^{d} \lfloor (2^k x_i) \rfloor 2^{-k} \right ]$

Where, in this exercise, $d = 32$ .

Here, $n$ represents the number of dimensions and $x_i \in [0, 100]$ for $i=1,...,n$ .

Two-dimensional Katsuura function

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

class go_benchmark.Keane(dimensions=2)¶

Keane test objective function.

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

$f_{\text{Keane}}(\mathbf{x}) = \frac{\sin^2(x_1 - x_2)\sin^2(x_1 + x_2)}{\sqrt{x_1^2 + x_2^2}}$

Here, $n$ represents the number of dimensions and $x_i \in [0, 10]$ for $i=1,2$ .

Two-dimensional Keane function

Global optimum: $f(x_i) = 0.673668$ for $\mathbf{x} = [0.0, 1.39325]$ .

class go_benchmark.Kowalik(dimensions=4)¶

Kowalik test objective function.

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

$f_{\text{Kowalik}}(\mathbf{x}) = \sum_{i=0}^{10} \left [ a_i - \frac{x_1(b_i^2+b_ix_2)}{b_i^2 + b_ix_3 + x_4} \right ]^2$

Where:

$\mathbf{a} = [4, 2, 1, 1/2, 1/4 1/8, 1/10, 1/12, 1/14, 1/16] \\ \mathbf{b} = [0.1957, 0.1947, 0.1735, 0.1600, 0.0844, 0.0627, 0.0456, 0.0342, 0.0323, 0.0235, 0.0246]$

Here, $n$ represents the number of dimensions and $x_i \in [-5, 5]$ for $i=1,...,4$ .

Global optimum: $f(x_i) = 0.00030748610$ for $\mathbf{x} = [0.192833, 0.190836, 0.123117, 0.135766]$ .

N-D Test Functions K¶

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